Full text of "Double counterpoint and canon"

See other formats

AUGENER'S EDITION, No. 9184.

DOUBLE COUNTERPOINT

AND CANON.

EBENEZER PROUT, B.A. Lond

Hon. Mus. Dot. Trin. Coll. Dublin and Edinburgh, and
Professor of Music in the University of Dublin.

SIXTH IMPRESSION

LONDON :

AUGENER LTD.

Printed in England

AUGENER LTD.,
287 Acton Lane, London. W. 4.

PRE FACE.

VARIOUS causes have conduced to the somewhat long delay in
the appearance of the present work. Since the second volume
of this series (Counterpoint ', Strict and Free) was issued, the
author, at the request of the publishers, has compiled four small
hooks supplementary to Harmony and Counterpoint. This for
some time prevented his commencing the present volume. But
the chief cause of the delay has been the difficulty of the task
itself. A book which, like the present, deals with many of the
most abstruse problems of musical theory, required a great deal
of preliminary work, not only in examining and comparing
existing treatises, but in writing a very large number of examples
to illustrate the various points touched upon. Such a book, if
produced in a hurry, would be of little or no value. The author
is by no means unaware of its shortcomings ; but he can at
least honestly claim for it that he has spared neither time nor
trouble in its preparation, and that he has done his best to make
it practical and useful, especially for those who are studying
without the aid of a master.

As mentioned in the preface to Counterpoint, it was originall)
intended to include the subject of Fugue in this volume. To
have done so, however, would have necessitated the omission of
so much which it is desirable that the student should know, and
would have compelled the author to treat of Fugue itself in such
a cursory, not to say perfunctory manner, that he soon decided
to confine the present volume to Double Counterpoint and Canon,
and to deal with Fugue in a separate work, which shall follow
this as soon as he can find time to write it.

In treating of double counterpoint, it has been thought
advisable to begin with it, as with simple counterpoint, in the
strict style. It must, of course, be borne in mind that this i>
merely preliminary technical work to such double counterpoint

PREFACE.

as is used in actual composition It has been necessary in some
respects to relax the strictness of the rules when applying them
to double counterpoint — especially in the tenth, the most difficult
interval to work. The fundamental principles of strict counter-
point are, nevertheless, observed ; and the author believes that
writing under restrictions will be of great value to the student,
as giving him freedom in the later stages of his work. The
whole of the examples to the strict double counterpoint have
been written expressly for this work

In treating of free double counterpoint, the plan pursued in
the preceding volumes of this series, of taking the examples, as
far as possible, from the works of the great masters, has been
adhered to. It will be seen that the quotations are both more
numerous and longer than in Counterpoint. This is because the
student now approaches more nearly to actual composition, which
can be better learned from the study of good models than in
any other way. It is impossible to teach the invention of melody,
though the general principles of its forms may be made in-
telligible enough; but the exercise of the imagination may be
stimulated by the study and analysis of existing masterpieces ;
and though it is not to be expected that the student will ever
acquire the skill of a Bach, yet, from the examination of that
composer's works, he can at least discover many general principles-
to guide him in his own efforts. The chapters on double counter-
point on a florid subject, and with free parts, largely consist of
analyzed extracts from the works of the great composers.

The subject of double counterpoint in the rarer intervals is
passed over in silence by most theorists. Though far inferior in
importance to those more frequently employed, these double
counterpoints are not without interest ; and, as they are more
often used than is generally supposed, a chapter is given to this
subject, in which some curious examples will be seen. In the
last chapter of the first part of the book, an attempt has been
made to simplify the difficult study of triple and quadruple
counterpoint.

The second half of this volume, which deals with Canon,
presented more difficulties to the author than the first, chiefly
because of the impossibility of giving on many points any beyond
the most general directions In one respect, it is believed, the
present book differs from most of its predecessors. A great part
of the instructions on canon to be found in many treatises has

PREFACE. v

reference to matters which are not of the slightest practical use
to the student. It is doubtful whether it is worth while for any-
body at the present day to trouble himself about writing an
infinite canon by augmentation, a canon cancrizans, or a riddle-
canon. Yet the old text-books give elaborate instructions for the
composition of these musical puzzles, for they are nothing better.
As the object of these volumes is to teach what the student
may really need, these subjects are not dealt with at all,
though, for the sake of completeness, specimens of all the
varieties are given. Only such canons are treated of in detail as
possess true musical value, and the learner who masters these
will find that he knows all that is really necessary for him. The
study of double counterpoint, and of the various forms 01
imitation is an invaluable and indispensable introduction to the
higher branches of composition, and amply rewards the musician
for the somewhat severe labour necessary for its acquirement.

The author has to acknowledge his obligations for assistance
from several quarters. He is indebted to a series of articles by
Mr. J. S. Shedlock, in the Magazine of Music, for calling his
attention to some of the examples of counterpoint in the rarer
intervals in Bach's " Wohltemperirtes Clavier." He has to thank
Mr. F. Corder for the canon in § 453, and Herr E. W. Fritzsch,
of Leipzig, for permission to reprint the canons in §§ 452, 468
from the Musikalisches Wochenblatt. His warm thanks are again
due to Dr. C. W. Pearce, not only for valuable suggestions, but
for his kindness in revising the proof-sheets of the volume — a
more than usually troublesome work, owing to the large amount
of music type.

LONDON, July. 1891.

NOTE.

The references throughout this volume, to " Harmony : Its Theory and
Practice," refer to the Revised Edition.

For the convenience of those who may desire to continue to use copies
of the First to the Fifteenth Editions, inclusive, the following table is
inserted :—

Revised

First to Fif-

Revised

First to Fif-

Edition.

teenth Editions.

Edition.

teenth Editions.

Chapter II

Chapter III

Section 228

Section 192

«• III

" IV

44 239

" 202

" IX

" X

" 251

" 207

•« X

" XI

44 258

11 211

•• XI

" XIX

44 294

44 243

•• XVI

" XVIII

44 307

" 248

44 313-321

44 253-258

41 3U

44 254

44 318

44 257

Section 75

Section 103

44 325

44 263

44 77

44 105

44 336

44 504

44 93-95

" 126-128

44 341

44 517

" 104

« 117

•• 411

44 381

44 133

44 137

" 418

41 75

" 169

«« 156

" 428

41 410

44 173

" 159

44 433

44 404

" 188-189

'« 164-166

44 440

44 432

44 490. 49L

" 426 («)(*)

•' 208

M 170-171

505. 507,

433, 439

520

& 460 (</)

.. j24

" 190

•• 645 647

44 562, 564

TABLE OF CONTENTS.

[N.B.— The numbers refer in every instance to the sections, not to the pages. \

PART I.— DOUBLE COUNTERPOINT.

CHAPTER L— INTRODUCTION page i

Definition of Counterpoint, i — Double, Triple, and Quadruple Counterpoint de
fined, 2 — Enlarged meaning of Inversion, 3 — Inversion at small distances not
employed, 4 — Limit in the distance of subjects to be inverted, 5 — How to find
the inversion of an interval at any distance, 6, 7 — How to find the interval of
inversion of two subjects, 8, 9 — Example of double counterpoint in the octave,
tenth, and twelfth, 10, n — Strict double counterpoint, 12.

CHAPTER II.— STRICT DOUBLE COUNTERPOINT IN THE OCTAVE AND

FIFTEENTH page 6

The difference between double counterpoint in the octave and the fifteenth, 13—
Inversion in the octave ; table of intervals, 14 — Limited use of the perfect
fifth, 15 — Employment of the unison and octave, 16 — Inversion of both parts
in double counterpoint in the fifteenth, 17 — First species : prohibition of the
fifth; cadence, 18— Implied harmony, 19 — How to write the exercises, 20 —
Examples in a major key, 21-23 — The s'xt^ above the dominant of a minor
key allowed in double counterpoint, 24 — Examples in a minor key, 25-27 —
Second species : use of dissonances ; employment of the perfect fifth, 28 — The
cadence free, 29 — Examples, 30-33 — Third species : the fifth of a chord in the
bass, 34 — Treatment of the octave ; the cadence, 35 — Examples, 36-39—
Fourth species: its difficulties; the cadence, 40 — Examples, 41, 42 — Fifth
species : the cadence ; the fifth as a harmony note, 43— Examples, 44-46—
Examples of double counterpoint on a subject with notes of equal length, 47 —
Subjects for double counterpoint in the octave, 48.

CHAPTER III.— STRICT DOUBLE COUNTERPOINT IN THE TENTH page 33

Various methods of inversion in the tenth, 49, 50— Table of inversions, 51— The
effect of inversion in the tenth, 52, 53— No consecutive intervals allowed, 54—
Similar motion forbidden, 55— Unavailable intervals in harmony, 56— Melodic
progressions which must be avoided, 57, 58— Implied root-progressions may
be disregarded, 59— Choice of subject; an uncomfortable subject worked,
60-65— A &ood subject, 66—Ftrsf species: in the major, 67— Ditto in the
minor, 68 — Second species: case in which similar motion is possible, 69 —
Examples, 70, 71 — Third species: examples, 72, 73 — Fourth species: rising
suspensions allowed, 74— examples, 75, 76— Fifth species: examples, 77, 7*—
The subject not to be transposed, 79— Two counterpoints employed simul-
taneously, 80— Subjects for double counterpoint in the tenth, 8x.

viii CONTENTS.

CHAPTER IV. — STRICT DOUBLE COUNTERPOINT IN THE TWELFTH page 36

Double counterpoint in the twelfth defined, 82 — Much easier than in the tenth,
83 — Effect of transposition in the twelfth, 84 — Table of intervals ; employment
of the sixth, 85 — Unavailable harmonic combinations and melodic progres-
sions, 86 — Choice of voices, 87 — The cadence always free, 88— Any subjects
available, 89 — The first note, 90 — First species : examples, 91 — Second species :
examples, 92, 93 — Third' species : examples, 94, 95 — Fourth species: its diffi-
culty ; examples, 96, 97 — This species of little use, 98^ — Fifth species : examples,
99, loo— Farewell to strict counterpoint, 101.

CHAPTER V.— FREE DOUBLE COUNTERPOINT IN THE OCTAVE, TENTH,

AND TWELFTH ON A CHORAL page 45

All harmonic resources available ; prohibition of the bare fourth, 102 — Fifth species
only need be practised, 103 — Dissonant notes, when sounded together, 104 —
Examples from Bach, 105-107 — The resolution of dissonances to be con-
sidered, 108, 109 — Employment of fundamental discords ; how used in double
counterpoint of the tenth and twelfth, no, m — Limitations in melodic pro-
gression, 112 — Free double counterpoint on a choral, 113 — Example in the
octave; the implied harmony to be considered, 114 — Necessity of considering
root-progressions in free double counterpoint in the tenth, 115 — Possibility of
similar motion in ditto, 116 — Example explained, 117-120 — Example in the
twelfth 121-126— How to choose subjects, 127.

CHAPTER VI. — FREE DOUBLE COUNTERPOINT ON A FLORID SUBJECT page 58

Definition of "florid subject," 128 — Subject and counterpoint should be con-
trasted, 129— Relative importance of double counterpoint in the octave, tenth,
and twelfth, 130— Various uses of double counterpoint, 131 — Double counter-
point in the octave : examples from Bach, 132-135 — Ditto from Handel,
136-139— Examples from Haydn's quartetts, 140-142 — Ditto from his sym-
phonies, 143, 144 — Examples from Mozart, 145-147 — Ditto from Beethoven,
148-150 — Example from Cherubini, 151 — Ditto from Mendelssohn, 152, 153 —
Ditto from Brahms, 154 — Double counterpoint in the tenth : examples from
Bach's "Art of Fugue," 155 — Ditto with added thirds and sixths, 156-158 —
Simultaneous double counterpoint in the octave, tenth, and twelfth, 159 — In-
cidental employment of double counterpoint in the tenth, 160 — Example by
Handel, 161 Ditto by Jomelli, 162 — Ditto by Haydn, 163 — Ditto by Mozart,
164 — Ditto by the author, 165— Extended extract from Bach's "Art of Fugue"
analyzed, 166-168 — Double counterpoint in the twelfth: examples by Bach,
169-172 — Ditto by Handel, 173, 174 — Ditto by Mozart, 175 — Ditto by
Beethoven, 176 — An entire piece written in this counterpoint by Kirnberger,
177 — Analysis of the same, 178 — Spurious double counterpoint, 179 —
Examples, 180, 181 — The importance of the study of good models, 182 —
Working exercises, 183.

CHAPTER VII.— DOUBLE COUNTERPOINT WITH FREE PARTS ADDED page 89

Double tounterpoint in only two parts comparatively rare, 184— Two simultaneous
cantifermi, 185— Their harmonic possibilities, 186-188— Selection of harmony
notes, 189 — Auxiliary notes, 190 — Free parts added to a short subject, 191—
One added part in th& middle, 192 — A free part above, 193, 194 — Ditto

CONTENT*. ix

below, 195, 196— Two free parts added in the middle, 197— Two upper parts
added 198 — One part above and one in the middle, 199— Two parts below,
200— One above and one below, 201 — Summary of these examples, 202— The
added parts themselves in double counterpoint, 203— Adding plain chords,
204— Additional parts improving weak progressions, 205, 206— Added parts
to examples previously given ; from Bach, 207, 208— Example from Handel,
209— Ditto from Beethoven, 210— Ditto from Cherubini. 211— Double counter-
point in the tenth with added parts, 212, 213 — Ditto in the twelfth by Bach,
214 — Ditto by Mozart, 215 — Directions for working exercises, 216.

CHAPTER VIII.— DOUBLE COUNTERPOINT IN THE RARER INTERVALS page 105

The rarer double counterpoints only employed incidentally, 217 — I. Double counter-
point in the ninth: Table of inversions, 218 — Example by Marpurg, 219 —
Ditto by Lobe, 220 — Ditto by Beethoven, 221 — II. Double counterpoint in the
eleventh : Table of inversions, 222— Example by Cherubini. 223 — Ditto by
Bach, 224, 225 — Ditto by Beethoven, 226 — III. Double counterpoint in the
thirteenth : Table of inversions, 227 — Example by Cherubini, 228— Ditto by
Bach, 229 — Simultaneous double counterpoint in the octave and thirteenth ;
example by Handel, 230 — Ditto by Beethoven, 231, 232 — IV. Double saunter-
point in the fourteenth : Table of inversions, 233 — Example by Marpurg,
234 — Ditto by Bach, 235 — Double counterpoint in the ninth and fourteenth,
by Bach, 236 — Ditto in the thirteenth and fourteenth, by Beethoven, 237—
Simultaneous counterpoint in the twelfth and fourteenth, by Beethoven, 238—
The cause of the difficulty of these rarer counterpoints, 239.

CHAPTER IX. — TRIPLE AND QUADRUPLE COUNTERPOINT ... page 116

Triple and Quadruple Counterpoint defined, 240 — Triple counterpoint : the pos-
sible combinations, 241 — Triple Counterpoint in the strict style useless, 242 —
The addition of thirds to a double counterpoint in the octave, 243 — A faulty
example, 244 — The only legitimate kind, 245 — Treatment of the fifth of a
chord, 246-250 — Consecutive chords of the sixth unavailable, 251 — Selection
of chords, 252 — Each part to appear once in the bass, 253 — Example fronr.
Cherubini, 254 — Examples from Bach, 255-257—01110 from Handel, 258 —
Ditto from Mozart, 259 — Ditto from Cherubini, 260 — Ditto from Beethoven,
261 — Ditto from Haydn, 262— Quadruple Counterpoint : its possible positions,
263 — General principles, 264 — Example by Cherubini, 265 — Ditto by Bach,
266-268 — Ditto by Cherubini, 269 — An example by Haydn analyzed, 270-273 —
Conclusion, 274.

PART II.— CANON
CHAPTER X.— IMITATION page 133

Frnitation defined, 275, 276 — Strict and free imitation, 277, 278 — Direct imitation,
279— Ditto by inversion, 280-282 — Ditto with reversed accents (per arsin et
thesin), 283 — Ditto by augmentation and diminution, 284 — Invertible imita-
tion, 285— Interrupted imitation, 286— Retrograde imitation, 287— Partial
imitation, 288— Close imitation, 289— Maybe accompanied by free parts, ^go-
Examples : in the unison and octave, 291, 292— In the fourth below, 293— At
various intervals, 294— Sequential, on a pedal bass. 295— At various intervals.
296, 397 — Direct and inverted ; by diminution and augmentation, 298 — By

CONTENTS.

augmentation, 299 — By diminution, inverted and direct, 300 — Close imitation
by contrary motion, 301 — By inversion in a major key, 303 — "Per arsin et
thesin"; partial imitation, 303 — Double imitation by inversion, 304— Canonic
imitation in four parts, 305 — Directions for the practice of imitation, 306 — The
use of imitation in actual composition, 307.

CHAPTER XL— THE ROUND page 145

Definition of Canon, 308 — Alteration of intervals, 309 — Finite and infinite canon,
310— Varieties of canon, y.\—The Round, 312-314— Two methods of com-
posing a round, 315, 316 — Importance of melodic interest, 317 — A round
composed ; the first phrase, 318 — The second phrase, 319 — The third part, 320—
The cadence, 321 — A fourth part added, 322 — Example by Dr. Hayes, 323—
Ditto by Mozart, for four voices, 324 — Ditto for six voices, 325 — Ditto by
Heethoven for three voices, 326 — Ditto for four voices, 327 — Ditto for six
voices, 328 — The round with instrumental accompaniment, 329 — Ditto for
mixed voices : example by Cherubini 330 — Employment of this kind of canon
in opera, &c. , 331.

CHAPTER XII.— TWO-PART CANONS page 160

Canons employed incidentally, 332 — How other canons differ from rounds, 333^
Varieties of two-part canon, 334 — Finite canon by direct imitation, 335 — An
example worked, 336— Choice •>( interval of reply, 337 — Interval of time, 338—
Strict and free imitation in canon, 339, 340 — Infinite canon; "making the
join," 341 — Examples, 342-345 — Importance of symmetry and form, 346—
Canons in the fourth and fifth, 347, 348 — An infinite canon by inverse move-
ment, 349- Canons by augmentation and diminution, 350 — Infinite canon by
augmentation ; example by C. P. E. Bach, 351 — The use of two part canon
in actual composition ; examples by J. S. Bach, 352, 353— Ditto by Mozart,
354 — Ditto by Schubert, 355 — Ditto by Haydn, 356 — Ditto by Mozart, 357—
Ditto by Dussek, 358— Ditto by Clemen ti, 359.

CHAPTER XIII.— CANONS WITH FREE PARTS— ACCOMPANIED CANONS page 179

Definition of " free parts," 362 — Various positions, 363, 364 — The free parts should
be composed simultaneously with the canon, 365 — Example worked, 366 —
A canon with two free parts, 367 — Examples, 368— -By Bach, 369-371—1)1110
by Mozart, 372-374 — Accompanied canon by Mendelssohn, 375 — Ditto by
Haydn, 376 — Example by Schumann, 377 — Directions for work, 378

CHAPTER XIV.— THE CANON ON A CANTO FERMO, OR CHORAL page 191

A more difficult variety of Florid Counterpoint, 379 — Cause of the difficulty, 380 —
Examples, 381, 382 — How to begin a canon on a canto fernto, 383 — Examples;
a canon in the seventh, 384 — Ditto in the fifth, 385 — Ditto in the octave, 386 —
A minor canto fermo; canon in the fourth, 387— Ditto in the octave at half a
bar's distance, 388 — Ditto by inverse movement, 389 — The advantages of this
kind of work, 390 — Two kinds of canon on a choral, 301 — The choral itself
treated in canon, 392— Example analyzed, 393-396— Examples by Bach,
397-399 — The choral treated as a canto fermo with an independent canon upon
it, 400 — Examples from Bach's " Canonic Variations,'1 401-406.

CONTENTS. n

CHAPTER XV.— CANONS ON ONE SUBJECT, IN MOKE THAN TWO PARTS page 206

Canons for more than two voices, 407 — The most usual intervals, 408 — The method
of composing, 409 — Illustration of ditto, 410 — General principles, 411 — Infinite
three-part canon by Byrd, in the octave and fourth, 412 — Ditto by Friedemann
Bach, 413— Three-part canons by Mozart, 414, 415 — Three-part canon on a
canto fermo, by Azopardi, 416 — Four-part canon in the fifth and octave by
Albrechtsberger, 417 — Ditto with close imitation, by Moeart, 418 — Four-part
canon by dementi, 419 — Infinite four-part canon in the unison, by F. Bach,
420 — Four-part canon with free parts, by Mozart, 421 — Infinite canon for six
voices, by Kirnberger, 422 — Ditto for nine voices by Marpurg, 423 — The
nomenclature of canons, 424 — "Open" and "close" canons, 425— Examples
of close canon 426-428 — An infinite canon for twelve voices, 429.

CHAPTER XVI.— CANONS WITH MORE THAN ONE SUBJECT ... page 290

Double and triple canons, 430, 431 — General directions, 432 — Infinite canon. 4 in 2,
by Mendelssohn, 433 — Finite canon, 4 in 2, by Bach, 434 — Example by Mozart,
435 — Ditto by Schumann, 436 — Canon by Mozart, 4 in 2, by inversion, 437—
Ditto, 4 in 2, with a free bass, by Bach, 438— Ditto, 6 in 3, by Mozart, 439 —
Ditto, 6 in 2, by Raff, 440— Ditto, 8 in 4, by Mozart, 441— Ditto, bv
Cherubini, 442— Ditto, 12 in 4, by Mozart, 443— The practical use of such
canonic writing*, w\

CHAPTER XVII.— CURIOSITIES OF CANON l>age 235

Ingenuity of the old theorists, 446— Canons with double and triple augmentaiion,
447, 448 — The Retrograde Canon (Canon Cancritans) explained ; example by
Bach, 449— Retrograde Canon, 8 in 4, by Byrd, 450— Reverse Retrograde
Canon ; example by Lobe, 451— Ditto by Bolck, 452— Ditto bv Corder, 453—
Canonic imitation by Inverse contrary movement ; example by Chembini, 454 —
The Circular Canon; example by Bach, 455— The Polymorphous Canon, 456—
Example by Stokei analyzed. 457-463— A subject to work, 464— The Riddle-
Canon ; examples from Martini, 465, 466— Ditto by Bach, referred to, 467—
Curious riddle-canon, by Link. 468 Solution of ditto, 469, 470— Canon, 4 in
a, on a canto fermo, with free part, by Byrd, 471 — Canon, 6 in 2, with frer
part, by Tallis, 472 — Infinite canon. 7 in i, on a ground bass by Bach, 473-
Canon, 3610 i, by Romano. 474— Conclusion, 475.

DOUBLE COUNTERPOINT
AND CANON.

PART I.— DOUBLE COUNTERPOINT.
CHAPTER I.

INTRODUCTION.

1. Before commencing the study of the present volume, the
student will be presumed to have completed his course of
Harmony and of simple Counterpoint, both in the strict and free
styles. He will therefore be fully aware that by the word
Counterpoint in its general sense is meant the art of combining
two or more independent melodies so as to make correct
harmony.

2. If two melodies which are to be played or sung together
are so written as to be capable of inversion, that is, if either of
them may be above or below the other, and the harmony still be
correct, we have Double Counterpoint, a term which simply means
"invertible counterpoint." The word "double" is appropriate,
because each of the two parts has a double function; it may
serve either as an upper melody, or as a bass. If three or four
melodies are combined, any one of which can be a highest,
lowest, or middle part, we have triple or quadruple counterpoint,
according to the number of voices. We shall deal first with
Double Counterpoint, reserving Triple and Quadruple for a later
part of this volume.

3. The first thing necessary for the student in commencing
this branch of work is to enlarge his conception of the meaning
of the term Inversion. Hitherto the word has always been used
in one sense — that of changing the relative position of notes by
putting one of them one or more octaves higher or lower than
before, or sometimes by placing one note an octave higher and
another an octave lower. Thus, when we speak of a sixth as
being the inversion of a third, we mean that one of the two notes
of the interval is placed an octave higher or lower than before.
Similarly, the inversion of a chord means the changing the relative
position of some note or notes of that chord, one of them being
the root, thus altering its pitch by one or more octaves. But in

* DOUBLE COUNTERPOINT. (Chap. i.

double counterpoint the inversion may be at any interval, though
inversion in the octave is the most common, and the most useful.
4. It is important that the student should be able to calculate
with ease and accuracy what intervals are produced by the in-
version of other intervals at any given distance, and also, when
two counterpoints are inverted with respect to one another, at
what interval the inversion is made. He already knows
(Harmony, § 25*) that the number of the inversion of an in-
terval in the octave is found by subtracting the number of the
interval itself from 9. The reason we subtract from 9 and not
from 8 is, of course, because the note of the interval which does
not change its position is reckoned twice. Thus a sixth is the
inversion of a third ; and 3 + 6 = 9. We do not usually make
inversions at a less interval than an octave, because if either
melody were of any considerable compass, it is probable that
some of the notes would cross, and there would be no inversion.
If, for instance, we write two subjects,

., iii ! i i i

=P J J J I * J .

and then try to invert them in the fifth, either by placing the
lower part a fifth higher or the upper part a fifth lower, it is
evident that the parts will cross, and that at the # there will be
no inversion, the part which was the higher still remaining so.

* Or *

For this reason inversions at a less distance than the octave are
not used ; but any distance beyond the octave may be taken.

5. It is a general rule that two subjects which are to be
inverted must not be at a greater distance from one another than
the interval of their inversion. In the example just given there
was no inversion in the fifth at * because the two notes were
originally more than a fifth apart. But it would have been quite
possible to invert the passage in the octave,

or in the tenth,

because these intervals are nowhere exceeded in the distance
between the two parts. The practical objection to inversion at a

* The references to "Harmony" and " Counterpoint " throughout this volume
are in all cases to the author's books on those subjects. (Augener Ltd.)

chap, i.j INTRODUCTION. 3

less distance than the octave is, that it restricts the range of the
melodies too much.

6. We saw just now that the inversion of an interval in the
octave was found by subtracting the number of the interval from
9, and we gave the reason for this. The same reason applies to
inversion at any other distance. Hence we get a simple rule of
universal application :

To find the inversion of an interval at any distance, subtract the
number of the interval itself from the next number above that of the
distance at which it is to be inverted.

7. An example or two will make this perfectly clear. We
wish to know what a fifth becomes when inverted in the tenth.
The next number above 10 is n, and 11—5 = 6. The pupil can
verify this at once. In the first example in § 4 the last crotchet
of the first bar is the fifth above G. In the inversion in the
tenth in § 5, the D has become B, the sixth below G. Similarly,
to find the inversion in the twelfth, subtract from 13 ; in the
fourteenth, subtract from 15, and so on in every case. It ought
to be added that the only intervals commonly used forTnverttng'
are"the octave or riiteentn (the double octave) — the latter being
necessary if the two melodies are more than an octave apart —
the tenth, and the twelfth. Inversions at the other intervals are
very rare ; we shall give a few examples later in the volume.

8. In analyzing compositions containing double counterpoint,
such as fugues, it is often useful to be able to ascertain the
interval in which two subjects have been inverted. The process
here is exactly the converse of that in the preceding case.
Observe the two intervals in their different positions ; add their
numbers together, subtract i from the total, and we get the
interval of inversion. For instance, if we find that a third by
inversion has become a sixth, 3 + 6 = 9. Take i from 9, and we
see that the inversion was in the octave. If the third by inversion
had become an octave, 3 + 8 = 11, the interval of inversion was
the tenth ; if it had become a tenth, 3 + 10= 13, and the inversion
was in the twelfth.

9. It sometimes happens, in double counterpoint other than
the octave, that the two voices will be in the same relative
position to one another, but the counterpoint will be at a different
interval, as in the following passages from Bach's Fugue in B flat
(No. 45 of the " Wdhlternperirtes Clavier")—

Here the themes are the same in both passages, but the intervals
are different. To find the nature of the counterpoint in such a
case, invert the smaller of the two intervals in the octave, add the
other interval to the inversion, subtract i, and the remainder

DOUBLE COUNTERPOINT:

[Chap. I

gives the interval of inversion. In the above example, if we take
the third quaver of the first bar, we see a tenth at (a\ an octave
at (£). Of these, the octave being the smaller, we invert it ; it
becomes a unison; 1 + 10=11; therefore the double counter-
point is in the tenth. If we take the first note of the second
bar, we obtain the same result. The third at (£) being the
smaller of the two intervals, we take its inversion, the sixth, and
add it to the fifth at (0), 6 + 5 = 11; and here again the rule we
have just given holds good.

10. We shall now give an example of double counterpoint in
all the usual intervals, taken from No. 40 of Bach's Forty-Eight
Fugues in the " Wohltemperirtes Clavier," which will illustrate
the rules we have given. We shall not quote the full harmony
where it is in three or four parts, but shall merely extract those
voices which are in double counterpoint with one another. At
the 5th bar of the fugue we find the following passage —

J. S. BACH. " Wohltemperirtes Clavier," Fugue 40.

At the 1 3th bar we see it inverted thus —

To find the interval of inversion, we take any of the notes in
both passages, and add their intervals. Let us take the first note
in the second bar. At (a) the interval is a sixth, at (b) it is a
tenth, 6+10 = 16; the inversion is therefore at the fifteenth or
double octave. We know that in harmony a tenth and a third
are practically the same interval. If we call the interval at (&) a
third, we get 6 + 3 = 9, therefore double counterpoint in the
octave, which is virtually identical with that in the fifteenth.

n. Later in the same fugue, we meet with some different
inversions. At the 28th bar is the following —

(The alteration of the first note of the lower part here is the

Chap. I.)

INTRODUCTION.

result of the construction of the fugue, and has nothing to do
with the double counterpoint.) It will be seen that the rest is
identical with the upper part of (a). Here we notice that the
lower part of (a) is transposed a twelfth higher, and we have
double counterpoint in the twelfth. Let us apply our test, as
before, to the first interval in the second bar. At (a) it was a
sixth, it is now a seventh, and 6 + 7 = 13. Lastly, at bar 36, we
get another inversion —

Here the upper part is almost the same as at (c), but the bass is
a sixth lower. To be certain of the distance of the inversion
here, we must once more apply our rule. The first interval in
the second bar at (a) was a sixth ; its inversion is a fifth. As
6 + 5 = 11, the double counterpoint here is in the tenth. It
the student has understood these examples, he will have little
difficulty in analyzing any combination he may meet with in the
works of the great masters.

12. In treating Double Counterpoint we shall begin, as with
simple counterpoint, by working it in the strict style. It must be
understood that this is simply the preparatory technical work ^ to
the free Double Counterpoint used in actual composition. To
those who have conscientiously worked at strict counterpoint, it
will present but little difficulty. We take first double counter-
point in the octave and fifteenth, as being the most used and the
most useful.

DOUBLE COUNTERPOINT: [Chap, n

CHAPTER II.

STRICT DOUBLE COUNTERPOINT IN THE OCTAVE AND

FIFTEENTH.

13. We have incidentally said in the last chapter (§ 9) that
double counterpoint in the octave and in the fifteenth were
virtually identical. The only practical difference between the
two is, that in the former the two parts may not be more than an
octave apart, and in the latter they may. But with this exception,
all the rules for the one apply equally to the other ; and, in fact,
double counterpoint in the fifteenth is, far more often than not,
spoken of as being in the octave.

14. It is evident that inversion in the octave changes neither
the names of the notes of the inverted part nor the 'intervals
between the successive notes of the melody, the only alteration
being that of pitch. We shall see later that this is not the case
in any other species of double counterpoint. But while the
names of the notes of the inverted part remain the same, its
relation to the part with which it is inverted is entirely different.
This will be clearly seen by placing under one another in two

columns all the intervals up to the octave with their inversions.

./
INTERVALS: 12345678

INVERSIONS: 8 7 6 5 4. 3 2 i

Notice that, as mentioned in the last cnapter (§ 4), the number
of the interval added to that of its inversion amounts to 9 in
every case.

15. On examining the above table, we shall see that the
dissonant intervals (the second and seventh) are also dissonant
in their inversions, while perfect and imperfect consonances also
do not change their nature by inversion in the octave. But, as
we are now writing in two parts, the lower part must always make
a correct bass to the upper one (Counterpoint, § 113). The
perfect fifth when inverted, becomes a perfect fourth ; and, in
the strict style which we are now studying, a fourth with the bass
is always a dissonance (Counterpoint, § 29). The perfect fifth,
therefore, although a consonance, can only be employed in strict
double counterpoint in the octave under special limitations, which
we shall explain as we proceed.

1 6. It will further be noticed that an octave when inverted
becomes a unison. Though the octave may be used freely in
simple counterpoint, the unison is only allowed on the first and

Chap. II.]

STRICT, IN THE OCTAVE.

last notes of an exercise. A little more liberty may be permitted
in this respect in double counterpoint ; it will nevertheless be
well for the student to avoid the octave and unison as far as
possible on an accented note, excepting at the beginning or end of
a counterpoint, though their employment is not absolutely pro-
hibited. If the double counterpoint is in the fifteenth, this
caution, as regards the octave, need not be attended to, as the
octave below then becomes the octave above, and vice versa.

17. In double counterpoint in the fifteenth, it will often be
found convenient, instead of changing the position of one voice
by two octaves, to place one part an octave higher, and the other
an octave lower than before ; in many cases, indeed, this may be
necessary in order to keep the parts within a reasonable compass
Take for example the following passage —

(a)

®\>i r j

h—

As these two subjects are in two places more than an octave
apart, it is clear that they must be inverted in the double octave.
But if we place the upper part two octaves lower, the first note

of the second bar will be @' .— , which is too low ; and if we

transpose the bass two octaves higher, the last note will be (fy '

It will therefore be best to put the upper part an octave lower,
and at the same time the lower part an octave higher, thus —

-^nhr^

It will be seen that the relative position of the two parts to one
another is precisely the same as if one had remained stationary,
and the other had been transposed two octaves, but that both are
now in a convenient position.

1 8. We will now proceed to double counterpoint of the first
Species. The chief point to notice here is, that it is impossible
to use the interval of the fifth at all, because by inversion it
becomes a fourth, which in strict two-part writing is unallowable.
Care must also be taken not to exceed the compass of the octave
between the two voices, unless the inversion is in the fifteenth.
If the subject leaps much, it will often be impossible to avoid the
overlapping of parts (Counterpoint, § 31); this is less objection-

8 DOUBLE COUNTERPOINT; [Chap. 11

able in double than in simple counterpoint. The only available
form of cadence in this species is

Inversion.

I II

19, One point remains to be mentioned. Owing to the
necessity of retaining the same melody above and below the
subject, we can allow ourselves rather more liberty than in simple
counterpoint as to implied harmony. This refers more particu-
larly to the interval of the third above the mediant, which in the
majority of cases will represent I£, and not Ilia. The inversion
of this interval can only represent 1 1 L£, and many cases will
occur in which this chord has to be followed by one of those
which, in the "Table of Root Progressions" (Counterpoint, p. 32),
is marked as only "possible." We may be content now if we
avoid the absolutely bad progressions.

20. The best way of writing exercises in Double Counterpoint
is to use a score of three staves, placing the Canto Fermo in the
middle, and writing the counterpoint above, and its inversion
below. It will, of course, be understood that no three-part
harmony is implied ; but this method gives the best opportunity
of observing the two melodies in their dual relation to one
another. It will be well to indicate this by a double brace at
the beginning of the lines, thus —

21. We will now take two of the subjects so often treated in
Counterpoint^ and work on them double counterpoints of all
species, beginning with the first. We take first a subject in a
major key —

Double Counterpoint.

U' S

=^=E^E)I

ii i, Inversion in the 8ve.

6 66

It will generally be found convenient to write the counterpoint

CM), ii.] STRICT, IN THE OCTAVE.. 9

and its inversion (when in the octave) for two voices, the compass
of which is an octave apart ; either (as here) the subject in the
alto with the counterpoints in treble and tenor, or (as in some of
the examples we shall presently give) the subject in the tenor
with the counterpoints in alto and bass. The two basses — the
subject when the counterpoint is above, and the counterpoint
when the subject is above — should always be figured. The only
point requiring notice in the example just given is that in the
third and fourth bars of the inversion the parts overlap (§ 18).
The only way to have avoided this would have been to take the
unison A as the fourth note of the upper part, and this would
have been far more undesirable than the course we have
adopted. The repetition of the nqte C would have been extremely
weak ; no repetition of a note should be allowed in two-part
counterpoint.

22. In our next example we will, for the sake of variety, write
the counterpoint in the

"7T

Z3 =2 & „ H

(a)

(*)

6 6
Inversion in the isth.

6
<')

6
(<t)

We here write the inversion in the bass, as it would be too low
for a comfortable tenor part. Notice at (a) the chord I IL*, and
observe its progression. The third interval (b) is marked as a
chord of the sixth. The tenth (or third) g might also represent
Ilia ; but if we so consider it here, the progression to the
following chord (Ilia to IV 'b) is one of the bad ones, while if
we regard it as Ib, the progression from lllb to 13, though not
one of the strongest, is at least possible. Here is an illustration
of what was said in § 19.

23. Now let us look at the inversion of this counterpoint.
At the third bar we find lllb again. The following chord (c)
may equally well represent IVtf and lib; but (as at (b) just
noticed), we regard it as lib here, because lllb to lib is a good
progression, while lllb to IVa is only a possible one. At (d) we
see another of the weaker progressions, lllb to V£. These will
often be necessary in strict double counterpoint of two parts.

24. We will now take a subject in a minor key. As with
simple counterpoint, this will be found more troublesome than a
double counterpoint in the major, because of our smaller choice
of harmony. We therefore, as usual in cases of difficulty, permit
ourselves a little more liberty. The octave and unison may be

DOUBLE COUNTERPOINT;

[Chap. II.

somewhat more freely used than in a major key. But the most
important concession refers to the harmonizing the mediant of
the scale. In simple counterpoint the sixth below the mediant
is forbidden (Counterpoint, § 118), because of its implying a chord
which is unavailable. But if we disallow it here, we shall be
also prevented from using the third above the mediant, repre-
senting \b\ and the only possible notes to place above the
mediant will be the octave or unison and the sixth. It will often
happen that neither of these will be good ; therefore, as the sixth
below the mediant is itself a consonance we can use it, if neces-
sary, in double counterpoint, though it implies no available chord,
because here the claims of melody are superior to those of
harmony. In figuring this interval it will be well to put the 6 in
brackets, thus — (6) — to show that it implies an interval only,
and not a chord. An illustration of this point will be seen in the
example now to be given —

£36

|iR| ^ tj

(6)

25. Here we have intentionally taken a subject which is not
very easy to work. Let it be noticed that the upper melody is
here almost, so to speak, compulsory. We can take no other
note than the octave to commence with ; as the lower counter-
point should not begin with a first inversion. The fourth note
of the subject is the only one which allows a choice of harmony,
and if instead of D (representing IL£), we take the chord of IVar,
every note of that chord gets us into trouble. F in the upper
part gives either a unison or bad hidden octaves ; C would be a
repetition of the preceding note ; if we take the upper A, we have
a seventh with one intermediate note ; while the lower A must
either be followed by the unison, or by a leap of an augmented
interval. Up to the fifth bar, therefore (to borrow a metaphor
from the chess-board), every move is virtually forced.

26. Now look at (a), bar 6. The E flat of the subject can
only bear a first inversion above it ; the only possible notes of
the counterpoint are C, E, and G, the root, third, and fifth of
the tonic chord. Our cadence is already fixed (§ 18) ; if we
take C here, we shall not only have the very weak repetition
X-^g? | ra j ff^~| i--* |? but the whole counterpoint will consist,

with the exception of the fourth note, of nothing but C and B tj

chap, ii.] STRICT, IN THE OCTAVE.

E flat will not do here ; we cannot come down by similar motion to
a unison, to say nothing of the impossible leap of an augmented
fifth ; and if we take the upper octave we break a law of melodic
progression (Counterpoint, § 19), for after the leap of a diminished
fourth we do not return within the interval. We have therefore
absolutely no good note here but G, and we consequently take it
in spite of the fact that its inversion will represent an interval, and
not a chord. It will not often be needful to use this interval in
any other than the first, and sometimes the fourth species, because
of the larger resources at our disposal. Observe that it would
have been possible to take G also at the third bar of this example ;
we did not do it then, because there was no necessity for it.

27. For the sake of getting more variety in the melody, we
will make our next counterpoint at the i5th.

Lfh 4t 1

jSSJ 1

b s

(«)

£16

6 6 Q6 H 6

After the full explanation given of the last example, but few
remarks are needed for this. At (a) we have taken the unison
as the best note available ; B fl would have led to the same posi-
tion as in the last example ; we cannot repeat the D ; and if we
take E (the interval of the sixth which we have just been dis-
cussing), we shall have four consecutive sixths between the two
parts. Though these are not so strictly prohibited in double as
in simple counterpoint, it is well to avoid them if possible.
Besides this, we wished to show that a counterpoint could be
written on this subject without using the sixth above the dominant
at all.

28. In double counterpoint in the octave, of the second
species, the dissonant intervals (the seconds, fourths, and
sevenths) can be introduced as passing notes on the unaccented,
and even occasionally on the accented, parts of the bar. The . /,
fifth, being the inversion of the fourth, can also be thus used ;

but it is important to remember that it can only be taken in an oil -
upper part in a descending, and in a lower part in an ascending,
passage. A moment's thought will show the student the reason
for this.

29. The cadence of the second species differs in an important
respect from that of any other. It is impossible in this species

12 DOUBLE COUNTERPOINT: (Chap.n

to make a cadence which can be properly inverted. The usual
cadences for simple counterpoint in the upper part are

=£f=F.

v/_

*K^

-«

t- c

9 -^

If we invert (a) in the octave, we shall have a fourth below the
subject on the accented beat; while (b) will not invert in the
octave at all; and though possible to invert it in the 15th, it
would still not be good to do so, because of the weak progression
of the harmony with inversions of two chords in the bar. On
the other hand, of the usual cadences in the lower voice

S(«) (b) (c)

(&} exceeds the limit of an octave, and all three when inverted
give a fourth taken as a harmony note, instead of as a passing
note. For this species, therefore, the cadence is always free, that
is to say, no attempt is made to invert it, but the last three or
four notes of the two counterpoints are quite different. It should
be noted that the forms (b) of both upper and lower cadences
are available for double counterpoint in the octave, though con-
taining the interval of a tenth, as they have not to be inverted.

30. We now give some examples of the second species, taking
the same subjects as before —

i — M — i — 1 \ — 1 r \ ^ 1 ! — EE

s _ w J

r ' "

in'

66 6
Free.

r 1 r_ i I r M r 1 r r . r a». i -

6 -

At (a) we see the fifth introduced on the second beat of the bar
as a passing note. The last four notes in the inversion show the
free cadence spoken of in the last paragraph. As with simple
counterpoint of the second species, it is best to take passing
notes, where practicable, on the unaccented beats, in order to
secure a smoother melody.

31. We now write another double counterpoint, also in the
octave, on the same subject, endeavouring to get as much variety
as possible.

Chap. II.]

STRICT, IN THE OCTAVE.

At (a) and (<r) will be seen the unison on the unaccented beat.
At (b) is the perfect fifth used as an auxiliary note, and not (as in
the last example) as a passing note.* At (d) we have an auxiliary
note quitted by leap of a third (Counterpoint, § 165). This
device should be sparingly used in strict writing ; it is introduced
here to obtain a better melody. The only other notes available
would have been A, which would have been weak here (compare
the preceding and following bars), or F, which would have given
in the next bar a seventh with one intermediate note.

32. We next take our minor subject, and first write to it a
double counterpoint in the octave —

- (6)

n -

The only point to notice here is that the inversion of the third
bar at (a) gives the interval of the sixth above the dominant
(§ 24). Here, however, we distinctly have a chord implied in
the second half of the bar, viz. : I£. We have, therefore, figured
the interval with (6) and exceptionally marked the implied
harmony under the second minim of the bar.

33. One more example of the second species will suffice, and
this shall be in the fifteenth.

6 6 6 - 6 -

*For the distinction between auxiliary and passing notes see " Hanmmy,

r4 DOUBLE COUNTERPOINT: [Ch*P. n

At (a) the bass evidently implies two chords in the bar. Though
we rejected this progression for the cadence, it may occasionally
be introduced in the course of a counterpoint. Special attention
should be given to (b). Here the sixth above the dominant at
the beginning of the bar does not of necessity imply the bare
interval, as in the third bar of the last example. On the contrary,
it is better here to regard it as an accented passing note ; and the
passage shows us the one exceptional case in which the fifth may
be taken as a harmony note. It must be on the second half of the
bar, preceded by an accented passing note, and (as its inversion
will be a fourth) it must be, as here, quitted as well as approached
by step. It is but seldom that opportunity will occur for its use ; it
is introduced here to show under what circumstances it is possible.

34. In the third species the general rules for simple counter-
point of the same species are mostly to be observed. A fifth can
still only be taken as a passing or auxiliary note ; the exceptional
treatment of this interval shown in our last example is seldom
practicable with four notes to one. But the rule which restricts the
employment of the fifth of a chord in the lowest part ( Counter-
point, § 223) is considerably relaxed in double counterpoint, as its
use in an upper part would otherwise be extremely limited. It may
in double counterpoint be taken on any part of the bar except the
first ; it is, however, better not to use it below the root when this
note is present, unless the fifth is so treated as to have something of
the character of a passing note, or come between other notes of the
chord. Illustrations of these points will be seen in our next example.

35. Excepting on the first beat of the bar, an octave in this
species should never be approached by conjunct motion, as its
inversion will give the unadvisable progression from a second to a
unison. This, though possible when the second is a passing note,
should be avoided altogether in strict double counterpoint. The
only good cadence for this species is, with four notes to one,

i I . i S(Inversion.)

With three notes to one, g (inversion.)

a J J ' I " • II

sr r

36. Our first example of this species with four notes to one

a r ' r ' i i i r i i

(«)

i i r i

Chap. II.J

STRICT, IN THE OCTAVE.

shows at (a) how little variety is sometimes possible in com-
mencing one of these counterpoints. When the subject leaps, as
here, from tonic to dominant, there is no other good commence-
ment than that given here, or in the following example, which it
will be seen is almost identical. A double counterpoint is mostly
harder to write on a subject that leaps than on one that moves
chiefly by conjunct degrees. At (b) in the inversion is seen the
fifth of the chord below the root. Notice that here the root is
not present, and we do not trouble ourselves, as in simple
counterpoint, about the implied |. At (c) the root is present, but
here the G, coming between F and A, though a note of the
chord, acquires the character of a passing note.

37. In our next example, written on the same subject,

i r j f i i r i • 'i

6
(*)

it will be seen that except the cadence no bar is the same as in
the last counterpoint. At (a) the fifth is taken below the root,
but (as at (b) of the last example) the root is not present in the
upper part. At (&) the A in the bass is figured as a sixth,
because the implied harmony (VII£ to la) is good, while
Ha to la would be bad.

DOUBLE COUNTERPOINT:

I Chap. H

38. We next give two examples of four notes against one in a
minor key —

-M m — •> •*
t S

' r r

— i —

— i ' — i

Jin

— J — «

(a)

ikJl b

i r r r r

1 |» f r p .

f*4 r ' r HTJ r r =

— E_a — ! — i — — i — a — i —

• r r

frh " -t—

P !

8 6

After the explanations already given, the only remarks to be
made on these counterpoints are, that at (a) in the inversion of
the first example two chords in the bar must be implied ; and
that, as with the examples in the major, every bar in the two
exercises is different, except at the cadence, for which there is no
other good form.

Chap. II.]

STRICT, IN THE OCTAVE.

39. Counterpoint with three or six notes to one being so
much less frequently used than that with four, it will suffice to
give one example of each.

Bigfe J ' r rrrr i rrnr rrTr"rirrj - rn

• 6 -

r-rrri'

1 I I T l I E

=>==•

These counterpoints require no explanation.

40. The fourth species is very difficult to work in strict
double counterpoint, owing to the limited means at our disposal.
The suspensions 9 8 and 4 3, with their inversions are available ;
but 6 5 is evidently inadmissible, as also is 7 8 in an upper part,
though its first inversion 5 6 can be taken. We are also greatly
restricted as to our syncopations by the prohibition of the fifth.
The only good cadence for this species is

(Inversion.) S

If the suspension cannot be prepared, the cadence must be free,
as in the second species.

DOUBLE COUNTERPOINT;-

[Chap. II.

41. It would be easy to write subjects against which it would
be absolutely impossible to put a double counterpoint strictly of
the fourth species. The student therefore, though it will be well
for him to work a few exercises of this kind, need not devote
much time to it; the fifth will repay him much better for his
labour. We give two examples, as specimens —

rr-fl

. •§ ZZ22Z

&—

P^F

— 1 (2_

(S> —

— f-^-^ — H

|llj i|

1—

M=d

-1 G^-l

<«)~

rj 1

-1 1

— -H

:ci!
fe

4
-(*)

>*•

5 6

-X X-

"•»

^ '

7 6

-v /•

-v

' P-

-P

— 1 1 — 1

^~ — r~

~^~ '

— 1 1

-P —

It so happens that this subject can be very easily worked ; it has
not been needful to break the syncopation at all.* Note that at
(a) the unison is taken, because otherwise we should have had
five consecutive sixths between subject and counterpoint. Let
the student also mark at (b) the fifth as a prepared suspension in
the lower voice, and ask himself why it could not be equally so
taken in an upper part.

42. Our minor subject is much more difficult to treat
satisfactorily.

'"^ —~s

&- t^""^,-,

3 I I I "Li

\ ' I' ' IJEEE*

(*)

JJ5 6 6

(6)

6 7 96

6 5 2

5 --

Observe that at (a) we are forced to break the suspension, as the
G, if tied, would in the bass have become a fourth below the
subject. At (b} it is only possible to continue the suspension by
taking the rather unsatisfactory interval of the sixth above the
dominant. Notice also that the inversion of (a) must evidently
imply two chords in the bar.

* The author has, since writing this counterpoint, discovered that it is identical
with the example he had given in § 268 of Counterpoint. The coincidence is
purely accidental ; the earlier book was not referred to at all while writing these
exercises.

Chap. II.) STRICT, IN THE OCTANE. 19

43. The fifth species will be found not only less difficult, but
more interesting to work than the fourth. The best forms of
cadence are

^n Ji .. •"• _n - ji ,.-.

The student will by this time be quite able to see the inversions
for himself. The form of cadence at (a) is on the whole best ;
but (b) will be needful if it be impossible to prepare the sus-

„_/>

pension, as, for example, if the subject ends ||— ^ \~~===-

1U S3

It should also be noticed that it will now be possible to take the
fifth of a chord as a harmony note in the upper voice, though this
could not be done in the fourth species. Our next example will
show how this is to be managed.

44. We now give some specimens of the fifth species —

e*=3^=

Hfl s

^-~
(jO:it4 > 1 — p" —

4 3

E- + m *

i
_pi — p — 1_^ — L^r r — F— i

U 4 J 1

M '

n '

6
(*)

At (a) we have the fifth of a chord taken as a harmony note in
the upper voice. As it becomes a fourth by inversion, it must
of course be prepared, and it must descend ; the fourth, to which
it descends, becomes a fifth in the inversion. It will be seen that
those notes which in the upper counterpoint are harmony notes,
become passing notes in the lower, and vice versa. At (b) of the
lower counterpoint we have the implied bad chord progression
I la to la. We have more than once said that in double counter-
point, considerations of root progression, though not to he wholly
disregarded, are of less importance than a good flow of melody.

DOUBLE COUNTERPOINT:

CChap. II.

45. For the sake of getting more melodic variety, we will
write our next example in the fifteenth, instead of the octave —

pit i j*. . . j c=a • |- j | rnj

) IUI S & ^

6 6
In the i5th. ^-^

fl1 f r r r ' ' r ^

fa — ^ r^

— -"

— CJ 1|

0 6
<*)

At (a) in the upper counterpoint, there are no consecutive
octaves with the subject, because the tied E at the beginning
of the bar is a note of the harmony. The lower counterpoint
at (ft) must clearly imply two chords in the bar.

46. We now give two examples of the fifth species in a minor
key-

I J •> H'

I -£ fg—p ^

<., r r

^ —

— .

Chap. II.]

T, IN THE OCTAVE.

]• 1

• —

1* h|» i

r—j: IT r " p »

b ^

«>

x—

- tr r P i

b* ' r '

4 ft «ft

Inl1^ ' :

1 F b=J — ' 1—1 «*

1 1 1=11

6 56

E=5 H

1 1— 1 ^*i 1 *^
(f\ - fi

r ' i a

Very little explanation is needed here. At (a) of the first
example, we see the same treatment of the fifth as a consonance
which was noticed in § 44. At the first bar of the second
example at (a) it looks as if the suspension were prepared by a
crotchet. Though this would not be absolutely forbidden, it is
best in general that the preparation should be a minim. In the
present case the G has been already sounded as the first note of
the counterpoint, and the mental effect is therefore quite satis-
factory. In the two examples just given, an effort has been
made to obtain as much variety as possible, both in melody and
rhythm.

47. It is comparatively seldom that in actual composition we
find double counterpoint written against a subject, the notes of
which (as in the canti fermi given in this chapter) are of equal
length. Here are two examples, as specimens —

HANDEL. " Judas Maccabeus.

&c.

MOZART. Mass in C minor.

w^> •_

Far more commonly both the parts which are in double counter-
point are in notes of unequal length — two parts of the fifth

DOUBLE COUNTERPOINT .-

[Chap. IL

species, so to speak. This kind will be dealt with later, when
we come to treat of free double counterpoint.

48. We conclude this chapter by giving a few subjects for
double counterpoint in the octave. It will hardly be possible
to write ten or twelve different exercises on the same canto fermo^
as with simple counterpoint ; to do this well, would require the
mastery of resource of a Bach. But with patience and per-
severance, the student will generally be able to invent two or
three counterpoints in each species, except the fourth. When
he has worked all the subjects given here, he may take any of
those to be found in any treatise on counterpoint, or, if he
prefers, he may write canti fermi for himself. As soon as he has
acquired fluency in strict double counterpoint, he will be ready
to commence the far more interesting study of the free double
counterpoint of Bach, Handel, and the great masters who have
followed them.

SUBJECTS FOR DOUBLE COUNTERPOINT IN THE OCTAVE.
(i.)

(ii.)

(in.)

(IV.)

I I

(V.)

rf^

(VI.)

(VII.)

i •• i

(IX.)

(X.)

chap, IH.J STRICT, IN THE TENTH. a 3

CHAPTER III.

STRICT DOUBLE COUNTERPOINT IN THE TENTH.

49. Double Counterpoint in the Tenth is that in which a
counterpoint to a given subject is inverted a tenth higher or
lower, as the case may be. But, as the interval of a tenth con-
sists of an octave and a third added together, there are two other
ways of inversion in the tenth. One of the two parts may be
raised an octave, and the other lowered a third ; or one may be
lowered an octave and the other raised a third. It is important
that the student should clearly grasp the fact that, whichever of
these four methods of inversion be chosen, the relative position of
the two notes to one another will remain the same, though their
absolute pitch will be different in each case.

50. An example will make this perfectly clear.

(«) (*) (c) (d) (e)

At (a) is the interval of the perfect fifth. Let us invert it in the
tenth in the various ways just described. If we keep the upper
note in its place, and put the lower note a tenth higher, we
have (b) ; if, on the other hand, we put the upper note a tenth
down, we get (r). Now move both, but in opposite directions,
just as we did in § 17 when inverting in the fifteenth. If we
raise the lower part a third, instead of a tenth, and bring the
upper part down an octave, we have (d\ which it will be seen is
the same as (^), but an octave lower ; and lastly, if we lower the
upper part a third, and raise the lower part an octave, we get (e),
which is the same as (c) an octave higher. The important point
to notice is that in each case the inversion of the fifth gives the
same interval — the sixth, though in two cases it is a major, and
in two a minor, sixth.

51. The above example shows one of the chief differences
between double counterpoint in the tenth, and that in the octave.
When a note of any interval is inverted in the octave, its name
always remains the same; but inversion in the tenth always

24 DOUBLE COUNTERPOINT: fchap. in

changes the name of a note. To find the inversion of any
interval in the tenth, we subtract the number of that interval
from 1 1 (§ 6). This gives the following table —

INTERVAL : 12345678910

INVERSION IN THE TENTH : 1098765432 i

Of course no interval larger than a tenth can be used in this
counterpoint (§ 5).

52. It will be seen from this table that every consonance
when inverted in the tenth remains a consonance, and every dis-
sonance remains a dissonance. It will further be noticed that
the perfect consonances (the unisons, fifths, and octaves) become
imperfect consonances (tenths, sixths, and thirds) when inverted,
and vice versa. The perfect fourth is of course a dissonance in
two-part counterpoint. The only exceptions to the general rule
given above are that the sixths above the subdominant, both in
the major and minor key, and above the submediant and leading
note in the minor key, become dissonant fifths by inversion.
This point we shall notice later.

53. It must be further remarked here that inversion in the
tenth changes not only the names of the notes but their position
in the scale, and their consequent relation to one another. For
example, if we take the first three notes of the scale of C, and
invert them either in the tenth above or in the tenth below, we
change the position of the semitones —

(a) <*) Q (c)

At (a) we see a tone between the first and second notes, and
another tone between the second and third. If we invert the
passage a tenth higher, as at (b\ we have a semitone between the
first and second, and a tone between the second and third notes ;
while the inversion a tenth lower, as at (c\ gives a tone between
the first two notes, and a semitone between the second and third.
It will thus be seen that the whole character of a melody is
changed by inversion in the tenth, unless we add accidentals to
take it into another key.

54. If we now turn to the table of inversions given in § 51,
we shall be able to draw some inferences which will assist us in
making rules for writing double counterpoint in the tenth. In
the first place, we notice that as the third by inversion becomes
an octave, and the sixth becomes a fifth, it is impossible to have
consecutive thirds or sixths, such as we are accustomed to in
simple counterpoint, or in double counterpoint in the octave.
Hence we get our first general rule :

No consecutive intervals of any kind are allowable.

55. Now we go one step further. We know that in strict

Chap, in.] STRICT, IN THE TENTH.

counterpoint hidden fifths and octaves are altogether forbidden.
But if, in double counterpoint in the tenth, we approach either
a third or a sixth by similar motion, the inversion of the passage
must give an octave or a fifth also approached by similar motion.
Therefore, in this counterpoint we cannot employ similar motion
at all, and our second rule is :

Only contrary and oblique motion are available.

As no repetition of a note is allowed in the first species, we
are in this evidently restricted to contrary motion.

56. The limitations to which we have to submit in writing
double counterpoint in the tenth are by no means exhausted yet.
There are various intervals, both in harmony and melody, which
we shall now see are unavailable, because they cannot be in-
verted without breaking rules. If the subject contains the sub-
dominant, this note cannot have a sixth above it, because its
inversion in the tenth below will give the interval of the
diminished fifth. The leading note in the subject can in a major
key take no interval but a sixth above it; because the third
when inverted would give a doubled leading note, while the fifth
is a diminished fifth, and the octave is obviously impossible. In
a minor key, there is no note that can be placed above the lead-
ing note— at all events in the first species; because here the
inversion of the sixth gives the augmented fifth. In other species
the difficulty may be evaded in some cases by treating the
augmented fifth as an accented auxiliary note.

57. There are also several pitfalls to be avoided in the
melodic progressions. The leap in an upper part from the super-
tonic down to the submediant, and its converse, the upward leap
from submediant to supertonic, give in the tenth below the
interval of the tritone.

Tenth below.

" I " =£**=

If the counterpoint is in the lower part, it is clear that the leap
up from supertonic to dominant, and its converse, will also give a
tritone when inverted in the tenth above.

58. In the minor key we are even more hampered, in conse-
quence of the four augmented intervals to be found between
various degrees of the scale. The simplest way to show the
intervals to be avoided will be by the table here given —

Unavailable

< ,>> (*>

in upper part.
Augmented

Intervals.
Unavailable

in lower part.

26 DOUBLE COUNTERPOINT. ichap.ni.

The middle staff shows the four augmented intervals of the minor
key. The upper staff shows the intervals which if inverted in
the tenth below will give augmented intervals, and the lower staff
shows intervals which become augmented when inverted in the
tenth above. We have omitted the lower intervals at (b) and
the upper ones at (c) because there is no danger with these, as
they are tritones themselves, and the student will of course avoid
them. Some of these intervals, however, can be occasionally
saved, as will be shown later, by the use of the melodic forms of
the minor scale.

59. It will be seen that double counterpoint in the tenth
requires so much to be avoided that its rules may be compared
to the laws of the Decalogue, nearly all of which begin with the
words, " THOU SHALT NOT." Consequently, this kind of counter-
point is far less frequently met with, and much less useful than
that in the octave. It is nevertheless important that the student
should be able to work it, and he will find its practice very
beneficial. But, as he will have to work under such difficulties,
he need not now trouble himself at all about implied root-
progressions, and may content himself if his melodies are good.

60. It is not every subject which is suitable for double
counterpoint in the tenth, especially in the strict style. In actual
composition, where we are free to make our own parts, we should,
of course, take care to write the two melodies with special
reference to their inversion in the tenth ; but with many of the
ordinary canti fermi it will be found all but impossible to write
a satisfactory counterpoint of this kind. To illustrate this, we
will take the major subject which we used in the last chapter, for
double counterpoint in the octave, and try to write a double
counterpoint in the tenth , on it. We give the subject, numbering
the notes, for convenience of reference —

We shall take the first species, as being the simplest, and also
because the harmonic progressions will almost always be the
same in the other species. We shall not have the same freedom
of choice here that we should have were our double counterpoint
in the octave.

6 1. As our exercise should begin with the tonic chord in
root position, our first note above the C of the subject must be
E ; for the octave C, if inverted in the tenth, would give A in
the bass, while G would give E, representing the first inversion.
We must, of course, take E as the tenth above C — not the third,
or the parts will cross on the next note. Of course if the second
note of the subject is below the first, we must begin with the

Chap. III.J

STRJCT, IN THE TENTH.

third, or else the contrary motion with the second note will make
us exceed the allowed interval of the tenth.

62. For our second note we have not much choice. We
already know that we must move in contrary motion to the sub-
ject (§ 55). If the upper part falls to B, the inversion will give
octaves by contrary motion. The unison, though sometimes
necessary, should not be used if it can be avoided. Here D is
obviously the best note, giving B as its inversion. If for the
third note we rise to G, its inversion gives the unison ; we there-
fore return to E.

63. Thus far we have had no difficulty at all ; but now our
troubles begin. The best note to put above 4 will clearly be C ;
but then what shall we do with 5 ? As this note is the sub-
dominant of the key, we cannot have D above it (§ 56), nor can
we rise to A, because the inversion will give octaves by contrary
motion The only possibility is F, the octave of the subject.

£_.

<^z

BE3__

4 5

64. The subject (5, 6) now falls from F to E ; our counter-
point must therefore rise to G— the only possible note (§ 51).
But from E the subject again falls to D ; and the counterpoint
must either exceed the limit of a tenth, or move in similar

DOUBLE COUNTERPOINT:

[Chap. 111.

motion, both of which are forbidden. We are therefore in a fix,
and must " try back."

65. We will now for our fourth note take the unison A instead
of C. This will alleviate our sufferings somewhat, but not much.
The student will see that we can now take C for our fifth note,
and E for the sixth ; we can even go up to F for the seventh —

fir*

1
i

=^=

— ^— \
=3

We are obviously no better off than we were before. But we
have a loophole for escape. In double counterpoint in the tenth,
as with the second species in the octave, the cadence may, if
necessary, be free, i.e., the last two bars need not be invertible.
Sometimes it is possible to continue the inversion to the very
end ; sometimes only the last note need be free. In the exercise
we have been working the best close would have been this —

Here the lower counterpoint at 6 7 moves in thirds with the
subject; but this does not matter, as the second third is not
intended for inversion.

66. We have entered in some detail into the difficulties in-
cidental to this kind of counterpoint, so that the student may
know what points are to be more particularly attended to in
working his exercises. We said above that subjects for double
counterpoint in the tenth should be specially adapted for that
purpose. Instead, therefore, of treating the same subjects as in
the last chapter, we shall write two, one in a major and one in a
minor key, and work them in each of the five species. For our
major subject we choose the following —

It will be noticed that we have avoided all large intervals. A

Chap. III.)

STRICT, IN THE TENTH.

subject which leaps much will often be found somewhat trouble-
some for treatment in the tenth.

67. A double counterpoint of the first species against this
subject gives very little difficulty.

The only points to notice in this example are that at (a) we see
in the upper counterpoint the false relation of the tritone, which
we disregard here (though we would not allow it in double
counterpoint in the octave), because we do not trouble ourselves
about root-progressions provided our melodies are good; and
that, except the last note of the bass, at (£), the inversion is kept
up throughout. It would have been possible here to keep the
inversion to the very end by writing the last half of the counter-
point thus —

Efl

In our next example we shall see the close strict.
68. We now take a subject in the minor —

-*" " 1 =H

ffn

=*==^=:±z=-»— =^zfl

Observe that at (a) we use in the lower voice the melodic form of
the minor scale. This is often advisable ; in the present case the
G of the subject takes the fifth in the upper counterpoint ; if we
write the third, B, we must either make it B fcj, as we have done
below, or have an augmented second from B|? to Cf, which we
must obviously avoid. Moreover, the use of the B y here allows
us to continue the strict double counterpoint to the last note.

69. In the second species of double counterpoint in the
tenth, we shall evidently have oblique motion on the unaccented
half of each note of the subject. It is best, where possible, still

DOUBLE COUNTERPOINT:

[Chap. HI.

to approach the accented notes by contrary motion, though occa-
sionally, in other species than the first, similar motion may be
employed, provided that the progression between the two accented
notes is contrary. (See below the examples to §§ 76, 77.) It is
often possible to continue the double counterpoint strictly to the
very end ; but this need not be insisted on, and a free cadence
will often produce a better effect, as in the example to § 71 below.

70. In the following example of this species

— IT— fc — n

^=fl

(a)

l PF

EO —

i — H

1 ^_

i r i •

i — p*-t-

will be seen at (a) a passing note quitted by leap of a third
(Counterpoint^ § 165), and the same procedure is repeated in
the next bar. This is done to obtain a better melody. In the
third bar we cannot take the upper G instead of E, or we shall
have a seventh with one intermediate note ; and, apart from the
similar motion, we cannot have either B, or the lower G — the
former because its inversion gives consecutive octaves, and the
latter because of the leap of a tritone to the following CJf. At
(b) we give two forms of cadence ; that in small notes preserves
the double counterpoint to the end ; but because of the leap of
the leading note, it is musically less commendable than the free
cadence given in large notes. The inversion of this latter
(marked in small notes in the tenor part) would not be sr
advisable as a cadence, and the last note must still be free, as
we obviously cannot finish on the submediant.

71. Our minor counterpoint of the second species

2_*_1_

(«)

1 * " r

1 "' T

r-a
Hh-M

—

Jill 5 L-

Li — i

H '

-i — \-

! i SB

i — —

— HI ^

shows at (a) an effective employment of an accented passing
note. In this species, these may be used without hesitation if
the melody is improved thereby. In this example, the strict
close is even less good than in the major given above. A free
close here is decidedly preferable.

Chap, in.] STRICT, IN THE TENTH. 31

72. We now take the same subjects for the third species —

1*4. r , TI

r r r r

ir r r n

ft*

>'t >• r J r

-i F — ' r—

.r. f. i — F-

)

Hb^

1— ^f — h — i — Hd

— i — i-

At (a) we have apparently implied the commencement of the
lower counterpoint with a first inversion. But if the student
will try for himself, he will see that the only other possible
commencements here were in the upper part,

(a)

J,- 0)

The two leaps of an octave at (a) are certainly not desirable;
neither would the sixth preceded by a third in the same direction,
as at (£), be good. We have, therefore, preferred to begin as
above, which gives both the third and fifth of the chord in the
upper part, and neutralizes, as far as possible, the effect of the
third in the bass by putting the root of the chord both before
and after it. The final note of the bass is, of course, free.

73. It will be seen that in commencing our counterpoint on
the minor subject

we can use a passing note at the end of the bar, which was

32 DOUBLE COUNTERPOINT: [Chap, in

impracticable with the other canto fermo. The only point re-
quiring remark in this example is the bar of counterpoint at (a).
The fourth note, FJf, in the upper part must either imply a false
relation with the harmony, or a major chord on the subdominant.
Its inversion in the bass cannot be satisfactorily explained, from
a harmonic point of view, according to the laws of strict counter-
point. The progression is here, nevertheless, perfectly good, and
it is introduced to impress upon the student's mind that in this
difficult branch of his art he need not trouble himself at all
about the implied root-progressions, if only his melodies and the
counterpoint in itself are correct. It must be remembered that,
in actual composition, double counterpoint in the tenth is only
written in the free style. Even in this there are restrictions
enough in all conscience ; and if in addition we hamper our-
selves by limitations very useful in simple counterpoint, but quite
needless here, we shall lade ourselves with burdens grievous to
be borne, and good writing will become almost impossible.

74. The force of what has just been said will be seen when
we attempt this counterpoint in the fourth species. Here it is
absolutely necessary that the rules for simple counterpoint of
this species must be to some extent relaxed ; because, as similar
motion is not permitted, whenever the subject descends, the
suspension must ascend. The student will remember that the
only ascending suspension allowed in simple counterpoint is that
of the tonic by the leading note ; but when inverted in the tenth,
above or below, we shall no longer have the suspension of the
same degree of the scale. We therefore so far modify the rule
as to allow the use of all rising suspensions (just as in harmony),
provided they move by step to a consonance. The fourth species
then becomes tolerably easy to work.

75. In our first example of this species

, ¥=^

(«)

•v x-

*s x-

-v .X

^W — (<:)

we are able, with the liberty we are now allowing ourselves, to
avoid breaking the syncopation at all. At (a) is the upward
suspension 5 6, while at (b) the suspension of the tonic by the
leading note has become by inversion that of the submediant by
the dominant. At (c) we make the close free for the same reason
as in § 70.

Chap. III.] Sm/CT, IN THE TENTH.

76. In our example in the minor,

I I

I ' II

the progression at (a) must be specially noticed. It has been
needful here to break the syncopation, because if we held the D
from the last bar, it could not be followed by E (§ 56) ; neither
could we rise to G, because the inversion would give a tritone
(§ 58). If we fall from D to B, we must break the suspension
here, or we shall have an augmented second ; and the third bar
from the end is a much better place than the last but one
for breaking the syncopation. It will be seen also that the
suspension 7 6 in the upper part gives as its inversion 4 5 ; that
is, the fifth is approached by similar motion. This would have
been wrong had the B of the tenor been the only harmony note
of the fifth bar ; but here G is also a harmony note, and we
therefore have contrary motion from accent to accent (§ 69).

77. The fifth species is by no means the most difficult to work
in the tenth.

s ^

M *4 , , *T

— ^

1 r Cj-f '

) H

1 f r r 1 i uf

J J ^ _F O p

We need only say of this example that at (a) is a precisely
similar instance in the lower part to that explained in our last
counterpoint, and that the small notes at the end of the upper
part show, as with some of the preceding examples, the possibility
of continuing the counterpoint strict to the end.

34 DOUBLE COUNTERPOINT: [Chap, m.

78. Our last example

M—

™

1 [J 1 — •yl

S~"

•v

requires no explanation. The progression to the cadence is the
same as in the two examples last given. To keep strictly to
contrary motion here, we must either have had an augmented
second or a major third above the subdominant.

79. It was said in § 49 that inversion in the tenth could be
effected in various ways. The student will see that in all the
examples given we, have never changed the position of the canto
fermo by placing it a tenth higher or lower. This is because
if we did so, we should so alter its character as to render it
unsuitable for our purpose ; for it would not then end on the
tonic, and we should either have to finish with an inversion of
the tonic chord, were the transposition of the subject upwards,
or with the submediant chord if we transposed downwards. In
writing exercises on a canto fermo, the student should follow the
same plan. We may again remind him that these exercises are
nothing more than the technical preliminaries for actual composi-
tion ; in free writing, as we shall see later, the other methods of
transposition can also be employed. We probably never find
double counterpoint in the tenth used throughout a piece ;
and when it is met with, it is mostly accompanied by free parts
(that is, parts not in double counterpoint), filling up the harmony.
None the less will working in the strict method prescribed in this
chapter be of great value to the student.

80. A peculiarity of this variety of counterpoint, in which it
differs from all other kinds,* is that the upper and lower counter-
points can be employed simultaneously against the subject,
excepting, of course, where the close is free. They will be evi-
dently in tenths (or sometimes in thirds or sixths, by transposing
one of them an octave), throughout. Examples of this will be
seen when we come to speak of free double counterpoint.

8 1. We have already said that it is not all subjects which are
adapted for double counterpoint in the tenth. We therefore give
a few canttfermi written specially for the purpose. The student
should work on each subject one counterpoint in each of the
five species. It will be difficult to invent more than one, as his

* Excepting the very rarely used double counterpoint in the thirteenth (see
Chapter VI II.).

Chap. III.]

STRICT, IN THE TENTH.

resources are so limited, but it will not be necessary in any case
to break the strict rules obligatory , upon him, and working at
these unquestionably troublesome exercises will go far to lighten
his subsequent labours.

SUBJECTS FOR DOUBLE COUNTERPOINT IN THE TENTH.

(III.)

(IV.)

(V.)

(VI.)

I I

(VII.)

(VIII.)

N.B.

N.B.~From this ooint the close most be free.

36 DOUBLE COUNTERPOINT: ichap. iv

CHAPTER IV.

STRICT DOUBLE COUNTERPOINT IN THE TWELFTH.

82. Double counterpoint in the twelfth, as its name implies,
is that which is designed for inversion in the twelfth above or
below. As the twelfth is the octave of the fifth, it is evident that
we can also invert two parts at this interval by transposing one of
them an octave, and the other in the opposite direction in the fifth,
just as in double counterpoint in the tenth we can transpose one
part an octave and the other a third (§ 49).

83. The double counterpoint we have now to consider is far
more used, and far more useful, than that in the tenth. It is also
so much easier to work that the student who has laboured
through the last chapter will, on arriving at this one, experience
something of the same feeling of relief as a mountaineer, who,
after toiling up a terribly steep and rough place, comes to a piece
of comparatively level ground. He must not, however, expect to
find double counterpoint in the twelfth, especially in the strict
style, quite so easy as that in the octave.

84. A moment's thought will show us that inversion in the
twelfth (or fifth) changes the character of a melody far less than
that in the tenth; for, with one exception, the semitones will
remain in the same place in the scale. Thus, if we transpose the
scale of C a fifth upwards, we shall have the scale of G without
an Fj{ ; while if we transpose it a fifth downwards, we get the
scale of F without a B |j.

85. We will now make our table of inversions in the twelfth,
as we did with the octave and tenth —

INTERVALS: . ... 12 3456789 ion 12
INVERSION IN THE 1 2TH : 12111098765432 i

This table shows us that octaves become fifths, and fifths
octaves, while thirds become tenths, and tenths thirds. This
latter fact will obviously facilitate our work greatly, since we shall
be no longer prohibited (as in double counterpoint in the tenth)
from the use of consecutive thirds, or of similar motion. The
interval requiring special care is the sixth, which, as by inversion
it becomes a seventh, can in the strict style only be used as a
passing note, or as a prepared discord. We shall give an
example presently (§ 100) of its employment in this way.

86. As by transposition in the twelfth the position in the
scale of every note is altered, we shall find, as we did with the

Chap. IV.]

STRICT, IN THE TWELFTH.

tenth in the last chapter, certain harmonic combinations and
melodic progressions which it will be needful to avoid, because
their inversion will produce prohibited intervals. For harmonic
reasons, we cannot take an octave above the subdominant in either
the major or minor key, because the inversion of the upper note
a twelfth will give us the diminished fifth above the leading note.
Neither, for a similar reason, can we in a minor key take the
octave above the submediant, or the octave below the supertonic
or mediant. The melodic progressions to be avoided will be best
shown by a table similar to that which we made for double counter-
point in the tenth (§ 58).

Unavailable

/Up k — ^^ r^> ||

U ..

Bh o 5 <s*-fl

in upper part.
Augmented

gH2 1 H
P IHIK' II ^C

Intervals.
Unavailable

in lower part.

r, \ ^A

We have given the progressions in the key of C minor, to show
all the augmented intervals ; but the student must observe that
those given at (a) — from tonic up to subdominant, and its con-
verse, in the upper part ; and from leading note up to mediant,
and its converse, in the lower part — are also unavailable in a
major key, as the tritone between subdominant and leading note
exists in that key also.

87. In writing double counterpoint in the twelfth it will be
best for the student to use the treble and bass voices for the
counterpoint and its inversion, and to put the subject in either
the alto or tenor — it is immaterial which. He must, of course,
be careful not to exceed the compass of a twelfth between
subject and counterpoint.

88. An important point to notice in this kind of counterpoint
is, that the cadence must always be free. The reason for this
will be evident with a moment's thought. A subject for strict
counterpoint always ends on the tonic, and generally has the
supertonic as the penultimate note. Let us take these two notes,
and try to write either above or below them a cadence which will
invert in the twelfth —

38 DOUBLE COUNTERPOINT: ichap.iv

The cadence at (a), which is that usually employed in simple
counterpoint and in double counterpoint in the octave, is clearly
out of the question here, not only because the sixth by inversion
becomes a seventh, but because if the upper counterpoint ends
on the tonic (whatever precedes it) its inversion must end on the
subdominant. To end the upper counterpoint on the mediant,
as at (£), is no better; for the inversion will end on the sub-
mediant. If, on the other hand, we write a good cadence in
the bass, and attempt to invert it, we get a cadence which is
simply atrocious in the treble, as at (<:). In two-part writing, the
student will remember, a third should not be followed by a fifth
when both parts move by step, except in going from submediant
to dominant (Counterpoint, § 157). The counterpoint against
the last two notes of the subject will therefore always be free.

89. As double counterpoint in the twelfth is so much easier
than that in the tenth, it will not be needful to write special
subjects for working it, as we did in the last chapter. We shall,
for the sake of comparison, take the same two subjects which in
Chapter II. we employed for double counterpoint in the octave,
and will give one major and one minor example in each of the
five species.

90. As every exercise should commence with the tonic chord,
unless the subject begins with the dominant, it is evident that our
upper counterpoint should always have the fifth (or twelfth) of the
subject for its first note ; otherwise the inversion will begin on the
subdominant or submediant. If the subject should happen to
begin with the dominant, we can put either the fifth or octave
above it ; in the former case the lower counterpoint will com-
mence with the dominant, and in the latter with the tonic.

91. We will now give double counterpoints of the first
species —

•J

^—

(a)

^•3

— &

1 U

1 «»

U—

_____

Chap. IV.J

STRICT, IN THE TWELFTH.

These counterpoints are so intelligible that we need only direct
the student's attention to the free close at (a) in each, of which
we spoke just now.

92. Counterpoint of the second species is mostly not very
difficult in the twelfth. As with the octave and tenth, passing and
auxiliary notes may be introduced, occasionally even on an
accented beat, as in the following example —

=11

• [g

At (0) we have taken a seventh as an accented passing note in
the upper part, which allows us to use a sixth as a harmony note
in the bass. Let the student compare the introduction of the
fifth as a harmony note in double counterpoint of the octave in
the examples, §§ 44, 46. It is necessary to begin to make our
cadence free at (b) in the third bar from the end. If the student
will try, he will find no second note for this bar which can be
properly inverted.

93. We next take our minor subject —

At (a) we have made on the unaccented notes of the second and
third bars consecutive octaves with the subject. These should
mostly be avoided, but here it was important to have A as the
second minim of the third bar, and there was no other good note
but G for the second minim of the second bar. The only way to
avoid the octaves here would have been to introduce a transient
modulation, by taking E flat as the first note of the third bar

thus—

This would have saved the effect of octaves (see Counterpoint^ 17 5;,

DOUBLE COUNTERPOINT:

[Chap. IV.

but as a modulation would have been unadvisable, we have
preferred to allow ourselves a small license here. In the inversion
there is no effect of consecutive fifths, because of the decided
mental impression of two chords in the third bar. We were
anxious to keep A in the third bar, so as to be able to intro-
duce the sixth as a harmony note in the following bar in such
a manner as that its inversion shall be an accented passing
note — see (b). Here we have the converse of the progression at
(a) in our last example, where the passing seventh was in the
upper part, and the sixth in the lower. At (c\ as at (£), of § q2
we have to begin to make our cadence free.
94. Our next example (third species)

J ^ J J

£&£

J J J J

1— +

J J r r 1

P r J •» =3=

' J- •

1 « 0 H

requires few remarks. At (#) two chords in the bar are clearly
implied. Had we begun as we shall in the next counterpoint,

we should have consecutive fifths, and in the inversion consecu-
tive octaves, by contrary motion. We need never hesitate about
taking two chords in a bar, if we find it advisable. At (b) we are
able to use the sixth again as a harmony note, because we can
ake its inversion in the bass as a passing seventh.

95. The only points to be noticed in our next example

P H, s

^ •» J

J J J «>

r r J Ji

Chap. IV.]

STRICT, IN THE TWELFTH.

4»

•+

I I.J II- „

'<J ' » '

are that at (0) the descending form of the minor scale with the
major sixth is used because the leading note is a harmony note ;
that at (b) we have the same treatment of the sixth as in the last
example ; and that at (<r) we can equally well use the major and
minor sixth and seventh of the scale.

96. The fourth species is mostly very troublesome in double
counterpoint in the twelfth, because we are unable to employ one
of the most useful of all the suspensions — viz., 7 6, as its inver-
sion will evidently give 6 7 in the bass. The difficulties of this
species will be illustrated in the examples we shall now give —

I " I ^ Ull ^ I " Ull

ii rir ri

Until the counterpoint becomes free, at # it is impossible to
obtain any conjunct motion against this subject, as the student
will soon find if he tries it. At * it is best to break the syncopa-
tion. We have given an alternative version; but this is distinctly
less good, first, because we have four consecutive thirds between
the subject and the upper counterpoint, and secondly, because
we end on the third of the key instead of the tonic, and our
cadence contains no leading note.

97. Against our minor subject the counterpoint will be rather
more flowing, but not much better.

Our progression for the first three bars is forced , but the effect
of the fourth in the bass going to the fifth, at (a\ can certainly

42 DOUBLE COUNTERPOINT: [Chap.iv

not be called satisfactory, though there is really nothing else to
be done here. It will be seen that at the beginning of the fourth
bar it is absolutely necessary to break the syncopation. The leap
of the octave at (b) is virtually compulsory. In the fourth
species the syncopation should never be broken for two consecu-
tive bars (Counterpoint, § 263) ; if we take B for the second note,
we get the unavailable suspension 7 6, besides a tritone in the
inversion (§ 86) ; if we take the only other possible note, D, the
inversion of the 9 8 suspension gives the 4 5 in the bass, which
we were forced to take in the third bar, but which should be
avoided wherever possible. At (f) we have again broken the
suspension in the upper part, to avoid the weak close,

*p= I I ' . M i I ' II II

98. Taken altogether, the fourth species of double counter-
point in the twelfth is so unmanageable (to say nothing of its
being of hardly any practical use), that the student is not recom
mended to spend much time over it. It is given here for the
sake of completeness, and a few exercises may be worked on it as
a preparation for the fifth species ; but there its utility ends.

99. Lastly we take our two subjects for counterpoint of the
fifth species —

s^—

\ r J r ;

-'.. J? J * Ir c r.c.e.l.J_jL.U

r i r j i-jEfeEj EEt

u ' i i r • r i r r ' '

It will be seen that this example is far more satisfactory, from a
musical point of view, than those of the fourth species. At (a)
we have apparently consecutive octaves in the upper part and
consecutive fifths in the lower, with the first note (which is here a
harmony note) of the second bar. Observe the way in which

Chap. IV.)

STRICT, IN THE TWELFTH.

they are saved by interposing two other notes of the harmony,
both of which are beyond the first note of the third bar, so as to
return by contrary motion. Such a procedure is often used in
simple counterpoint to save hidden octaves and fifths (Counter-
point, § 178); we extend the principle here, so as to save actual
consecutives. We must, however, add that it would have been
bad to do this had not the first note in the second bar been a tied
note, held on from the preceding chord, so that the note on the
second crotchet has the character of the real harmony note of the
bar. We have to begin our free cadence at (b\ in the third bar
from the end, as in the second and fourth species.

100. For our next example we transpose our subject to
G minor, so as to keep our counterpoints in a better compass.

b »' J r

r jigtj J

ffj J^fl,

J. J J f""

— »*

=^l

In this example we have illustrated at (a) the use of the sixth as
a harmony note. It should be, as here, prepared in the bass ; it
must then continue to descend by step to the harmony note of
the next chord. By comparing the upper and lower counter-
points in this bar, it will be seen that the harmony notes of the
one become passing notes of the other, and vice versa. We have
already met with something similar in the examples in §§ 44, 46,
92, and 93. At (£) we have used a sixth again in the upper part,
as we did in the counterpoints of the third species (§§ 94, 95).
This enables us to continue the inversion to the end of the bar,
by using the form of cadence shown at (c\ Had we taken the
same cadence as in the last example, we must have broken the
inversion in the middle of the third bar from the end.

101. The student may now proceed to work double counter-
point of the twelfth in the different species on any of the subjects

44 DOUBLE COUNTERPOINT : [Chap iv.

given at the end of Chapters II. and III., or on any of those
contained in Counterpoint. When he has acquired a mastery of
this kind of writing, he may congratulate himself that his labours
in the domain of strict counterpoint are at an end. He will find
double counterpoint in the free style comparatively easy, if he
have prepared himself for it by a conscientious course of hard
work at the preliminary and technical part. The fundamental
principles to be borne in mind will be the same by which he has
hitherto been guided, and with which he may reasonably be
supposed to be now familiar ; but the strict study he has been
through will give him a command of free writiner, without fear of
his abusing his liberty, which, it may be confidently affirmed, can
be obtained in no oiher way.

chap, v.] FREE, ON A CHORAL. 45

CHAPTER V.

FREE DOUBLE COUNTERPOINT IN THE OCTAVE, TENTH AND
TWELFTH, ON A CHORAL.

102. In writing double counterpoint, at whatever interval, in
the free style, the student will have at his command (as with
simple free counterpoint) all the resources of harmony. Chro-
matic chords may be introduced or implied ; auxiliary notes may
be taken by leap ; in fact, all the additional freedom which was
allowed in Chapters XIII. and XIV. of Counterpoint may be-
taken advantage of now. There is, however, one important
difference to be borne in mind. All the exercises in simple
counterpoint in the free style were in four parts. For the present
our double counterpoint will be in two parts only ; and for this
reason, as the lower part has always to be considered as the bass,
we must avoid the interval of a bare fourth between the two
voices, unless it is used as a prepared suspension or a passing
note. When we come, in a subsequent chapter, to treat of
double counterpoint with added parts, we shall find it possible to
employ the fourth more freely.

103. As the work on which we are now about to enter is in
reality a species of actual composition, we are no longer restricted
as to the length of our notes. It is therefore unnecessary to
practise writing in any other than the fifth species; and in this
we allow ourselves in one respect greater liberty than heretofore.
In strict counterpoint it is rarely good, with the fifth species, to
have two or more consecutive bars of the same pattern (Counter-
point, § 310). But we may now employ the same figure —
especially sequentially — for several bars, if desirable, provided
that the treatment does not become monotonous. It is no
uncommon thing in the works of the great masters to find a
double counterpoint constructed almost entirely on one pattern.

104. An important modification of our previous rules, and
one that will greatly facilitate our work is, that in free double
counterpoint dissonant notes may be sounded together, not only
when (as in the strict style) one of them is an accented passing
or auxiliary note, but also wh&n the two taken together clearly
represent a fundamental discord. Numberless illustrations of this
might be given from the words of Bach ; a couple will suffice
here, both taken from his " Fifteen Inventions in two parts " — a
work which contains some admirable examples of double counter-
point in two parts only.

DOUBLE COUNTERPOINT:

(Chat. V.

105. In the first of these Inventions the opening bars are
subsequently inverted thus —

J. S. BACH. Inventio I.
Bars T, 2. vf won __*— _ **"

At (a) in the first bar an augmented fourth is sounded between
the two parts, implying a third inversion of the dominant seventh.
In the seventh bar the passage is inverted in the octave (or more
strictly speaking, in the fifteenth), and at (c) the augmented
fourth becomes a diminished fifth, which implies the first inver-
sion of the dominant seventh. The first note of bar 8 is free,
that is, it is not an inversion of the corresponding note of bar 2 ;
but the second half of this bar is an inversion of bar 2 in the
twelfth. The student will see this at once, by comparing the
intervals at (b) and (d). The fifth at (b) becomes an octave at
(d) ; 5 + 8=13; the inversion is therefore in the twelfth (§ 8).

1 06. Now look at the dissonant notes in this passage.
Against the second quaver of (b) a seventh (an accented passing
note) is struck in the bass ; this by inversion becomes a sixth — a
harmony note (compare the progression at (b) in our example to
§ 93). The semiquavers C and A against the third quaver of
(b) represent with the F above a chord of the sixth ; in the
inversion at (d) the fourth becomes a ninth, and the sixth
becomes a seventh against the bass note, which is itself the
auxiliary note here, the implied harmonic progression being

7
6 5

4 ft

This example shows how possible it now becomes to use many
progressions hitherto forbidden.

107. Our next illustration is taken from the same work —

Bars 3, 4. "* ^ J.S.BACH. Inventio IL

FREE, ON A CHORAL.

Here the inversion is in the fifteenth throughout ; but, as this is
virtually the same as the octave, we shall in future adopt the
more usual name, and speak of such double counterpoint as
being in the octave, unless there is any special reason for
describing it otherwise. This example illustrates three different
uses of dissonant notes. The se\enth at (a) is a 7 6 suspension,
inverting at (d) as 2 3 ; at (b) the dissonant fifth is a harmony
note, and we have the first inversion of a dominant ninth ; the
augmented fourth at (e) gives the third inversion of the same
chord ; at (c) and (/) we see accented passing notes. The
examples at (b) and (e) are those bearing more immediately on
the point we are now discussing. Let it be also noticed that the
passage just given exemplifies what was said at the end of § 103,
the lower counterpoint being in semiquavers almost throughout.

1 08. We must not, however, rush to the conclusion that
every dissonance may now be freely used, if it can only be
explained as part of a fundamental discord. It is needful also
carefully to consider the resolution which is to follow. If, for
example, we take the interval of the diminished seventh, and give
it its ordinary resolution,

it is evident that if this be inverted in the octave, the perfect fifth
will become a bare fourth, which we know (§ 102) is unallowable.
If we employ a diminished seventh at all, we shall have to give it
some other resolution ; e.g. —

, <«> i J r*i i <*>i J i i !

The student will see for himself the inversions of these passages.
At the fourth crotchet of (a) we have taken a fourth as an
accented passing note ; and the inversion of (b) will give a fourth
as an unaccented passing note; both these examples are quite
correct. It must further be observed that the usual resolution of
the diminished seventh, though unavailable for inversion in the
octave, is quite practicable for either the tenth or the twelfth —

inversion in the loth. Inversion in the xath.

48 DOUBLE COUNTERPOINT; [Chap. v.

109. The interval of the augmented sixth shows a somewhat
similar case. In double counterpoint of the octave it should
not be used at all — at all events in two parts — because of the
harsh effect of its inversion, the diminished third ; but its inver-
sion in the tenth gives a diminished fifth, and in the twelfth a
diminished seventh, both of which may be freely used, provided
they are properly resolved —
-.*.,'

_ Bad. Good. Good.

Inversion in 8ve. Ditto in ibth. Ditto in i2th.

no. The admission of fundamental discords further lightens
our labour in another way, especially with double counterpoint
in the tenth and twelfth, by allowing us to use certain combina-
tions which in strict counterpoint were unavailable because their
inversions produced dissonances. For example, we saw in § 56
that in double counterpoint in the tenth we could not take a sixth
above the subdominant, because its inversion gave us a
diminished fifth. But in free counterpoint we can use this
interval, provided that the next note of the subject will allow us to
resolve its inversion correctly. An example will make this clear.

\U.) VJWt

i n n ^ ]

fc^1 - • " 1

Inversion in loth.

At (a] the diminished fifth in the inversion receives its proper
resolution, at (b) it does not ; the former progression is therefore
good, the latter bad. In the same way, we may take a
diminished fifth above the leading note in double counterpoint in
fhe tenth (representing the first inversion of the dominant
seventh), always provided that we can follow it properly.

in. Under similar limitations, we can employ some of the
hitherto prohibited intervals in double counterpoint of the
twelfth. A sixth can be used as a harmony note above either the
subdominant, the tonic, or the submediant, because its inversion
will give us in the first two cases the root position of either the
dominant or supertonic seventh, and in the last the first inversion
of the dominant ninth. For the same reason a sixth may be
employed below the dominant, supertonic, or. leading note ; we
can also take the octave above the subdominant, in either a majoi
or minor key, and the octave above the submediant, or below the
supertonic, in the minor key. But we cannot too strongly impress

Chap. V.]

FREE, ON A CHORAL.

on the student the fact that these intervals can in no case be used
when their dissonant inversions cannot be properly resolved.
With double counterpoint in the octave there will be no danger ;
because the inversion of a dissonance in the octave changes only
the position, not the nature of the chord. With the tenth and
twelfth, as we know, it is quite different.

112. The restrictions as to melodic progressions in double
counterpoint of the tenth and twelfth (§§ 57, 58, 86) are still to
be enforced, with the one exception, that it is occasionally
possible in the free style to take an augmented second or an
augmented fourth in the melody, when both notes are part of the
same harmony. This, however, will very seldom be necessary,
and the progression should be most sparingly used, if at all.

113. Free double counterpoint can be written either against
a subject in notes mostly of equal length, such as a choral or
hymn-tune ; or two florid parts can be written in double counter-
point with one another. In actual composition the latter is by
far the more common ; but it will be best for the student to begin
by practising double counterpoint on a choral. In doing this,
he should always endeavour to make his counterpoint as flowing
and melodious as he can, and remember that he is writing music,
and not solving mathematical problems on intervals. We shall
now give, as patterns, double counterpoints in the octave, tenth,
and twelfth, on a well-known German choral.

114. We first write a double counterpoint in the octave —

kpir r Q»|J J r CncFfl"J rir Q»J -N

Mm s r

JgJW «P|

7 fl fi ft A 7 6

3 5
2

.7 — 6 5 9 6-

4 2 5

5°

DOUBLE COUNTERPOINT:

[Chap. V

*£$» tz.

— 5

\t r J J7

6 D5

u r

Lfl 1 — 1

6 7 $6

lab, r *n

3 $6

T riJ J3l

76 6 48

6 76

In|g T- ' — E

J — J J * ** ^

11 ' ' rn 'j1 — E

Lj ! — 1

2 —

7 6

5 6

6 —

As it is important to think of the implied harmony, it will be
well for the student to figure his basses throughout, as we have
done here. These counterpoints will repay close examination,
though they require but few explanations. At (a) will be seen in
the upper counterpoint an augmented fourth, and in the inversion
a diminished fifth, both representing the dominant seventh of
B minor. The transient modulation to the key of the relative
minor is here of excellent effect. At (&) a sharp is put in brackets
over the second quaver, G ; this note can be either G J or G H,
according to the harmony intended. As the basses are here figured^
we have an inverted cadence (Counterpoint, § 505) in B minor;
in this case the G must be natural. But it would be also pos-
sible, though less good here, to regard the chord as the submediant
of A major, not making a modulation to B minor at all ; and in
this case G, being an auxiliary note above the harmony note, would
be sharp. At (c) and (d) we have indicated two chords as implied
above the bass note, so as to avoid the bad harmonic progression
lla to la.

Chap. V.j

FREE, ON A CHORAL.

115. Double counterpoint hi the tenth, though easier in the
free style than in the strict, will be found considerably more
difficult than that in the octave. We are, however, as has been
shown above, much less restricted in our harmony, owing to our
ability to use fundamental discords ; and the permission given in
§ 59, to disregard implied root-progressions, must, therefore, now
be withdrawn ; for it will be quite impossible to write really good
musical double counterpoint in the tenth unless we consider the
harmony that is implied. We shall, therefore, for our next
examples figure our basses, as we did with the last.

1 1 6. It must further be observed that we are now no longer
absolutely debarred, as in the strict style, from the employment
of similar motion ; for we can now approach a third or a sixth
by similar motion whenever its inversion will produce one of the
allowed hidden fifths or octaves (Harmony, §§ 7S~77).

117. We now give a double counterpoint in the tenth on the
choral we are treating. It will be seen to furnish much more
material for comment than that in the octave —

n j i 1 «• ^^^n

IKy J r cnn rrr r

IfePV" P |f- f

(d) (*)

* 6 6
to _ -p...

(fr1 *« r T — rri i i — r-r—

U £_, ^-^ ,

r mm+ m -J rl- m *f ' m» mm

i — U-1 — I—1 — ?— 1 — I '' I" I 1 — U— '

6 —

8 «

DOUBLE COUNTERPOINT; [Chap. v

fir* r " j i i (°

J6 56 9 8
6 —

r - -r r r f- f= -r

6 6 56

2 —

At (a) the progression from a third to a fifth, which is mostly
bad in two parts moving by step, cannot well be avoided.
It must be noticed that this double counterpoint is seldom used
in two parts only ; as soon as a third part is added the bad effect
disappears. At (ft) we use the hitherto prohibited sixth above
the subdominant, the inversion of which is a diminished fifth,
because the next note of the subject allows us to resolve it cor-
rectly (§ no). At (c) we meet with a case of frequent occurrence.
Here we take the inversion in the key of B minor, while the
upper counterpoint remains for four bars longer in the original
key of D. It would have been possible, by the omission of the
sharps to G and A, to keep the lower counterpoint in the key of
D ; but this would have been far less satisfactory, because the
harmony would have consisted almost entirely of the weaker
chords of that key, II. , III., and VI., whereas by going into
B minor, we use all the strong chords, I., IV., and V. It may
be laid down as a general principle that the larger the proportion
of the three strong chords of the key (the tonic, subdominant,
and dominant), the firmer and better the harmonic progression
will be.

1 1 8. At (d) we save the approach of a sixth (becoming in the
lower part a fifth), by similar motion, by going beyond the interval
and then returning. It must be observed that it would have been
quite possible here to approach this sixth by similar motion, had
we desired it, because the hidden fifths in the lower parts are among
those that are permitted. At (e) we take the sixth above the
leading note, instead of the here possible diminished fifth (§ no)

Chap. V.J

FREE, ON A CHORAL.

because its inversion gives us a stronger harmonic progression in
the lower part.

119. The figuring of the bass at (/) should be particularly
noticed. We have indicated the second inversion of the dominant
eleventh, because this chord is much better than a plain chord of
the sixth on the supertonic to precede the chord of the dominant
seventh. Observe also at (g) how the third inversion of the
dominant seventh in the upper part gives the root position of the
same chord when inverted in the tenth below. At (fi) we have
taken the dissonant Gj in the bass instead of Gfi, not only to
avoid the augmented second, but to introduce the second inver-
sion of the fundamental ninth on the supertonic, resolving it on
the first inversion of the dominant seventh.

120. At (i) in the upper part we do not necessarily imply a
doubled leading note. We may quite well consider the key of
this bar to be A major, when the C sharp will be the doubled
third of the tonic chord approached by step. At (K) we see the
reverse procedure to that which we noticed at (c). Here the
upper part modulates, while the lower remains in the original
key. From (/' the first two lines of the choral are repeated.
We have endeavoured here, as in the preceding counterpoint, to
obtain as much variety of harmony and melody as possible. At
(m) we make the cadence free in the bass, so as to get a better
close.

121. Lastly, we write a double counterpoint in the twelfth
against the same choral, and, in order to get more variety, we
will take it in triple time —

^ r

6 60

-p-S-Si.

J ns

1 — I^TT

«*

5 4
3

--i

5 6

DOUBLE COUNTERPOINT:

IChap. V.

\ "J • | ^ ' If- |

^±4

ff=»=s

<',,L

mf^

6 66

686
8

7 6
5

!••£? •. ^=7— :

6 5

r T - i" r!*r

6 6 6 —
4 3

m»mm , P- f [•(>

Chap. V.]

FRRE> ON A CHORAL.

(a) the sixth above the subdominant is taken as a harmony
(§

note (§ m), its inversion in the bass giving a dominant seventh.
As we happen to have approached it by step, it would be possible
here to regard the E in the bass as an accented passing note ;
but it is far better to consider it as a genuine seventh ; the
harmonic progression is stronger. At (/>) we indicate two chords
in the bar to save the bad progression lla to la ; the same thing
is seen two bars later.

122. The progression of the bass at (c) requires careful atten-
tion. Evidently the last quaver must be a harmony note ; for
if we think of the submediant ch^rd as being continued through
the bar, D will be a second gassing note returning in the
following bar to the first one. If we here take DQ instead of
DJ, we shall have the false relation of the tritone ; but by using
the first inversion of the fundamental seventh on the supertonic,
we get out of all our difficulties. We have marked the f under
the DJf for the sake of clearness ; but the harmony should of
course change at the beginning of the third crotchet, the E
being an accented passing note. In four parts we should fill up
the harmony, thus —

gj •

r •
J .

That no modulation is implied to the key of E is, of course
shown by the contradiction of the Djf in the following bar.

123. At (d) is a somewhat, though not precisely, similar point.
Here we are just going into the dominant key ; if we take, t>B
here, we have the false relation of the tritone, and the effect is
extremely harsh. As the preceding crotchet E is here a harmony
note, it would be possible to consider D as an auxiliary note ;
but if we do this, it must still be D sharp t because it is below the
fifth of the chord (Harmony, § 307). At (e) we see a diminished
fifth in the bass, resulting from the inversion of the octave
above the subdominant The apparently free treatment of the
dominant seventh here will be seen to be fully justified if we give

DOUBLE COUNTERPOINT:

(Chap. V

the harmonic outline, omitting the passing notes in the bass, and
filling up the chords —

The chord of the seventh first changes its position ; and the
seventh rises because the bass moves to the note of its resolution
(Harmony §§ 266, 240). We are intentionally introducing the
progressions which were forbidden in the strict style, to show how
to manage them properly.

124. The *| marked in the bass at (/) is a parallel case to
that which we have been examining at (c). The last quaver must
imply a change of harmony, which (as before) should, of course,
be taken at the third crotchet, DJ being an accented passing
note. The figuring here given represents the second inversion of
the supertonic major ninth ; we have taken this in preference to
the submediant triad, as the latter might easily imply consecutive
fifths in a middle voice with the following chord. Note also that
in this passage, as in some places of our double counterpoint in
the tenth, the two counterpoints are in different keys, the upper
one being in FJf minor, and the lower in E major. In all double
counterpoints in the tenth and twelfth, we may freely introduce
accidentals, if we thereby obtain better melodic or harmonic
progressions.

125. After what has been already said, the student will readily
see why at (g) we have marked Jf6 under B. In the bass the
modulation back from B minor to A is made one bar sooner,
at (h). Here we have, exceptionally, three chords implied in
the bar. On the repetition of the first part of the subject, we
have, as in our previous examples, varied the counterpoint.

126. At (i) will be found another sixth treated as a harmony
note — not one of the sixths, be it noticed, which we have allowed
in § in. We have introduced this one to illustrate our general
rule that any dissonance may be used in free counterpoint,
provided it clearly represents a fundamental discord (§ 104).
Here the seventh below the dominant represents the fourth
inversion of the dominant major thirteenth. The full harmony
of this and the preceding bar will be

•j

i&:y

r i • . i
J •* J ^ T

, r rjr r r |

chap, v.i FREE, ON A CHORAL. 57

At (/£) two chords are indicated below B, as before, because the
next chord is la ; and, lastly, from (/) the cade/ice is free (§ 88).

127. The student will now be prepared to write double
counterpoint for himself on hymn-tunes and chorals. He can
take any familiar tune for treatment, or, if he prefer chorals, he
will find a selection of fifty, which will be admirably adapted for
his purpose in the author's Additional Exercises to Counterpoint.
But he will do well to remember that it is not every subject that
is adapted for double counterpoint in the tenth. The choral we
have been treating proved very suitable, though it was not
selected for that reason. What we may describe as an un-
dulating melody — one that alternately rises and falls — will be
the easiest to manage in the tenth. But any melody can be
fitted with a double counterpoint in the octave or twelfth by
dint of patience and perseverance. It will be excellent practice
to write one of each kind of double counterpoint on the same
subject, as we have done in this chapter ; the student will thus
obtain some insight into the almost inexhaustible resources of
harmony.

58 DOUBLE COUNTERPOINT: tchap.vi.

CHAPTER Vi.

FREE DOUBLE COUNTERPOINT ON A FLORID SUBJECT.

128. By the term "florid subject" is here meant any subject
in which the notes are of no regular length, as distinguished from
the chorals which have been treated of in the last chapter. We are
not now restricting the word "florid" to the narrow sense in which
it is frequently employed, as meaning rapid, or highly ornamented;
but we are using it just as we do when we speak of the fifth
species as " florid counterpoint." We have already said that this
is by far the most common kind of counterpoint in actual
composition; and we shall therefore in this chapter have the
advantage of being able to draw our illustrations from the works of
the great masters, instead of being obliged, as hitherto, to write
all our own examples.

129. The general principles to be followed in writing this
kind of double counterpoint are exactly the same as those which
guide us in adding a double counterpoint to a choral ; but there
is one additional rule to be enforced, with which the student has
already made acquaintance in working combined counterpoint.
Let him remember the rule given in Counterpoint, § 407: "When
two parts are in the fifth species, variety should be sought by
taking longer notes in one of the parts against shorter notes in
the other." To apply this rule in the present case, the subject
and counterpoint should be contrasted as much as possible, both in
melody and rhythm,

130. Double counterpoint in the octave is by far the most
frequently employed and the most useful ; next in order comes
that in the twelfth, which is tolerably common. But double
counterpoint in the tenth (probably owing to its difficulty) is
extremely rare, and, excepting in pieces written expressly to
illustrate it, is only to be met with incidentally, and generally for
only a few notes.

131. The most frequent employment of double counterpoint
of all kinds is in fugues, of which (as will be shown in the next
volume of the present series) it forms one of the chief in-
gredients. But its utility is by no means restricted to this branch
of composition. It frequently plays an important part in large
instrumental works, such as symphonies and sonatas, and is even
to be met with in vocal music which is not fugal. Illustrations of
each kind will be given presently.

Chap VI.]

ON A FLORID SUBJECT.

132. It will be most convenient to give first examples of
double counterpoint in the octave, reserving those in the tenth
and twelfth for a later part of tne chapter; and we shall
commence with some examples from the works of the greatest
contrapuntist that the world has ever seen — Johann Sebastian
Bach.

(1) J. S. BACH. " Wohltemperirtes Clavier," Fugue 30.

— h

This passage requires hardly any explanation. At (2) we see the
inversion, in a different key, of the passage at (i). By the
simple rule given in § 8 it will at once be seen that the inversion
is in the octave. Notice the contrast in rhythm and melody
between the two subjects (§ 129).
133- Our n^xt illustration

(1) J. S. BACH. Organ Fugue in C minor.

shows a case frequently met with. Here the inversion is not
only in a different key, but in the major mode, instead of the
minor. This evidently alters the character of the music, but the

DOUBLE COUNTERPOINT:

[Chap. VI.

intervals of inversion are still exactly maintained. It should be
mentioned that at (2) the double counterpoint is in the two
middle parts of a four-part harmony. We have not quoted the
outer parts, as double counterpoint with added parts will be
treated of in the next chapter.

134. The following extract from the two-part fugue in
E minor of the " Wohltemperirtes Clavier "

J. S. BACH. "Wohltemperirtes Clavier," Fugue 10.

illustrate? "some fresh points. Observe, first, that there is much
less contrast in the character of the two subjects than in the
examples hitherto given. This is because the fugue is a kind of
moto continue^ a special feature of which is the persistence of the
semiquavers from the first bar to the last. This passage also
illustrates what was said in the last chapter (§ 104), as to the
sounding of dissonant notes together. The semiquaver E in the
first bar of the bass is clearly an auxiliary note ; but at (a) are
evidently notes of the chord of the dominant eleventh (second
inversion), and at (b) we have the dominant seventh in the key
of D. The third and fourth bars of this passage are the inversion,
in the key of the dominant, of the first and second ; and the
augmented fourth at (c) represents the fourth inversion of the
dominant eleventh in D.

135. A similar point is illustrated in our next example —

(1) J. S. BACH. Organ fugue in E minor.

ON A FLORID SUBJECT.

Here the inversion is really in the twenty-second, or triple octave,
the upper part being transposed two octaves lower, and at the
same time the lower part an octave higher. But just as in harmony
we speak of one note as being the octave of another, though it
may be three or four octaves away, so we speak of double counter-
point in the octave, whatever the actual distance, if the relation of
the two parts to one another harmonically is the same as if the
transposition were really only one octave. (Compare §§ 13, 107.)
At (a) in this example the augmented second represents the last
inversion of a supertonic minor ninth, and the diminished fifth at
(b) represents the second inversion of the augmented (French)
sixth. In the inversion at (2) these two intervals give us at (c)
and (d) the first inversion of the supertonic ninth, and the
uninverted French sixth, thus exemplifying what we said in
§ in — that the inversion of a dissonance in the octave changes
only the position, and not the nature of a chord. At (e) two
crotchets are substituted for four quavers for technical reasons ;
^ere the bass is played on the pedals, and Bach has simplified
the passage, as he often does in such a case.

136. We shall give more examples from Bach when we come
to deal with double counterpoint in the tenth and twelfth, and
also with added parts ; we will now take a few extracts from the
works of Bach's greatest contemporary — Handel. Our first
illustration

(1) HANDEL. " L'Allegro.'

(2)

shows a very simple double counterpoint in the octave in the
strict diatonic style. It is so straightforward as to require no
explanation.

DOUBLE COUNTERPOINT:

[Chap. VI

137. In our next example, taken from the grand chorus,
" From the censer,"

HANDEL. "Solomon."

the two subjects are first announced by the alto and tenor voices,
and immediately inverted by the soprano and bass, the former
entering before the end of the subject. This passage illus-
trates at (a) the occasional possibility, in free writing, of the
crossing of the parts for a moment. This, however, is a license
which the student is not recommended to imitate.

138. The following passage
(i)

'.niJ J J ilr- m

HANDEL. " Susanna.

ftJhry-J-

Chap. VI.]

ON A FLORID SUBJECT.

is given for the sake of comparison with the extract from Bach,
quoted in § 132. In both cases, the most important feature is
the descending chromatic scale. Notice what a different counter-
point is written against it by the two composers. At (a) will be
seen a chromatic FJJ in the alto, which in the inversion at (&}
becomes Ft). Such chromatic alterations, though by no means
infrequent in double counterpoint of the tenth and twelfth, are
rather rare when the inversion is in the octave.

139. The last example that we shall give from Handel

(1) HANDEL. Anthem, "O come, let us sing.'

1 Q g,* r>

1 —

1 ! —

1 1 — 1

[Hi fi ;{ —

— is> F—

— ^

5>—

1 J r II

&c.

— r — r i^j^ H

" I 1 —

(2)

eV r r r | A^' ' 1 r r | r ^ |

' ' |T i

shows how to obtain variety by contrast of melody and rhythm
when both subjects are in notes of comparatively slow time.
The DJ at (a) of the inversion is another instance of the
chromatic alteration of a note spoken of above. This will be
oftener met with in Handel than in Bach.

140. With the older composers such double counterpoint as
that which we have been quoting is mostly found in fugal writing.
All the extracts we have given as yet have been from fugues,
either instrumental or vocal. But in modern music the devices
of double counterpoint are frequently used to impart additional
interest to the development of the thematic material of a com-
position. Old "Father" Haydn, the founder of the modern
school of instrumental writing, was one of the first to employ
double counterpoint for this purpose ; we give a few extracts from
his works to illustrate the method of procedure.

141. One of his quartetts opens with the following subject —

-&c.

In the second part of the movement the first notes of this theme
are varied, and a double counterpoint in the octave added, thus —

64 DOUBLE COUNTERPOINT-. (ChaP.vi

Counterpoint HAYDN. Quartett in F, Op. 74, No. 2.

Subject varied.

j T

It will be seen that the fourth and fifth bars of this passage are
the inversion of the first and second. We omit the middle parts
of the harmony, as the progression of the outer voices is not
affected by them.

142. In the example just given, a counterpoint in quavers is
added to a subject in longer notes. Our next illustration will
show the reverse process, a counterpoint of long notes being
added to a subject in quavers. It is taken from the finale of
Haydn's quartett in A, Op. 55, No. i, the first theme 01 which is

In the course of the movement occurs the following passage, of
which, for the sake of clearness, we give the score in full,
numbering the bars for convenience of reference —

Viol. i.

HAYDN. Quartett in A, Op. 55, No. i.

Violoncello, mf

Chap. VI.]

ON A FLORID SUBJECT.

In bars i to 4 we see in the first violin a variation of the first
theme, to which in the violoncello is added a counterpoint in
semibreves and minims, the viola entering with a free part at the
second bar. From bars 5 to 8 we find the inversion in the
octave of bars i to 4. Note here that, in consequence of
the close position of the two voices, the first part of the counter-
point does not undergo inversion. We see it strictly inverted by
the viola and violoncello at bars 12 and 13. At bars 8 and 9 is a
curious and interesting variation of the counterpoint. In the
ninth bar the quaver figure which in bar i was used against the
first bar of the counterpoint is employed against the second, and
one bar of the continuation is consequently omitted. The whole
passage is a beautiful example of the application of double
counterpoint in practical composition.

143. The following passage from one of Haydn's earlier
symphonies shows the employment of double counterpoint in a
sequence —

(1) u /-> »_ f _ HAYDN. Symphony in G Minor.

DOUBLE COUNTERPOINT;

[Chap. VI

As in previous examples, we have omitted the middle parts of the
harmony.

144. Our last extract from Haydn is of an altogether different
kind. It is the commencement of the slow movement of a
symphony —

Andante ' HAYDN. Symphony in D.^

££

*"** ! 1 1 1 1 ~l ' I ^JF /

The first eight bars of this passage are in two-part harmony only,
each part being doubled in the octave. At (a) the whole phrase
is inverted, middle parts (not quoted) being added to fill up the
harmonies.

145. Our first quotation from Mozart

(1)

MOZART. Fugue for Orchestra (Fragment).
tr

:haP. vi.] ON A

(2)

/ Q »

FLORID SUBJECT.

Jj J

1 f* H

strikingly illustrates the difterence between strict and free counter-
point. In the strict style, the consecutive octaves between the
first notes of the first and second bars, and the arpeggio of the
chord of the dominant seventh in the fourth bar would be
objectionable. Here they are quite allowable.

146. Our next example

MOZART. Symphony in G minor.

iifr ..uri rayiM*-

shows a case of not infrequent occurrence. We have here a
counterpoint written on the first subject of the movement ; the
subject (as many readers will be aware) is that given in the bas?
of the first four bars. If we compare (b) of the inversion with
(a) of the original counterpoint we shall see that a slight change
in the figure is here made. This is to keep the parts in a com-
fortable compass. Had the counterpoint at (£) been an exact
imitation of (a) the bass would have been inconveniently high;
on the other hand, the figure of (b) could not have been
employed at (a) without crossing the parts. Such small modifica-
tions may always be allowed in free double counterpoint.

DOUBLE COUNTERPOINT:

[Chap. VI.

147. The following charming little piece of double counterpoint

MOZART.
Variations on " Unser dummer PSbel meint."

B.r fcf tf^r

lr ffff -

- ' • ' i.
f f €F,

gives in the first four bars an example of free imitation, a subject
to be dealt with later in this volume (Chapter X.). As in a
previous example, we have numbered the bars. The inversion
begins at bar 9. Compare the upper part of bars 6 and 7 with
the bass of bars 14 and 15. At bar 14 we see bar 5 varied by
the addition of accidentals (compare §§ 138, 139); and in bar 15
we find in the upper voice the bass of bar 7 with accented
auxiliary notes (appoggiaturc)* The eight and sixteenth bars of
this passage are free.

* As we have not had occasion to use this word before, it may be well to
remind the student that an appoggiatura (Italian — a "leaning note") is an
accented auxiliary note placed at the distance of a second from a harmony note.
and of not less than half its value.

Chap, vi.] ON A FLORID SUBJECT.

148. The following example

BEETHOVEN. Symphony in D, No. a.

shows a different treatment of a sequence from that seen in
§ 143. Here the pattern is set in the first two bars, and inverted
in the third and fourth; the whole passage is then repeated a
tone lower.

149. Our next extract requires few words —

( 1 ) BEETHOVEN. Quartett in C minor, Op. 18, No. 4.

The upper part of (i) shows the first subject of the slow move-
ment, to which, on its resumption in the latter part of the piece
the counterpoint here quoted is added.

150. It is comparatively seldom that a long passage of double
counterpoint is to be found in a scherzo; the following is an
excellent example —

DOUBLE COUNTERPOINT: rchap. vi.

_. _ BBETHOVKN. Sonata, Op. 26.

Notice at (b) the alteration of one note of the subject at (a).

151. A charming combination of two melodies strongly
contrasted in character is to be seen in the following extract
from the first movement of Cherubini's second string quartett —

(11 tr ff. CHERUBINI. Quartett in C

*jj -MITT

1 — ' ' — r

(2)

Chap. VI.)

ON A FLORID SUBJECT.

As before, we omit the middle parts of the harmony.

152. Among the composers of the first half of the present
century Mendelssohn and Schumann were probably the greatest
contrapuntists. It is seldom, nevertheless, that we find double
counterpoint developed at any great length in their works,
especially in those of Schumann. We give two examples from
Mendelssohn, totally differing from one another in character —

MENDELSSOHN. "St. Paul."

The overture to "St. Paul," from which the above passage is
taken, is in the contrapuntal style throughout ; but there are not
many bars which are strictly inverted. Even in this extract it
will be noticed that the last group of semiquavers is free, and the
inversion only lasts for a little more than two bars. •

153. Every one who has heard the " Scotch " symphony
will remember the beautiful effect produced in the first movement
by the combination of two of the chief themes.

(i)

MENDELSSOHN. 3rd Symphony.

r err

(2)

5 — 3 — 4-1 —

- -P f* i

-: j

J J ^

DOUBLE COUNTERPOINT:

[Chap. VI.

154. Our last example of double counterpoint in the octave
will be from a work by a living composer, Johannes Brahms —

BRAHMS. " Dcutsches Requiem."

IJ-J

#*?

~^ri^--pr-^-^— ,

' — ^ ^ <a

)

' J •

(2)

i^rrrr

^>br 'r^r

r-i^flj ,1

r^ J = ^J g^-J ^J

irr^rrbj^| uW^

s — i J ^

pj f i i

This fine passage illustrates the tendency of the modern school
toward freedom in contrapuntal writing. It will be seen that in
the second bar of this extract, the fifth crotchet, at (a) is altered
in the inversion at (b). (Compare § 150.) The last bar of the
passage, also, though maintaining a general resemblance, is not a
strict inversion at (2) of the model. It is difficult to give a satis-
factory reason why any change should have been made here ; but
it may be said, once for all, that in modern compositions good
specimens of -strictly inverted double counterpoint are far more
rarely to be met with than in the works of the old masters.

155. We have several times noticed the rarity of the employ-
ment of double counterpoint in the tenth, as compared with
those in the octave and twelfth. In the whole of Bach's " Forty-
Eight Preludes and Fugues " it is very seldom met with, e.g., in
fugues Nos. 29, 40, and 45 ; though double counterpoint in the
twelfth is tolerably frequent, and that in the octave is to be
found in every fugue. In our first chapter we quoted passages
from two of the fugues we have named (§§9, n) ; we will there-
fore give for our present illustrations some extracts from Bach's
" Art of Fugue," the tenth number of which is specially written
to exemplify double counterpoint in the tenth, of which it is
probably the finest specimen ever composed. The two subjects
to be combined are in the first instance worked separately,
and are not found together until the 44th bar, when they appear
thus —

Chap. VI.]

(*)

ON A FLORID SUBJECT. 73

J. S. BACH. " Art of Fugue," No. 10.

x--— —

ff &c"

We first see them inverted in the following manner in bar 52 —

The alteration of the first note in this passage is necessitated by
the laws of fugal construction. If the student will compare
these two examples, he will see, with the guidance given him in
§§ 8, 9, that the inversion here is in the tenth. The next inver-
sion (bar 66) will make this even clearer —

y ' J ^ j=t=

^y (^ 0 *"*" """

&c.

go r El ! — i

r j r 1 1 —

By comparing (<r) with (0) it will be seen that the lower part of
(c) is the upper part of (a\ an octave lower ; while the upper part
of (c) is the lower part of (a), a tenth higher.

156. When treating of double counterpoint in the tenth, we
mentioned (§ 80) that it differed from all other kinds, in the fact
that it could be used simultaneously in two parts, which would
move in thirds, sixths, or tenths, according to their positions.
Our next extracts will illustrate this (bar 75) —

&C.

Compare this with (c\ and we shall observe that the two outside
parts are inverted in the octave, and therefore in the tenth of the
original model at (a). The middle part makes with the bass
exactly the same intervals which the two parts made at (a) ;
and were it inverted with the bass, we should have double

DOUBLE COUNTERPOINT;

[Chap. VI

counterpoint in the octave. In our next example, the thirds are
added to the other subject (bar 86) —

157. Let the student compare this passage with (a). He will
see that the middle voice gives the inversion in the tenth, and
the thirds above it now give the inversion in the twelfth. In
example (d) the added part would have inverted in the octave.
Thirds or sixths added to a double counterpoint in the tenth will
produce double counterpoint in the octave or twelfth, according
to the position.

158. The last passage we shall quote (bar 104) —

shows the inversion of (e). By comparing it with (a), it will be
seen that here, as at (d), we have simultaneous counterpoint
in the tenth and octave. These various counterpoints show how
both the subjects can be transposed. It will be remembered
that in § 79 the student was told not to transpose his canto fermo ;
but when, as here, the two parts are of equal importance, it is
possible, as we see, to transpose either of them.

159. It will be noticed that in none of the passages we have
given are the thirds or sixths added to both subjects at once.
This, however, would have been possible, and is not infrequently
done. An excellent example of this will be seen in the great
fugue, in G minor, of the "Wohltemperirtes Clavier." The
subject and counterpoint, with their original inversion in the
octave, were given at (a) and (b) of § 10. At the 59th bar of
the fugue, we find the following —

J. S. BACH. " Wohltemperirtes Clavier," Fugue 40.

i^-

tj.*

Here thirds are added both to the subject and counter-subject
By comparing the voices with those of the model in § 10,

Chap. VI.]

ON A

we shall see that we have simultaneously double counterpoint
in the octave between the alto and tenor, in the tenth between
soprano and tenor and alto and bass, and in the twelfth between
soprano and bass. Such thirds can be added to any double
counterpoint in the octave, provided that similar motion and
an unprepared sixth are avoided, the former being, as we know,
unavailable in double counterpoint in the tenth, and the latter in
the twelfth.

1 60. In the third volume of Albrechtsberger's theoretical
works will be found two fugues written in double counterpoint
in the tenth ; but as extracts from these would show little or
nothing which has not been already illustrated in the examples
we have just given from Bach, we content ourselves with re-
ferring students to Albrechtsberger's treatise. We will now
give a few passages in which double counterpoint in the tenth
is incidentally used. These will mostly be very short, as it is but
seldom that an opportunity occurs for the employment of this
device.

1 6 1. Handel scarcely ever writes double counterpoint at
this interval. A fragment, of only half a bar's length, will be
seen in the following passage —

HANDEL. Anthem, "Have mercy upon me, O God."

At (b) is shown the inversion in the tenth of (a).

162. Our next example shows how double counterpoint in the
tenth can be obtained by adding thirds or tenths to that in the
octave —

. (1) JOMKLLI. Mass in D.

----- ,„

(2)

DOUBLE COUNTERPOINT:

r r F «J ' J L J " i

[Chap. VI.

®n=3

* * ^.

Bass.

r r

Here the regular double counterpoint of the fugue is, as usual, in
the octave. This inversion is shown in the small notes of (2) ;
but the bass is here accompanied by the treble in tenths, the
latter thus giving double counterpoint in the tenth against the
subject.

163. In the following passage,

HAYDN. Mass, No. 12.

Ffr

hMI

(2)

^ rrrr-u

the two subjects are mostly worked in the octave ; but an
incidental double counterpoint in the tenth is introduced, as
shown at (2). If the student will examine these two subjects,
he will see that they would also be capable of inversion in the
twelfth.

1 64. In instrumental music, double counterpoint in the tenth
is even rarer than in vocal. We give a very short specimen by
Mozart —

( 1 ) . MOZART. Sonata in D.

Chap. VI.]

ON A FLORID SUBJECT.

It will be noticed that only the first eight notes of this counter-
point are really inverted in the tenth. By a slight modification of
the figure, Mozart changes the interval of inversion to the octave.
165. The scarcity of illustrations must be the author's
apology for introducing an example from his own pen —

E_. PROUT. Symphony in D, No. 4.

1 66. We conclude our illustrations of this counterpoint with
a portion of the masterly canon in the tenth in Bach's " Art of
Fugue "—

J. S. BACH. " Art of Fugue."

DOUBLE COUNTERPOINT: (Ch*P.vi

Pf£^=

r ' |T=

f-" V -f-

Wtari

' 1

Chap, vi.i

ON A FLORID SUBJECT.

This wonderful movement is too long to quote in full ; but a
somewhat extended extract is required to make it intelligible.
We number the bars for the sake of reference, The subject of
canon will be spoken of later in this volume ; it will suffice now
to say that a canon is a composition in which one part continuously
imitates another at any given interval. In the present case it will
be seen that the upper voice imitates the lower at a distance of
four bars, and at the interval of a tenth above. We give only the
first half of the canon, which is continued strictly to the 39th bar.
From the 44th bar to the end of the piece, with the exception of
the last four bars, which are free, the whole canon is inverted in
double counterpoint of the tenth.

167. Let the student first examine the model, bars 5 to 21.
The counterpoint begins at bar 5, and it will be seen that in the
harmony considerable prominence is given to the intervals of the
fifth and sixth, which in the inversion become respectively the
sixth and fifth. Observe that in the exposition, so to speak, of
the canon, Bach, at bars 9 to 12, inverts bars 5 to 8 in the tenth.
In bar 18, at the second crotchet, is a diminished fifth struck by
the two parts. This is because it clearly represents the first
inversion of a dominant seventh (§ 104). In bars 13 and 14
will be seen sixths approached by similar motion, from accent to
accent ; the inversion produces hidden fifths (bars 52, 53).
Though these would be objectionable in strict counterpoint,
they may be allowed in free, especially in double counterpoint
in the tenth, in accordance with the often mentioned general
principle, that in proportion as the difficulty of the task increases,
the strictness of the rules in less important points is relaxed.
Similar hidden fifths will be seen in the example to § 165.

1 68. Now let us look at the inversion, bars 44 to 60. If we
compare the commencement of the canon (bars i to 4) with the
upper of the two subjects of the fugue given in § 155, we shall
see that the former is a variation of the latter. Both are, in fact,
varied forms of the canto fermo, on which, either direct or by
inverse movement, the whole of the "Art of Fugue" is
composed.

Subject Inversion.

<»:— rr P I g? 1 1 m-

*y II

1 — f5" —

F | r r r i ^

1 rlr

In the canon, as in the fugue, it is the inverted form of the
subject which is taken for treatment ; but it will be noticed thai

DOUBLE COUNTERPOINT:

[Chap. VI.

in all the examples quoted from the fugue (§§ 155 to 158), it is the
lower part which is transposed a tenth higher, while in the canon
it is the upper part which is transposed a tenth lower. Moritz
Hauptmann, in his Analysis of the " Art of Fugue," has called
attention to this, and given, undoubtedly, the true reason, that
the original subject (the upper part at § 155 (a\ and the lower
part in the canon) is really the canto fermo, which therefore must
not be transposed (§ 79). At bars 48 to 51, we see again the
inversion in the tenth of the four preceding bars, with the
curious result that bars 48 to 51 are identical with bars 5 to 8
of the model. The whole of this extract deserves close examina-
tion, comparing the inversion bar by bar with the model.

169. Double counterpoint in the twelfth, though somewhat
less rare than that in the tenth, is yet far from common,
especially with modern writers. The examples now to be given
will sufficiently illustrate its practical use —

( 1 ) Bars 5, 6. BACH. " Wohltemperirtes Clavier," Fugue a.

In the inversion, we have added the upper voice, which is in
thirds with the bass, because it does not obscure the clearness of
the counterpoint. Notice at (a) and (b) sixths in the pattern, the
inversions of which at (c) and (d) become fundamental sevenths,
and compare §§ 104, in.

170. Our next illustration is very similar in character —

(1 ) Bars 48, 49. BACH. " Wohltemperirtes Clavier," Fugue 28.

Bars 55. 56.

Chap. VI.J

ON A FLORID SUBJECT.

Here, again, the sixth can be taken at (a), because the inversion
at (b) gives us the dominant seventh on Cf.
171. The following passage

(1) Bars 77-30. BACH. " Wohltemperirte. Clarier," Fugue 47.

(2j Ba« 49—45.

shows how the inversion in the third bar produces a modulation
to the key of the relative minor. Similar cases were seen in the
counterpoints on a choral in the tenth and twelfth which we gave
in the last chapter.

172. Our next example is taken from the fugue, written by
Bach, in his "Art of Fugue," expressly to illustrate double
counterpoint in the twelfth —

(1) BACH. " Art of Fugue," No. 9.

DOUBLE COUNTERPOINT;

[Chap. VI.

At (i) we see in the upper voice the direct form of the subject
quoted in § 168, the inversion of which was used for double
counterpoint in the tenth. At (2) is the inversion of the
counterpoint in the twelfth. Compare (a) with (c\ and (b) with
(d\ and notice particularly the alterations in the position of the
semitones. Had the scale passage at (a) been exactly imitated
at (b\ there would have been a modulation to the key of
A minor, which Bach did not wish. Such chromatic alterations
of notes will be more often met with in a minor than in a major
counterpoint. In the last three bars of this example, where the
music modulates to the key of the relative major, the only note
altered in the inversion is the leading note (§ 84).

173. The following passage, taken from one of Handel's
"Chandos Anthems,"

HANDEL. Anthem, " My song shall be alway."
(2)

exemplifies a somewhat different point. Here the inversion
begins in the twelfth. The sixth above the subject at (a)
becomes in the inversion, not a fundamental seventh, as in the
examples to §§ 169, 170, but an accented passing note (§ 85).
Observe that at (b\ in the third bar, further inversion in the
twelfth becomes impracticable ; the rest of the passage, from (d),
is therefore inverted in the octave.

174. Sometimes more than one part may be inverted at once—

(1) HANDEL. "Jephtha."

r I I | , J 1 , ! „

<S)

:Q£±F

&C.

In this example, we have at (2) omitted the tenor part, which is
free, to show the inversions more clearly. The subject, which at
(i) is in the treble, is at (2) in the bass. The alto, which at (i)

Chap. VI.]

ON A FLORID SUBJECT.

began on the fifth below the subject, begins at (2; on the octave
above, and it is therefore inverted in the twelfth. At the same
time the tenor, which in the model began on the tenth below the
subject, begins (now in the treble) in the inversion on the tenth
( = the third) above, and is therefore likewise in double counter-
point in the twelfth.

175. The double fugue in Mozart's "Requiem" has probably
been quoted in every book on double counterpoint published
since it was composed. It is, nevertheless, too fine an example
to be omitted here —

(1) MOZART. "Requiem."

[AJ Jj^lj-Tf

The two themes announced at (i) are first inverted in the octave,
as at (2), and subsequently in the twelfth, as at (3). Notice how,
as in previous examples, the inversion of a sixth becomes a
dominant seventh. We omit, as in many other examples, the
accompanying parts.

176. Double counterpoint in the twelfth is not very frequently
employed by Beethoven. The following example will require no
explanation —

(1)

Bar 4.

BBBTHOVKN.
(2) Bar 19.

Sonata, Op. no.

177. For our last example we give an entire movement,
by one of Bach's most distinguished pupils, J. P. Kirnberger,

DOUBLE COUNTERPOINT:

[Chap VI

which is written throughout (excepting the free close, four bars in
length) in double counterpoint in the twelfth. This very fine
specimen is taken from dementi's " Practical Harmony " —

J. P. KlRNBERGER.

• »i m*V ' T~,T'r+ . km i _<•

Chap, vi.] ON A FLORID SUBJECT.

Inversion in the lath.

24 25 26 ^-" 27

86 DOUBLE COUNTERPOINT: [Chap. vi.

That the student may the more readily compare the inversion
with the model, we have numbered the bars in both, from
i to 36. The bars at the end, from 37 to 40, give the free
close, which (§ 88) is always a necessity with this counterpoint.

178. This example illustrates nearly all the rules we have
given for double counterpoint in the twelfth. Observe that it
begins with the interval of the octave — not with the fifth, as we
have done in our examples. This is because the upper part is
considered as the subject, and the lower as the counterpoint.
This is clearly shown by the inversion, in which the lower part
is transposed a twelfth higher, and not the upper part a twelfth
lower. Notice also the great prominence given to the interval of
the third, not only on the accented notes of the bar, but in
successions of thirds (bars 17, 18, and 29 to 31). At bars 9 to 14
will be seen a good example of the sixth as a harmony note,
prepared in the lower voice (§ 100), which in the inversion
becomes a regularly prepared seventh. It will be seen that
accidentals are freely used in the inversion (§ 124), if the
melodic or harmonic progressions are thereby improved, as,
for instance, in bars 29 to 31. The smooth flow of the whole
counterpoint, in spite of the restrictions under which it is written,
shows the composer to have been a worthy pupil of his illustrious
master.

179. There is a spurious kind of double counterpoint, fre-
quently to be found even in the works of the great masters,
which must not be confounded with that which we are now
treating of. It is not uncommon to meet with a theme accom-
panied by a more or less elaborate counterpoint above it (or
below it, as the case may be), and then to find the theme
accompanied in the reverse position by another counterpoint,
bearing a general resemblance to the first, but not identical
with it. Such counterpoint may be described as "electro-
plated " double counterpoint — very useful, often even artistic
and beautiful ; but after all only an inferior substitute for the
genuine metal. It is naturally far easier to write, and very often
for all practical purposes quite as effective ; but it must be taken
for what it really is — counterpoint — good counterpoint, in most
cases, but not double counterpoint.

1 80. That we may not be suspected of intending to disparage
the counterpoint of which we are now speaking, we give two
excellent examples of it —

(1) HAYDN. " Creation."

Chap. VI. J

ON A FLORID SUBJECT.

It will be seen that the counterpoint of the second bar of (2) is
not an inversion of the corresponding bar of the model, though
it sounds sufficiently like it to be mistaken for it by a casual
hearer. It is quite as effective as the real inversion would be —
perhaps even more so ; and Haydn, doubtless, had good reasons
for making the alteration, since nobody could write better double
counterpoint than he when he chose.

1 8 1. Our second example is from Mendelssohn, indisputably
one of the finest contrapuntists of the present century —

Viol. i. MENDELSSOHN. 4th Symphony.

Viol. 2.

Here the figures of quavers, playing round the subject, are equally
effective and charming in the first and second violins ; but they
are not identical. It will be seen that each time the melody is
above the counterpoint; we have, nevertheless, quoted the
passage, because, as the student will easily see, both phrases
are really written in double counterpoint in the -octave, though
neither is strictly inverted in the course of the movement.

182. It will be seen that nearly the whole of this long
chapter is taken up with the analysis of examples. The
resources of double counterpoint are so exhaustless that it is
quite impossible to lay down precise rules as to every point
that may occur. All that can be done is to indicate the general
principles which should guide the student in writing, and to
inculcate their observance by showing the practice of the great
composers. Experience will do the rest. We have here taken
but a few gleanings from the ample field of musical literature ;
he who will take the trouble to explore it more thoroughly will

88 DOUBLE COUNTERPOINT: [Chap. vi.

reap a large reward for his pains. In double counterpoint, Bach
is the unrivalled master of all masters. Let those who would
thotoughly understand this branch of their art, follow the advice
of Schumann, and " let the * Forty-eight Preludes and Fugues '
be their daily bread." Careful analysis of these works will teach
the earnest student far more than he can learn from this, or any
similar book.

183. The student should now begin to write free counterpoint
in the octave, tenth, and twelfth for himself. If he can, it will
be best for him to invent both melodies ; but if he at first feels
himself unequal to this, he will find a selection of 'subjects which
will serve his purpose in the third and fifth sections of Part I. of
the author's " Additional Exercises to Counterpoint"

Chap, vn.1 WITH FREE PARTS ADDED. 89

CHAPTER VII.

DOUBLE COUNTERPOINT WITH FREE PARTS ADDED.

184. We have frequently referred to the addition of other
parts to two parts which were written in double counterpoint;
and in many of the examples given in the last chapter we have
said that such parts were in the original, though they are omitted
in our illustrations for the sake of showing the double counter-
points themselves more clearly. As a matter of fact, by far the
larger number of double counterpoints are accompanied by what
are called " free " parts, that is, by parts written only in simple,
not in double, counterpoint with the others. When actual two-
part double counterpoint is found without such additions, it is
mostly at the beginning of a fugue, or in old music intended for
the harpsichord, as in the examples from Bach and Kirnberger
given in §§ 166, 177. We now proceed to show how one or
more free parts are to be added to an existing double counter-
point.

185. The task now before the student differs from those
which have hitherto engaged his attention in one important
respect. In all the counterpoint he has yet worked, whether
simple or double, he has always had one given subject to which
to add one or more parts. Now, however, the two voices of the
double counterpoint may be regarded as two cantifermi, to which
one or more florid counterpoints have to be added. His work
will now be very similar to that which he has had in some of the
varieties of combined florid counterpoint.

1 86. It will be seen at once that the fact of there being two
given subjects instead of one considerably limits the choice of
harmony. In his very first attempts at counterpoint the student
was told (Counterpoint^ § 62) to consider the harmonic possi-
bilities of each note of his subject. Thus, in the key of C, the
note C may be the root of the tonic, the third of the submediant,
or the fifth of the subdominant triads. As we are now dealing
with free counterpoint, the same note may also be the seventh of
the fundamental discord on the supertonic, or the eleventh of
the chord of the dominant eleventh. Similarly F may be the
root of the subdominant chord, the third of the diatonic triad on
the supertonic, or the seventh of the dominant. But if the two
notes of the double counterpoint, which are sounded together
are F and C, several of the chords just named are at once

90 DOUBLE COUNTERPOINT: [Chap. vn.

excluded. We can neither take the tonic nor submediant triads,
nor the fundamental seventh on the supertonic, because F is not
a note of any of these chords. So also, C does not belong to
the triad on the supertonic, nor the chord of the dominant
seventh. With these two notes given, we are therefore practically
restricted to the subdominant triad or some position of the chord
of the dominant eleventh as our only available harmonies.

187. It is from such limitations as these that the student's
chief difficulties will arise, especially when he is adding a new
bass below both the given parts. But in this case he must be
careful not to hamper himself needlessly. Supposing, for
example, that the two notes of his double counterpoint in the
key of C, are E and C, making a sixth between them, and that
he is going to add a bass, or perhaps a tenor and bass below
them. The notes given suggest, of course, the first inversion of
the tonic chord : but it would be a great mistake for the student
to imagine that the tonic harmony is the only one possible. The
following example will soon convince him of this. We give the
sixth in the upper parts, and add a tenor and bass below —

(«) (*) (?) (d) (f) (/)

(e)

(A)

1 88. At (a) we give the tonic chord, as that which would
naturally occur first to the student. This might also be taken in
the first inversion, and even in the second, if the context allowed
of its proper treatment. At (b) and (c) are the root position and
first inversion of the submediant triad. All these positions can
be used freely, and there would be little or no difficulty about
their introduction. But it would be also possible to treat the
given notes, under certain circumstances, as parts of fundamental
discords. At (d) (e) and (/) are shown various positions of the
supertonic major ninth. These, of course, would be only avail-
able if the seventh and ninth could be properly resolved, and the
third of the supertonic chord were able to move a semitone
(Harmony ', § 474). At (g) and (h) are shown two positions of
the dominant thirteenth, and at (/) the tonic minor thirteenth,
any one of which would be at least possible. It will be seen
that here are nine combinations of which the notes C and E
could form a part, and this list is not exhaustive. The student's
resources, therefore, are not so limited after all as would appear
at first sight.

189. A very important consideration will always be, which
notes of the subject are to be treated as harmony notes, and

Chap. VII.]

WITH FREE PARTS ADDED.

9»

which as auxiliary or passing notes ? It is impossible to lay
down any absolute rules on this point ; experience will be the
student's best guide ; but he must remember that he is now
not restricted to one chord in a bar, or even to one chord against
each note of his subject. He may change his harmony as often
as he finds it expedient ; and it will frequently happen that the
interposition of a second chord will save a weak or bad root-
progression.

190. It must also be borne in mind that we can now treat
any note, accented or unaccented, whether approached by step
or by leap, as an auxiliary note, provided it is quitted by step. In
many cases this permission will be found extremely useful.
The rule prohibiting the sounding of dissonant notes together,
except when taken by step, is also to a very considerable extent
relaxed; though judgment will need to be exercised in this
matter, so as to avoid harshness as much as possible.

191. We shall now give a series of examples, to illustrate the
principles we have laid down ; from the careful analysis of these
the student will, it is hoped, learn all that it will be needful for
him to know. Following our usual method, we will take one
short subject, and treat it in many different ways. For this
purpose we will choose the first eight bars of the double counter-
point in the octave which we worked on a choral in § 114.

192. We will first add a free middle part to the counterpoint.
In order to do this, it will be necessary to transpose the lower
voice an octave, to make room —

D. c.

(a) „

B.C.

In all our examples we shall indicate by a " D. C." the two
parts which are in double counterpoint with one another. We

DOUBLE COUNTERPOINT:

[Chap. VII.

choose an added middle part for our first example, as being
on the whole the easiest to work. The only remarks to make
upon this example are, that at (a) there are not consecutive
octaves between treble and alto, because the crotchet A is a note
of the harmony; and that at (&) two chords are used on the
bass note.

193. We now take the double counterpoint in the two lower
parts of the harmony, and add a free part above —

&tf^ .

r ' Jl

W 4m> '

yff./'V

m — P — r *r

,rrrr

s*~

> rr- — ,

1 Jf r i

D.C

J-

| J p

-*4 —

(')

Fn*~

•HP

We have here transposed the upper part of the double counter-
point an octave to get a better position for the harmony. This
example might have been written, like the last, in the key of D,
giving the lower parts to tenor and bass, and the new part, a fifth
lower than written, to the alto. We have preferred, for the sake
of comparison, to give all our three-part examples for treble, alto,
and bass voices. Observe that the EJf at (a) makes no false
relation with the Efl of the alto of the preceding bar, because
the latter is a passing note.

194. We next add an upper part to the inversion of this
counterpoint —

D.C

r i

D.C

CTir r r

Chap. VII.)

WITH FREE PARTS ADDED.

A florid bass, such as is seen here, usually necessitates more
changes of the harmony. In the second bar of this example,
there are distinctly four chords. At (a) is a point worthy of
notice. It looks at first sight as if the bass leapt to a second
inversion from the inversion of another chord. So it does, if we
choose to consider the third and fourth quavers of the bass as
each representing different harmony. But in the free style in
which we are now writing, we are not bound to assume this. We
may look at the four quavers of the first minim as representing
the supertonic chord— B and E being harmony notes, and C and
D passing notes, as also would be the C in the treble ; or, instead
of this, the A in the bass may be regarded as an ornamental note
interposed between the E and D. That we have not, in spite of
appearances, a true second inversion here, is shown by the fact
that it is neither followed by another chord on the same bass
note, nor by a chord on the next degree of the scale (Harmony,
§ 165). The only other harmony possible above the A and F
would have been the first inversion of the mediant chord, from
which the progression to the tonic at the fourth crotchet would
have been weak. This passage illustrates the liberty which is
allowed in free counterpoint, such as this.

195. It will generally be found more difficult to add a new
bass below a double counterpoint than to add upper or middle
parts. The following examples will show how this is to be
managed —

r r,U J

fr*

D.C.

. , ET,f r*

--^/rp

DOUBLE COUNTERPOINT;

[Chap. VII

At (a) is seen a somewhat, though not precisely, analogous
example of the introduction of a second inversion to that which
we have just been considering. Here we can either call the F in
the bass a passing note, in which case the root progression is IVa
to "VY ; or we can look on the G as a passing note, both in treble
and bass, and regard the F as the harmony note, the progression
then being \b to W.

196. We now put a bass below the inversion of the
counterpoint —

\) 1 * 1 1

1 1

^ 4 ^

i ' =

r r r Cri

ii i =
E£g% r

•>.

gV* j.r t f=

E!EE£L£==

r^rr jai

=F==^==^^

>• I £=-

Ur ' uj i — ri r i r i =n

This example requires no explanation ; but the student should
notice that, though the two upper parts are (except in relative
position) identical with thost, of the last example, every bar of
the bass — except, of course, the final tonic — is different from the
preceding. It would have been possible to make the same bass
do duty below the counterpoint in its new position. We have
written a fresh bass, to show that variety can be obtained, even
with the restrictions under which we are now writing.

197. We will now add two free parts to our counterpoint.
This is naturally rather more difficult than adding only one. We
will begin with the easiest position, in which the additional parts
are the middle voices —

|-^s 4 ^

ij J r rp

1 — r* J r^Ti* ^1 — 1

p jr r JT;

ir rr f7 if • frrrr|

P'*'" ' ^

feU4 IP., j-g_

— ^
1 r ^

D. C.

i1 r r

Chap Til.]

WITH FREE PARTS ADDED.

At (a) there is no second inversion of the submediant chord ; if
there were, the bass could not leap to the next note. There
is here the resolution of a double suspension on the tonic
chord, and the B in the treble is only a passing note. At (ft) the
alto and tenor parts cross for one crotchet, to save the octaves
with the bass on the unaccented beats which would result if the
alto part were —

(Compare Counterpoint, § 175.)

198. We next add two upper parts —

SEES

D. C.

D. C

F=T

At (a) the G J of the tenor, being an auxiliary note, makes no
false relation with the Gfl of the alto. A somewhat similar

DOUBLE COUNTERPOINT:

fChap. VII.

example from Bach will be seen in Harmony, § 324. Here, how-
ever, the effect is much less harsh than in the passage cited,
partly because it is here in a middle voice, but chiefly because it
returns to the harmony note before the chord changes, which in
the extract from Bach it does not. The cadence at (b) is not
very comfortable ; this cannot be helped, as it arises from the low
position of the two given parts.

199. In our next example the double counterpoint is in the
alto and bass, with one free part above, and one in the middle —

D. C.

t=«

The only remark to make on this example is that at (a) we have
varied the cadence given in the example to § 193, and inten-
tionally sounded two dissonant notes together in the treble and
alto, to show that they can be thus taken, because they form part
of the chord of the dominant thirteenth (§ 104).

200. Our next examples illustrate the most difficult com-
binations—
P.C.

P-C. „

ll i*

Chap. VII.]

WITH FREE PARTS ADDED.

Here two parts are added below the counterpoints. Note the
firm progression of the new bass, resulting from the almost
exclusive employment of the strong chords, I., IV., and V., of
the keys of A and Fj minor (§ 117).

20 1. Lastly we take the double counterpoint in the two
middle voices, adding a treble and a bass —

D. C.

£=

crieeCTr nr

.J ra

! JTfj

This example needs no explanation.

202. We have now added free parts in ten different ways to
the same phrase of eight bars ; and if the student will compare
them, he will see that no two are alike. There is necessarily a
general resemblance ot character about all, because our choice
of harmony is so restricted ; but there is quite sufficient diversity

08 DOUBLE COUNTERPOINT: (Cbap.vn.

of detail to show how much variety is possible, even with simple
harmonies. It will be seen that no chromatic chords are em-
ployed in any of the examples ; this is because the melody of the
choral is so diatonic that chromatic harmony would have been
entirely out of keeping with it. But to those who have the
resources of harmony at their fingers' ends — and it may be pre-
sumed that no others will essay such advanced work as that which
forms the subject of this volume — there never need be the least
danger of monotony. Even the seven diatonic notes of the key
furnish an exhaustless supply of harmonies to those who know
how to use them.

203. It is possible to write free parts which shall themselves
be in double counterpoint with the two given parts, and, in at
least one standard work of counterpoint, exercises of this kind
are set. In reality, however, these are of little practical use ;
because if a composer wishes to add such a part, he will most
probably (and had better) make triple counterpoint at once.
How this is to be done will be shown in a subsequent chapter.

204. It would have been considerably easier to add plain
chords to our double counterpoint instead of florid parts. We
have chosen the latter as being not only musically more in-
teresting, but far more instructive to the student The more
complete command he can acquire of all the varieties of florid
counterpoint, the better equipped he will be for the work of
practical composition.

205. The addition of free parts will often improve a progres-
sion that would be harsh or stiff in two parts only. For example,
in our double counterpoint in the tenth in § 117 is seen between
the first and second bars of the lower counterpoint a third followed
by a fifth, with both parts moving by step. The uncomfortable
effect disappears as soon as a free part is added, either above, in
the middle, or below —

' -AgJf(a)i n -' ii(g)J d n(c)

:F=S^

206. In the same way that which in two parts looks like the
false relation of the tritone can often be saved. Thus, in § 67,
the third and fourth bars of the example can be mended, in free
counterpoint — and we need no longer confine ourselves to strict —
as follows —

Chap. VII.J

WITH FREE PARTS ADDED.

207. We shall conclude this chapter with a few examples of
the addition of free parts to double counterpoints by the great
masters ; and, as the most instructive course we can pursue, we
will show how several of the double counterpoints quoted in the
last chapter were filled up by their composers. We first take
the two passages from Bach's Organ Fugue in C minor given in
§ 133. Bach fills them up thus —

(1) J. S. BACH. Organ Fugue in C minor.

At (i) the added voices are the soprano and tenor, and, as the
bass remains, the general effect of the harmony is much the same
as in the outline previously given. But at (2) outside parts are
added, and the new bass gives a totally different character to the
music. Notice, especially, how in the third bar at the first
crotchet the weak effect of the two-part harmony is improved by
the additional voices.

208. In the extract from Bach's Organ Fugue in E minor
given in § 135, the passage (i) is in two parts originally. The
inversion at (2) appears in the following form —

D. C.

Here nothing but plain chords are added ; the filling up is less
contrapuntal than in our last example. Observe the fine and
unexpected effect of the chord of the dominant thirteenth in the
third bar.

209. The example from Handel's "Susanna" in § 138 is
somewhat similar. Here (i) is also in two parts only; at (2) a
middle part is added —

D. C. HANDEL. " Susanna.

D.C

f-g/2

Note that the sequential character of the treble and tenor is
maintained in the added alto part. The upper part, as here
given, is obviously too high for a chorus. Handel modifies it
somewhat in the voice part ; it is the first violin part, in which
the inversion is exactly retained, which is quoted here.

210. We next show the filling up of the example from
Beethoven in § 149 —

Chap. VII.J

WITH FREE PARTS ADDED. 101

BEETHOVEN. Quartett in C minor, Op. 18, No. 4.

This passage illustrates what was said in § 203. If the added
middle part of (i) and upper part of (2) be compared, they will
be seen to be identical, except that the last three notes in the
second passage are lowered an octave. The added part is in
double counterpoint with the upper part of (i). Notice that it
is possible to use the fourth here in the second bar (becoming a
perfect fifth in the inversion), because there is another voice
below it. We have no triple counterpoint here, because the
added part cannot be used as a bass ; if it be, we shall have at
the beginning of the third bar a | taken by leap from the
inversion of another chord.

211. Our next illustration is somewhat similar: it is the com-
pletion of the passage from Cherubini given in § 151 —

Jl)

D.C

-r- *3-n

L-HER

UBINI. vjuarteu in ^
tr

-=— - — €f J — *L-

*J' £J

-P — !lp_

• •* . ' ^l*1

[ fJ

1 — JL J iJ 1

D.C

»rf r fi

% 1 j -.*-,

^x. *

I OS

DOUBLE COUNTERPOINT;

[Chap. VH.

I -, \ -\

tr tr

i -r P n.» r, i» r r r

(S)

1 J ' T =f r i

D.C.

" 1 J ' J -1 flj ^J A

± j. ^ j-~j — r~-p

^—. -

•^ ^ f f r r r j-

ll^rl^ r VVJ

. £*• * * - -^

-N D.

^/
/j^*^

j j bj j j 4 j.

,/ , J . J^j

JU"

r r r ur V-P- T

">JLJ i ' J"

W^f

^i /r F

' ir ^.i»

In both these passages we have three-part harmony only; in (i)
the first violin is silent, while in (2) the two violins play in octaves
throughout. The added middle part is here written in double
counterpoint to both the others ; the slight modifications at (2)
are evidently made for the sake of getting more complete
harmony.

212. We now give two examples of the filling up of a double
counterpoint in the tenth. We first take that shown at (a)
of § iSS—

J. S. BACH. " Art of Fugue," No. 10.

D. C

3±=

chap, vii.j

WITH FREE PARTS ADDED.

103

Here, as in (2) of § 207, two outer parts are added. The
passage requires no explanation, but, like all the other examples
we are giving, will repay close examination.

213. In the inversion shown at § 155 (b) the added parts are
the treble and tenor —

J.S.BACH. "Art of Fugue, No. 10.

& 1

D.C.

214. Lastly we shall give two examples of added parts to
double counterpoint in the twelfth. We first take that quoted in
§ 171 from Bach's " Wohltemperirtes Clavier." At (i) the
counterpoint is as we have given it, in two parts only ; but at (2)
it is completed thus —

D.C.

J. S. BACH.

' Wohltemperirtes Clavier," Fugue 47.

„ I J

r *

r "IT cu

! i E, i i —I *a*:

Here the added parts are the tenor and bass.

215. Our concluding example will be from Mozart's "Re-
quiem " ; the outline of which was seen in \ the three passages
given in § 175. Of these that marked (i) is, as there shown, in
two voices only ; numbers (2) and (3) are filled up in the
following manner —

104

DOUBLE COUNTERPOINT; tchap.vn.

MOZART. " Requiem."

In the first of these passages the double counterpoint is in the
bass and alto; in the second it is in the treble and bass, two
middle parts being added. Observe in the last bar but one
of (2) how the inversion in the twelfth of the interval of the sixth
in the pattern is used as a dominant seventh.

216. The student will now be prepared to practise the
writing of additional parts to double counterpoints for himself.
He will find it a very useful exercise to complete the counter-
point given in § 114 in all the ten ways in which we have worked
the first eight bars for him in §§ 192-201. He can then take the
double counterpoints we have given in §§ 117, 121, and treat
them in the same way. After this, if he desires more practice,
he may take any of the two-part examples from the works of the
great masters given in Chapter VI., and try to add free parts of
his own to them in all the various positions. Or, if he prefer it,
he can first write for himself two parts in double counterpoint in
the octave, tenth, or twelfth, and then add free parts. He will
find ample material in the examples we have given for as many
exercises as he is likely to want.

Chap. VIII.J /y THE RARER INTERVALS. IO>5

CHAPTER VIII.

DOUBLE COUNTERPOINT IN THE RARER INTERVALS.

217. Although, as has been more than once said, the only
double counterpoints in common use are those in the octave,
tenth, and twelfth, it is also possible to write counterpoints which
will invert at the other intervals. There arc, however, as will be
seen directly, such difficulties connected with all these, as to
render them practically useless, except incidentally. So far as
we know, no compositions exist in which double counterpoints
in the ninth, eleventh, thirteenth, or fourteenth are used
systematically, as are those which have been already considered.
It is, indeed, quite probable that where they are to be found,
their occurrence is the result of accident, rather than of design.
It would, therefore, be useless to give the student any rules for
writing such counterpoints ; but, for the sake of completeness,
and as musical curiosities, we shall in this chapter give a few
examples of each variety.

218. I. Double counterpoint in the ninth. One of the rarest
and most unmanageable of all. The table of inversions in the
ninth will evidently be the following —

INTERVALS: 123456789
INVERSIONS 1987654321

An examination of this table shows at once where the difficulty
lies. Every consonance except the fifth becomes a dissonance
when inverted; and although we have several times seen, in
working our double counterpoints in the octave, tenth, and
twelfth, how a harmony note may become a passing note in the
inversion, and, vice versd, it is very evident that if we have to
treat every harmony note, except the fifth, in this way, our
difficulties will be enormously increased. That it is, nevertheless,
possible to write short passages in this counterpoint will appear
from the specimens now to be given

io6 DOUBLE COUNTERPOINT: [Chap, vm

219. Our first example is by Marpurg —

Counterpoint. MARPUI

i yt , ^TTr= ,

V. Jj

rd — r~i 1

s J.

^_ip 1 1— J-+

Inversion in the ninth below.

t-4-
»< — :j— |

1 T=&
ij j j i

i'iM nr IT =L

In this very ingenious example, it will be seen that, excepting
with the first and last notes, every consonance in one part is a
prepared dissonance in the other. This counterpoint can also be
inverted by transposing the subject a ninth higher, in which case
it would be better to sharpen the Cs, and thus take the music into
the key of D.

220. Our next illustration is taken from Lobe's " Composi-
tion," Vol. III.—

J. C LOBE.

^rir LriLr^-r^m^r

The inversion in the bass is in the original an octave lower than
here given, so as to leave room for the subsequent addition of
free parts. It is here put at the real interval of a ninth below the
model.

221. We shall give later in this chapter (§ 236) a very fine
example from Bach, in which double counterpoints in the ninth
and fourteenth are employed simultaneously. We now give a
specimen by Beethoven —

Chap. VIII.]

IN THE RARER INTERVALS.

107

Mass in D.

Here the counterpoint at (b) is clearly a repetition of that at (a\
but at a different interval ; for the first interval, which before was a
seventh, is now a sixth, and so on with the others. Whenever
both counterpoints are above (as here), or both below, it is
necessary to invert one of them in the octave, in order to find
the interval of inversion (§ 9). If we thus invert (b) we shall
get this form —

Now let us compare this with (a). We see that the seventh has
become a third, the third a seventh, the sixth a fourth, and so on.
In each case, the addition of the two intervals gives the
number 10 ; the counterpoint is therefore in the ninth (§ 8).

222. II. Double counterpoint in the eleventh. This counter-
point is, on the whole, much less difficult and troublesome than
that in the ninth. The inversions are the following —

INTERVALS: i 2 3456789 ion
INVERSIONS: nio 9876543 2 i

Here, as with the ninth, there is only one consonance which does
not become a dissonance when inverted ; but there is this advan-
tage that, as the consonance in question is the sixth, we can use
it more than once consecutively. We shall see presently that in
no other of these rarer counterpoints is there any interval which
can be used in the same way.

223. Our first example is by Cherubini, and is taken from his
" Treatise on Counterpoint and Fugue " —

CHERUBINI.

[\Jf „ _ xlTm m + 1*1

-1 <9-\ 1 — rsr-

1 p (•

s i — ;

Inversion in the nth.

•-r-

Various other transpositions are, of course, possible, e.g.t the
subject might be transposed a fourth higher, and the counterpoint

loS

DOUBLE COUNTERPOINT.

[Chap. VIII.

an octave lower; or the subject might be transposed a fifth
lower, while the counterpoint keeps its place, &c. Notice the im-
portance given in this example to the interval of the sixth.

224. The next illustration is taken from that exhaustless
mine of counterpoint, the immortal " Forty-Eight Preludes and
Fugues "—

Bars 25, 36. J. S. BACH. " Wohltemperirtes Clavier," Prelude 7.

It will be seen that the counterpoint is not strictly carried on to
the end of the inversion, to which Bach has added a lower part.
It is very seldom that these rare counterpoints are continued for
more than a few notes.

225. The following curious, though very fragmentary, passage
from Bach deserves quotation —

/ . rN -x „ I. S. BACH. Cantata, " Ich, elender Mensch.

W ^ ^ ^ (J) (e)

r ir r •

The first bar of (a) is inverted at (b) in the eleventh, and at (c) in
the ninth. It should be said that there are not really consecutive
fifths at (£), as would appear from this extract ; the accompanying
harmony, which we have not given, shows that the fifths are
accented passing notes.

226. Our last example of this double counterpoint will be
from Beethoven's Mass in D, in which marvellous work examples
of all the rarer counterpoints are to be met with —

(a) BEETHOVEN. Mass in D.

I J

rEfJLrStrr!

f^n

Chap VIII.]

IN THE RARER INTERVALS.

109

It is cunous that in this example the sixth, the only consonant
interval, is not used at all except as a passing note.

227. III. Double counterpoint in the thirteenth. The table of
inversions for this counterpoint will be —

INTERVALS: i 2 3 4 5 6 7 8 9 10 n 12 13
INVERSIONS: 1312111098765 4 3 2 i

The thirteenth is a compound interval, an octave and a sixth ;
and it will be seen from the above table that these two intervals
are the only consonances which remain consonant when inverted.
Though we have here two such intervals, while in double counter-
point of the eleventh we had only one, we are in reality no better
off — rather worse, in fact, because we cannot use consecutive sixths.

228. Our first specimen of this counterpoint is from
Cherubini —

CHERUBINI.

3 — i r Ji ^ '

-L=L — - — 1 1 ^ r ri
• Hj itj u ^q

Inversion in i.3th.

r1 '' ±=*

lupr r r r i

x-^—

l"j f-r-l

Cr •*

^ —

j j j

•-^

h ' r ri

' 1

'•-^ -f-M

L__JI

As with other double counterpoints, various other methods of
inversion are possible. We give the commencement of two —
(«) <*)

&C.

no DOUBLE COUNTERPOINT: [Chap. viii.

The student will notice that, although we call these two positions
" inversions," the counterpoint is still above the subject, but at a
different interval from before. At (a) the subject is a third lower,
and the counterpoint is unchanged ; but at (b) the subject is
unchanged (except as to its octave), and the counterpoint is a
sixth lower. We have purposely chosen these positions to make
clear to the student's mind a point which has more than once
been incidentally touched on. In any double counterpoint, other
than that in the octave, the counterpoint is frequently seen in the
same relative position to the subject, but at a different distance.
In such a case, inversion, followed by re-inversion in the octave, is
always implied. In the present examples, if the student will
invert (a) and (b} in the fifteenth, he will see that the intervals
they will make with the subject will be the same as in the fully-
worked example.

229. Our next illustration will further exemplify this point —

(a) Bars 49— 51. BACH. " Wohltemperirtes Clavier,' Fugue 4.

Bars 79-81.

***fr i r

r i* 1 1* EJ* r \

=*M

In these two passages, the relative position of the two subjects is
unchanged; but the intervals are quite different. To find the
interval of inversion, re-invert (b) in the octave —

L» j J J. J. .1-3 j
^f.ffff.f\...f . ==p

Now compare (^) with (0). The eighth has become a sixth, the
fifth a ninth, the third an eleventh, and so on. The two
numbers added together amount in every case to 14 ; the double
counterpoint is therefore in the thirteenth. We met with a
similar case in § 221 ; but as this matter may cause the student
some trouble if not properly understood, it was as well to give a
second illustration.

230. In treating of free double counterpoint in the tenth, we
saw (§§ 156, 157) how thirds added inside a double counterpoint
in the tenth would give double cou-nterpoint in the octave.

Chap. VIII. 1

IN THE RARER INTERVALS.

Similarly, if we add thirds inside a double counterpoint in the
octave, we get double counterpoint in the sixth, or (which is the
same thing) in the thirteenth, as in all probability the octave will
really be at the distance of a fifteenth. We give a very good
example of this device —

(«) HANDEL. " Chandos Tc Deum."

gL_4 — i i 1 II 1 ' i i rr-f i 14— ll
EvrjIV- rrJ.f f f .f : ,

Here the treble and bass of (b) are the inversion in the octave of
(a) ; and the student will be easily able to discover by calculation
that the thirds added above the bass give the inversion, in the
thirteenth with the upper part, of the preceding counterpoint.

231. Our next example is somewhat, though not precisely
similar —

(a) BEETHOVEN. Mass in D.

^ J- —

(iff r- - &=& — ]

r^ — i — T~T — i

i.Sp..- - ..^ *?— n

<y r * =J
j j j i j

\l— 5. f—J p_| L^

' r r^d

Here, as in our examples to §§ 221, 229, the relative position of
the parts is unchanged. The tenor of (b) is the same as the bass
of (a)-, and thirds are added outside the counterpoint in the
octave. But if the two lower parts are inverted with the subject,
thi

ii2 DOUBLE COUNTERPOINT; [ChaP.vm

it will bring the added thirds inside the octave; and it will be
readily seen that we have here, as in the last example, double
counterpoint in the thirteenth.

232. From our last two examples it will be clear that the
double counterpoint now under notice shares with that in the
tenth (§ 80) the peculiarity that it can be employed against
the subject in its two positions simultaneously. In both these
passages the two counterpoints were below the subject ; for our
final illustration we give a passage in which one of the counter-
points is above, and the other below. In § 176 we quoted the
subject and counter-subject of the fugue in Beethoven's Sonata
in A flat, Op. no. Al the 27th bar of the same movement the
following is seen —

BEETHOVEN. Sonata, Op. 110.

Here there is not the slightest difficulty in discovering the nature
of the counterpoint, as the two positions are exactly a thirteenth
apart Just as the two counterpoints of the tenth move in thirds
or sixths according to their position (compare examples to §§ 156,
158), those in the thirteenth move in sixths or thirds. It must
be noticed that here the upper counterpoint could not be used
without the lower, because of the consecutive fourths which
would result with the subject. A similar case will be seen
later (§ 238).

233. IV. Double counterpoint in the fourteenth. The last of
these rare counterpoints. Its table of inversions is —

INTERVALS: i 2 3 4 5 6 7 8 9 10 n 12 13 14
INVERSIONS: 14 13 12 n 10 9 8 7 6 5 4 3 2 i

This counterpoint resembles that last noticed in containing two
consonances (the third and fifth, with, of course, their octaves,
the tenth and twelfth) which remain consonances when inverted ;
but, just as in double counterpoint of the thirteenth, we were
unable to use consecutive sixths, we are now unable to employ
consecutive thirds, as they become fifths by inversion.

234. We first give an example of this counterpoint by
Marpurg —

Chap. VIII.]

IN THE RARER INTERVALS.

MARPOKG.

Inversion in 1410.

£y r — u

y r » J .

1 1 J . Jllr r J J 1 =1

Note how in this example fifths are taken by contrary motion on
the accented notes, and the bad effect saved by the interposition
of other notes. As in previous cases, other methods of inversion
are possible; these the student ought now to be able without
difficulty to discover for himself.

235. Our next example is from Bach —

(a) Bars 49—51. J. S. BACH. " Wohltemperirtes Clavier," Fugue 4.

We have already quoted the passage (a) in § 229 ; it is curious
that the same counterpoint of which we there showed the inver-
sion in the thirteenth, should here be inverted in the fourteenth —
an inversion which the student will have no difficulty in verifying.
236. The following most interesting combination,

J. S. BACH. " Wohltemperirtes Clavier," Fugue 41.

(a) Bars 6, 7.

DOUBLE COUNTERPOINT;

(Chap. VIII

(3) Bars 22, 23.

. 1

is that which we referred to in § 221. If the two upper parts of
(a) are compared, beginning at the third crotchet, with the two
lower parts of (£), it will be seen that they are there inverted in
the ninth. At the same time the two lower voices of (a) are, as
the treble and tenor of £ inverted in the fourteenth.

In our next example

BEETHOVEN. Mass in D.

we have the pattern (a) inverted at (3) partly in the thirteenth and
partly in the fourteenth. The variation arises from the modifica-
tion at (ft) of the upper part of (a).

238. Our last illustration is taken from the very interesting
fugue from which we have more than once quoted —

BEETHOVEN. Sonata, Oj

t£y'^jj&±'^±&^^

If the counterpoints here be compared with that of the pattern
given in § 176, it will be found that the lower voice is the inver-
sion in the twelfth, and the middle voice that in the fourteenth,
of the original counterpoint. Here the twelfth and fourteenth
are used in thirds, just as the octave and tenth can be. We
assume that the student is by this time sufficiently familiar with
the necessary calculations to be able to verify the intervals of
inversion for himself.

239. If the examples and explanations given in this chapter
have been fully understood, it will easily be seen why these rarer
double counterpoints can only be used incidentally. Whereas
in double counterpoint of the octave, tenth, and twelfth, there

Chap, vi n.i IN THE RARER INTERVALS. 115

are never less than four consonant intervals which remain con-
sonant when inverted, there is only one which so remains in
double counterpoint of the ninth or eleventh, and only two in
double counterpoint of the thirteenth and fourteenth, and, except
in double counterpoint of the eleventh, no consecutive intervals
are possible. It will nevertheless be useful practice for the
student to try to invent short passages, similar to those that we
have given in §§ 219, 220, 223, 228, and 234, which shall be in
double counterpoint with one another at these various intervals.
This will be found profitable, because complete mastery of free
part-writing is best acquired by much practice of all styles; it
will also be interesting, and even amusing, for there is no greater
delight to the earnest student than that of overcoming some
formidable difficulty. It is for this reason that we have devoted
a whole chapter to a subject which most treatises (except
Cherubini's) either pass over in silence, or dismiss in a few
contemptuous words, as unworthy of serious attention.

n6 TRIPLE AND QUADRUPLE COUNTERPOINT.

CHAPTER IX.

TRIPLE AND QUADRUPLE COUNTERPOINT.

240. By Triple and Quadruple Counterpoint are meant those
varieties in which three or four combined melodies are capable
of being inverted in the octave, so as to be taken in any possible
relative position to one another — that is to say, that each of the
voices may be either an upper part, a middle part, or the bass ;
and in all positions the voices considered together shall form
correct harmony.

241. Speaking first of triple counterpoint, it must be remarked
that three parts are capable of combination with one another in
six different ways. Supposing we call the three parts A, B, C,
the possible combinations will be the following —

A A B B C C
B C A C A B
C B C A B A

It does not necessarily follow that if we are writing in triple
counterpoint, we are compelled to employ all six positions ; it is,
as a matter of fact, rather the exception than the rule to meet
with a piece in which all six combinations are actually used ;
but in any correctly written counterpoint of this kind they will
all be possible, and the composer will select whichever he chooses.

242. If we take the three notes of the common chord of C,
and place them in their six possible relative positions to one
another,

it will be seen that in two cases out of the six, the fifth of the
chord must be in the bass, giving us a second inversion. For
this reason, if we attempt to write triple counterpoint in the strict
style, we shall be unable to use the fifth of a chord at all, except
as a passing note. We are thus so restricted that writing music,
in the true sense of the term, becomes impossible under such
conditions; therefore, though some text-books give rules for the

chap, ix.] TRIPLE AND QUADRUPLE COUNTERPOINT. 117

writing of such counterpoints, the student is not recommended to
waste his time over them.

243. The simplest, but also (so to speak) the cheapest and
least valuable kind of triple counterpoint is obtained by adding
thirds above or below a double counterpoint in the octave. But
a moment's thought will show the student that it is not every
double counterpoint in the octave which is capable of being thus
treated. If, for example, we have consecutive thirds or sixths,
the additions of thirds outside these intervals will give con-
secutive fifths or octaves. We can, therefore, as with double
counterpoint in the tenth, employ only contrary and oblique
motion. But even then, we shall not always be comfortable.
If, for example, our double counterpoint contains the interval
of a third left by leap, and we add a third above it, we shall get
a fifth, also left by leap, which in some of the positions will
certainly get us into trouble.

244. An example will make this clear. Let us write a simple
counterpoint of the first species against a few bars of the subject
which we have so often treated in the strict style. As we are
proposing to add thirds to it, to make a triple counterpoint, we
will take care to use nothing but contrary motion.

gy J .^ ' ^ ' ~ ' +

&C.

This counterpoint is correct enough, though not very interesting.
Now we add thirds above it —

fh*=h

1 « 1 1 ^=H

&C.

' i " i ~-r— i

If this is a good triple counterpoint, it must be capable of being
taken in any position. Let us put the added upper part in the
bass —

&C.

More atrocious counterpoint than this cannot be conceived. As
we are writing in the free style, second inversions are not pro-

ii8 TRIPLE AND QUADRUPLE COUNTERPOINT. [Chap, ix.

hibited, but they must be properly treated. Here the second
inversion in bar 2 leaps to the inversion of another chord, which
in its turn leaps to another second inversion, which is just as
badly quitted as that in the second bar.

245. It is seldom, if ever, that a triple counterpoint made by
the addition of thirds will be available in all positions. Even in
the example of this kind, given by Cherubini, in his " Treatise on
Counterpoint," we find such progressions as the following —

/ 1 \ / A %

(3)

_J J» n J B « i -^ ii >•

— 1 H

r - y n r r ^

n '

-&-

| J

r i •
^ i j

If we put the upper or middle voice in the bass,

(3)d) n (2) i3)i

Bm \ e> ^ _ \\
f i ^ i r i* -&- fg» ^ ' p » F- "

\f p ii f i r i - n i i i

we see in all examples a second inversion wrongly treated;
Cherubini, in fact, does not give these inversions. The truth is,
that this is at best a very inferior kind of triple counterpoint ; the
only legitimate sort, and that which we shall treat in this chapter,
is the combination of three independent melodies.

246. The examples we have given in the last two sections
show us wherein the real peculiarity of triple counterpoint con-
sists. It is in the treatment of the fifth of the chord. Apart from
this, it differs very little from ordinary combined counterpoint.
To paraphrase a well-known proverb, we may say, Take care of
the fifths, and the roots and thirds will take care of themselves.

247. If the student will look at the six possible positions of
a common chord given in § 242, he will see that, as there pointed
out, the fifth will in two positions be in the bass, giving a second
inversion. He has also seen in §§ 244, 245, the disastrous effect
of a careless or injudicious treatment of the fifth. What he has
now specially to attend to is, the approaching or quitting the fifth
of a chord, in whatever part, only in a manner in which the- bass
of a second inversion could be approached or quitted. The
rules for the employment of second inversions are given in full in
Harmony, §§ 188, 189. Let us apply these rules to the present
case.

Chap, ix.i TRIPLE AND QUADRUPLE COUNTERPOINT. 119

248. I. The fifth of a chord can only be approached by leap
from another note of the same chord (when the harmony must
remain the same, except as to its position), or from the root of
the preceding chord. If it be approached by leap from the third
or fifth of the preceding chord, it is clear that when taken in the
bass, the second inversion will be approached from an inversion
of another chord.

249. II. If the fifth be approached by step, the preceding
note may be either the root or the third (very rarely the fifth) of
another chord. It is also possible, though seldom advisable in
triple counterpoint, for the fifth to have appeared as one of the
notes of the preceding chord.

250. III. The fifth of any chord may not be quitted by leap,
except to the root or third of the same chord, the harmony
remaining unchanged. If the fifth of a chord be either the tonic
or the dominant of the key, it may remain as the root of the
following chord, provided that the first of the two chords be on
the stronger accent. In all other cases it must move by step to
the following note.

251. It is evident that in triple counterpoint it will be
impossible to make use of consecutive chords of the sixth,

because in some of the inversions they will make consecutive
fifths. It will be well in any case to use the fifth sparingly,
especially in the weaker chords of the key (II., III., and VI.) :
many theorists prohibit the employment of the fifth of a chord
altogether, except as a prepared discord, or a passing note ; but
if the rules we have given be observed, it may be safely used, as
no objectionable 4 chords will result from its inversion.

252. It will be advisable in writing triple counterpoint to use
chiefly the strong chords of the key (I., IV., V., and VII.) ; and
in the weaker chords it will often be well to omit the fifth alto-
gether, as the second inversions of these chords, though not
impossible, can rarely be employed with good effect.

253. It will frequently happen that some positions of a triple
counterpoint will sound more satisfactory than others. As it will
be seldom necessary, or even advisable, to use^ all six positions,
the student can select for himself those that are best; but he
should at least introduce each of the three voices once in the
bass. This is the general practice of Bach.

254. It is but seldom that a composition is written in triple
counterpoint throughout. Before proceeding to give examples
from the works of the great masters, we quote from Cherubini's

120 TRIPLE AND QUADRUPLE COUNTERPOINT. [Chap. ix.

Treatise a short specimen, giving all the six positions, for the sake
of comparison —

to j

JL. — i 1| : 1 j]

j» ^

^ \» • •/

J J * J.

** J a, II

ll =

in uj.i i

\ w *•/ f^

H . , j ' .j

" m

iirr^' r=^

4-i

Concerning this very simple and intelligible example, it is only
needful to remark that the fifth of a chord only appears once, at

, when it leaps to the root of the same chord, according to the

e given in § 250.

255. Oar first examples of triple counterpoint shall be taken
from the inexhaustible Bach —

J. S. BACH. " Wohltemperirtes Clavier," Prelude 19.

chap, ix.] TRIPLE AND QUADRUPLE COUNTERPOINT. 121

We shall give all our examples in score, that the student may
more easily fojfow the progression of the several voices. Let the
student first notice the strongly contrasted character of the three
subjects. This should always be borne in mind as an important
essential in writing triple counterpoint. Four of the six possible
combinations are used in this prelude, and No. I. is also met with
in the key of the relative minor. It will be seen that the end of
the alto of No. I. is modified in II. and III.; such slight changes
can always be made, if desirable, provided that they do not too
far alter the character of the subject In the middle of the
theme they would be objectionable.

122

TRIPLE AND QUADRUPLE COUNTERPOINT, ichap. ix.

256. In two of the fugues in the same work (No. 4, in
C sharp minor, and No. 21, in B flat), triple counterpoint is very
extensively employed. The latter fugue is, with the exception of
two short episodes, written in triple counterpoint throughout
Instead of quoting from these, we give a short episodical passage
from another fugue in the same work, which is particularly
interesting from t?ie fact that Bach has here used all the six
possible inversions. We mark the subjects with A, B, C, for
their easier identification —

J. S. BACH. " Wohltemperirtes Clavier,' Fugue 37.
(IL)B

chap, ix.] TRIPLE AND QUADRUPLE COUNTERPOINT.

123

It will be seen here, as in our last illustration, that the theme
which at (I.) appears in the bass is slightly altered when given to
the other voices, the rise of a fourth being substituted for the fall
of a fifth. Observe also the way in which the fifths of the chords
are treated, and compare the rules given earlier in this chapter.

257. Our last extract from Bach is taken from the "Art of
Fugue " —

(I.) Bar 147. A

J. S. BACH. " Art of Fugue, No. 8.

r t, »r i - r T r i . r r r

J * r I r I I ' If*' ' I

(II.)

* j:

124 TRIPLE AND QUADRUPLE COUNTERPOINT, ictap. ix

(IV.) Bar 170.

To assist the student in examining these passages, the entries of
the three subjects are marked as above, with A, B, and C, in
each inversion. Notice here the curious changes in the harmony
produced by the chromatic alteration of notes in the subjects.
In (I.) for instance the DJf in the upper voice is treated as the
augmented fourth of the scale, and is resolved on the first inver-
sion of the dominant seventh. But at (TI.) the third note of the
subject C which before was a tone (GJf) above the preceding
note, is now only a semitone (D) ; this alters the harmony, and
here the Ajf of the middle voice is shown by its resolution on the
chord of D minor to be really B |>. If the student will analyze
this example carefully, he will find other interesting points for
himself.

258. In our next example

HANDEL. " Hercules.

n"ciP" ' i' r i- &=4

we give only the model. After the full illustrations and explana-
tions already given, it will be profitable for the student to write
out the six positions for himself. In the chorus from which the
above passage is taken Handel employs four of the six positions.

Chap, ix.j TRIPLE AND QUADRUPLE COUNTERPOINT.

259. We next give a little-known specimen by Mozart —

MOZART. Mass in C, No. ia.

It is worth mentioning here that this extract is taken from
Mozart's i2th Mass. The work usually known by this name is
now universally admitted by all authorities to be spurious. The
Mass here quoted was published for the first time in 1878 in the
complete edition of Mozart's works. Note, as in previous
examples, the contrasted character of the three themes.

260. The following example, from Cherubini's " Counterpoint
and Fugue," requires no explanation —

^ CHERUBINI.

261. In modern compositions real triple counterpoint is very
rare. The reason for this is probably to be found (at all events
in some cases) not so much in the inability, or even indolence, of
the composers, as in the general tendency in the direction of free
part-writing which is characteristic of most recent music. For a
specimen of this kind of counterpoint we give a short passage by
Beethoven —

BEETHOVEN. Mass in D.

rri r i

r i i r

This illustration differs from those already given in the fact that
the upper part is an imitation of a part of the tenor subject. IP

126

TRIPLE AND QUADRUPLE COUNTERPOINT. [Chap. ix.

one sense it may be called an independent melody, as it differs
from both the others in rhythm, having rests on the accented
beats ; but it has less individuality than we have seen in the
examples previously quoted.

262. Our last example of triple counterpoint is taken from
one of Haydn's quartetts, and is an admirable illustration of the
composer's skill in combining three totally different melodies —

^ ^ HAYDN. Quartett, Op. 20, No. 6.

The small notes at the commencement are those which do not
form part of the triple counterpoint.

263. If to a Triple Counterpoint a fourth voice be added
which, like the others, is available in any position, we obtain
Quadruple Counterpoint. If the student will refer to the table
of positions of triple counterpoint given in § 241, he will
readily see that a fourth part may be added to each of the six
positions in four different ways. It may be the highest voice ; it
may be put between the first and second, or between the second
and third ; or it may be below all three. It is evident that this
gives twenty-four possible positions for a quadruple counterpoint.
It need hardly be added that if all were made use of, a composi-
tion would become intolerably long, and most probably extremely
tedious. Composers therefore only select from this large number
such as they consider advisable.

264. No new rules are necessary for writing quadruple
counterpoint. As with triple, it is the fifth of a chord which
needs special attention ; and the student must carefully observe
the directions for the management of this note given in
§§ 248-250. He must also endeavour to make his additional
voice distinct in character from the others. This will call his
powers of invention into play.

265. Quadruple counterpoint, from its complexity is naturally

Chap, ix.] TRIPLE AND QUADRUPLE COUNTERPOINT.

127

much rarer than triple, and we shall not be able to give many
good examples of it. Our first will be a short specimen by
Cherubini, very similar in character and themes to the triple
counterpoint that we gave in § 254 —

j CHERUBINI.

i r ==*

y r r 1

' u r L II

It will be a useful exercise for the student who wishes to obtain
an insight into the subject to write out all the possible positions
of this little example.

266. It is very seldom that genuine quadruple counterpoint
is to be found excepting in fugues ; and even in these it is more
often met with incidentally than as an essential part of the
structure of the piece. Such is the case in the illustrations from
Bach next to be given. In the fugue in F minor, in the first
book of the " Forty-Eight," is found the following passage-

Bar 13.

J. S. BACH. " Wohltemperirtes Clavier," Fugue 12.

£28

TRIPLE AND QUADRUPLE COUNTERPOINT. [CbaP. ix.

Notice in the second and third bars, the crossing of the alto and
tenor parts. In quadruple counterpoint, the crossing of parts is
neither infrequent nor objectionable.

267. If the above extract be examined, it will be seen that
the parts can be taken in any position. Bach, however, only
inverts it once in its complete form —

(2) Bara7.

Here two slight modifications are made. At (a) the rests in the
pattern are filled up ; and at (b) the note B enters one semiquaver
earlier than before, to avoid the repercussion of the note im-
mediately after it had been sounded in the treble.

268. Our next example is simpler in construction —

J. S. BACH. " Wohltemperirtes Clavier," Fugue 33.

'i r • *i

r Jfag* • >«*l F ' P=F

ix.] TRIPLE AND QUADRUPLE COUNTERPOINT. 129

Here the inversion follows immediately on the model. (Compare
the example in § 256.) That the student may more readily trace
the inversion, we have marked with letters the entries of the four
subjects.

269. We now give two examples in which quadruple counter-
point is employed more systematically. The first is the opening
of a fugue on four subjects, taken from Cherubini's " Counter-
point and Fugue " —

CHBRUBINI.

1 r j r r u

-1 -1 — 1 1 N

K,. j j .. ==b=3=

&c.

*sTrr r r

It is usual for the voices to enter in succession, as here, and it is
by no means necessary, either in triple or in quadruple counter-

130

TRIPLE AND QUADRUPLE COUNTERPOINT. [Chap. ix.

point, that all should be moving together throughout. For
illustrations of this point see the examples in §§ 258, 260, and
262.

270. For our last specimen we give a very fine example by
Haydn, taken from a fugue on four subjects which forms the
finale of his Quartett in C, Op. 20, No. 2. As we have not yet
shown a quadruple counterpoint in its various positions, we shall
here quote the different inversions made use of in the course of
the fugue —

HAYDN. Quartett, Op. 20. No. a.

ist VIOLIN.

and VIOLIN.

VIOLA.

VIOLONCELLO.

(II.)

(HI.)

Chap, ix.] TRIPLE AND QUADRUPLE COUNTERPOINT.

J J J

132 TRIPLE AND QUADRUPLE COUNTERPOINT. [Ch«p.ix.

We see here six different positions of the four voices, out of the
possible twenty-four. To assist the student we have lettered the
four themes, so that he can readily trace them in all their
inversions.

271. At (I.) we have the first presentation of all the subjects
together, on which immediately follows at (II.) an inversion of the
same. The slight alteration of the theme D at (a) is a result of
the construction of the fugue, and will be explained in our next
volume. At (f) is a transposition of the last notes of theme B an
octave lower — roost likely to avoid the awkward overlapping of
the parts which would otherwise have occurred. Notice, both in
the model and in the inversions, how very freely the parts cross.

272. The only point to notice in the position marked (III.) is
the consecutive unisons at (c) between the first and second
violins. An examination of the passage that follows shows that
we have not a slip of the pen here. The first violin is beginning
an ascending chromatic passage, and Haydn apparently thought
the unisons better than changing the last note of the subject in
the second violin. It is perhaps such passages as this that the
old composer referred to, when he said that " the rules were all
his very obedient humble servants ! "

273. Position (IV.) is interesting as showing, not only a new
disposition of the parts, but a fragmentary presentation of some
of the subjects. Only the first bar of theme A appears ; and the
violoncello does not complete theme B, the conclusion of which
is taken up by the viola. Position (V.) needs no special remark.
In the last position (VI.) it will be seen that the violoncello,
instead of completing its theme, D, takes up at (d) the theme
C, in imitation of the first violin, being itself imitated by that
instrument in the following bar.

274. This masterly specimen of quadruple counterpoint
deserves the most careful study. One of the most striking points
about it is the apparent ease and perfect fluency with which the
four parts move together. It is related of one of the old Italian
contrapuntists that one of his pupils, greatly admiring a piece of
very elaborate counterpoint, remarked to the master, " Ah !
quanto e facile ! " (" Ah ! how easy it is ! ") His master
replied, " Ma quanto difficile e questo facile ! " (" But how
difficult this ' easy ' is ! ") Such counterpoint as that which we
have treated of in this chapter can only be mastered by great
patience and perseverance. The great composers were unre-
mitting in their studies, and any one who can write the more
elaborate varieties of counterpoint with correctness and fluency
may justly say, like the Roman captain of old, "With a great
price obtained I this freedom." Rules and examples will do
something, but not all ; abundant practice and steady hard work
are the true requisites for success.

Chap. X.j

IMITATION.

'33

PART II.— CANON.

CHAPTER X.

IMITATION.

275. If in a piece of music the same melodic figure occurs
twice or more in succession, in the same part, either beginning
each time on the same degree of the scale, or each time on a
different degree, we have in the former case a repetition, and in
the latter (if the interval of entry of each new recurrence of the
figure be regular) a sequence. (Harmony ', § 133.)

BHBTHOVKN. Pastoral Symphony.

In the above extract the first four bars are a repetition of the
same figure; from the fourth bar to the end we have a rather
free sequence.

276. If, however, the repetition of the figure, whether at the
same or at a different pitch, be in a different part of the harmony,
we get IMITATION.

BEETHOVEN.

V. i mo. __ p i^ Pastoral Symphony.

V. ado.

Viola.

'Cello.

'•J^'Cr

In this passage, the figure announced by the first violins is
imitated by the violas and violoncellos in the lower octave ; these
are in turn imitated by the first violins, and these last by the
seconds. It will be noticed that in the last two bars of the
passage just quoted, the imitation is far from exact ; it is, in fact,
only a rhythmic imitation. This point we shall refer to presently

134 CANON. [Chap. x.

277. Imitation may be either strict or free. It is said to be
strict when not only the names of the intervals between the
various notes are the same in the imitation as in the pattern, but
when the character of the intervals is also unchanged, that is to
say, when a tone is always imitated by a tone, a semitone by a
semitone, and so on. In actual composition this is very rarely
possible except when the interval of imitation is either the
unison (or octave), the perfect fourth or the perfect fifth, above
or below. A little thought will show the student that if the
imitation be at any other distance, and we preserve the intervals
exactly, we shall modulate — probably into some remote key.

278. Suppose, for instance, that we announce a simple phrase
in C major for imitation,

J_ «l ^ I « — * • ^ «l I J

and attempt to imitate it exactly at the second below, the imita-
tion will be —

which, obviously, destroys the feeling of tonality at once. In
such a case we should use free imitation, and write the latter
passage without the sharps, altering the character of the intervals,
but keeping the music in the key of C. At the distance of a
fourth or fifth it would be possible to use strict imitation, because
any modulation that might be effected would be only to one of
the most nearly related keys.

279. There are many different varieties of imitation with
which the student should be acquainted. By far the most
frequently used is direct imitation, that is to say, that in which
every ascending or descending interval of the pattern is answered
by a corresponding ascending or descending interval, whether
strict or free depending, of course, as just shown, on the distance
of the imitation.

280. Imitation by inversion is obtained by imitating every
ascending interval of the pattern by a corresponding descending
interval, and every descending by the corresponding ascending.
It is possible in a major key to make this kind of imitation strict
as to intervals. For this purpose the following scheme is used—

The position of the semitones, it will be seen, is identical in botn.
series of notes, and to get strict imitation by inversion it is only

Chap. x. ] IMITA TION.

135

necessary to answer each note of the one series by the note
immediately over (or under) it of the other. Thus tonic and
mediant will answer each other, dominant and suhmediant, and
so on. On this method the following passage

tr
would be imitated thus —

(B) §

where it will be seen that every interval is of precisely the same
quality in the imitation as in the pattern.

281. Though this strict imitation by inversion is, as we see,
quite possible, it is seldom that it is actually used by the great
masters. The probable reason for this is, that (as will be
observed) all the most important notes of the key — the tonic,
dominant, and subdominant, are imitated by the less important
ones. It is therefore far more usual, either to imitate tonic by
tonic, and dominant by subdominant, or to imitate tonic by
dominant, and dominant by tonic. Instead, therefore, of the
imitation shown at (B) of the last section, one of the two follow-
ing would mostly be employed —

^:t * I f r j J I " - II r I r r r r I ~ ' I

282. In a minor key strict imitation by inversion is not
possible. That which is to be found is constructed according to
the following scheme —

If the melodic forms of the minor scale be used, the major sixth
and seventh of the ascending scale will be answered respectively
by the minor seventh and sixth of the descending. It will be
seen that in this scheme two of the three semitones correspond in
both scales, while the important interval of the augmented
second, and its inversion, the diminished seventh, will be retained
in the imitation —

Imitation by inversion.

It is worthy of notice that imitation by inversion is far more
frequently met with in a minor key than in a major.

136 CANON. lCh«p. x.

283. A third kind of imitation is that with reversed accents —
that is to say, that the notes which in the pattern are on accented
beats are in the imitation on unaccented, and vice versa ; e.g. —

r ir r r

&c.

Here it will be seen that the lower part is a strict imitation of the
upper at the distance of a fourth below, but with all the accents
reversed. This kind of imitation is frequently called by the
name given to it by the old theorists — imitation per arsin et thesin.
Arsis is a Greek word, meaning "raising," and Thesis another
Greek word, meaning " putting down," or, as we say nowadays,
the "up-beat" and the "down-beat," in other words the un-
accented and accented parts of the bar.*

284. Two other varieties of imitation are not infrequent — that
by augmentation, in which the notes of the imitation are double
the length of those of the pattern, and that by diminution, in
which the pattern is imitated in notes of half its length. It is
evident that more than one of these varieties can be combined ;
for instance, an imitation may be by augmentation and contrary
motion, or by inversion and with reversed accents, and so on.

,285. Invertible imitation is that which is written in double
counterpoint with its pattern, so that it can appear either above
or below it It will be seen that the little example given in § 283
is written in double counterpoint in the octave.

286. One of the most important kinds of imitation — canonic —
will be dealt with in subsequent chapters ; but, in addition to the
varieties already spoken of, there are two others, of little or no
practical use, which should be mentioned for the sake of com-
pleteness. These are interrupted and retrograde imitation. The
former is made by putting rests between each note of the imita-
tion, as in the following example by Cherubini —

[sr'i J|» r1' ' ir ;*M- «MJ ^EW*

fS»- fS> ^ -^ ^ 4=2.

^<? " ' 1" M- ' 1- [ 1- ' 1- 1- 1-

&C.

This is evidently a mere curiosity ; and we are unable to give any
instances of its employment by the great masters.

287. Retrograde imitation, which is occasionally met with in
canons, is that in which the notes of the subject propounded are

* In old music the term "per arsin et thesin" is also occasionally used ax
equivalent to " by contrary motion"— i.e., the one part rising as the other l.ills.
See Hawkins's "History of Music," Chapter LXVII.

Chap. X.]

IMITATION.

137

given in reversed order — that is to say, the last note of the model
becomes the first note of the imitation, and so on, c.g. —

Model Retrograde Imitation.

It is clear that contrary motion can also be combined with this
species of imitation. It is sometimes called imitatio cancrizans —
"crab-like" imitation, which walks backwards, as a crab is
popularly supposed to do.

288. Partial imitation is when only a part, and not the
whole, of the model is imitated ; the name may also be appro-
priately applied to such imitations as those seen in the third and
fourth bars of our example to § 276, where the intervals of the
model are not exactly reproduced, but the rhythm is preserved,
so that the general resemblance of the imitation to the model is
clearly maintained.

289. Close imitation is that in which the imitation enters (as
in the example to § 283) immediately after the commencement
of the model. With the ordinary use of words, it would seem as
if to speak of a passage being " closely " imitated were much the
same as to speak of its being " exactly " imitated ; but it is very
important for the student to notice the distinction that exists in
the technical use of these two terms. " Exact " imitation is the
same as "strict" (§ 277), and refers to the nature of the intervals;
" close " imitation has nothing to do with the intervals, and refers
only to the distance of time at which the reply commences.

290. Though the examples we have hitherto given have
mostly been in two parts only, the student must understand that
imitation may be in any number of parts. This will be seen in
the illustrative passages we shall give directly from the works of
the great masters. Very frequently, also, imitation between some
of the voices is accompanied, like double counterpoint, by free
independent parts.

291. We now give a series of examples from the works of the
great composers, containing specimens of the various kinds of
imitation described in this chapter; it would, of course, be
superfluous to illustrate every possible variety. We begin with an
example by Bach of imitation in the unison and octave —

, V. i. J. S. BACH. Concerto in G.

The subject announced by the first violin is here imitated half a
bar later by the second, and this at the beginning of the second
bar by the third violin and the viola, all these entries being in
unison. The final entry is in the lower octave.

292. It is not very often that a figure is thus imitated twice in
succession in the unison. More frequently each new entry is in
a different octave, as m the following passage —

BEBTHOVEN. Quartett, Op. 18, No.

4t • :

&f*T_f

n v

. | j*j n

ola.

if ferr^T^-—^
' q ^ **p=g

Here, as in our last example, each imitation begins on the same
degree of the scale, but no two consecutive entries are in the
same octave. In the second violin part will be noticed an
example of imitation per arsin et thesin (§ 283).
293. Our next example is of a different kind —

HANDEL. " Solomon."

../Taajga u- JB.J
y r n— 'U^-e£.P^^^

yrt r i • • — • 'i^y ^J-^i -uj ||

Chap. X.I

IMITATION.

Here we have in the two upper parts imitation in the fourth
below; the imitation of the upper by the lower of these two
voices commences at (a). The student will learn later that the
imitation here is so continuous as to make the passage into a short
canon; we have quoted it in this place as a good example of
imitation with a free bass part added (§ 290).
294. In the following passage

CHBRUBINI. " Medea.'

•__^CiT

f'^_r

ft^r

JL7

f^f

if r r |

\J pfr ^

/ 1

— 1 1

1 r r

-f — 1—\

a short figure, first announced in the bass, is imitated at various
intervals; it will be seen that with some of the entries the
imitation is only partial.

295. We next give a sequential imitation in the second above
upon a pedal bass —

y> le HAYDN. Symphony in G, No. 51.

296. The following extract from one of Beethoven's quartetts

BEETHOVEN. Quartett, Op. 18, No 3.

J _rJ

shows imitation at various distances. It will be seen that there is
no regularity in the intervals of entry of the different parts.

140

CANON.

,Ch»p. X.

297. Our next illustration is somewhat different —

BEETHOVEN. Sonata, Op. 10, No. 3.

Here the figure announced in the first bar is imitated in the fifth
below; it is then repeated in the upper part, on which follow
three successive imitations, each in the fifth below the preceding,
while the last entry is at the sixth below.

298. Ali the examples hitherto given have been of direct
imitation ; we now give specimens of other varieties —

J. S. BACH. Organ Prelude on "Ach Gott und Herr.'

This passage deserves close examination. The subject itself
appears in the tenor at the commencement, accompanied by itself
in diminution in the bass, first direct and then twice in an
inverted form. In the alto we have the subject first direct, and
then inverted and diminished ; lastly, the treble enters with the
subject in augmentation. It would also be possible here to
regard the treble as the original model ; in this case the other
parts would give us examples of diminution and double diminu-
tion. It should be added that the frequent treatment of the
phrase by inversion doubtless here results from the fact that the
next notes of the melody of the choral are themselves the inver-
sion of the first notes —

&c.

Chap. x.i IMITA TION. 141

299. Our next illustration

ft*

SCHUMANN. "Faust"

^ I I IX-K I

T i r 'a*

shows a simple imitation in direct motion by augmentation, and
requires no further explanation. The middle voice is, of course,
free.

300. The following example ;> s BACH «(Artof Fugue,,, No< «,

* r

^k. r > a*1 1*'11

' c Ir ^L^le-l

shows in the second bar the imitation of the subject announced
in the bass by diminution and inversion ; while at the third bar
we see the subject diminished, but in its direct form. Observe
that the imitation by contrary motion is according to the scheme
given in § 282.

301. We next show close imitation by contrary motion at half
a bar's distance —

HANDEL. Judas Maccaoeus.

Though the imitation begins only half a bar after the model, we
have not here an example of the "per arsin et thesin " spoken of
above, because, in consequence of the rather slow time of the
movement, there are really two accents in the bar; thus the
second notes of the model and imitation alike come upon
accented beats, though the one is a stronger and the other a
weaker accent.

302. It was said above that imitation by inversion was rarer
in a major than in a minor key. The following passage is an
example of the former —

MENDELSSOHN. Overture, " Melusina."

r r r

142

CANON.

[Ch«p.X.

It will be seen that here the imitation is not strict as to intervals,
a tone between the sixth and seventh quavers of the model being
imitated by a semitone, and a semitone between the two last
quavers being imitated by a tone. But the general resemblance
of the two passages is much better preserved by answering tonic
by dominant and dominant by tonic, as here, than would have
been the case had Mendelssohn followed the scheme shown in
§ 280, and made the imitation strict, in which case the reply
would have suggested the key of C sharp minor.

303. In the following well-known passage from the "Messiah"

HANDEL " Messiah."

is seen a good example of close imitation per arsin et thesin in
triple time, and also of partial imitation (§ 288). It will be seea
that from (a) the treble ceases to imitate the tenor, though the
composer, had he chosen, could have continued the strict imita-
tion to the end of the passage.

304. In our next illustration

J. S. BACH. " Wohltemperirtes Clavier," Fugue 46.

there is a double imitation per arsin et thesin. The two upper
parts are imitated by contrary motion, at a distance of orre minim,
by the two lower.

305. The great fugue which forms the finale of Mozart's
so-called "Jupiter" symphony is full of masterly specimens of
close imitation. For our final example we select one short
passage from this movement, giving only the string parts —

IMITATION.

Here the imitation is so continuous in all the parts that (as we
shall see in the next chapter) the passage is really a canon. The
subject given by the first violins is imitated in the octave below,
at one beat's distance (per arsin et thesin\ by the second violins ;
the basses then take the theme in the twelfth (fifth) below, being
imitated in the octave above by the violas ; and this close imita-
tion, at one minim's distance between all the parts, is continued
not only to the end of our extract but beyond it. The whole
movement deserves to be carefully studied.

306. The student who has thoroughly mastered the contents
of this chapter will have a sufficient insight into the subject of
which we have been treating, to begin to practise writing imita-
tions for himself. His exercises for this purpose should be of
two kinds. He should first try to invent short figures for himself,
and to write phrases of from eight to sixteen bars, introducing
the model in turn in the different voices, at various intervals, and
by contrary motion, augmentation, diminution, &c, He will
then find it extremely useful to take some of the chorals given in
the Additional Exercises to Counterpoint, and write imitations
above or below these, according to the voice in which they are
placed. The way in which he should commence this is explained

144 CANON. (Ch«p.x.

in Counterpoint, § 475. He can either use figures of his own
invention, or he can take material from the choral itself, as in the
example given in § 298 of this chapter. (See also the fine
example from Bach in Counterpoint, § 567.)

307. The study of imitation is not only a valuable preparation
for that of Canon, but it is absolutely indispensable for any one
who wishes to write a fugue, of which it is a most important
ingredient. This will be clearly shown in the next volume of
the present series. Its utility, however, by no means ends here.
The examples we have given, especially those from more modern
works, show its great value in imparting unity of character to a
composition by means of thematic development. This point wi
be dealt with in a later volume of this series.

Chap. XL 1 THE ROUND, 145

CHAPTER XI.

THE ROUND.

308. Among the examples of imitation given in our last
chapter were some which were continued for a considerable
length. Imitation which is maintained continuously, either
throughout a whole piece, or at least through an entire phrase, is
said to be canonic; and if a composition is so written that the
various parts imitate one another throughout, such a piece is
called a CANON.

309. The student will remember that in Double Counterpoint
at any other interval than the octave, while the names of the
intervals remain the same in the inversion as in the pattern, their
quality is frequently changed — minor being substituted for major,
and vice versa. The same is found with regard to canons, as
indeed was seen in the last chapter with imitation in general

(§ 278).

310. Canons may be either Finite or Infinite. A finite canon
is that in which the imitation is discontinued as soon as the
pattern has been once repeated in each of the voices taking part
in the canon. If, however, the close of the pattern is imme-
diately followed by the repetition in the same voice of its com-
mencement, so that the last part of the subject in the imitating
voice or voices accompanies the first part of the subject in the
leading voice, the canon is infinite. We shall meet with examples
of both as we proceed.

311. As a canon is nothing more than continuous imitation,
it is evident that there can be as many different varieties of canon
as of imitation itself. Thus it may be strict or free as to interval,
direct or inverted, augmented or diminished, or even retrograde.
It is also possible to combine the different varieties ; but in such
cases the imitation is often so totally unlike the pattern that the

[Chao. XI.

canon becomes one merely for the eye, and not for the ear. For
instance, in the following example by Kirnberger,

J. P. KIRNBERGER.

a comparison of the lower voice with the upper shows that we
have here an infinite canon by augmentation and contrary
motion. But will anybody who listens to this composition (if it
deserves the name) maintain for a moment that he can hear any
resemblance between the two parts ? Music is meant for the ear,
not for the eye ; and, however ingenious these puzzles may be,
they are not music. As this volume is meant to be practical in
its aim, we shall not waste time over the discussion of such
problems as these. Life, at all events in the nineteenth century,
is not long enough ; and students who wish to study such
subjects must be referred to some of the old and curious treatises
on theory; our space will be far more profitably employed in
teaching what may be found actually useful in composition.

312. A canon may not only be at any interval, but at any
distance of time. The simplest and easiest kind of canon to
compose is that which in this country is known as a Round ; we
shall therefore begin by showing how this is to be written. A
Round is a canon in the unison — that is to say, that each voice
in turn begins upon the same note— which differs from other
canons in two respects. In the first place, the parts all enter at
equal distances of time ; if the second voice commences four bars
after the first, the third will commence four bars after the second,
and so on. With other canons there is no such restriction ; the
distances of entry may be, and very often are, irregular. The
second distinctive characteristic of the round is, that each voice
completes a musical phrase or sentence, before the next one
enters, the phrases being usually of two, four, or eight bars'
length, though they are occasionally even longer.

313. Let us suppose, by way of illustration, that the round is
for three voices — this being one of the commonest forms. The
whole of the music must then consist of three phrases of equal
length. The first voice begins by singing the first phrase alone ;

chap, xi.] THE ROUND. 147

having finished this, it goes on to the second phrase, while the
next voice enters with the first phrase, thus making two-part
harmony. The third voice then takes the first phrase, while
the second voice has the second, and the first voice the third.
The three-part harmony is now complete. The first voice,
having now sung the whole music, returns to the first phrase,
while the second takes the third phrase, and the third the
second. We here see why the name of "Round" is given to
this form of canon. It can be continued as long as desired;
but this should be at least until the voice that last enters has
sung the whole of the music once.

314. A diagram will help to make this clearer to the student.
Let us call the three phrases of the Canon A, B, and C, and put
the part for each voice on a separate line, placing the phrases to
be sung simultaneously under each other—

ist Voice . . A, B, C, A, B, C,^
2nd Voice . . A, B, C, A, B, ( &c.

3rd Voice . . A, B, C, A,J

The music of a round is occasionally written out in full, as it
would be in the above scheme ; but it is more usual to write out
the harmony in score, placing the phrases one above another, and
indicating at the beginning and end of the lines the order in
which they are to be sung, thus—

JOHN HILTON

J ,r-JJ J J J

+ • *

* | i' rgr i ir E'g^l1

The figure at the end of each line here shows which line of the
round is to be sung next. It will be seen that here the parts
cross freely ; to this there is no objection in rounds, which do not
really contain upper and lower parts, as each singer is in turn
performing the highest, lowest, or middle part of the harmony.

315. There are two different ways of composing a round.
We may write the three parts simultaneously, as if we were
writing a three-part florid counterpoint ; and to the student who
is sufficiently advanced to undertake the study of canon at all
this method of composition would probably present no great
difficulty. It is, however, open to the objection that he would
be very apt to think only of the three-part harmony, and to
forget that before the third voice has made its first entry, the

i48

CANON.

[Chap. XI.

two part harmony between the first and second voices must be
correct, though of course not complete. For example, if he
were writing a series of sixths in three parts, he would naturally
arrange them thus —

This would be the usual disposition for the voices ; but it will be
seen that when only the two upper parts are singing together —
that is to say, before the entry of the third voice — there will be a
most atrocious series of fourths ; it will therefore be necessary to
arrange the parts in the following way —

It is mostly advisable in rounds to give the real bass of the
harmony in the second, rather than in the third or fourth line.

316. A preferable method of procedure is the following.
Begin by writing the first phrase of the round. In composing
this it will evidently be necessary to carry in the mind at least
the outline of the accompanying harmonies. It by no means
follows that the phrase first written will be the highest part
throughout when the piece is completed ; because in a canon in
the unison the parts are always allowed to cross freely. To the
phrase first composed must now be added a second part, which
will make a correct bass to it. It would also be possible, though
less usual, to write the first phrase in such a way as that it would
form a correct bass to a melody which would be subsequently
written above it in the second part. A third part can then be
added, filling up as far as possible the harmony of which the first
and second parts necessarily give only the outline. The number
of parts may sometimes be increased to four, or even more ; but
every additional voice above three makes the composer's task

Chap. XL]

THE ROUND.

149

more difficult, owing to the limited range of the harmony, and
the resulting close position of the voices.

317. It will be seen that this species of composition is in
reality a variety of free counterpoint — that is to say, it is a com-
bination of as many different and independent melodies as there
are parts; and this leads us to impress on the student the
especial importance of giving melodic interest to each phrase of
the round. It is not sufficient that the harmony should be pure ;
if this be all that is aimed at, the music will probably be as
uninteresting to sing as to listen to. Look at the excellent
example by Hilton given in § 314, and observe, with all its
simplicity the absolute individuality of each part. This is a
point which in the composition of a round should never be
overlooked.

318. We will now write a round, in order to show the student
how he is to set to work. We first compose a simple sentence
of eight bars to commence with —

The only point to notice about this melody is, that we have made
it end on the mediant, instead of on the tonic, so as to get two-
part harmony for the last note when the second voice is added,
instead of finishing on the unison.

319. Our next process is to add a second part. The principles
by which we should be guided in selecting our harmonies have
been explained in Counterpoint, Chapter XVI. Obviously
various harmonizings are possible ; we select a simple and natural
one —

J I J

i J I

J I

At (a) we have given the root rather than the third of the
dominant chord, because the latter would have necessitated a
rather low position for the third voice, which we intend here to
carry above the first, as will be seen directly.

CANON.

[Chap. XL

320. We now add a third voice, filling up the harmony, and
the round is complete —

— i — i — r^H — i — P — *-r»-

| J Jill

j jp b j-J *| •— • — J —
sJ

-ft j — i M j — n — i — e

Jf u PT 1^ 1

-^ H
( H

d)b J J r = r r r

In the first and second bars will be seen an illustration (inten-
tionally introduced) of what was said in § 315. Had the third
part been written as the second, the harmony here would have
been horrible, as the student will readily see. But as the fourths
below the upper part are never heard without the thirds below
them, the effect is unobjectionable. It should also be remarked
that passages in thirds for two parts, as here between the second
and third voices, are often to be met with. Care must be taken
that they are not so continuous as to destroy altogether the
independent character of the two parts.

321. On examining the cadence the student will now see why
C, and not E, was written for the second voice. Had the latter
note been chosen, C must have been the note for the third part,
and the cadence, however written, would have been less satis-
factory. We will try —

C)r * — •< — =^ — li •* — E

-^ — 0

The cadence at (a) is evidently bad, because of the hidden
octaves, That at (b) is somewhat better, but the repetition of
the note C, and the ending with the fifth of the chord at the top

Chap. XL]

THE ROUND.

can hardly be recommended. If we alter the end of the third
voice, so as to keep it below the others,

•J J- -I '

the cadence, though not wrong, is weaker than that we have
chosen, "because of the repetition of the B in the lowest part.
The melody of the third voice is also far less good than as we
have written it.

322. If we now try to add a fourth part to this already
complete piece, our difficulties will be considerably increased.
There is not much room left for a new voice. Clearly it cannot
be either a new upper or a new lower part throughout, or it will
exceed the range of the voice. It will have for the most part to
thread its way in and out among the others, and some care will
be required to give it an independent melody. In many places
it must necessarily be in unison with one of the other parts. It
is, however, by no means impossible to add such a part : here is
one way of doing it —

ir r r !J J =

|ipi

r-m

J J

^ J J IJ^J ^

— U

H i-1

j r i

^3E

I*
ii

152 CANON. [Chap.xi

The compass of the voice part is here rather large, but it is not
beyond that of a mezzo-soprano, and therefore allowable. Note
the new character given to the third bar by the addition of a
fresh bass to the harmony.

323. We shall now give a few examples of this form of canon.
Our first will be a well-known specimen by Dr. Hayes —

> DR. HAYES.

tut *~

5™

1 : — 1 1 h

CT~I — ll i

J J 1 J Jl

ffli ^

The only point to notice in this round is that at (a) we see
motion from the second into the unison. This is by no means
uncommon in such compositions where the second (as here) is
an auxiliary note; indeed, owing to the close position of the
voices, it is often almost unavoidable.

324. We next give two examples by Mozart. In their
published form they are printed at full length ; in order to save
space they are here given in the condensed form already
described. The first is for four voices —

^ MOZART.

•g- i III i I i

UTI- J 4 |j«Bg:

Chap XI.]

THE ROUND.

153

£=£

J N

Notice in this piece the very free way in which the parts cross.
There is not one of the voices which does not, in the course of
the canon, cross with each of the others. The pause (^\) at the
beginning of the last bar does not here indicate, as usual, that
the notes over which it is placed are to be dwelt upon ; it is a
very common way of showing the notes on which the final close
is to be made.

325. The following example for six voices

MOZART.

if-

=$=

S^—

>- —

_^-

1-.. J I =B A

1 J J

j=£=

-i

[—4-

i-J-J-

=^=

I L ' ' ' ' 11 °

— t—

i* r r •

is instructive as showing the management of a large number of
parts in a close position. The first four bars form a kind of canto
fermo. Observe the little piece of imitation in the second above
in the fifth and sixth lines.

154

CANON.

(Chap. XI

326. Our next illustration, by Beethoven, requires no remarks —

BEETHOVBN.

=fr

T r i r r i

^=J=

1 r r

1 i r UT' rj "

$
^

!=i=J

— i — =1

liij j JJ^J j=U^I3

— =1 1 1 • — 1 1 H «

327. The round we shall next give is especially interesting as
containing the germ of the favourite Allegretto of Beethoven's
eighth symphony. It was composed for Maelzel, the inventor of
the metronome —

BEETHOVEN

i r

Chap. XI.)

THE ROUND.

155

328. The last example of this species of canon which we
shall give 'is an excellent specimen by Beethoven —

BEETHOVEN.

j*r J J J J * I

"'JIJJ

Here we see in the second part a free imitation of the first, and
in the fifth a partial imitation of the fourth. The consecutive
unisons between the fifth and sixth voices are most probably an
oversight.

329. Sometimes a canon of the kind now under notice is
written with instrumental accompaniment of a more or less
independent description. A familiar illustration of this variety is
Cherubini's popular canon, "Perfida Clori," which is accom-
panied by arpeggios on the piano throughout. The piece being
so well known and readily accessible, it will be sufficient here to
refer to it.

CANON.

[Chap. XI.

330. We occasionally meet with a round written for mixed
voices— that is to say, as a canon in the unison and octave. In
such a case it will be evidently necessary that the parts shall be
written in double counterpoint in the octave. A very fine
example of this kind is to be found in the second act of
Cherubim's "Faniska"; as the opera is very little known, we
present the movement here. It has an independent orchestral
accompaniment throughout ; but, to save space, we shall merely
give a figured bass, to indicate the harmonies. The canon is pre-
ceded by a symphony of eighteen bars, which it is not necessary
to quote —

CHBRUBINI. " Faniska. '
— ' Andantino.

SOPRANO

SOPRANO 2<k>.

TENOR.

ACCOMPANIMENT.

iSE

^£u Eu-r

f I

-1 rj"1!^ *

q4 6 7

£=£

Chap. XL]

THE ROUND.

r jjj

Q. u ^~

/b r-

7 8
3

. . *&QT\G N - j

p g-Sa^^ lu J -1 J

6667

4 0

•58

CANON.

[Chap. XL

gJSrSrlja'E - 3TIC £a'J Vl-T'i

* e

5 6

1 ^i |S_ _] nl rr^^

67J5 656 67

/ " "ic 5=rg ^ IP c r i

r |

^ =tE± ^

j "i

8 t?7 9 6 6 7

4 0

xi.] THE ROUND. 159

Let it be noticed that the canon is here continued until the voice
that last enters (the second soprano) has sung the whole of the
music (§ 313). Observe also the great increase of variety
resulting from one of the parts being an octave lower than the
others, the consequence being that at each repetition a different
position of the harmony is obtained. The full effect of the
music is also much enhanced here by the varied orchestral
accompaniment, which is not given in our quotation, and which
is different on each fresh entry of the theme.

331. Though it is not often that we find a complete round of
the kind just given introduced in the course of a large work, there
is a somewhat similar species of canon by no means uncommon,
a description of which will appropriately conclude this chapter.
In this form the music is carried on strictly as a canon in the
unison and octave, mostly with a free orchestral accompaniment ;
but the canon ceases as soon as the voice which last enters has
completed the first phrase. In this form the voice that com-
mences is the only one by which the entire canon is sung. As
familiar examples of this kind of canon, may be named that in
the first act of Beethoven's " Fidelio," and " Mi manca la voce "
in Rossini's " Mose in Egitto." Excellent specimens may also be
found in Schubert's Masses — the " Benedtctus " of the Masses in F
and G, and the " Et incarnatus " of the Mass in E flat. We do
not quote these, as they would only be further illustrations
similar to that which we have given from Cherubini.

160 CANON. [Chap, xn

CHAPTER XII.

TWO-PART CANONS.

332. The canons treated of in the last chapter, though not
infrequently met with as' independent compositions, are seldom
employed incidentally. We have now to speak of other varieties
of the canon, more often used, and perhaps on the whole more
useful. Though rounds are seldom written in less than three
parts, other canons are very often in only two ; and as these are
simpler in construction, and therefore easier to compose, than
those with a larger number of voices, we shall treat of them first.

333. The most important difference between the form of the
canons now to be noticed and that of rounds is that in the former
the imitation generally enters at a much shorter interval of time
after the pattern — almost invariably before the close of the first
phrase. The entry is for the most part about one or two bars
after the commencement, and not infrequently the canon com-
mences in the course of the first bar.

334. We said in the last chapter (§ 311) that there were
many possible varieties of canon ; by far the most useful is that
l>y direct imitation; the canon by inversion is also not un-
common ; but canons by augmentation and diminution are of
little practical utility. We shall therefore confine our attention
chiefly to the first kind, adding a few words on the others for the
sake of completeness.

335. To the student who is fairly skilful in writing florid
counterpoint, the composition of a finite canon by direct imita-
tion offers not the slightest difficulty, whatever be the interval of
imitation, or the distance of time of entry. The method of
procedure is simplicity itself. All that is necessary is to write
the two parts in short sections alternately. An example will
make this perfectly clear.

336. Supposing that we wish to write a canon in the octave
at one bar's distance. It is immaterial whether we begin with
the treble or bass ; in the former case the canon will be in the
octave below, and in the latter in the octave above. We will
commence with the bass, and, as the canon is to be at one bar's

Chap. XI I.I

TWO-PART CANONS.*

161

distance, we write only the first bar in the bass, copying this aa
octave higher for the treble of the second bar, thus —

We next add a counterpoint in the bass to the second bar,
placing it then an octave higher as the treble of the third bar —

w » r 1 1 — 1 1 f 1 1

It will be seen that all this is as easy as possible. We give a few
bars of continuation for this canon, which, it is evident, could go
on for a thousand bars, if desired —

* -I I

JJ^ I

tS5t

S3E

337. It makes little, if any, difference in the difficulty what-
ever be the interval of reply ; a canon in the fourth or fifth, for
example, is just as easy to write as one in the octave ; but there
are a few points to be noticed with regard to the interval selected.
It must be remembered that if the canon is at a very close
interval — e.g., in the unison or second — the parts are sure to
cross. We saw this repeatedly in the case of the lound in the
last chapter. '1 here, however, it was not objectionable, because the
leading voice enunciated a complete phrase before the next part
entered, and the subject could therefore be clearly distinguished.
But in the canons of which we are now speaking, where the
second part mostly enters very soon after the first, it will be
L

1 62 CANON. [Chap, xn

difficult, if not impossible, to distinguish the subjects if there is
much crossing. A canon in the octave or ninth is therefore
much more usual than one in the unison or second, and even a
canon in the third would be more often written in the tenth.

338. It is also advisable in general not to make the canon too
straggling, by having too long an interval of time before the
entry of the imitation, because in this case the canon is much
more difficult for the hearer to follow; the clear recollection of
the passage which is being imitated will probably have become
blurred, if not altogether effaced by the interposition of other
matter. An interval of two, or at most three, bars will generally
be quite enough ; many of the best canons are at a bar's distance,
or even less.

339. It is important to remember that a canon in any othei
interval than the unison or octave will be free as to intervals
(§ 277) > if not> tne music will De m two keys at the same time
The only exception to this general rule is in the case of canon?
in the fourth or fifth above or below ; in these it is possibh
(though by no means necessary) to use strict imitation. But it is
needful to observe that if this be done we shall probably intro-
duce transient modulations into the key of the dominant or
subdominant, as the case may be ; and care must be taken to
restore the original key by the subsequent introduction of the
necessary accidentals.

340. An example will make this clear. We will write a short
piece of canon in the fourth below, strict as to intervals —

la -

L r r«r i

£=£j

p^=^

_^ ^^

=d=l^-LL-^^=^MI

^-t^=^^l

At (a) the interval of the minor third requires to be answered
with the same interval at (b). The FJJ here evidently takes the
music into the key of G. To return to the original key, it is
necessary to introduce B^ in the upper part, as at (c\ that
its imitation at (d) may give us Ft), restoring the key of C.
Similarly, had the canon been in the fifth below, instead of the
fourth, F, the subdominant of C, would have been answered by
B£ ; it would then have been needful to introduce a chromatic

Chap. XII.]

Two- PART CANONS.

FJf in the upper part, that its imitation might restore BQ, the
leading note of the key. Such devices as these, though some-
times practicable, are often difficult to manage, and in any case
hamper the composer so much that it is usually better to write
canons in the fourth or fifth free as to intervals. It would be also
possible to retain the strictness of the imitation by carefully
avoiding the introduction of the leading note in the pattern of a
canon in the fifth above or fourth below, and the subdominant
for a canon of the fourth above or fifth below.

341. The composition of an Infinite canon (§ 310) is con-
siderably more difficult than that of a finite one. Till we reach
the point where the leading part is about to begin the repetition
of the subject, all is plain sailing ; we proceed exactly as with a
finite canon ; but what the student will mostly find troublesome
is what may be termed "making the join" neatly. For the
conclusion of the subject must be so constructed as to form a
good counterpoint to what has preceded it, and also, when
placed in the answering voice, it must be fitted for accompanying
the first part of the subject in the leading voice. To write such a
passage sometimes requires a good deal of planning. To illus-
trate this, we will make the little canon in the octave given at (f)
of § 336 infinite.

342. We give the last two bars of the canon as they stand.
The bass is the leading part, and will therefore be the first to
commence the repetition. As we intend this to take place in the
following bar, we add this bar to our quotation —

By examining this sketch, the student will see exactly what is the
problem that he has to solve. The second bar of the upper part
is fixed, as also is the thiru bar of the lower part ; and he has to
invent a counterpoint which will equally well serve as a bass to
the second bar and as an upper part to the third. In the present
case the task is very easy —

The double bar with the marks of the repeat shows that here lUe

i64

CANON.

[Chap. XII.

piece recommences ; but it will now be needful also to give the
sign for repetition before the second bar of the canon, thus —
CO

I H: J f J -J 1

frr ' r ||: r frN

Very frequently a few bars of d^fe, not in canon, are added after
the repeat in order to bring the piece to a close.

343. It will be seen that the completion of an infinite canon
in the way just described somewhat resembles the writing of a
counterpoint on two cantifermi at once. If, however, the canon
be at any other interval than the octave, the problem becomes
rather more complicated; for it is then necessary to invent as
the last part of the subject a counterpoint which not only fits
the preceding, but which, when transposed at the proper interval,
will fit the commencement of the subject. There is, un-
fortunately, no royal road for the attainment of this end, nor can
any definite rules be given for the purpose. It is here that the
student's contrapuntal knowledge and inventive skill will be
found most valuable, and it is precisely those who are most
at home with free counterpoint who will most easily overcome
the difficulties here to be met with.

344. We will now give a few short examples of infinite
canons at various intervals, to show the student how to write
them. We will first write a canon in the third (tenth) above—

As the imitation is here at two bars' distance, it is clear that the
canon must be composed in alternate sections of two bars each,
and not of one bar, like that which was given in § 336. The
bars forming the " join " are seen at (a). These had to be so
contrived as that they would also serve when transposed a tenth

Chap. XII. J

TWO-PART CANONS.

higher, as a counterpoint to the subject, as at (£). Attention
will also be required to the flow of the melody ; here the second
bar of (a) had to connect naturally with the C on which the
subject begins.

345. Our next canon is in the ninth above at one bar's
distance —

It is not always necessary to make the join exactly before the
commencement of the repetition; sometimes it will be more
convenient to work from both ends, and make the connection
somewhere in the middle. This was done in the present case ;
it was decided to have eight bars within the repeat, so as to
form a complete musical sentence; the first five bars of the
canon were then written as they stand ; the first bar of the bass
was repeated as the last bar, and a counterpoint written above
this which would make a good connection with the next bar of
the upper part. The process hitherto followed was then reversed.
The last bar of the upper part was transposed a ninth lower for
the penultimate bar of the bass ; and so the canon was worked
from both ends (like the piercing of the Mont Cenis tunnel), the
actual join being made at (a). In writing an infinite canon the
join may be made wherever it is found easiest.

346. We spoke just now of having eight bars within the
repeat, for the sake of making a complete musical sentence. It
will be well for the student to try to get some kind of symmetry
and form in his canons, and not to allow them to go meander-
ing aimlessly along. At the same time, it is only right to add
that the examples we are now giving must not be judged as
compositions ; they are only exercises, and have no claim to any
higher musical merit than that of correctness.

347. The following canon is in the fourth below —

ay. A _

J c II: r j^J Ji'i r I

1" J r fllir C-tf-gfMH

i66

CANON.

(Chap XII

•Ql J f ill

^ r ' r J' ill

j . j

i 1—

— u
=1

It will be seen that this canon is strict as to intervals throughout.
This has been effected here by avoiding the leading note in the
upper voice (§ 340), and not, as in the canon given in the section
just referred to, by its subsequent contradiction by the minor
seventh of the key. A few bars of free close are added to this
and the following canon.

348. As the last example was strict, we will write the next, in
the fifth below, free as to intervals—

3 a _• — * — ^~

S2 — i r ' — U-

?=&

^3*=&i
r r r en

Here the tone between D and C at (a) is answered by the
semitone between G and FJf at (b\ and throughout the canon
the note C is answered by its diminished fifth below, while every
other note of the scale is answered by its perfect fifth. The
example needs no further remark.

349. A canon by inverse movement is not in general much
more difficult to write than in direct. The different methods of

Chap. XII.]

TWO-PART CANONS.

inversion were shown in Chapter X., §§ 280-282. But it will
generally be found more troublesome to make a canon by
inversion infinite, because now the join has to be so contrived
as that its own inversion will fit the commencement of the
subject. This will often require considerable calculation as well
as much patience. Though seldom employed in actual com-
position, the construction of an infinite canon of this kind will
be valuable practice for the student. We give a short specimen
of this variety —

In writing this canon the join was made at (a) ; its inversion is
seen at (£).

350. It is also possible to write canons by augmentation and
diminution, and finite canons of these kinds are not at all
difficult to compose. But it will be seen that in the former, the
imitation, being in notes of double the length of those of the
subject, can only at most give the first half of it ; while a canon
in diminution must very soon come to an end, as the shorter
notes in the imitation must speedily overtake the subject. We
give two short examples —

(a) * W. FR. BACH.

r r r r i» |J ' M^crpgL£rirrrrcsTQgir r^i

CANON.

[Chap. XII.

J. C. LOBE.

r rrJ J*Jf=F=

1 — : 1 — — H

The method to be followed in writing such canons as these is
the same as that described in § 336, except that the alternate
sections written in the two parts are not of the same length.
The asterisks in the above examples show where the canon ends.

351. The composition of an infinite canon by augmentation
or diminution is, on the other hand, so extremely difficult as not
to be worth the labour it requires. The special difficulty arises
from the fact that the part which is moving in shorter notes has
to be repeated against the latter half of the part which is moving
in longer notes, that is to say, against both the first and second
half of itself taken by augmentation. Those who have plenty of
time to spare, and wish to amuse themselves with problems of
this kind, will find full instructions as to how to proceed in the
large works of Marpurg and Lobe; the aim of the present
volume being purely practical, and designed to teach the
student what is likely to be useful to him, we shall content
ourselves here with giving a very neat specimen of a canon of
this kind —

C. P. E. BACH.

3£

It will be seen that at (a) the subject in the upper part recom-
mences, as an accompaniment to the augmentation of its own

Chap, XII.]

TWO-PART CANONS.

169

latter half. The alteration of the last bar of the subject at (£)
is evidently necessary here to avoid consecutive octaves; but
this does not detract from the merit of the canon, which
Marpurg (from whose work it is taken) describes as a " real
masterpiece."

352. Some further varieties of two-part canon will be. noticed
in the last chapter of this volume, on " Curiosities of Canon " ;
we shall conclude the present chapter with some illustrations
from the great masters showing the use of two-part canon in
actual composition. We have , already, in treating of Double
Counterpoint in the Tenth, given an example of a canon at that
interval from Bach's " Art of Fugue" (§ 166); we now give the
commencement of an infinite canon in the octave from the same
work ; the piece is too long to quote in its entirety —

J. S. BACH. " Art of Fugue.

170

CANON.

[Chap. XII

353. Our next illustration, again from Bach, is very interesting.
It is a finite canon in the ninth ; in the first half the bass leads,
and the canon is in the ninth above ; in the second the treble
leads, and the canon is in the ninth below. Notice especially
the smoothness of the progressions, and the beautiful flow of the
melody —

J. S. BACH. " Thirty Variations.

chap, xi i. j TWO-PART CANONS.

171

172 CANON. [Chap. xn.

354. A two-part canon in the unison is rather rare, though
those in the octave are common enough. The following
example of an infinite canon in the unison, by Mozart, is
curious —

MOZART.

ir r -1 ' c.

I:-1 r' Cj=g=

- - •* ^

hS' J ^bJ ' £=

' > J — =

It will be seen that in the first eight bars of this canon the
imitation is per arsin et thesin ; at (a) a minim rest in the second
voice replaces a semibreve rest in the first, thus causing the
accents in the two parts to correspond for the next three bars ;
at (b) an extra minim's rest is added in the second voice, to lead
back to the repetition at the original distance — three minims
after the leading part.

355. In the canon just given the distinctness of tne two
voices is chiefly preserved by the reversal of the accents. In the
following, clearness is obtained by contrast of tone colour — the
subject announced by the piano being answered at a bar's
distance by the strings.

Chap XII 1

VlOLINO.

VIOLONCELLO.

TWO-PART CANONS. 173

SCHUBERT. Triojn E flat, Op. 100.

The continuation of this beautiful movement contains other
canons in the unison and octave, sometimes at one bar's, and
sometimes at two bars' distance, and deserves to be carefully
studied.

356. Our next example, from one of Haydn's quartetts, is
somewhat similar in character, though presenting points of
difference —

HAYDN. Quartett, Op. 76. No. a.

'74

CANON.

iChap. XII.

Here we have a two-part canon in the octave, at one bar's
distance, and each part doubled in the octave, as in the extract
from Schubert given in the last paragraph. But the general
effect is quite different, owing to the increased distance between
the outside parts. Notice at (a) the modification of the lower
part, arising from the compass of the instruments. The lower
A (the third below C sharp) would have been impracticable for
the viola and violoncello.

557. The first movement of one of Mozart's sonatas (that in
time) furnishes some excellent examples of canons in the
octave —

MOZART. Sonata in D.

357
D, i ti,

-* "fr g ff-

In all these passages the canon commences with the same
theme (the first subject of the movement), though each time

Chap. XII.]

TWO-PART CANONS.

175

with a different continuation. At (a) the canon is in tlie octave
below, at one bar's distance ; at (b) in the octave above, at half
a bar's distance ; and at (c) in the octave below at one quaver's
distance.

358. The following example, from Dussek's sonata known as
" L'ln vocation," is no less remarkable for its musical beauty than
for its neat workmanship —

DUSSBK. "L'Invocation.

CANON.

fChap. Xll

The first part of this canon is in two parts only, at the seventh
below. It is written in double counterpoint in the octave, for a
reason which will immediately appear. At (a) a free middle
part is added, to fill up the harmony ; and at (b) the opening
phrase is inverted in the fifteenth, the canon in the seventh
below thus becoming one in the ninth above.

359. Our last example is a canon by inversion, strict as to
intervals —

CLEMENTI. Gradus ad Parnassum.

TWO-PART CANONS.

177

'78

CANON.

[Chap. XII

The inversion is made according to the scheme given in § 280.
The piece is very clever, but unmistakably dry, and it is given
here for the sake of completeness. Whether its effect is worth
the trouble involved in writing it, is at least an open question.
It is extremely doubtful whether any one hearing it without any
previous acquaintance would have the least idea that it was a
strict canon by contrary motion !

360. The student should now try to invent canons at aU
intervals, after the model of those given in this chapter.

xin.i CANONS WITH JPREE PARTS.

179

CHAPTER XIII.

CANONS WITH FREE PARTS. ACCOMPANIED CANONS.

361. In addition to the canons treated of in the last chapter,
in which the harmony is only in two parts, we frequently find
compositions in more than two parts, of which two are in canon,
while the others are free ; indeed, canons of this kind are probably
more common than the others. It is of these that we shall now
treat.

362. When we speak of the addition of a "free" part or parts
to a canon, it is not meant that the added parts are to be merely
a filling up of the harmony by plain chords ; the very essence of
canon is contrapuntal writing ; and unless the additional parts be
also in florid counterpoint there will be little unity about the
composition. The new voices should be of nearly, if not quite,
equal importance with the parts that are in canon ; when they are
subordinate, and merely serve to complete the harmony, we have
an accompanied canon — a somewhat different thing from a canon
with free parts.

363. To anyone who is well practised in counterpoint, the
canon with free parts offers little more difficulty than those
already treated of — indeed it is sometimes easier, because the
addition of another voice, especially when this is the bass, will
allow progressions (e.g., consecutive fourths) between the parts
that are in canon which could not otherwise be introduced.

364. As with the added free parts to a double counterpoint
dealt with in Chapter VII., those which are added to a canon
may be in any position ; that is to say, the canon may be in the
outer voices, in an outer and a middle, or in two middle voices.
The forms most frequently met with are those in which the canon
is either in the two upper, or in the two outer parts. It is impos-
sible to give any precise rules for writing the free parts ; the
method of doing this is best taught by examples, which we shall
proceed to give, adding such notes upon them as may be likely
to assist the student.

iSo

CANON.

[Chap. XIII.

365. If the parts which are in canon be the two outer parts of
the harmony, it is generally possible to add one or two free parts
in the middle after the first sketch is completed ; but if the canon
be in any other two voices, it will be necessary to write (or at least
to think of) the added parts at the same time with the canon
itself. To illustrate this, we will take the little infinite canon at
the fourth below, given in § 347, and add a free middle part
to it—

1% J3J

**>* •

~d M

Qr r Q* r J • — m* - ur " **\ ** ' 1

Though the new counterpoint here flows fairly well, it perhaps
scarcely moves as freely as it might have done had it been
originally written at the same time with the canon.

366. If now we were to take the two parts of the canon,
transpose the bass an octave higher as an alto part, and endeavour
to write a new bass underneath, the music would most likely
sound stiff and forced. It will be remembered that it is generally
more difficult to add a new bass than a new upper or middle
part (§ 195). It would therefore be better to write a new canon
at the same intervals, and on the same general harmonic outline
as the last.

Cliap XllI.J

CANONS WITH FREE PARTS.

181

=£

r CCP '' i

7> U * — 1»* I* — W5-

f— i — nrn i

II K f^* P m^~^

^.j+. .pA

<«)

r f- -r

i — p J

Lccri

Crossing of the parts such as is seen at (a\ is very common in
canonic writing, and quite unobjectionable. Here it was neces-
sary in order to keep the upper part in a comfortable position.
At (b] will be noticed what appear like consecutive sevenths.
The first note, A, however, is here a passing note, and the rule is
not actually broken. It is better in general to avoid even such
sevenths as these, though Bach continually uses them ; they are
introduced here because the quaver figure as it stands gives,
when it appears in the alto of the next bar, a neater counterpoint
than it would have done had the bar in the treble been written

as the student will easily see for himself. It may be said in
general, that in these higher branches of composition greater
freedom of treatment as regards minor points is allowed than in
the more elementary stages of his work. A student who has
been thoroughly grounded in counterpoint will be in little danger
of letting his liberty degenerate into license.

182

CANON.

IChap. XI 11

367. We will now give a canon with two free parts, and
choose a rather more difficult combination than the last, making
the canon in the octave between alto and bass, and adding free
counterpoint for treble and tenor. As our last canon was infinite,
this shall be finite —

& l}* r T t=

fei==z= — -£ 1 : ' ' ' d

r frr r

frfr** — ^ H

— 1 1 1

E r *M

=F=FF

4^=«

J h» [

If the second and third bars of the alto part are compared with
the corresponding passage of the bass, it will be seen that, while
the notes are the same (except, of course, as to their octave), their
harmonic significance is entirely changed. In the bass the notes
form parts of chords in C minor, and in the alto they are no less
distinctly in E flat. This is a frequent device in writing canon ;
its employment often helps to prevent the monotony which would
be likely to occur, especially with canon in the octave, from too
great prevalence of the same harmonic progressions.

368. The examples we have given will, it is hoped, sufficiently
show the student how to write similar ones for himself. We now
add a series of illustrations selected from the works of the great
masters ; because far more can be learned from the study and
analysis of good models than in any other way.

369. A most interesting work, from the canons which it con-
tains, is Bach's " Thirty Variations for the Clavier." Among these
are to be found canons in every interval from the unison up to

Chap. XIII.l

CANONS WITH FREE PARTS.

the ninth, which will well repay careful examination. We quoted
one of these in our last chapter (§ 353), it being the only one of
the set in two parts ; all the others have free parts added. We
give the commencement of a few of them.

J. S. BACH. " Thirty Variations."

It will be seen that this canon is in the third below. In con-
sequence of the closeness of the interval of imitation, we find the
parts crossing here at the fourth and fifth bars, in the same way
\n which we have seen them cross at (a) in our example to § 366.

370. We next give the first eight bars of a canon in the fourth
below in contrary motion —

J. S. BACH. " Thirty Variations.

1 84

CANON.

[Chap. XIII

This example requires no annotations ; but it may be as well for
the student to compare it with the canon in contrary motion by
dementi, given in § 359, and to observe how laboured and dry
dementi's workmanship is, as compared with Bach's. Merely
technical skill will never produce really artistic results in the
solution of such elaborate musical problems as those we are now
considering.

371. Our last example from this work is the commencement
of a canon in the sixth above, which needs no explanation —

J. S. BACH. " Thirty Variations."

-- --

J J3

•^

J G ^

^j -*-^ -«*-

-+-*+• j <~j

»J I t

1* »-*•-*-

^*^_

1 1 —'01

&c.

372. Next to Sebastian Bach, no one has shown greater
mastery of scientific resources than Mozart. We give two
charming specimens by him of canons with free parts. The
first is well known — the opening symphony of the " Recordare "
in the " Requiem "—

MOZART. " Requiem.

Chap, xiii.] CANONS WITH FREE PARTS.

185

The first six bars of this passage show a canon in the second
above, at one bar's distance. At (a) begins a canon in the unison
at one crotchet's distance, on a pedal bass, and with a free middle
part ; at (b) the canon is in the octave, instead of the unison.

373. Our next example by Mozart is as little known as the
extract from the " Requiem " is generally familiar. It is a remark
ably neat and beautiful canon in the fifth above, by contrary
motion, and, as it is only short, no apology is required for quoting
it in its entirety —

*•> • MOZART. Canonic Adagio for 2 Corni di Bassetto and Fagotto.

[A , r i 1 li f ibJ b

J J i.J I

^;L — ^ — 1 — ^ — J . m . — *— : — >—. i»r 1 — ar — m ar~

• — jp ^ i ^~l

^P *\>»» J j '^' 1 •< '* r-^ g;^J :

bf-litP

186

CANON.

IChap. XIII.

Chap. XIII.]

CANONS WITH FREE PARTS.

187

374. It is impossible to draw any hard and fast line of dis-
tinction between canons with free parts and such canons as are
merely accompanied. All the examples hitherto given unques-
tionably belong to the former class ; in our next illustration we
have more of the character of an accompanied canon —

MOZART. Serenade in C minor.
tr

I I I

-,—J-

1 J J

fc3rr rj-r-d=

17 L^-Ld^Ld^

Here the canon is in the octave between the outer parts, and
although the middle parts are to a certain extent contrapuntal,
they have none of the individuality of character which can be
seen in the free parts of the examples previously given. To save
space, we have arranged the passage on two staves, taking no
notice of the crossing of the parts : it must not be supposed that
the consecutive octaves in the fifth and seventh bars are to be
found in Mozart's score.

i88

CANON.

[Chap, xni.

375- The following passage is a good example of an accom-
panied canon —

MENDELSSOHN. 95th Psalm.

CHORUS.

ORCHESTRA.

_^J~

P*. p ii.

WhU V

fe£#

&c.

1 r r

j^.

k |pgg

Here the instrumental parts are strictly subordinate to the vocal,
and do nothing more than fill up the harmonies.

376. Our next illustration shows an accompanied canon of a
different kind —

(g)

HAYDN. 3rd Mass.

-J-^

CHORUS.

A J A ^rj. A

A _ '

\ \ i i

ORCHESTRA.

^Rj

Chan. XIII.]

CANONS WITH FREE PARTS.

189

T- ' f^r F^T -M— — r"

J yjj J- JTjJ. J^v, J

I i_ L

r r~-r

r— r

' r r r

Here the voices have a canon in the fifth below, both parts being
doubled in the octave ; the orchestra not only fills up the middle
parts of the harmony, but gives a florid embellishment of the
canon itself. This form of accompaniment is maintained to the
end of the movement, which is in canon throughout ; and in
the latter half the instrumental parts become more independent,
as in the following passage —

JS J^ J^ i j

"' " I i f • rrr.

R35

i go

CANON.

[Chap. XIII

In both the above examples holding notes for the wind, which
merely fill up the harmony, are omitted for the sake of clearness.
377. Our last example will be a more modern one —

SCHUMANN. AlbumblStter, Op. 124, No. 20.

f — i ""

J r - r

-*• *>.. J..
r r

p' ° "" •»*-
i p

r1 r r

in — J J J i

J j J-

1 *^ m

.j-^y-^, J^^

^ **^

— i — ' — i —

m\ m =

j — i — i

f f

^J^p1f
^ r r

r r r f1

^Ij J 1

;?^j j i^j ^

i J. A -S. J J J. J^

\ •> \ *>

^|iJ^r|

r • Y

J r r^F r r F F r

| ^ « \Qm -J | j J J J J

r r *r

Here the canon is in the octave at one bar's distance between
the upper part, and what we should call the first tenor were the
piece for voices instead of piano. In consequence of the crossing
of the middle parts, it will require a little attention to follow the
canon in some places ; it is carried on to the last note. It will
be seen that the harmony is in some places very free.

378. The student should now practise writing canons in
different intervals, and with free parts, putting the canon into
the different voices in turn. He will find this more interesting,
and little, if at all, more difficult than writing canons without
accompaniment.

chap, xiv.) ON A CANTO FERMO* OR CHORAL. 191

CHAPTER XIV.

THE CANON ON A CANTO FERMO, OR CHORAL.

379. The last kind of two-part canon which it will be advisable
to study is that in which two voices which shall be in canon are
to be added to a given subject — either a simple canto fermo^ or a
choral. This is in reality a variety of florid counterpoint ; but it
is far more difficult than any that the student has yet attempted.
He is, however, strongly recommended to devote some time to
it, because it will materially lighten his labours when he comes
later to deal with more elaborate canons.

380. The special difficulty of this species of composition arises
from the fact that every note of the leading voice of the canon
has, as soon as the imitation has begun, to be regarded from a
threefold point of view. It must form a good counterpoint both
to the canto fermo and to the phrase in the imitating voice which
it accompanies ; in this there is no very great difficulty ; but it
must also be so written that when transposed into the imitating
voice it will form a good accompaniment to the following note of
the canto fermo. It will often happen that either the interval or
the distance of time of the reply will be such that it will be quite
impossible to continue the canon ; in that case, a fresh attempt
must be made at some other interval, or some other distance,
until success rewards our efforts.

381. A short example will best illustrate what has just been
said. Let us take as a subject the familiar theme —

&c.

put it in the bass, and try to write above it a canon in the ninth
at one bar's distance. Supposing we begin with a very simple
figure —

t'fi -

&c.

19*

CANON.

[Chap. XIV

The first note in the second bar of the alto must be one which
will not only go with the D of the subject, but which, when
transposed a ninth higher in the treble will harmonize with the
F in the third bar. A little thought will show us that B is the
only note fulfilling both these conditions ; this will become C in
the treble. So far so good ; but now comes the knotty point.
How can we complete the second bar in the alto in a way which
will suit, when it is transposed, for the third bar of the treble?
We cannot write thus —
CQ

because of the consecutive octaves with the bass ; neither can we
take this —

for this will give us a most unpleasant mediant chord in root
position in the fourth bar. If we try —

which at first sight looks more promising, we find that when
transposed it gives us consecutive fifths with the canto fermo. In
fact, we are in "a tight place," and there is no ically satisfactory
continuation. We therefore try again, altering the first bar, and
making the canon in the seventh, instead of the ninth, above —

W x-^

fr*

i r*

-^-: — J— «-* j"^, j

&C.

The canon now goes smoothly enough.

382. It would also have been possible here to continue in the
original interval by introducing the imitation at two bars' distance,
instead of one —

&C.

— 1 — *-J — ij 1

iftl -.1 1

J J^ - — '

Chap, xiv.i ON A CANTO FERMO, OR CHORAL.

193

383. The student will form a fair idea from the above
examples of the kind of difficulty which he will meet with. In
writing exercises of this kind, he should first try to work mentally
the outline of a canon upon a canto fermo^ when he has decided
on the interval and distance of reply. He need not do this for
the whole of the subject ; but only for enough of it to make sure
that he has a promising commencement; though even then he
will not seldom come across a progression in the middle that
stops further progress. He may find it useful in his earlier
attempts to avail himself at starting of some of the models for
imitation given on pages 8, 9, of the Additional Exercises to
Counterpoint. He must also be prepared for many disappoint-
ments ; at first, indeed, he will probably have at least two or
three failures to every success.

384. We will now give some examples of canons in various
intervals, using the same two subjects as canti fermi that we have
so often treated for counterpoint. We first take our major
subject in the bass, and write above it a canon in the seventh
below —

X\. *T — 1= ft ? f —

— & — ' p 1*

h«-r r r

— fZ* — ; j*,.j_...J

IM 1 ' *=

i r CJ :

H • 1 1

-1 UJ— 1

s
®i ~ —

^H5 — r — * — 5-^

—& P r r p

. r J rt=

-» — H

gi ! — : — p —
<«)

-4- rr r r

H— i Eas

1 ' ' f =

Hj r_r..i r

H
=1

It is only needful to remark of this, as of the other examples we
shall give, that they are written in free, not in strict counterpoint.
Thus at (a) we have two chords in the bar, the second (at the
fourth beat) being the third inversion of the dominant seventh.
The seventh is not really doubled, as the F in the alto is only a
passing note. It must further be said that in all the examples
we shall find a certain amount of stiffness. Under such limita-
tions it is almost impossible to make the parts move freely.
Correctness must be the first thing aimed at ; but in general it
would require a genius little less than Sebastian Bach's to make
such exercises very interesting musically. The student must be
content if he can make them fairly melodious.

IQ4

CANON.

[Chap. XIV.

385. We now take the subject in the treble, writing below it
a canon in the fifth, again at one bar's distance —

_,2_l4 «s> " 1

fl> * -

INI 4 • i u * - - i ! - -i * I » -— n

/* — ^ 1

-«».

ft — ! — • — •

JUI

iffli r ' s^=

^ — i 1 —

— i — '- — E—

— ^

r Cr

==ll

There is no special objection here to the consecutive octaves by
contrary motion between the treble and alto in the first and
second bars, for two reasons. In the first place, they are between
tonic and dominant, and are, therefore, allowed occasionally in
the free style ; and besides this, as we are not restricted to one
chord in a bar, we are fairly justified in assuming that the E at
the end of the first bar represents a submediant chord. The
harmony at the entry of the tenor is, of course, the third inver-
sion of the dominant seventh. The close of the canon is free ;
this is always allowed where necessary.

386. For our last example in a major key we put the subject
in the middle, and write a canon in the octave above, at two bars'
distance —

lr J^J Jlr eft ' I

— 1 — r^v a — »-

H» m-^. — ! 1 1 i 1—

J ^r r

Chap, xiv.] ON A CANTO PER MO, OR CHORAL.

'95

This requires no explanation ; it will be seen that the canon is
here continued to the last note.

387. We next take our minor subject, placing it in the bass,
with a canon for treble and alto in the fourth above —

lib'i 4 h r r r

.•err*

i=b

J Cj r r r d

Hi 4 —
S

^t ^ '" '

•4 F =^ — 3

J> J J ^ )=F=q

> u

>J J J I J Jjjjl ^ ^

N^__

Here again the canon is continued to the end. Notice at (a)
how the consecutive fifths between the first notes of the treble
and bass of the fourth and fifth bars are saved by the clear
indication of a new chord at the half bar.

388. Our next canon shall be in the octave, at only half a
bar's distance —

In order to preserve the canon, it has been needful here to make

196

CANON.

(Chap. XIV

the harmony in the sixth and seventh bars somewhat free. The
harmonic framework of these bars is —

but without the addition of free parts it is impossible to make it
complete.

389. To write a canon by inverse movement upon a canto
fermo is even more difficult than to write one in direct movement,
and the result will in most cases not be worth the trouble
involved — that is to say, from a musical point of view. Such is,
at any rate, our own opinion of the following specimen, which
has been produced with some little labour, merely to show the
possibility of a canon of this kind —

390. After working several canons on a iait/0 fermo, the
student will find what we have described as " making the join "
in an infinite canon (§ 341), considerably easier; for here he has,
so to speak, to make a join in every bar. The work is also very
interesting for its own sake; and the fluency in part writing
acquired by its practice will well repay the learner for the hard
work that it demands.

391. There are two methods of writing a canon upon a choral.
The simpler, and easier, is to treat the choral itself as the subject
of the canon, and to add one or more free parts to it. To do

Chap, xiv.] ON A CANTO FERMO, OR CHORAL.

197

this, it is first necessary to find a choral which will work in
canon — that is to say, which can be imitated by itself at some
given interval throughout. This is the chief difficulty, for it is
by no means every chorfcl which is capable of being treated in
this way ; but when this is once done, the addition of free parts
will be fairly easy for students who have mastered the preceding
chapter.

392. As an illustration, we will write a canon on the choral,
"Jesu, der du meine Seele." An examination of the melody
shows that it can be made to work throughout as a canon in the
seventh below. At this interval the canonic voices will evidently
be too near to one another for both to be treated as outside
parts. We therefore take them as treble and tenor, adding free
parts for alto and bass —

(^'l "m
rr

I;J r-^l *fl

77 r,r7

-H

t— h

-lrr fyip-LJlj

r r r ' i r » r ' r r r^ ' » _

~ ' ~ r r ' ' ' i iff"

J. iigj. j- -^ hd. ^ J. ..._

!• - ^*^ I ^ Ji I - ^ ^ I (• ' ^

^^f

,J J =^=g=

^ JV ^ '

!* ^ J

Wf^

J J

* r r r
j j

i r r nr r^pj

rr rr

^-r^

\ I

ir r f fir r=r**

I I I rn I T^

r ' • i r i r
± A ^. >

v , r^ 1 — i — -j -j

f)r r,"r u<" r r ^' ' r^

(^ ^ ^ J J J

^b ' r-ifr f 1 r - —

— ~] ~J '

'T ^r r F

j. J.

— i — —
r T

1 98

CANON.

[Chap. XIV.

rr'rrr

i i

rr IT r

T=F*?*P

A «•

JL J. A

393. This example has several points calling for remark.
Note first that, in order to give unity of character to the music,
the moving figure of crotchets is maintained persistently, either
in the alto or the bass, till the last bar. The canon in the tenor
commences at two bars' distance ; but, though it is desirable to
retain the same distance of interval throughout a canon of this
kind, there is no objection to altering the distance of time\
indeed, this is sometimes absolutely necessary (as we shall see
directly with this very choral), if we are to continue the canon at
all. At (a) the D in the tenor is made natural for harmonic
reasons ; as a canon in the seventh is never strict as to interval,
such chromatic alteration as this can always be made where found
expedient.

394. At (b) the first eight bars of the melody are repeated ;
to avoid monotony, it is desirable, where practicable, to change
the harmony. As the canon in the tenor has to be retained, we
are, of course, somewhat limited in our choice of chords ; but it
will be seen that, notwithstanding this, considerable variety is
possible.

395. The entry of the tenor at (c) should be particularly
noticed. If the student will try, he will find that there is no
other point of entry at which the canon in the seventh can be
continued. Here we have three consecutive fourths with the
treble. It will now be seen why we gave the imitation to the
tenor and not to the bass. Had the canon here been in
the outer parts, or in the alto and bass, it could have been
carried no further; but the addition of the other voices here
makes the harmony quite correct (§ 363).

396. In the next line of the choral it is necessary to vary the
distance of time of the imitation. Had we not inserted a bar's
rest in the tenor before (d), it is evident that we should have had
consecutive fifths with the treble, Here the canon is at three

chap, xiv.] ON A CANTO PER MO, OR CHORAL.

199

bars' distance. At (e), by omitting in the tenor the two bars'
rest of the treble, we reduce the distance of time between the
two parts to one bar.

397. It is hoped that these explanations will sufficiently show
how a canon of this kind is to be written. We now give a few
short examples from the organ works of Bach. The first is the
commencement of a canon in the octave * —

J. S. BACH. Organ Prelude on " Gottes Sohn 1st Kommen.

A.t (a) it will be seen that the imitation of the preceding bar of
the pattern is not exact. In consequence of the difficulty of
writing such canons as these, slight modifications of detail are
allowed, provided they do not obscure the imitation.

* In the original, the lower voice of the canon, which we have here printed on
the middle staff, is given to the Pedal, and marked " Trompete, 8 ft." As it
is the middle part of the harmony, we have altered the relative position of the
two bass lines in order to make it easier for the student to read.

200

CANON.

[Chap. XIV.

398. In the example just given the canon is between treble
and tenor. In the next it will be between the outside voices —

J. S. BACH. Organ Prelude on " Erschienen ist tier herrliche Tag."

J r Cr r err Q»

f 1*^ i

ft j J f=|

U| ^

J=l

IT^ — I '

_5i ,

r p* r Cr ^ —

— 1 K — p — kl —

^^^

The /TN marked here indicates, not a pause on the notes
as usual, but simply the end of a line of the melody. This
example is quoted to show how much freedom Bach occasionally
allows himself in the treatment of a canon of this kind. Let
the student compare the melody of the upper part from (a) with
that of the bass from (b).

399. Our last example of this species of canon is in the
fourth below, and in five parts —

J. S. BACH. Organ Prelude on " Lieb»ter Jesu, wir sind bier."
" ! i ."-^ ^

MANUAL.

PEDAL.

ma-

forte.

Piano.

ch«p. xiv.] .ON A CANTO PER MO, OR CHORAL.

fi«_fl . , i ^^3 i /TV

201

j j i

rgJ3£P3

>):8 tf * -- m £ F J ffi ff i S r

This little piece is written for two manuals and pedal, the right
hand part being marked forte, to bring out the canon distinctly,
and the left hand piano, containing merely the accompanying
counterpoints. The form of the melody is here somewhat
altered, to enable the composer to treat it in canon ; the more
usual form can be seen in the Additional Exercises to Counterpoint,
page 12. Notice, also, the curious harmony of the cadences —
free, even for Bach, but necessitated here by the restrictions
under which he is working.

400. The second, and more difficult, way of writing a canon
upon a choral is to treat the choral itself as a canto fermo, and
to write upon it two parts in canon, with or without the addition
of free parts. The material of the canon may be taken from the
choral itself, but this is optional. It will generally be found con-
venient to commence the canon alone, and to let the choral
enter later ; it is usual, also to separate the diifr rent lines of the

2O2

CANON.

[Chap. XIV.

choral by rests, during which the canon must, 01 course, be con-
tinued. In a canon of this kind the choral is usually given in
long notes ; this allows more frequent changes of harmony in the
canon itself, and to some small extent lightens the student's
labours.

401. The general principles by which we should be guided in
attempting a canon of this kind are the same that have been
explained in the earlier part of this chapter in connection with a
canto fermo ; but the difficulty of obtaining a really artistic result
is so great that but few specimens of the kind are to be met with.
Probably the finest example in existence is Bach's Canonic
Variations for organ on Luther's Christmas Hymn, "Vom
Himmel hoch da komm' ich her." The whole piece is worthy of
careful examination ; we give the opening bars of the first four
variations.

402. The first variation is a canon in the octave, at a quarter
of a bar's distance, with the choral on the pedals —

J. S. BACH. Canonic Variations on "Vom Himmel hoch da komm 'ich her.

PEDAL.

Here the subject of the canon is quite independent of the choral,
each line of which enters, like the first, after a bar and a half's
rest. The florid canon, which, though in the octave, is not
entirely strict as to intervals, is kept up to the last note of the
variation. It ought to be mentioned that the piece is written for
two manuals ; this will explain the frequent crossing of the parts
to be met with in some of the extracts.

403. In the second variation, the choral is again in the pedals,

Chap, xiv.] ON A CANTO FERMO^ OR CHORAL.

203

and the canon is now in the fifth below, at half a bar's distance,
the opening theme being the commencement of the choral —

J. S. BACH. Canonic Variations on " Vom Himmel hoch da komm 'ich her.'

404. With all their ingenuity, the two variations of which we
have quoted the first bars are simple, compared with those that
follow, which show what is possible in the way of scientific device
to a composer with the genius of Bach. The third variation
begins thus —

J. S. BACH. Canonic Variations on " Vom Himmel hoch da komm 'ich her."

204

CANON.

[Chap. XIV.

Here we have a canon in the seventh above at half a bar's dis-
tance, accompanied with a florid free part. It will be seen that
the first four bars of the canon are made of sequential repetitions
of the first phrase of the choral. This phrase is resumed in the
last bar of our quotation, and continues to be a prominent
feature in the whole variation. The choral, treated as a canto
ferine^ is here in the upper part.

405. The fourth variation is perhaps even more astonishing.
It is a canon in the octave by augmentation, continued for forty
two bars, with a free middle part, the choral being again on the
pedals —

J. S. BACH. Canonic Variations on " Vom Himmel hoch da komm ich het."

chap, xiv.] ON A CANTO FERMO* OR CHORAL.

406. The last variation, which we only refer to here, as it is
not an illustration of the species of canon we are now treating,
introduces the choral in canon by contrary motion in the sixth,
third, second, and ninth, and winds up with a most marvellous
tour de force, the four lines of the choral being simultaneously
introduced as counterpoints to one another ! Let the student
carefully examine this wonderful piece, and then go and do like-
wise— if he can ! In any case, the study and analysis of the
scientific masterpieces of Bach cannot fail to be of the greatest
benefit to him,

2o6 CANON. (Chap. xv.

CHAPTER XV.

CANONS ON ONE SUBJECT, IN MORE THAN TWO PARTS.

407. There is practically hardly any limit to the number of
parts in which it is possible to write a canon; though if the
number of voices be very large the composition becomes con-
fused, owing to the continual crossing of the parts, which renders
it impossible to hear the separate melodies distinctly. In a later
chapter (§ 474) we shall give an example of this kind. But
canons for three, four, or five voices are by no means unusual,
and it is of those that we have now to speak.

408. As with the canons in two parts, treated of in preceding
chapters, those in more than two parts may be at any interval.
But it is most usual to write canons in three and four parts,
either in the unison and octave (as we have already seen with
the Rounds in Chapter XI.), or, if they are for mixed voices, to
combine canon in the octave with that in the fifth. Thus, if a
canon were for four voices — treble, alto, tenor, and bass — the
general arrangement would be that the canon in the alto would
be in the fourth or fifth below the treble, that in the tenor would
be at the octave below the treble, and that in the bass an octave
below the alto. This, however, is by no means obligatory, and
we shall give presently examples of canons at other intervals than
these.

409. No fresh directions have to be given for writing canons
in more than two parts. The method of procedure is the same
as before. (See § 336.) Having decided on the number of parts,
their interval, and distance of entry, we write the first part down
to the point where the second voice enters. We then copy the
subject, as far as we have written it, into the line of the second
voice, adding a counterpoint for the first voice, and continue as
with a two-part canon until the entry of the third voice. We
then write the commencement of the subject in the third voice,
adding to it the counterpoint in the second voice which was
before in the first — of course transposing to the proper interval
if the canon is not in the octave or unison. In this order we
continue to the end, writing the parts which are already fixed
first, and adding the new counterpoints later.

Chap, xv] IN MORE THAN Two PARTS. 207

410. As an example of this system of working, we will take
the commencement of Byrd's universally-known canon, "Non
nobis, Domine," for three voices, and show how such a piece is
to be composed The number over each bar indicates the order
in which it would most probably have been written down —

U~~fc

i ' — ^ — ~a~

1 ^ — I

2 4 ,

16 &c.

L_L_ L — )

The canon here is led by the alto, answered a fourth below (in
the dominant), by the tenor in the second bar, and an octave
below by the bass in the fourth bar. We do not, of course,
maintain that Byrd certainly wrote down the parts in the order
which we have shown here; because a composer for the most
part carries on his work to a considerable extent in his head
before he commits anything to paper ; but what we do point out
by the figures we have given is the order in which the different
voices must be added. Thus, after the first three bars were
written for the alto and tenor (i to 5), the next three bars were
fixed, so far as the bass was concerned, therefore, 6, 7, and 8
would most likely be the next to be put down ; 9 is also fixed, as
the imitation of 5 ; 10 follows naturally as the accompaniment of
9; and this in its turn is transposed as n. The student will
easily follow this analysis to the end of the extract.

411. It will now be readily understood that the general
principles which guide us in this kind of composition are in the
main the same as those which we indicated in the last chapter,
when treating of the writing of canons on a canto fermo. Every
note added in the leading voice has to be considered, not only
in its relation to all the parts which it is actually accompanying,
but in its relation to those which when transposed into another
voice it will have to accompany in some subsequent bar. It will
also be seen how impossible it is to lay down any absolute rules
for our guidance ; because a rule which might apply perfectly
well for a transposition at one interval, might be (and probably
would be) quite useless at some different interval, or distance of
time. The whole thing is a matter of practice, of calculation,
and of facility in counterpoint ; and it is in this respect that the
working of canons on a canto fermo is so valuable as a preparatory
step. The student will find a three or four-part canon little, if at
all, more difficult than the exercises prescribed for him in the
preceding chapter. All we can do now to assist him is to furnish

208 CANON. tchap. xv.

him with models of various styles for analysis and imitation.
His own industry and perseverance must do the rest.

412. As a particularly neat specimen of an infinite three-part
canon, we first give the whole of the " Non nobis," of which we
have been examining the opening bars —

W. BYRD. " Non nobis, Domine."

i i j H. ,T ^

IE «* =

r r \

(•— 1» — — | "^ 1 ~^~ frf5* | -

(9 — :

lid! b

(*)

?=&

r-3

IUI b

| i 1 J :ll

==f

, ,

3 — i —

— ca ni

The pause in the fourth bar indicates, as in some of our preceding
examples, the place at which the music ends. As the canon is
only in the octave and fourth, it is strict as to intervals (§ 277);
at (a) the minor seventh of the scale is therefore introduced
(§ 34°)> as otherwise, the imitation of the leading note by the
tenor in the next bar would have induced a modulation into the
key of the dominant.

413. Our next specimen is by Friedemann Bach —

W. FR. BACH.

U,. . r r|T r ^ i=r r f=r=

hr £.r i* <• 1

7?~n ™ P P m \\: — 1 r — p —

BM=± 1

-7^ /•» u Ij-i 1 1 — t-

1 — r^~

«))"r j J J j * J|= J J j j3

Chap. XV.]

IN MORE THAN Two PARTS.

209

. r i r r f

Here the canon is in the fourth and octave above ; in our last
example we saw the fourth and octave below. Here, also, the
imitation in the octave precedes that in the fourth, and the latter
is not, as in Byrd's canon, strict as to interval.

414. The two following canons — both finite — are by Mozart —

MOZART.

J •* J

J •*

' ir r r ri

ir r r ri

*=*f

In this example, in the unison and octave, it is only needful to
call attention to the frequent crossing of the two upper parts
(§ 337)- Here the clearness of the imitation is preserved by the
contrasted rhythm of the two parts which lie close together.

210

CANON.

[Chap. XV.

415. In our next example, the second voice enters at the
distance of a second above the first, and the third at the sixth
below the first — of course, the inversion of the third above —

MOZART.

In both this and the preceding canon the close is free.

416. It is far from easy to write a canon in more than two
parts upon a canto fermo ; the following example, taken from
Cherubini's work, will show that it is not impossible —

> F. AZOPARDI.

Chap. XV.]

IN MORE THAN Two PARTS.

211

£s

*^=E

1 ' r'

417. It has been already said (§ 408) that in four-part canons
it is very common to find the imitation at the fifth and octave.
Such is the case in the first specimen of this kind to be given —

G. ALB»ECHTSBERGER.

Lj — p — S — gfd

1 1 r i

r,JJJ|.' — 4H

if-

^gE&

rrr1

r r -t-&=

212

CANON.

[Chap. XV.

r ' i

The occasional introduction of rests, as here in the sixth bar, is
frequently advisable in canons, as it renders the next entry of
the leading part, as also of its later imitation, more clearly per-
ceptible to the hearer. In the above canon the close from (a) is
free.

418. Our next illustration is a very beautiful specimen of a
four-part canon in the fifth and octave at only one crotchet's
distance —

j MOZART. Mass No. 10.

*&v . •» i'f • J^eic,^- ^^t

Let the student compare with this the somewhat similar example
of a four-part canon with close imitation from the "Jupiter"
symphony, quoted in § 305. Something analogous will also be
seen in the " Amen " chorus of Handel's " Messiah."

419. Another variety, as regards interval of entry, is seen in
the following —

CLEMBNTI. Gradus ad Parnassum.

J-r-E

Chap. XV.]

IN MORE THAN Tiro PARTS.

213

The canon, which extends to 64 bars, is too long to quote in its
entirety. Though founded only on one subject, the piece has in
form some resemblance to the double canons to be spoken of in
the next chapter. The treble, which leads, is imitated by the
alto in the fifth below; the tenor and bass stand in the same
relation to one another, the tenor being a fourth below the treble,
and the bass a fourth below the alto.

420. As all the four-part canons hitherto shown have been
finite, we give next a short infinite canon in the unison —

, W. FR. BACH.

y ~-T -

rp r

. ^ * f ~ ji _

This is so clear and simple as to require no explanation.

214

CANON.

[Chap. XV

421. Like two-part canons, those for more than two voices
can be accompanied by free parts. We give an example by
Mozart —

MOZART. Mass No. xa.

SOPRANO. /
ALTO.

TENOR,
BASS. v

ORCHESTRA.

m 'r g c cj- cj»

Viol. a. 1 n

Bassi e Org.

Here each voke enters a fourth higher than the preceding. (The
bass enters at the fifth below, which is practically the same as the
fourth above.) On the two lower staves we give the orchestral
accompaniment exactly as it stands in the full score. The figured
bass indicates the harmony that is to be filled up on the organ.
It was not the custom in Mozart's time to write out the organ
part in full, excepting where it had solo passages.

422. It was said in commencing this chapter that canons
could be written in almost any number of parts. We now give

Chap. XV.]

IN MORE THAN Two PARTS.

215

two specimens of canons for a larger number of voices. The
first is by Kirnberger —

KlRNBERGBR.

14 ~ . i r '

1* ill

r J ill

Here we have an infinite canon for six voices in the fifth and
octave; the entries are alternately half a bar and a whole bar
behind one another.

2i6 CANON.

423. Our last example is more curious —

[Chap. XV.

MARPUKG.

ft" I

II" ** \*t

\y *] I

r i I

i i

\-J " \'J J I^=F4

J J Jl I I I I -I IrJ ^ I

ihr ^i- r ir r in rrir^ *!. J

irr*jNj ji j i i

Chap, xv.i IN MORE THAN Two PARTS. 217

This is an infinite canon for nine voices, the peculiarity of which
is that each successive voice enters a third lower than the pre-
ceding. Such canons as these require an amount of ingenuity
and patience to invent which can generally be much more pro-
fitably employed in other directions.

424. A canon is usually described according to the number
of parts and the number of subjects which it contains. On the
continent the general plan is to speak of a Canon "a 2," "d 3,"
"d 4," and so on; in this country a rather different nomenclature
is adopted. A two-part canon, such as those given in Chapter
XII., is described as a " Canon 2 in i " — that is to say, having
two voices and one subject; similarly, those we have been
treating of in this chapter would be said to be canons " 3 in i,"
"4 in i," and so on, according to the number of voices. If
there be more than one subject, the first of the two figures shows
the number of voices taking part iti the canon, and the second
shows the number of subjects. Thus a double canon, with four
parts and two subjects, would be spoken of as a canon " 4 in 2,"
and similarly in other cases. Canons of this kind will be spoken
of in our next chapter.

425. All the canons we have given in this chapter have been
written out in full, either in score, with each part on a separate
staff, or in "compressed score"— *>., on two staves, as for the
piano. (See examples to §§ 419, 421.) A canon written in this
manner is called an " open canon." But there is another method
of writing a canon, which has now to be explained. It was
formerly the custom to write only the theme of the canon on one
staff, indicating at the commencement the number of voices, and
placing signs to show where and at what intervals the other
parts entered. A canon written in this way is called a "close
canon." The student must not confound this meaning of the
word " close " with that which has been made use of in speaking
of imitation (§ 289). As applied to canon, "close" simply
means not written out in full.

426. The usual method of indicating the later entries of the
parts in a canon was to place the sign § over the notes on which the
parts were to enter. If the canon was in the unison, no further
indication was necessary. Thus the canon by Friedemann Bach,
given in § 420, would be written as a close canon, thus —

(1) "• "PL --

i— I i

If, however, the entries of the other parts are at some other
interval than the unison, it became necessary also to show at
what interval these other voices entered. This was effected by
adding figures to the sign §, placing the sign and figures above

218

CANON.

[Chap. XV.

the canon when the entry was for an upper voice, and below when
it was for a lower. The figure gave the interval above or below
the first note of the subject ', and did not refer to the particular note
over or under which it was written. Two examples will make
this clear. , We will write the little canon by Mozart, in § 418, as
a close canon —

(2) a 4.

§8 §12

By comparing this with the open canon, it will be seen that, as
all later entries are below the first, the figures are all under the
subject, and that they are all reckoned from D, the first note,
and not from the notes under which they are written. Occa-
sionally, however, this method is departed from, somewhat to the
perplexity of the student.

427. We now give the nine-part canon of §423 in the same
notation —

§3

§5

§7

§9 §11 §13

§15

§17

After what has been said, this example will be quite intelligible.

428 There is another method of indicating in a close canon
the number of voices and the order of entry. This is, to prefix
to the canon the various clefs of the different voice parts. Un-
fortunately, no uniform system is adopted as to the order in
which these clefs shall stand. Albrechtsberger says that "when
a canon is answered on the fifth or octave above, or on the fifth
or octave below, it is usual to place the clefs of the voices, in
the order in which they are to succeed, before the clef used for
the commencement of the canon and before the signature is
marked ; then either the sign § or a figure indicating the distance
of the interval shows the note on which the successive voices are
to enter." On this method, the notation of the canon in § 417
would be —

.. !! .1 § § § &C.

q! lfti&:(!v) (t !

g I -r- •

' IT r * c/i i ^ I- r

»~j

Sometimes, however, the clefs are all put in reversed order ; e.g.,
for the same canon —

while Marpurg, in some of his examples, gives the clefs in the
regular order of entry of the voices. As close canons are seldom

chap, xv.] IN MORE THAN Two PARTS. 219

written now, the matter is not of much practical importance;
but it is well that the student should understand these signs if
he meets with them in old music.

429. We will conclude this chapter with a specimen of a
twelve-part infinite canon in the unison, taken from Marpurg.
We shall give it as a close canon only ; it will be interesting for
the student to put it into open score for himself. He will find
that the harmony is extremely simple, consisting of nothing but
alternations of tonic and dominant chords —

r § r § r - 5

$ * 5

9 9 r f

tJ •*•

2 ao CANON. icaap xvi

CHAPTER XVI.

CANONS WITH MORE THAN ONE SUBJECT.

430. Hitherto we have treated exclusively of canons which
have had only one subject ; but it is quite possible to work two,
three, or even more voices simultaneously in canon. It will be
evident that the number of parts in the harmony must be at least
double the number of the subjects to be treated canonically. On
the continent a canon with two subjects is usually described as
a Double Canon, one with three as a Triple Canon, and so on ;
but in England it is more usual to speak of them after the
method explained in the last chapter (§ 424) ; and this method
will, therefore, be that which we shall now follow.

431. The canons to be described in this chapter are the most
elaborate, and in many cases the most difficult, that it will be
necessary to study. It is true that there are other varieties which
are more complicated; but these are of so little practical use
that we do not recommend the student to trouble himself over
them at all. We shall speak of them in our final chapter on
" The Curiosities of Canon." Such double and triple canons as
we are now about to notice are, on the other hand, of real artistic
value; and it will be well worth the student's while to spend
some time in trying to write them.

432. It is impossible to give any exact rules for the composi-
tion of a double or triple canon. -Like the various kinds with
which we are already acquainted, it must be worked in small
sections, the length of which must depend on the distance of the
time of entry of the different voices, and those parts which are fixed
(that is to say, which are the canonic imitations of the themes
given by the leading voices) must always be written in the
imitating voices before the counterpoint is added to them in the
other parts. Beyond these general directions it is not easy to
assist the student, who will learn best how to work by the careful
examination and analysis of the examples we are about to give
him, which we shall accompany by such remarks as may be likely
to be helpful.

433- We shall commence with a number of canons 4 in 2,
that is to say, canons with two subjects, each subject being
imitated in one other voice, thus making four parts in all. The

Chap xvi.]

WITH MORE THAN ONE SUBJECT.

least difficult canon of this kind is one which resembles a Round
(Chapter XI.), inasmuch as the two leading voices complete an
entire phrase before the following voices enter. In this case the
imitations will be almost always in the octave — either above or
below, according to the voices which are selected to lead.
Canons of this description are not very common ; the following
is a very good specimen by Mendelssohn —

MENDELSSOHN. " Lerchengesang," Op. 48, No. 4.

»/ j» f u i UJ ' U- r u-i

*— •" i £j

-s-1 — ef

t^-J^-

fW Wj

5' r e/

JJjJ J3J

222

CANON.

[Chap. XVI.

It will be seen that the above is an infinite canon ; it concludes
with seven bars of free coda, which it is not needful here to quote.
As with a round for mixed voices (§ 330), it is of course necessary
that the parts should be written in double counterpoint in the
octave.

434. Our next example is taken from the organ works of
Bach—

J. S. BACH. Organ Prelude on " In dulci jubilo.'

r r r r r r

chap, xvi. j WITH MORE THAN ONE SUBJECT.

223

rrccrucr

rrir

As in the canon from Bach that we quoted in § 397, we have
re-arranged the score, to make it easier to read. The choral
printed on the middle staff is in the original given to the pedals.
We have here the commencement of a finite canon, 4 in 2, in
the octave ; it is curious that Bach has throughout the movement
written triplet quavers instead of crotchets in the counterpoint.
For the slight alteration of the melody at (a) compare § 398.

435. In the two canons last given, the two upper voices have
been imitated in the lower octave by the two lower ones. In our
next examples other methods of procedure will be shown —

MOZART.

224

CANON.

[CUp.XVI.

*j -&- -» ill W~t

1 j

^1 ta

Pb!M C^r cr1 ' > ' cj'^a'1

1 "T

In this very neat infinite canon, 4 in 2, the subject announced by
the alto is imitated in the fourth above by the treble, while the
other subject, given to the tenor, is imitated in the fifth below
by the bass.

436. Another specimen of an infinite canon, 4 in 2, of a rather
different kind will be seen in the following —

SCHUMANN. " Die Capelle," Op. 69, No. 6.

MI.- r c * j i ^ r • c i

r err i

r*it

M J J r 1

^''' r
jtt \ j^j •

^ ^

J ^K^l
u- j jhJVJ N

rj-'

A , j./

h j JJR"1*1^

S2— =J — !L-!L

i T £^

•^ — -

| J • J J — _ j

r rr^

gi_J — J

•

^ , h> jj

'^•^

Chap, xvi.j WITH MORE THAN ONE SUBJECT.

2*5

•£

=Efep

3£

M'^JIJ

g=Fg

A -., ... J 1— 1 =T7T

i r — i

p — h

fl)br cJJ N re

/b , , j M , , i*

Li; J-J^ — J Jl ^ ^ —

SEES

1^ ^ Jl

4=1

. J[ ' _

This little piece is the last of a collection of part-songs for female
voices. The two subjects of the canon are announced in the first
bar by the two treble voices, and imitated in the following bar in
the fourth below by the two altos. Owing to the close position
of the harmony, arising from the use of female voices only, it
will be seen that the parts cross very freely. The canon, it
should be observed, is not strict as to interval. There is a free
coda, which we have quoted here, as, though not strictly in canon,
it contains a good deal of free canonic imitation.

437. In § 374 we gave the commencement of the Minuet
from Mozart's Serenade in C minor for wind instruments, as an
example of an accompanied canon. The trio following this

226

CANON.

{Chap. XVI.

Minuet is a very beautiful example of a canon 4 in 2 by inversion,
which is worth quoting in its entirety —

• — — ^ MOZART. Serenade in C minor.

Oboe i mo.

Oboe ado.

mezzo, voce.

ess;

Fagotto imo.

Fagotto ado.

mezzo, vocc.

mezta voce.

=fr*=*

J r r ;-||

&3 1 rf-

itt==|=^±=

«7

Sy : J * ^' -^:^' —

^ :• 1 1 r

r ^ 1 J

Chap. XVI ]
t-*' ^ '"""

-«v

—

VlTH

s—

•^

RE THAh

'E .

k s~

BJECT. 22

§ '

rfr

-T

. s-

-»v

• &^

^
•^

^^-**.

p- -

5f*^ r

^*'

~J —

p__ | J r r

Here the second oboe is imitated in contrary motion by the first,
and the first bassoon (with a different theme) in contrary motion
by the second. In the first half of this piece, the variety of the
imitation is that shown at (D) in § 281. But in the second half,
while the oboes still retain the same imitation (answering tonic
by dominant, and dominant by tonic), the second bassoon imitates
the first after a different manner, answering tonic by tonic, and
dominant by subdominant. (Compare (C) in § 281.) The last
two bars before each double bar are free. The whole movement
is a very fine illustration of the way in which, in the hands of
such a master as Mozart, the most elaborate scientific contrivances
can be employed without producing the least stiffness of effect ;
the music flows as naturally as if it were nothing but simple four-
part counterpoint.

438. Our last example of a 4 in 2 canon is taken from Bach —

J. S. BACH. Cantata, " Ein'feste Burg."

VIOLIN.

ALTO SOLO.

TENOR SOLO

x- «

CONTINUO. jgff * ^

228

CANON.

[Chap. XVI

Pr — r

,g r*r » L^

ffl" U Uj^*41

•r rrfrft

Here we have a canon in the fourth below between the alto and
tenor, and a second canon in the fifth above between the violin
and the oboe da caccia* the first note of the violin part being free.
The difference between this and the examples previously given is
that here, in addition to the double canon, we have an inde-
pendent free bass part.

439. We now give some examples of canons in more than
four parts. Our first is a very charming little specimen of an
incidental canon, 6 in 3, from one of Mozart's little-known
Masses —

Viol. x.

MOZART. Mast in D, No. 7.

Here we see three canons, each in two parts, proceeding simul-
taneously; an interesting point to notice is that each of the

* The oboe da caccia, frequently used by Bach, was a now obsolete species of
oboe, tb i compass of which was the same as that of the modern cor anglais.

Chap, xvi.] WITH MORE THAN ONE SUBJECT.

229

canons is at a different interval. That between the treble and
alto is in the fifth below ; between the tenor and bass it is in the
seventh below; while the canon between the instrumental bass
and the first violin is in the sixth above.

440. Our next illustration is the beginning of a canon 6 in 2
in the octave from a sextett for strings by Raff —

J. RAFF. Sextett, Op. 178.

PP Schenoso.

~~* - ' \*^ ^-^

PP Schertoso.

230

CANON.

fChap. XVI

&C.

The canon, which continues strict for 27 bars, is too long to
quote entire. It will be seen that the harmony is at times some-
what free. It must be remembered that in pieces of such
elaborate and artificial construction as a canon of this kind
greater liberty will always be allowed than under ordinary cir-
cumstances. At the same time, such liberty must never degenerate
into license.

441. The following short extract from the " Rex tremendae "
of Mozart's " Requiem," is a fine example of a quadruple canon —
8 in 4—

S.A.

MOZART. " Requiem.

I ^ h v >

T.B.

1 - -

chap, xvi.i WITH MORE THAN ONE SUBJECT.

231

Here there is a canon for the treble and alto in the fourth above,
another for the tenor and bass in the fifth below ; while the two
lower string parts in thirds are imitated by the two upper parts,
also in thirds, there being altogether eight parts, all moving in
canon.

442. A different kind of canon (also 8 in 4) will be seen in
our next illustration, which is taken from Cherubim's magnificent
" Credo " for a double choir —

CHERUBINI. Credo & 8 voci.

. J

T.B.r ^

-jr.,;

S.A.

j) 3- Ir r rl • I

I _. «M

•^ ' .

T.B. J J.

r'*1
r J **
. i»-i»- U.J •*• i

r *

5HHJ-

J J I I

I • I • t

J J

r J ^Jj

I U I

We give only the beginning of the canon, which extends over 38
bars, during the whole of which the first choir is imitated note for

232

CANON,

[Chap. XVI.

note by the second at two bars' distance. The latter part of the
canon, which we have not room to quote, is even more elaborate
and intricate than the first. Every student should make the
acquaintance of the masterly work from which this extract is
taken.

443. Our last example is the most complex we shall have to
give. It is a short infinite canon, 12 in 4, for three choirs, and,
like the other examples by Mozart which we have quoted, is
remarkable for the ease with which the master moves, in spite of
his self-imposed fetters —

MOZART.

r r i — r— ' i ' i — i r r — r- —f- —\
J J J J-Qj i . . j rrnJ J. J.

J i

•t-T

J J J

i r r

i i

q— r i — r
=L ^3.^ JT .

j. j.

J I

J J"2 J i

r r r

Chap, xvi.j

WITH MORE THAN ONE SUBJECT. 233

r L-f1 j j

J J J J

sMfr ,» f*

j- r j. .

1 r r r r ' f '

J J J J J"3j j

/ #«-J , c_, r

Lr ...T r , _, —
1 J J J - 1

* r fr/r • - 4'J
_j ^y j. j. j j

r r r r
^r f r ^ *

^ f r r
r#- -1 - — ^-

i- ; j j '» p=i

p^ r ^_

J. J.

a ^j^~,j J^j j

if— i — 1

s t

.UJJ3J i , ,J

r r '

*^ i^.~ j jjj

I I

444. It will be readily seen that the composition of such
canons as we have been treating in this chapter is a task of no
ordinary difficulty, the more so because of the impossibility of
laying down any definite rules as to their construction. More
can be learned by the examination of good models than in any
other way; and it is for that reason that such numerous and
lengthy examples have been given. But to succeed in this
department of work, not only considerable natural aptitude is

234 CANON. ichap. xvi

requisite; but a very large amount of practice. This will be
found most beneficial, not only (nor perhaps even chiefly) for its
own sake, but for the freedom of imitative contrapuntal writing
which it will give. It is quite possible that a student may never
want to write a 4 in 2 canon as long as he lives ; but the ability to
do so, if necessary, would be invaluable to him in such a depart-
ment of composition as the thematic developments of a symphony
or quartett Besides this, the more thoroughly a composer is
equipped at all points for his work, the greater his chance of
producing something which is likely to live.

445. With this chapter we complete our instructions on the
subject of canon. There are other varieties which we have not
yet touched upon ; but these are of so little practical use to the
student that it will not be worth his while to waste time in
writing them. We speak of them, giving examples of some of
the more curious varieties, in our next chapter.

XVII.]

CURIOSITIES OF CANON.

CHAPTER XVII.

CURIOSITIES OF CANON.

446. The old theorists exercised an enormous amount of
ingenuity in the invention of canonic devices, many of which
were of not the slightest practical use. We have already given
one specimen of this kind in the canon by augmentation and
contrary movement quoted from Kirnberger in § 311; and we
shall now briefly describe some of the chief varieties to be met
with, though, because of their mostly unpractical character, we
shall not give directions as to the method of writing them.
Those who are curious in such matters will find instructions on
the subject in the works of Marpurg and Lobe.

447. A favourite device of these old writers was to compose a
canon with double, and even triple augmentation, such as the
following —

CANON.

[Chap. XVII

r r I r i rj

3^=^

Here the second treble shows the augmentation, the alto the
double, and the bass the triple augmentation of the subject given
to the upper part. This is merely a canon to the eye, and not
to the ear ; it is of no musical value, and utterly useless except as
an exercise of ingenuity.

448. Our next example is even more elaborate —

J. P. KlRNBERGBK.

Chap. XVII.]

CURIOSITIES OF CANON.

Here again the subject is given in notes of four different lengths ;
but the alto and the first treble are in contrary movement to the
other parts.

449. A species of canon frequently to be met with is a
RETROGRADE CANON (Canon Cancrizans). In this, as explained
in § 287, the notes of the subject are given in reversed order —
that is, the answer of the canon is the subject read from right to
left, instead of in the usual manner from left to right. Many
very ingenious specimens of this kind of canon exist. The fol-
lowing is from Bach's " Musikalisches Opfer " —

I S. BACH. " Musikalisches Opfer.

flr1* J J 1

<«>

J I J J -1 J73 1 J

CANON.

iCnap. XVII.

~tf — ¥

J ?D

v^-

— J J-

' * JL _ihj.

[J.bJ tiJ •* — * •* tH ^^

:aW\ 1 1 ' t~~P

— 1 1 1 — I

WJ> J flrJ =J^zz y L

- J 1 J =

If this piece be examined, it will be seen that the lower part
read backwards from the last bar to the first is the same as the
upper line read in the ordinary way. In a canon of this descrip-
tion it is usual for the two voices to commence together, as in
the present example. The upper part from (a) to the end is the
same as the lower part read backwards from this point to the
beginning, and vice versa.

450. Our next example, quoted from Hawkins' "History of
Music," is much more complex —

W. BYRD.

lit

Chap. XVI I.]

CURIOSITIES OF CANOIT

239

J I \ ^ J - \ €» fg fg

^ -J| J "| p^R

ir r r

«gJ <g

tfj Ig P
^ P r i

F=t=*±

CANON.

[Chap. XVII.

Ctau. XVII.]

CURIOSITIES OF CANON.

i«j r i

=F^=F

I. •==!

rj ^ I

242

CANON.

[Chap. XVII

si ^ I

J J I ^

Chap. XVIL]

CURIOSITIES OF CANON.

243

We have here a most elaborate retrograde canon, 8 in 4. The
second treble part is the first treble read backwards, and the
second alto, tenor, and bass parts are also their respective firsts by
retrograde movement. Though rather long, it has been needful
to give the whole piece here to render it intelligible. Let the
student notice that in addition to the retrograde imitation we
have spoken of there is also almost continuous close direct
imitation between the voices. Truly the old masters of the
Elizabethan age possessed rare skill in contrapuntal writing !

451. Another even more intricate kind of canon is the
REVERSE RETROGRADE CANON. This is a canon so constructed
that when the book is reversed (that is to say, when it is turned
upside down), the music shall read the same as in the usual
position. This is, of course, a mere curiosity ; but a few examples
will be worth giving. The first is by Lobe —

J. C. LOBE.

^i r r 1 r * ^ 1 * ^ \ r ri rr J -J

1 ^ — i r r i

J ^3 1 j . j=

^-^ ^ to

To indicate a canon of this kind, the inverted signature is placed
at the end, as here.

452. Our next example is by a living German musician, and
was published in the Musikalisches Wochenblalt —

. . OSCAR BOLCK.

244

CANON.

[Chap. XVI I

453. We now give a specimen by a living English composer
of a reverse retrograde canon in four parts —

F. CORDER.

4— — i «! — j— 1 — 1 1 i 1 — i 1 — j— I

jL . (It — 1 i

fc gj w •* 1 ^J J J 1 gJ * J 1 g-
1 — i 1 i 1— — i 1 — 1 —

^•^—rs • J J .
c7

J' J. ^ J' J jj'j- J. ' Jr

| J J J

>x_

i r rr r ^

p=l

rrr

f° T*" "T" /*

ir_r - >-4

IT r -i

_| j j ff — ^«L

JJ Ij JJIJ

fan

I'M

Chap, xvn.j CURIOSITIES OF CANON. 245

/ f. 1 Jl 1 1 " J

^LJ ,, J|, , sL

^-3^

y J j 1 j j j
...r t> ,f f r.

1 j. j

Ir r r

This is a different kind of canon from those given above. Here
there is no canon by direct motion ; but it will be seen that when
the book is turned upside down the whole composition is exactly
the same as before. The accidentals are here printed under or
over the notes, instead of before them in the usual way, as they
are only wanted in one of the two positions.

454. Another highly ingenious, but, owing to its great
difficulty, very rarely-used device is that known as canonic
imitation by Inverse Contrary movement. This is a canon for a
double choir, in which a theme is announced by one choir and
answered by the other in the following manner — the movement
is inverse — that is to say, the voices of the one choir are imitated
by the other in reversed order, the treble of the first choir by
the bass of the second, the alto by the tenor, the tenor by the
alto, and the bass by the treble. Besides this, all the subjects
are imitated by contrary movement. The chief rule to be
observed in writing imitation of this kind is that none of the
lower voices must ever sound the fourth below the treble except
as a passing note. In Bach's " Art of Fugue " will be found a
whole fugue (No. 12) which is inverted in this way; but probably
the finest example of imitation of this kind for two choirs is in
Cherubini's " Credo " for a double choir, from which we quoted
a passage in our last chapter (§ 442). This great work contains a
canon of this kind 77 bars in length, which begins thus —

CHBRUBINI. Credo & 8 vocL

246

CANON.

[Chap. XVll.

\4 • I

I I

r rrii i i it=»p jg^

fry

5=p=

It will be seen that the first choir commences the imitation on the
last note of each phrase sung by the second choir, and that the
imitation is carried out in the manner just described. Towards
the end of the canon the imitation becomes closer and more
elaborate. We give the last nineteen bars —

IT rirrL

Chap. XVII.]

CURIOSITIES OF CANON.

247

• * *V J •

psh 1

r r r

, L-J

1 1

r r r

J J

A J-

&c.

-1 — l-f

Here we not only have the inverse contrary imitation carried on
as before, but also direct imitation between the different voices of
the same choir. Besides this, the second choir is not now silent
when the first enters. The whole passage is a masterpiece of
scientific contrivance.

248

CANON.

[Chap. XVII.

455. A CIRCULAR CANON is one which modulates so that
each repetition is in a different key.* The most common variety
is that in which each repetition is a tone higher than before;
hence the old name for this species of canon, " Canon per tonos"
Obviously after six repetitions, each a tone higher, we shall return
to the original key. If it is desired to pass through all twelve
keys, each repetition must be either a semitone, or else a fourth
or fifth, higher than the preceding. The following is a good
example of a circular canon —

J. S. BACH. " Musikalisches Opfer.

* Some writers use the word ' ' circular " as equivalent to ' ' infinite " ; but it is
more usual, and also preferable, to employ it in the sense we are now explaining.

Chap. XVII.)

CURIOSITIES OF CANON.

249

Here the upper part is a variation of the theme seen in our
example to § 449. Below this theme are two parts in canon in
the fifth. The music, beginning in C minor, modulates to D
minor. The signs §, indicating an infinite canon, show where
the repetition commences ; but this repetition will now be a tone
higher in all the parts. Evidently the two following repetitions
will begin in E minor and F sharp minor ; and so on to the end.

456. A POLYMORPHOUS CANON is one in which the same
subject is capable of being worked in many different ways. The
old theorists devoted much time and labour to the invention of
such things. Marpurg gives the subject of a canon by Valentini
which the composer worked in 2,000 different ways ! But the
most celebrated and best-known example of this kind of canon
is one by Stolzel, written by him to disprove the assertion of an
opponent that the possibilities of canon were exhausted. We
give an abstract of Marpurg's analysis of this canon, which will
show its chief features.

457. We first give the canon (which is an infinite canon 4 in i,
in the fifth and octave) in its original form —

STOLZEL.

0 i'"^ 1 1 1 J 1 1 1

^<l'' - 1 - 1 • .1

i • II: ' 1 r i '^

{$=£=

i j

ppg

r r r r r

' • • ' ' i r ' '

" T r — *

250 CANON. [Chap. xvn.

Marpurg remarks of this canon that it is so constructed that we
can begin with any one of the seven bars of which the subject
consists, or at any half bar. For example, if we commence at
the fourth bar, the subject takes this form —

§5 §8 §12

This evidently gives us fourteen forms of the subject. To save
space, we give this and the following examples as close canons
(§§ 425~427)- As the notation of a close canon has been fully
explained, the student will easily be able to write them out as
open canons for himself.

458. But further, each of these seven subjects can be equally
well treated per arsin et thesin. We give the original form thus
altered as a specimen of them all —

§5 §8 §12

This clearly gives fourteen canons more, making twenty-eight.

459. The next step is to treat the subject by inverse contrary
movement, as in the example by Cherubini in § 454. The canon
then assumes the following form —

(d) §5 §8 §12

Pursuing the same method as before — that is, beginning at any
half bar, and also treating the various forms of the subject per
arsin et thesin^ we obtain twenty-eight new canons, making alto-
gether fifty-six.

460. The subject in the form first shown can also be taken
by retrograde motion, altering the time values, where necessary,
and introducing passing notes, to obtain a better melody. This
produces the following —

§5 §8 §12

which can be varied in the same way as those preceding. This
last given theme can also be inverted —

(/) §5 §8 §12

Chap, xvii.]

CURIOSITIES OF CANON.

251

461. By beginning with one of the middle voices, and varying
the distance of time of entry, and the order of entry of the
voices, many new combinations are obtained —

§4

(*)

§3

§5
§4

*r \" rl:E=^±=

§8
(*) ^ ^

§12 §5
§8 §11 §4

i r r ^ i

lia

§11

§8

§4

The total possible combinations already given amount, according
to Marpurg, to 392.

462. A new series of canons is obtained if we make the
imitation closer. It will suffice to give one as an example —

J ^^-^^gE]

W^ = & II: ,v —
•j \ — 1 p"

J A- — -A |j

-^ = ^ 1

' l|: r =^^

§ r r r r r
^. • j-

r
i J j

-* r r r

-&-

A~~

Eighty-four combinations of this kind are possible.

CANON.

[Chap. XVII.

463. Lastly, the canon can be treated as a circular canon, by
recommencing on the fourth below and fifth above alternately at
each new repetition. The following example will show this
clearly. The * shows the note on which in each of the voices
the subject begins afresh —

-& —

X— s

1 1 — L_[I p — 1
*

IBJ —
gjnq

r ' i ' r r r i

• — _ _ « — :

^t: s s-s —
*

r r =j= ' -

r r ^ -* —

chap, xvii.j CURIOSITIES OF CANON.

* > — .

&c.

Evidently we shall go through the entire " circle of fifths," and
ultimately return to the key of C.

464. It need scarcely be said that but few subjects are capable
of such infinite variety of treatment as that which has just been
shown ; but it is by no means difficult to write short and simple
subjects for canon which are capable of many different treat-
ments ; and although a polymorphous canon is of but little use
for its own sake, the practice of writing such is very valuable as a
preliminary study for the stretto of fugues, as will be explained in
the next volume of this series. To afford the student an oppor-
tunity of exercising his ingenuity in this direction, we give the
subject of a polymorphous canon from Marpurg —

This simple scale passage can be treated as a canon at any
interval, above or below, either by direct, contrary, or retrograde
motion, per arsin et thesin, and by augmentation and diminution.
Marpurg shows that more than a hundred different canons are
possible on this subject in two parts only, while by adding
thirds, sixths, or tenths to either or both of the two parts, the
number of possible combinations is increased to over a thousand.
And all this can be done with a simple scale !

465. The last kind of canon we shall describe is the RIDDLE-
CANON. This is a variety of close canon (§425), in which the
usual signs to indicate the place and interval of entry of the
different voices are omitted. The number of voices is mostly
given, though sometimes not even this is done. The old
theorists wasted an immense amount of time and ingenuity in the
invention and solution of such puzzles as these, with regard to
which Marpurg pithily remarks that one fool can ask questions

254

CANON.

[Chap. XVII

which ten wise men cannot answer. We give a few curiosities of
this kind; the first is from Martini's "Storia della Musica"—

Plutonica subiit regna.
(a) Canon ad Diapason-Diapente.

MARTINI. " Storia della Musica.

Tertia pars, si placet.

§a

Here more clue to the solution is given than in some cases.
The " Diapason-Diapente " is the Greek name for the interval of
the twelfth ; and the Latin motto, " Plutonica subiit regna " (" He
went down to the realms of Pluto"), is an obscure method of
hinting that the canon must begin by descending. Here, there-
fore, is to be a canon in the twelfth by contrary motion, and the
puzzle is to find where the imitation is to commence. The
" tertia pars, si placet " — the third part, if desired — indicates a free
bass ad libitum. The canon, it will be seen, is infinite, and the
solution * is the following —

!T E •> I J

Of course the treble does not enter at first until the fourth bar.

466. A considerably more difficult example of a riddle-canon,
taken from the same work, is the following —

* Reprinted by permission from the Musikalisches Wochenblatt.

Chap. XVII.]

CURIOSITIES OF CANON. 255

MARTINI. " Storia della Musica."

Here we see that there are two subjects, before each of which
four clefs are placed ; each is therefore to be sung by four voices,
and the canon will be 8 in 2. The direct (w) put at the end of
each subject, and referring back to the first note, shows that the
canon is to be infinite. But no clue is given as to the order,
interval, or distance of time of entry of the different voices. To
solve such a riddle-canon as this, it would be needful to try the
subjects at all possible intervals, by direct, inverse, and retrograde
imitation, even by augmentation and diminution, until success
rewarded our efforts. In the present case the true solution was
given in the Musikalisches Wochenblatt for 1880. We reprint it,
as a remarkably neat specimen —

S.A.

JjJJUJ ^^g
•* H r P l*r^« _ i»-

T.B.

r ,
SN-'

S.A.

T.B.

:^h

CANON.

(Chap. XVII.

r r • LUJ- r r r

i Is J*J E !* •*• •**• -••

r GE

*/ r l

' '

r rr I

— r- —

Each subject is here treated as a canon in the fifth above, the
second choir entering four bars after the first.

467. Two very clever riddle-canons, one for two, and the
other for four voices, are to be found in Bach's " Musikalisches
Opfer." The former, to which the composer has prefixed the
motto "Quaerendo invenietis" ("Seek, and ye shall find"), is
capable of four different solutions. Both canons are unfortunately
too long to quote here.

468. That the art of composing riddle-canons is not yet lost
will appear from the following very ingenious example by Fr.
Link, which was published some years since in the Musikalisches
Wochenblatt—

AlUgretto,

DUO FOR TWO VIOLINS.

FR. LINK.

This veritable puzzle illustrates a variety of riddle-canon which
we have not yet seen — that in which the bars are not to be read
in the usual order ; it is somewhat similar to the canon cancrizans
(§ 449), though with a difference. The difficulty is to discover

Chap. XVII.]

CURIOSITIES OF CANON.

257

the order in which the bars are to be taken. The only clue
afforded is the double bar in the third line, which seems to
suggest that this is the final bar of the piece.

469. The author's solution of the riddle is as follows : " The
above piece in two parts is a special modification of the retro-
grade canon by contrary motion (canon cancrizans in motu
contrario). The first part begins above at the first bar, and takes
all the bars in the order indicated by the figures —

Allegretto.

1 2

VIOLINO Imo.
3 4

•opg ONIIOIA

when the notes of bars 9, 10, n, 12, 13, 21, 22, and 23, must be
read backwards — from right to left. The close of this part is bar
25 in the middle. The second part turns the music upside
down, as shown by the clefs, and performs the whole piece
simultaneously with the first part, beginning with bar 25 of the
first part. The second part takes the bars in the order indicated
above them, reading the notes of bars 3, 4, 5, 13, 14, 15, 16, and
17 backwards — from right to left. This part ends with bar i of
the first part, which is its 25th bar. The succession of the bars
is arranged as a spiral, which with the first voice runs from the
outside to the centre, and with the second in the reverse
direction ; the whole in its peculiar arrangement thus forming a
so-called musical labyrinth."

158 CANON. (Chap. XVH.

470. We now give the two voices together in score —

It will be seen that we have here a reverse retrograde canon
(§ 451). It is of little musical value except as a curiosity. Per-
haps even more surprising than the ingenuity displayed in its
invention is the fact that it should have been solved. The

Chap. XVII.

CURIOSITIES OF CANON.

259

solution was sent to the Musikalisches Wochenblatt by Hen F.
Bohme, of Leipzig.

471. The above examples will sufficiently illustrate the nature
of the riddle-canon ; we shall now in conclusion give a few mis-
cellaneous curiosities, which can hardly be classified under any
of the divisions we have spoken of. Our first is a fine specimen
by Byrd—

W. BYRD.

J J

6=3

2=3=

T r

*! J

260 CANON. [Chap. xvn.

r r • r i

ff gl

Here is seen an unusual kind of 4 in 2 canon upon a plain chant
(or canto fermo). The chant, which is seen in the second treble
part, is the melody of the old hymn, " O Lux beata Trinitas," on
which Byrd wrote many canons. Between the third treble and
first alto is a canon in the fifth below per arsin et thesin ; while
between the second alto and the bass is a canon in the octave by
irregular augmentation, some of the bass notes being four times
the length of the alto, others only double the length, while two
notes of the alto (the first and third minims of the third bar) are
only of the same length in the bass. The pauses in the third

Chap. XVII.)

CURIOSITIES OF CANON.

26?

treble and second alto parts do not indicate a rest on those
notes, but merely show how far the canon is carried in the
imitating voices. The first treble part is free.

472. Our next example, a "Miserere" by Tallis, is of an
extremely complicated description —

T. TALLIS.

=3=

UrflJ'

fit

262

CANON.

[Chap. XVII

I I I

i j. j

r r r

J * I ^ J-

Chap, xvii.j CURIOSITIES OF CANON.

263

Lrrr

i ' r

r * i i

264

CANON.

(Chap. XVII

Here we have a canon 6 in 2, with one free part. Usually in a
6 in 2 canon each of the two subjects is in three of the parts.
(See the example by Raff in § 440.) Here, however, the first
subject is only given to the two trebles, which have a canon in
the unison at two minims' distance. The other subject, which is
seen in the first tenor, appears in direct motion, and in double
augmentation, in the second tenor; the first bass (which in the

original is written in the very rare C clef on the fifth line, !nl )

is the free part ; the second bass gives the subject of the first
tenor in contrary motion, and by triple augmentation, while the
third bass takes the same subject in simple augmentation, and
also by contrary motion. The amount of labour required for
writing such a canon as this can hardly be conceived. The part-
writing is necessarily somewhat free. The whole piece deserves
careful examination and analysis.

473. Our next curiosity is by Bach. It is an infinite canon,
7 in i in the unison, on a ground bass (basso ostinato\ that is,
one figure continually repeated. The subject is given by Bach as
a close canon thus —

Chap. XVII.J

CURIOSITIES OF CANON.

-fr — IT

i '

• i^

ff=

^i»fg i

ffi fr a

^ . . r r

y_j — i

\r r u^

-^-i

The ground bass, which serves as a perpetual accompaniment, is
the following —

We give the canon in score —

(b)

J. S. BACH.

i i

feS

Ground Bass.

266

CANO*.

[Chap. XVIL

I I

| II: *=£

F3=^

lr r Q-ll: '• =

* i=

J — Hf-^l

J =j

Chap. XVII.)

CURIOSITIES OF CANON.

267

¥=*

1±=£

F=F=F

T=f*

r r r ir

r i

r r c,rl

*68 CANON. (Chap, xvn

nr r r

474. It was a favourite amusement with the old theorists to
practise writing canons for an enormous number of parts.
Marpurg gives the subject of a canon which Valentini wrote for
96 voices, arranged in 24 choirs, and he tells us that Kircher
discovered that the same canon could be performed by 512
voices, or 128 choirs. The subject itself consists of nothing but
the notes of the common chord. Evidently such a canon as this
has no claim to be considered as real music; in performance
one would hear nothing whatever but the common chord, with
the parts incessantly crossing, so that there could be no clear
effect. As a specimen of the kind of ingenuity that was ex-
pended over these curiosities, we give as our last example a
canon 36 in i, for nine choirs, by Michielli Romano, a composer
who lived at Venice about the beginning of the seventeenth
century. As it is impossible to get a score of 36 staves on our
page, we give each of the nine choirs in " short score }>—

Chap. XVII.J

CURIOSITIES OF CANON. 269

MlCH. RcHANOk

270

CANON.

[Chap. XVII.

r-— r r

Chap, xvn.] CURIOSITIES OF CANON.

271

272

CANON.

[Chap XVII.

i r

^ F"

chap, xvn.j CURIOSITIES OP CANON. 27$

Here the tenor part of the first choir imitates the bass, by con-
trary motion throughout; the alto and treble commence at the
half bar, taking the octave above the bass and tenor parts. Each
successive choir enters in the same way one bar later than the
preceding. The marks X indicate the crossing of the voices.
It will be seen that the musical effect of the whole is by no
means exhilarating ; such canons as these are not of the slightest
use when they are written. We have inserted this one simply as
a curiosity, and to show the student what these canons with a
multitude of parts were really like.

475. Here our task ends.* The student who desires to go
deeper into these curiosities can find further details in the works
of Marpurg and Lobe. Our object is simply to teach such
matters in connection with canon as are likely to be practically
serviceable. The great use of canonic writing is not so much for
its own sake as for the freedom that the study gives in fugal,
and to a considerable extent also in symphonic, composition.
Those who have thoroughly mastered the contents of this volume
will find their acquired knowledge invaluable if they proceed to
the next step in composition — the writing of fugue, which will
form the subject of our next volume

THE END

ANALYTICAL INDEX

"DOUBLE COUNTERPOINT AND CANON."

%* The numbers refer to the paragraphs, not the pages.

Accidentals, introduction of, in double
counterpoint, 124, 147, 178.

ACCOMPANIED CANON defined, 362, 374;
examples of, 374-377.

Added thirds and sixths in double counter-
point in the tenth, 80, 156-159.

ADDED FREE PARTS in double counter-
point defined, 184. Exercises worked,
192-201. Examples from old masters,
207-215 ; in canon defined, 362.
Exercises worked, 365-367. Examples
from old masters, 369-373.

All harmonic resources available in xree
double counterpoint, 102.

ALTERATION, CHROMATIC, of notes in
free double, counterpoint, 138, 139 ;
in triple counterpoint, 257 ; in canon,

393-

Alteration of subject in free double
counterpoint, 141, 146, 150.

Any number of parts possible in a canon,
407.

Appoggiatura defined, 147 (note).

Arsis et thesis, defined, 283.

ARSIN ET THESIN, imitation per, 292 ;
canon per, 354.

AUGMENTATION, imitation by, defined,
284 ; example of, 299 ; canon by,
defined, 284, 311 ; example of, 405.

Augmentation and contrary motion, canon
by, 311.

AUGMENTATION AND DIMINUTION, finite
canon by, 350 ; infinite canon by, 351.

AUGMENTED SECOND AND FOURTH IN
MELODY in strict double counterpoint,
57, 58, 86 ; in free double counter-
point, 112.

Augmented sixth, treatment of, in free
double counterpoint, 109.

AUXILIARY AND PASSING NOTES in
strict double counterpoint, at the
eighth, 28, 30, 31 ; at the tenth, 70,
71 ; at the twelfth, 92 ; in free double
counterpoint, 102, 189, 190.

•Cadence, free, defined, 29.

Cancrizans, meaning of, 449.

CANON, accompanied (see Canon IV.);
alteration of interval in, 309 ; any
number of parts possible in, 290, 407 ;
by augmentation defined, 284, 311 ;
example of, 405 ; by augmentation
and contrary motion, 311 ; by aug-
mentation and diminution, 350, 351 ;
by diminution defined, 284 ; by double
and triple augmentation, 447 ; by in-
version, 280, 334 ; by inverse contrary
movement, 454 ; ' cancrizans ' defined,
example of, 449 ; chromatic alteration
of notes in, 393 ; circular, defined,
example of, 455 ; close, defined,
425, 426 ; examples of, 426-429 ;
curiosities of canon (see Canon IX.) ;
defined, 308, 311 ; direct imitation in,
279; double (see Canon vin., a);
finite (see Canon n., a) ; free as to
intervals, 339 ; in two parts (see Canon
n. , a, ti) ; in three parts (see Canon
VIL, a, b)\ in four parts (see Canon
VIL, c, d) ; in six parts (see Canon
VIL, e)\ in seven parts, 473 ; in nine
parts (see Canon vn.,/); in thirty-six
parts, 474; in enormous number oi
parts, 474 ; infinite (see Canon n., £);
invertible, example, 166 ; nomencla-
ture of, 424 ; on a canto fermo (see
Canon v.) ; on a choral (see Canon
vr.) ; on a ground bass, 473 ; on one

274

ANALYTICAL INDEX.

275

subject in more than two parts (see
Canon VH.) ; open, defined, 425 ; per
arsin et thesin, 305, 354 ; polymor-
phous (see Canon IX.,/); quadruple
(see Canon vni. , c) ; rests, introduc-
tion of — why, 417 ; retrograde (see
Canon ix., g) ; riddle (see Canon ix.,
i) ; round (see Canon I.) ; strict as to
intervals, 339, 340 ; symmetry, and
form in, 346 ; triple (see Canon vni., b)\
varieties of, 311 ; with free parts (see
Canon ill.) ; with reversed accents
(per arsin et thesin), 283, 354 ; with
two subjects (see Canon vni., a); with
three subjects (see Canon vni., b) ;
with four subjects (see Canon vni ( *)
/. THE ROUND , crossing oi parts
in, 314 ; defined, 312 ; employment of,
in opera, etc. , 331. Exercise worked,
in three parts, 318-321 ; in four parts,
322 ; melodic interest in, importance
of, 317 ; methods of composing, 315,
316. Examples by the great masters,

(a) in three parts, by Cherubini, with
orchestral accompaniment, 330, by
Dr. Hayes, 323 ; by Beethoven, 326 :

(b) in four parts, by Mozart, 324 ; by
Beethoven, 327 ; (c) in six parts, by
Mozart, 325 ; by Beethoven, 328.

//. CANON IN TWO PARTS,
332; varieties of, 334; (a) Finite canon,
crossing of parts in, 337 ; defined, 310,
332. Exercises worked, in the octave,
336 ; in the fourth below, strict as to
intervals, 340 ; how to write, 335, 336 ;
imitation, strict and free as to interval,
339 ; interval of reply, 337 ; interval of
time of reply, 338. Examples bv the
great masters, by Dussek, at the
seventh below, in double counterpoint,
358 ; by Bach, at the ninth below,
353 I by Schubert, at the octave, 355 ;
by W. Fr. Bach, at the octave,
by augmentation and diminution, 350 ;
by J. C. Lobe, at the octave, by
augmentation and diminution, 350 ;
by Mozart, at various distances, 357 ;
(b) Infinite canon, coda in, 342 ;
defined, 310. Exercises worked, in
the octave above, 342 ; in the tenth
above, 344 ; in the ninth above, 345 ;
in the fourth below, strict as to
intervals, 347 ; in the fifth below,
free as to intervals, 348 ; by inverse

movement, 349 ; how to write, 341,
342; making the join, 341, 342.
Examples by the great masters, by
Bach, in the octave, 352 ; by C. P. E.
Bach, by augmentation and diminu-
tion, 351 ; by Clementi, by inversion,
359; by Haydn, in the octave, 356;
by Mozart, in the unison, 354.
///. CANON WITH FREE PARTS.
Free parts, consecutive fourths, 363 ;
defined, 362; how to write, 361-364;
must be contrapuntal, 362 ; no rules
can be given for, 364 ; to be composed
simultaneously with the canon, 365.
Exercise worked, with free middle
part, 365 ; with free bass part, 366 ;
with two free parts, 367. Examples
by the great masters, by Bach, in
the third below, with free bass part,
369 ; by Bach, in the fourth be-
low, by contrary motion, with free
bass, 370 ; by Bach, in the sixth above,
with free bass, 371 ; by Mozart, at
various intervals, with free bass, 372 ;
by Mozart, in the fifth above, by
contrary motion, with free bass, 373.

IV. ACCOMPANIED CANON.
Defined, 362, 374. Examples by the

great masters, by Haydn, in the
fifth below, with orchestral accom-
paniment, 376 ; by Mendelssohn, in
the octave, with orchestral accompani-
ment, 375 ; by Mozart, in octave, 374 ;
by Schumann, in the octave, 377.

V. CANON ON A CANTO PER MO.
Defined, 379 ; difficulty of, 380 ; how to
begin, 383. Exercise worked (Major
key), in the seventh below, 384 ; in the
fifth below, 385 ; in the octave above,

386 ; (Minor key), in the fourth above,

387 ; in the octave, 388 ; by inverse
movement, 389.

VI. CANON ON A CHORAL, two
methods of writing, 391.

(a) FIRST METHOD, explained, 391.
Exercise worked, 392 ; analysis of,
363-396 ; chromatic alteration of notes,
393 ; distance of time varied, 396.
Examples by the great masters, by
Bach, canon in the octave, between
treble and tenor, 397 ; by Bach, canon
in the octave, between outside parts,
398 ; by Bach, canon in the fourth
below, 399.

276

ANALYTICAL INDEX.

(6) SECOND METHOD, explained, 400.
Examples by the great masters, by
Bach, canonic variations, in the octave,
402 ; in the fifth below, 403 ; in the
seventh above, 404 ; in the octave, by
augmentation, 405.
VII. CANON ON ONE SUBJECT
IN MORE THAN TWO PARTS,
any number of parts possible, 407 ;
how to begin, 409 ; in the octave and
fifth, 408 ; in the unison and octave,
408 ; may be at any interval, 408 ;
method of composition, 409-411.

(a) THREE-PART CANONS. Examples
by the great masters, by Byrd, in the
fourth and eighth below, strict as to
intervals, infinite canon, 412 ; by W.
Fr. Bach, in the fourth and eighth
above, free as to intervals, infinite
canon, 413 ; by Mozart, in the unison
and eighth, 414 ; by Mozart, in the
second above and sixth below, 415.

(b) THREE - PART CANONS ON A
CANTO FERMO, by F. Azopardi, 416.

(c) FOUR-PART CANONS, imitation in
the fifth and octave, 417; method
of composition, 409. Examples by the
great masters, by Albrechtsberger, in
the fifth and eighth, 417 ; introduction
of rests, 417 ; by W. Fr. Bach, in the
unison, infinite canon, 420 ; by Cle-
menti, at various intervals, 419 ; by
Mozart, in the fifth and eighth, 418.

(d) FOUR- PART CANONS, WITH FREE
PARTS, example by Mozart, 421.

(<?) Six- PART CANONS, by Kirnberger,
in the fifth and eighth, infinite canon,
422.

(/) NINE-PART CANONS, by Mar-
purg, infinite canon, 423.
(/) CLOSE CANONS, defined, 425.
Examples by Friedmann Bach, in four
parts 426 ; by Mozart, in four parts,
426 ; by Marpurg, in nine parts, 427 ;
by Marpurg, in twelve parts, 429.
VIII. CANON WITH MORE THAN
ONE SUBJECT, defined, 430;
Double, Triple, and Quadruple canon,
430 ; general directions, 432.
(a) DOUBLE CANON (four in two),
defined, 430 ; the parts to be written
in double counterpoint, 433. Ex-
amples by the great masters, by Bach,
finite canon, 434 ; by Mendelssohn,

infinite canon, 433; by Mozart, in-
finite canon, 435 ; by Mozart, canon
by inversion, 437 ; by Schumann, in-
finite canon, 436 ; crossing of parts,
436 ; coda, 436 ; by Bach, with a free
bass part, 438.

(b) TRIPLE CANON, defined, 430.
Examples by the great masters, by
Mozart (six in three), 439 ; example by
Raff (six in two), 440.

(c) QUADRUPLE CANONS. Examples
by the great masters, by Cherubin1'
(eight in four), 442; by Moz'/
(eight in four), 441 ; by Mozart
(twelve in four), 443.

IX. CURIOSITIES OF CANON.

(a) Canon by augmentation and con-
trary motion, by Kirnberger, 311 ;

(b) Canonic imitation, by inverse
contrary movement, defined, 454 ;
example by Cherubini, 454 ; (c) Cir-
cular canon defined, 455 ; example by
Bach, 455 ; (rf) Canon by contrary
motion, 448 ; example by Kirnberger,

448 ; (e} Canon by double and triple
augmentation, example by S. Sechter,
447 > (/) Polymorphous canon de-
fined, 456 ; example by Stolzel, 457-
4^3 ; (f) Retrograde canon defined,
449 ; example by Bach, in two parts,

449 ; example by W. Byrd (eight in
four), 450 ; (A) Reverse retrograde
canon defined, 451 ; example by J. C.
Lobe, 451 ; example by Oscar Bolck,
452 ; example by F. Corder, 453 ;
(z) Riddle canon defined, 465 ; ex-
amples by Bach, referred to, 467 ;
example by Fr. Link, 468 ; solution,
469, 470 ; example by Martini, 465 ;
another example by Martini, 466 ;
(f) Various other forms — Canon, four
in two, on a canto fermo, with a free
part, example by W. Byrd, 471 ;
Canon, six in two, with a free part,
example by Tallis, 472 ;* Infinite canon,
seven in one, on a ground bass, ex-
ample by Bach, 473 ; Infinite canon,
thirty-six in one, example by M.
Romano, 474.

Canonic imitation by inverse contrary
movement, 454.

CANONIC imitation defined, 308 ; ex-
ample of, 305.

CANTO FERMO, canon on (see Canon v.) ;

ANAL YTICA L IND EX.

277

double counterpoint on (see Strict

Double Counterpoint).
CHORAL, double counterpoint on (see

Free Double Counterpoint, i.) ; canon

on (see Canon vi.).
CHROMATIC alteration of notes— in canon,

393 ; in double counterpoint, 138, 139 ;

in triple counterpoint, 257.
Circular canon (see Canon IX., c).
CLOSE CANON, defined, 425, 426; ex-
amples, 426-429.
CLOSE IMITATION, defined, 289 ; example,

305-

Coda in infinite canon, 342, 436.
Compass of subjects to be inverted in

double counterpoint, 5.
COMPOSITION, use of canon in, 352 ;

use of double counterpoint in, 131 ;

use of imitation in, 291-305.
Consecutive fourths in canon, 315,

363-

CROSSING OF PARTS— in canon* ^14, 337,
366, 369, 414, 436 ; in double counter-
point, 4, 137 ; in quadruple counter-
point, 266, 271.

Diminished seventh, treatment of, in
double counterpoint, 108.

DIMINUTION, canon by, 311 ; example
°f> 35° (*) '• imitation by, 284; ex-
ample of, 298, 300.

DIRECT CANON, 311, 334; example of,
336 ; imitation, 279 ; example of, 291.

Dissonances, resolution of, in double
counterpoint, 108.

Distance between two subjects to be in-
verted, in double counterpoint, 5.

Double canons (see Canon vni., a).

DOUBLE COUNTERPOINT, accidentals in-
troduced in, 124, 147, 178 ; chromatic
alteration of notes in, 138, 139 ; cross-
ing of parts in, 4 ; defined, 2 ; (a) In the
octave, defined, 13, 14 ; and fifteenth,
difference between, 13 ; example of,
10 ; table of intervals, 14 ; (6) In the
tenth, defined, 49 ; example of, n ;
inversion of both parts in, 50 ; table
of intervals, 51 ; (c) In the twelfth,
defined, 82 ; example of, n ; inversion
of both parts in, 82 ; table of intervals,
85 ; (d) In the fifteenth, and octave —
difference between, 13 ; defined, 13,
14 ; example of, 10 ; inversion of both

parts in, 17 ; table of intervals, 14 ;

(e) In the rarer intervals (see Free

Double Counterpoint, IV.).
Double counterpoint, free (see Free

Double Counterpoint).
Double counterpoint, strict (see Strict

Double Counterpoint)
Double diminution, imitation by, 298.
Double imitation, example of, 304.

Fifth, interval of, in strict double counter-
point, 15, 1 8.

FIFTH OF THE CHORD, treatment of,
in strict double counterpoint, 28, 30,
33. 34. 37. 40. 43. 44, 46; in triple
counterpoint, 246 ; in quadruple
counterpoint, 264.

Fifth species in strict double counterpoint
(see Strict Double Counterpoint).

Finite canon (see Canon n., a).

First species in strict double counterpoint
(see Strict Double Counterpoint).

Florid, meaning of, 128.

Four-part canons (see Canon vn. , c, d).

FOURTH, interval of, in strict double
counterpoint, 15 ; in free double
counterpoint, 102 ; in canons, 315,
363 ; species in strict double counter-
point (see Strict Double Counterpoint).

Free cadence defined, 29.

Free close, 385, 417.

FREE DOUBLE COUNTERPOINT, contrast
between subject and counterpoint, 129;
defined, 102 ; examples by Bach ana-
lyzed, 105-107; exercises for working,
183 ; fifth species only need be prac-
tised, 103 ; fourth, interval of, 102 ;
harmonic resources available in, 102 ;
how to choose subjects, 127 ; import-
ance of good models, 182 ; in two
parts rare in composition, usually
found with free parts added, 184 ;
melodic progression — augmented
second and fourth, 112 ; no restric-
tion as to the length of notes, 103.
Resolution of dissonances to be con-
sidered, 108-111 ; augmented sixth,
109 ; diminished seventh, 108 ; funda-
mental discords, no, in.
Sequential repetitions, 103 ; sounding
dissonant notes together, 104.
Spurious kinds of — defined, 179; ex-
ample by Haydn, 180 ; by Mendels-
sohn, 181.

278

ANALYTICAL INDEX.

Various employment of, 131 ; various
kinds of, 113.

/. ON A CHORAL, («) /« the
octave, 113 ; basses to be figured, 114;
exercise worked, 114; implied har-
mony, 114; modulation, 114; (£)
In the tenth, 117 ; exercise worked,
117-120; implied harmony, 115;
modulation, 117; root progressions
to be considered, 115; similar
motion possible, 116; (c) In the
twelfth, 121 ; exercise worked, 121 ;
introduction of accidentals, 124 ;
modulation, 124, 125.
//. ON A FLORID SUBJECT.
"Florid" defined, 128; in the
octave, tenth, and twelve, relative
importance of, 130 ; subject and
counterpoint to be contrasted, 129 ;

(a) In the octave, EXAMPLES BY THE
GKEAT MASTERS, by Bach, contrast
in rhythm and melody, 132 ; by Bach,
inversion in a different key and mode,
133 ; by Bach, sounding dissonant
notes together, 134 ; by Bach, in the
twenty-second or triple octave, 135 ;
by Beethoven , treatment of sequence,
148 ; by Beethoven, 149; by Beethoven,
alteration of subject, 150 ; by Brahms,
alteration of subject, 154 ; freedom of
modern writing, 154 ; by Cherubini,
two contrasted melodies, 151 ; by
Handel, 136 ; by Handel, crossing
of parts, 137 ; by Handel, 138 ;
chromatic alteration of notes, 138,
139 ; by Handel, 139 ; by Haydn,
alteration of subject, 141 ; by Haydn,
double counterpoint in a sequence,
143 ; by Haydn, 144; by Mendels-
sohn, 152 ; by Mendelssohn, 153 ;
by Mozart, 145 ; by Mozart, slight
alteration of subject, 146 ; by Mozart,
introduction of accidentals, 147 ;

(b) In the tenth, by Bach, 155; by
Bach, added thirds and sixths, 156-
158 ; simultaneous double counter-
point in the octave, tenth, and
twelfih, 159; by Bach, 166-168 ; by
Handel, 161 ; by Haydn, 163 ; by
Jomelli, added thirds, 162 ; by Mozart,
164; by E. Prout, 165; (c) In the
twelfth, by Bach, example of sixths
becoming fundamental sevenths in in-
version, 169, 170; by Bach, 170; by

Bach, modulation produced in the
inversion, 171 ; by Bach, alteration
in position of semitones, 172 ; by
Beethoven, 176 ; by J. P. Kirnberger,
177 ; use of accidentals, 178 ; by
Handel, 173 ; by Handel, sometimes
more than one part may be inverted at
once, 174 ; by Mozart, 175.
///. SPURIOUS DOUBLE
COUNTERPOINT, defined, 179; ex-
ample by Haydn, 180 ; example by
Mendelssohn, 181.

IV. WITH FREE PARTS ADDED,
added parts improving weak progres-
sions, 205, 206 ; added parts some-
times themselves in double counter-
point, 203 ; auxiliary and passing
notes, 189, 190; chords added to
double counterpoint, 204 ; free parts
defined, 184 ; harmonic possibilities,
186-188 ; how to work exercises, 216 ;
no restriction as to the number of
chords in a bar, 189 ; selection of har-
mony notes, 189 ; two canti fermi, 185.

(a) In the octave — Exercise worked in
three parts, with free middle part, 192 ;
with free upper part, 193, 194 ; with
free bass, 195, 196 ; Exercise worked in
four parts, with two free middle parts,
197 ; with two free upper parts, 198 ;
with free parts in treble and tenor,
199; with two free lower parts, 200;
with free bass and treble parts, 201.
Examples by the great masters, by
Bach, 207 ; by Bach, by adding plain
chords, 208 ; by Beethoven, added
part in double counterpoint, 210 ; by
Cherubini, added part in double
counterpoint, 211 ; by Handel, 209 ;

(b) In the tenth, by Bach, 212 ; by
Bach, 213 ; (c) In the twelfth, example
by Bach, 214 ; by Mozart, 215.

V. IN THE RARER INTER VALS,
217 ; (a) In the ninth, 218 ; table of
inversions, 218. Examples by the
great masters, by Beethoven, 221;
by J. C. Lobe, 220 ; by Marpurg,
219 ; (£) In the eleventh, 222 ; table of
intervals, 222 ; by Bach, 224 ; by Bach,
225 ; by Beethoven, 226 ; by CKerubini,
223 ; (c) In the thirteenth, 227 ; table
of intervals, 227 ; by Bach, 229 ; by
Beethoven, added thirds, 231 ; by

ANALYTICAL INDEX

279

Beethoven, 232 ; by Cherubini, 228 ;
by Handel, added thirds, simultaneous
double counterpoint in octave and
thirteenth, 230 ; (d) In the fourteenth,
233 ; table of intervals, 233 ; by Bach,
235 ; by Bach, double counterpoint
in ninth and fourteenth, 236 ; by
Beethoven, double counterpoint in
thirteenth and fourteenth, 237 ; by
Beethoven, double counterpoint in
twelfth and fourteenth, 238.

Free imitation (see Imitation).

Free parts, added to Canons (see Canon
in.) ; added to double counterpoint
(see Free Double Cour.terpoint iv.) ;
added to imitation, 290 ; defined,
184.

Fundamental discords, treatment of, in
double counterpoint, no, in.

GROUND BASS, canon on, example by
Bach, 473.

IMITATION, defined, 275, 276 ; directions
for work, 306 ; may be in any number
of parts, 290 ; sometimes accompanied
by free parts, 290 ; use of, in composi-
tion, 307 ; Varieties of— by augmenta-
tion, 284 ; by diminution, 284.
By inversion, 280 ; more frequently
met with in minor than in major key,
282 ; schemes for, in major key, 280,
281 ; schemes for, in minor key, 282.
Canonic, 286, 308 ; canonic, by inverse
contrary movement, 454 ; close, 289 ;
direct, 279 ; free, 277, 278 ; inter-
rupted, 286 ; invertible, 285 ; partial,
288 ; with reversed accents (per arsin
et thesin), 283 ; retrogade, 287 ;
rhythmic, 276 ; strict, 277, 278 ; with
free parts, 290 ; with reversed accents
(per arsin et thesin), 283. Examples
by the great master S, by Bach, direct
in the unison and octave, 291 ; by
Bach, direct, inverted and by diminu-
tion, 298 ; by Bach, direct and di-
minished, inverted and diminished,
300; by Bach, double imitation
(per arsin et thesin) by inversion,
304 ; by Beethoven, in the octave,
292 ; by Beethoven, at various dis-
tances, 296 ; by Beethoven, in the fifth
and sixth below, 297 ; by Cherubini,
at various intervals, 294 ; by Cheru-

bini, partial imitation, 294 ; by Han-
del, in the fourth below.with a free bass
part, 293; by Handel, close imitation
by inversion, 301 ; by Handel, close
imitation (per arsin et thesin), partial
imitation, 303 ; by Haydn, in the
second above, on a pedal, 295 ; by
Mendelssohn, by inversion in a major
key, free as to intervals, 302 ; by
Mozart, close imitation in the octave
and fifth (per arsin et thesin), canonic
imitation, 305 ; by Schumann, by aug-
mentation, 299.

Infinite Canon (see Canon ii., b).

INTERRUPTED imitation defined, 286;
example of, 286.

INTERVAL OF INVERSION in double
counterpoint, 3 ; how to find, 4, 6, 7 ;
of two subjects, how to find, 8, 9.

INTERVAL of time of reply in canon, 312 ;
in imitation. 289.

Intervals mostly used for inversion in
double counterpoint are the octave or
fifteenth, tenth and twelfth, 7.

Introduction of rests in canon — why, 417.

Inverse contrary movement, canon by
(see Canon IX., b).

INVERSION, canon by, 280, 334 ; example,
349 ; imitation by, 280-282 ; examples,
280-282, 298, 300. 302, 304 ; in double
counterpoint at small intervals not
used — why, 4, 5 ; at the octave, tenth,
and twelfth mostly used, 7 ; at the
octave, example, 4, 5, 10 ; at the
octave and fifteenth, difference be-
tween, 13 ; at the tenth, example, n ;
at the twelfth, example, n ; may be
at any interval, 3 ; meaning of. 3 ; of
two subjects in double counterpoint,
distance between, 5.

INVERSIONS, TABLE OF, in double coun-
terpoint at the octave, 14 ; at the
ninth, 218 ; at the tenth, 51 ; at the
eleventh, 222 ; at the twelfth, 85 ;
at the thirteenth, 227 ; at the four-
teenth, 233.

Limit in the distance of subjects to be
inverted, 5.

MELODIC progression in double counter,
point — in the tenth, 57, 58 ; in the
twelfth, 86 ; in the free style, 112.

280

ANALYTICAL INDEX.

Nine-part canons (see Canon vii., f).
Nomenclature of canons, 424.

Oboe da caccia, 438.

Octave and unison, use of, in strict double

counterpoint, 16.
Open canon defined, 425.
Overlapping of parts in strict double

counterpoint, 18, 31.

PARTIAL IMITATION, defined, 288; ex-
amples of, 294, 303.

PER ARSIN ET THESIN, defined, 283 ;
canon, 305, 354 ; imitation, 283, 303-

305.

.'OLYMORPHOUS canon, defined, 456 ;
example of, 457-463.

Quadruple canons (see Canon vm.,t).

QUADRUPLE COUNTERPOINT, 263 ; cross-
ing of parts in. 266, 271 ; defined, 263 ;
fifth of the chord, treatment of, 264 ;
its possible combinations, 263 ; much
rarer than triple counterpoint, 265 ;
Examples by the great masters, by Bach,
266, 267 ; crossing of parts, 266 ; by
Bach, 268; by Cherubini, 265; by
Cherubini, 269 ; by Haydn, 270 ;
crossing of parts, slight alteration of
them, 271.

Rests introduced into canon — why, 417.

RETROGRADE canon, defined, 287, 449 ;
examples of, 449, 450.

RETROGRADE imitation, defined, 287 ;
example of, 287.

REVERSE retrograde canon, defined, 451 ;
examples of, 451-453.

REVERSED accents (per arsin et thesin),
canon with, 354 ; imitation with, de-
fined, 283 ; examples, 283, 305.

RHYTHMIC imitation, defined, 276; ex-
ample, 276.

RIDDLE canon, defined, 465 ; examples,
465-470.

Round (see Canon i. ).

Round and other canons, difference be-
tween, 333.

SCHEME for inversion, in major key, 280,

281 ; in minor key, 282.
SECOND INVERSIONS, treatment of, in

triple counterpoint, 246-251 ; in

Quadruple counterpoint, 264.

Second species in strict double counter-
point (see Strict Double Counter-
point).
Sequence, employment of, in double

counterpoint, 143.

SIMULTANEOUS double counterpoint, in
the octave, tenth, and twelfth, 159 ;
in the octave and thirteenth, 230 ;
in the twelfth and fourteenth, 237.
SIX-FOUR CHORD, treatment of, in triple
counterpoint, 246-251 ; in quadruple
counterpoint, 264.
Six-part canons (see Canon vn., e).
SPURIOUS DOUBLE COUNTERPOINT, de-
fined, 179 ; example by Haydn, 180 ;
example by Mendelssohn, 181.
STRICT DOUBLE COUNTERPOINT:—
IN THE OCTAVE AND FIF-
TEENTH^ bass to be figured, 21 ;
difference between, 13 ; how to write
the exercises, 20, 21 ; implied har-
mony, 19 ; interval of fourth with
bass, 15 ; interval of fifth with bass,
18 ; inversion of both parts in double
counterpoint in the fifteenth, 17 ; keep
within the octave, or fifteenth, 5, 18 ;
octave and unison in, 16 ; overlapping
of parts in, 18 ; table of intervals in,
14 ; two-part writing in, 15, 18 ; voices
to write for, 21 ; First species, 18 ;
cadence in, 18 ; exercise worked — in
major key in the octave, 21 ; in major
key in the fifteenth, 22 ; in minor key
in the octave, 24-26 ; in minor key in
the fifteenth, 27 ; harmonic progres-
sion, 19 ; interval of fifth forbidden —
why, 18 ; mediant chord, in major
key, 19, 22, 23 ; in minor key, 24, 26 ;
no repetition of a note allowed, 21 ;
Second species, 28 ; auxiliary notes, use
of, 31 ; cadence free, 29, 30 ; disson-
ances, use of, 28 ; exercise worked,
in major key in the octave, 30, 31 ;
in minor key in the octave, 32 ; in
minor key in the fifteenth, 33 ; fifth as
a harmony note, how used, 33 ; in-
terval of fourth, how used, 28, 30;
of fifth, how used, 28, 30 ; mediant
chord in minor key, 32 ; passing notes,
accented and unaccented, how used,
28, 30 ; two chords in a bar, possible,
33 ; Third species, 34 ; cadence, four
notes to one, 35 ; cadence, three notes
to one, 35 ; cadence, six notes to one,

ANALYTICAL INDEX.

28T

39 ; exercise worked — in major key,
in the octave, four notes to one, 36,
37 ; in major key, in the octave, three
notes to one, 39; in minor key, in
the octave, four notes to one, 38 ; in
minor key, in the octave, six notes to
one, 39 ; fifth, treatment of, 34, 37 ;
octave, treatment of, 35 ; second to
a unison, treatment of, 35 ; six-four
chord, implied, 36 ; Fourth species, 40 ;
cadence, 40 ; exercise worked— in
major key, in the octave, 41 ; in minor
key, in the octave, 42 ; fifth of the
chord, prohibition of, 40; suspen-
sions available, 40 ; Fijlli species, 43 ;
cadence, 43 ; exercise worked — in
major key at the octave, 44 ; in major
key at the fifteenth, 45 ; in minor key
at the octave, 46 ; fifth of the chord,
treatment of, 43, 44, 46 ; in actual
composition, example by Handel, 47;
example by Mozart, 47.
IN THE TENTH, 49 ; character of
melody entirely changed when in-
verted, 53 ; choice of subject, 60, 66 ;
consecutive intervals forbidden — why,

54 ; contrary and oblique motion only
allowed, 55; emplcyment of both j
counterpoints simultaneously against
the subject, 80 ; false relation of the
tritone, 67 ; harmonic progression —
available intervals, 56 ; implied root
progressions may be disregarded, 59 ;
inversion, effect of, 52, 53 ; inversion
of both parts in, 49, 50; keep within a
tenth, 51 ; melodic progression in, 57 ;
similar motion not available — why,

55 ; table of intervals in, 51 ; table
of intervals to be avoided, 58; First
species, 60 ; cadence may if necessary
be free, 65 ; example of bad subject
worked, 60-65 ! exercise worked— in
major key, 67 ; in minor key, 68 ;
Second species, 69 ; accented passing
note, employment of, 71 ; cadence,
69-71; contrary motion between
accented notes, 69 ; exercise worked —
in major key, 70 ; in minor key, 71 ;
oblique motion, 69; passing note
quitted by leap of a third, 70 ; similar
motion, when possible, 69 ; Third
species, 72 ; cadence, 72 ; commencing
with implied first inversion, 72 ; exer-
cise worked — in major key, 72 ; in

minor key, 73 ; implied root progres-
sions may be disregarded, 73 ; Fourth
species, 74 ; cadence, 75, 76 ; exercise
worked — in major key, 75 ; in minor
key, 76 ; Fifth species, 77 ; cadence, 77,
78 ; exercise worked — in major key,
77 ; in minor key, 78.
IN THE TWELFTH, cadence al-
ways free, 88 ; harmonic progression
— unavailable intervals, 86 ; inversion,
effect of, 84 ; inversion of both parts
in, 82 ; keep within a twelfth, 87 ;
melodic progression in, 86 ; table of
intervals, 85 ; table of intervals to be
avoided, 86 ; use of the sixth, 85 ;
First species, 90 ; exercise worked — in
major key, 91 ; in minor key, 91 ,
Second species, 92; auxiliary and
passing notes, use of, 92 ; example
worked — in major key, 92 ; in minor
key, 93 ; Third species, 94 ; exercise
worked— in major key, 94 ; in minor
key, 95 ; two chords in a bar, 94 ;
Fourth species, 96 ; breaking the synco-
pation, 97 ; exercise worked — in major
key, 96 ; in mi nor key, 97 ; Fifth species,
99 ; exercise worked — in major key,
99 ; in minor key, 100.

6TRICT IMITATION, defined, 277, 278 ;
example of, 278.

SUBJECT, variation of, in free double
counterpoint, 141, 146 ; in triple
counterpoint, 255 ; in quadruple
counterpoint, 271.

TABLE OF INTERVALS in double coun-
terpoint, in the octave, 14; in the
ninth, 218 ; in the tenth, 51 ; in the
eleventh, 222 ; in the twelfth, 85 ; in
the thirteenth, 227 ; in the fourteenth,
233 ; to be avoided in strict double
counterpoint, in the tenth, 58 ; in the
twelfth, 86.

Third species in strict double counter-
point (see Strict Double Counterpoint).

Three-part canons (see Canon vii., a, 6).

TIME OF REPLY, interval of, in canon,
312 ; in imitation, 289.

Triple canon (see Canon vin., b).

TRIPLE COUNTERPOINT, 240; chromatic
alteration of notes in, 257 ; consecutive
chords of the sixth not available —
why, 251 ; contrasting the three sub-
jects, 255 ; defined, 240 ; each part to

282

ANALYTICAL INDEX.

appear once in the bass, 253 ; fifth of
the chord, treatment of, 246 ; its p- s-
sible combinations, 241 ; second in-
versions, rules for treatment of, 247-
250 ; selection of chords, 252 ; sim-
plest form of, by adding thirds to double
counterpoint in the octave, 243 ; slight
modifications allowable, 255 ; Examples
by the great masters, by Bach, con-
trasted character of the three sub-
jects, 255 ; by Bach, treatment of the
fifth of the chord, 256 ; by Bach, chro-
matic alteration of notes, 257 ; by
Beethoven, 261 ; by Cherubini, made

by the addition of thirds to double
counterpoint in the octave, 245 ; by
Cheiubini, 254; by Cherubini, 260;
by Handel, 258 ; by Haydn, 262 ; by
Mozart, contrasted character of the
three subjects, 259.

Twelve-part canon (see Canon VII., g).

TWO-PART CANONS, finite (see Canon
II., a) ; infinite (see Canon II., b).

Varieties of canon, 311, 334.
Varieties of imitation (see Imitation).
VOCAL music, double counterpoiar an,
137- *39. 161.

MUSICAL ILLUSTRATIONS.

ALBRECHTSBERGER.G. .Four-part Canon,

417, 428.
AZOPARDI, F., Three-part Canon on a

"Canto Fermo," 416.

BACH, C. P. E. , Infinite Canon, by Aug-
mentation and Diminution, 351.

BACH, J. S., "Art of Fugue," 155-158,
166, 172, 212, 213, 257, 300, 352 ;
Canonic Variations on " Vom Himmel
hoch da komm' ich her," 402-405 ;
Cantata, " Ein' feste Burg, "438 ; Can-
tata, "Ich, elender Mensch," 225;
Concerto in G, 291 ; Infinite Canon
(seven in one) in the unison, on a
ground bass, 473; Inventio L, 105;
Inventio II , 107 ; Musikalisches Opfer,
449, 455 ; Organ Fugue in C minor,
*33i 207 ; Organ Fugue in E minor,
I3S> 208 I Organ Prelude, ' ' Ach Gott
und Herr," 298 ; Organ Prelude,
" Erschienen ist der herrliche Tag,"
398; Organ Prelude, " Gottes Sohn
ist kommen," 397 ; Organ Prelude,
" In dulci jubilo," 434 ; Organ Pre-
lude, " Liebster Jesu, wir sind hier,"
399 ; Thirty Variations, 353, 369-371 ;
" Wohltemp-rirtes Clavier": Fugue 2,
169; Fugue 4, 229, 235 ; Fugue 10,
134 ; Fugue 12, 266, 267 ; Fugue 28,
170 ; Fugue 30. 132 ; Fugue 33, 268 ;
Fugue 37, 256 ; Fugue 40, 10, n, 159 ;
Fugue 41, 236 ; Fugue 46, 304 ; Fugue
47, 171, 214 ; Prelude 7, 224 ; Prelude

19. 255-

BACH, W. Fr., Canon, by Augmentation
and Diminution, 350 ; three-part In-
finite Canon, 413 ; four-part Infinite
Canon, 420, 426.

BEETHOVEN, Mass in D, 221, 226, 231,
237, 261 ; Quartett, Op. 18, No. i,
292 ; Quartett, Op. 18, No. 3, 296 ;
Quartett, Op. 18, No. 4, 149, 210;

Round for three voices, 326; Round
for four voices, 327 ; Round for six
voices, 328 ; Sonata, Op. 10, No. 3,
297 ; Sonata, Op. 26, 150 ; Sonata,
Op. no, 176, 232, 238 ; Symphony in
D, No. 2, 148; Symphony, Pastoral,
•275, 276.

BOLCK, O., Reverse Retrograde Canon,
452.

BRAHMS, " Deutsches Requiem," 154.

BYRD, W., Retrograde Canon (eight in
four), 450 ; Canon (four in two) on a
Canto Fermo, 471 ; " Non Nobis,
Domine," 410, 412.

CHERUBINI, Credo a eight Voci, 442,
454; " Faniska," 330; "Medea,"
294 ; Quartett in C, 151, 211 ; Trea-
tise on Counterpoint and Fugue, 223,
228, 245, 254, 260, 265, 269, 286.

CLEMENTI, "Gradus ad Parnassum,"

359. 419-
CORDER, F., Reverse Retrograde Canon,

453-

DUSSEK, "L'Invocation," 358.
\

HANDEL, Anthem, "Have mercy upon
me, O God," 161 ; Anthem, " My
song shall be alway," 173 ; Anthem,
"O come let us sing," 139; "Chandos
Te Deum," 230; "Hercules," 258;
" Jephtha," 174; " Judas Maccabaeus,"
47, 301 ; " L'Allegro," 136 ; " Mes-
siah," 303; "Solomon," 137, 293;
"Susanna," 138, 209.

HAYDN, " Creation," 180 ; Mass, No. 3,
376; Mass, No. 12, 163; Quartett,
Op. 20, No. 2, 270 ; Quartett, Op. 20,
No. 6, 262; Quartett, Op. 55, No.
i, 142 ; Quartett, Op. 74, No. 2,
141 ; Quartett, Op. 76, No. 2, 356 ;

283

284

INDEX TO MUSICAL ILLUSTRATIONS.

Symphony in G minor, 143 ; Sym-
phony in D, 144 ; Symphony in G,
No. 51, 295.

HAYES, Dr., Round for three voices, 323.

HILTON, JOHN, Round for three voices,
3*4-

JOMELLI, Mass in D, 162.

KIRNBERGER, J. P., Double Counter-
point in the twelfth, 177 ; Four-part
Canon by Double and Triple Augmen-
tation, with Contrary Movement in
two parts, 448 ; Infinite Canon by
Augmentation and Contrary Motion,
311 ; Infinite Canon for six voices, 422.

LINK, Fr., Riddle Canon, 468-470.

LOBE, J. C., Canon by Diminution, 350 ;
Double Counterpoint in the ninth,
220 ; Reverse Retrograde Canon, 451.

MARPURG, Double Counterpoint in the
ninth, 219 ; Double Counterpoint in
the fourteenth, 234 ; Infinite Canon for
nine voices, 423, 427 ; Infinite Canon
for twelve voices, 429.

MARTINI, Riddle Canons, 465, 466.

MENDELSSOHN, "Lerchengesang," Op.
48, No. 4, 433 ; 95th Psalm, 375 ;
Overture, " Melusina," 302; " St

Paul," 152 ; Symphony, No. 3. 153;
Symphony, No. 4, i8t.
MOZART, Canons, 354, 414, 415, 435,
443 ; Canonic Adag.'o for two Corni
di bassetto and Fagotto, 373 ; Fugue
for Orchestra (a Fragment), 145 ;
Mass in C minor, 47 (&) ; Mass in D,
No. 7, 439 ; Mass No. 10, 418, 426 ;
Mass No. 12, 259, 421 ; Requiem, 175,
215, 372, 441 ; Round for four voices,
324 ; Round for six voices, 325 ; Sere-
nade in C minor, 374, 437 ; Sonata in
D, 164, 357 ; Symphony in G minor,
146 ; Symphony, " Jupiter," 305 ;
Variations on " Unser dummer Pobiel
meint," 147.

PROUT, Symphony in D, No. 4, 165

RAFF, J. , Sextett, Op. 178, 440.
ROMANO, Canon for 36 voices, 474.

SCHUBERT, Trio in E flat, Op. too, 355.
SCHUMANN, " Albumblatter," Op. 124,

No. 20, 377 ; " Die Capelle," Op. 69,

No. 6, 436 ; " Faust," 299.
SECHTER, Canon with Double and Triple

Augmentation, 447.
STOLZEL, Polymorphous Canon, 457-463.

TALLIS, T., <: Miserere," 472.

T/5/1923

MUSICAL TEXT BOOKS

AUGENER'S EDITION

ENGLISH PRINTING & PAPER

9171
10123
10124
10125

9205
9206

9210

10091

10092
10093

9215

9199

10097

9179

10112

ALEXANDER, J. "Con Amore." Poetical Intro-
duction to Musical Instruction .........

ANTCLIFFE, H. The Successful Music Teacher.

Second Impression.

How to Pass Music Examinations. The Successful

Candidate, Words of Advice ..... Paper

The Amateur Singer. Words of Advice ......

BACH, J. S. Analysis of J. S. Bach's " 48 Preludes
and Fugues" (Wohltemperites Clavier). By
Dr. H. Eiemann. Fifth Impression : —
Part I. 24 Preludes and Fugues ...... Bound

Part II. 24 ,, Third Impression ...Bound

BEETHOVEN PIANOFORTE SONATAS. Letters to
a Lady, by Dr. C. Eeinecke, translated by

Net

8. d.

1 6
1 -

3 -
3 -

E. M. Trevenen Dawson ...... .. ... 3 9

CARSE, ADAM. Summary of the Elements of
Music, with Exercises and Instructions on

"How to Write Music"

Key to the above ... ...

Practical Hints on Orchestration

CLARKE, J. A. Catechism of the Eudiments of
Music. 384th Edition Paper

COCKING, F. The Composer's Vade Mecum.
(English-Italian)

1 6

CROKER, NORRIS. Handbook for Singers. Sixth
Impression Bound

DANNREUTHER, E. Wagner and the Reform of

the Opera ... ... Bound with Portrait

DAUGHTRY, 0. Ear- Tests and How to prepare for
Them. Fourth Impression...

EVETTS, EDGAR T. The Vocal Student's Practice

Eegister with Vocabulary

Modulator for use of Students of the Numeral and
Ehythmic Methods (Vocal)

GOODWIN, A. Practical Hints on the Technique
and Touch of Pianoforte Playing. With Illustra-
tions. Fifth Impression Bonud

HAMILTON. Dictionary of Musical Terms. Bound 2/3

HEALE, H. A Short Treatise on the Eudiments
of Music. 16mo ...

1 6

3 -
1 6

Printed in England

Net

10117 HULL, A. EAGLEFIELD. Organ Playing ; its Tech- s'd<

nique and Expression. 6th Impression. Bound 10 6

10118 Modern harmony : Its explanation and application.

Second Impression Bound 9 -

10120 Harmony for Students. 2nd Impression 7 6

10119 250 Questions on Musical Form in 25 tabulated

papers, with appendix * ... 1 6

10113 300 Questions on Pianoforte Teaching 1 6

10111 HUNT, E. M. Scales* Arpeggio Fingering. Classified - 9

10110 KNOWLES, C. H. G. Rhymes on the Rules of Har-
mony, founded onDr.Prout's"Harmony." Bound 2 3

9177 LA MARA. Thoughts of Great Musicians. Paper 2 -

6570 LANG LEY, G. Pianoforte Students' Chart of Tech-
nique and Theory Boards 5 -

10122 LINDO, ALGERNON H. Pianoforte Study for

Teachers and Students ...Bound 4/6 (Paper) 2 3

9178 LOBE, Catechism of Music. Translated and Edited

byC. Bache Bound 3 -

10140 McEWEN, JOHN, B. The Principles of Phrasing

and Articulation in Music 3 -

MACPHERSON, C. Harmonic Thought. Past and

Present 1 6

9214 MATTHEWS, J. A Handbook of the Organ. Fifth

Impression Bound 3 -

10114 100 Examination Questions for Organ Students - 9

10100 MOZART. Practical Elements of Thorough Bass,

with examples of Harmony and Counterpoint 2 3
9193 MUSIC AND ITS MASTERS. A Conversation by

A. Rubinstein. Third Impression. Bound 3 -

10096 MUSIC LOVER'S BIRTHDAY BOOK, THE.

(E. M. Trevenen Dawson) Bound (gilt edges) 5 -

MUSICAL CARDS for learning to read the notes

quickly at sight ... . Price per set in case 3 -

9180 NIECKS, PROF. F. A Concisp Dictionary of Musical
Terms to which is prefixed an Introduction to the
Elements of Music. Eighth Impression. Bound 4 -

9180a Introduction to the Elements of Music. Third

Impression ... ... ... ... Bound 1 6

8234a PAUER, E Harmonious Ideas. Mottoes for

Mendelssohn's " Sonj^s without words"... ... 1 -

PETERSON, FRANKLIN.

9191 Elements of Music. Twelfth Impression. Bound 1 6

9192 An Introduction to the Study of Theory. Seventh

Impression ... ... ... ... Bound 3 -

10101 A Theoretic Companion to Practice. Fourth

Impresssion ... ... ... ... Bound 2 3

10102 A Handbook of Musical Form. Sixth Impression.

Bound 1 6

10103 Catechism of Music. Sixth Impression. Bound 3 -

10104 The Student's Handbook of Musical Knowledge.

Second Impression Bound 2 3

Net

PRACTICE RECORD and Mark Register for Music *• d'

Pupils. Arranged for three terms 1 6

The same, arranged for one term ... - 9

10116 POCHHAMMER, A. Popular Handbook of Musical

Information. (H. Heale.) Bound 4 -

PROUT, PROF. EBENEZER:—

9182 Harmony : Its Theory and Practice Thirty-ninth

Impression Bound 7 8

9182c Analytical Key to the Exercises in the same.

Fifth Impression Bound 6 -

9183 Counterpoint : Strict and Free. Eleventh

Impression Bound 7 6

9183a Additional Exercises to "Counterpoint" with
Melodies and Unflgured Basses for harmonizing.
Ninth Impression Bound 3 6

9184 Double Counterpoint and Canon. Sixth Impression.

Bound 7 6

9185 Fugue. Seventh Impression Bound 7 6

9186 Fugal Analysis. Fourth Impression ...Bound 7 6

9187 Musical Form. Tenth Impression Bound 7 6

9188 Applied Forms. Eighth Impression ...Bound 7 6
The Orchestra :

9189 I. Technique of the Instruments.

Seventh Impression. Bound 7 6

9190 II. Orchestral Combination. Seventh Impression.

Bound 7 6
9181 PROUT, LOUIS B. Harmonic Analysis. Second

Edition 3 -

10106 Sidelights on Harmony 3 -

10107 Time, Khythm and Expression. Second Impression - 9

9210 REINECKE, C. The Beethoven Pianoforte
Sonatas. Letters to a Lady. Translated by
E. M. Trevenen Dawson 3 9

9198 RIEMANN, Dr. H. Harmony Simplified; or the
Theory of the Tonal Functions of Chords. Trans,
from the German. Second Impression. Bound 6 -
10115 L'Harmonie simpliflee, ou Theorie des fonctions
tonales des accords. Translated by Prof. Georges
Humbert 4 6

9201 Catechism of Musical Instruments (Guide to In-

strumentation). Third Impression. ...Bound 3 -

Catechism of Musical History : —

9202 Part I. History of Musical Instruments, and

History of Tone Systems and Notations. Third
Impression Bound 3 -

9208 Part II. History of Musical Form, with Bio-

graphical Notices. Third Impression ...Bound 3 -
9204 Catechism of Pianoforte Playing. Third Impression.

Bound 3 -

9207 Catechism of Musical .Esthetics. Second Impres-

sion Bound 3 -

9209 Catechism of Orchestration Bound 3 -

9208 Introduction to playing from Score Bound H -

RIEMAIflf, Dr. H.— (Contd.) Net

Analysis of J. S. Bach's "48 Preludes and Fugues.*1 s. d,

9205 Part I. 24 Preludes and Fugues. Fifth

Impression Bound 3 -

9206 Part II. 24 Preludes and Fugues. Third

Impression Bound 3 -

ROCKSTRO, W. S. Practical Harmony. ...Bound 3 -

Key to " Practical Harmony" ... ...Paper 2 3

Kulcs of Counterpoint ... ... ... ...Bound 3 -

A History of Music. Twenty-fifth Edition. Bound 3 -

9193 RUBINSTEIN, A. Music and its Masters. A Con-

versation. Third Impression Bound 3 -

9212 SCHROEDER, C. Handbook of Violin & Viola

Playing. Fourth Impression Bound 3 -

9211 Catechism of Violoncello Playing. Fourth Impr.

Bound 3 -

9213 Handbook of Conducting. (J. Matthews.) Fourth

Impression ...Bound 3 -

9194 SCHUMANN. Advice to Young Musicians - 9

10146 SHEDLOCK, J. S. Beethoven's Pianoforte Sonatas.
The Origin and Respective Values of Various
Headings 1 6

10148 SHINN, Dr. F. G. Elementary Ear-Training.

I. Melodic 6 -

II. Harmonic and Contrapuntal 6 -

A Method of Teaching Harmony based upon

Ear-Training :

10150 I. Diatonic Harmony 6 -

10151 II. Chromatic Harmony and Exceptional Progression 7 6

10152 Musical Memory and its Cultivation 6 -

10121 SIBLEY, C. The Voice and its Control ... ... 16

10131 SIMPSON, J. 300 Questions on the Grammar of

Music. Based on the Syllabus of the Associated
Board of the R.A.M and R.C.M 1 6

10132 Key to the above 1 6

10133 400 Questions on the Rudiments of Music 2 6

10134 Key to the above 2 6

10135 A concise textbook on the Rudiments of Music ... 2 6

9196 STEILER, J. The Great German Composers. Bio-
graphical Notices, with some account of their
Works. lUustrated Bound 5 -

SUTTON, R. Elements of the Theory of Music. Bound 3 -
10109 WARREN, J. Catechism of the Harmonium ... - 9

WEST, G. F. Hints to Young Teachers of the
Pianoforte 16

WHITTINGHAM, A. 200 Questions and Exercises
on F. Davenport's "Elements of Music" ... ... - 9

AUGENER Ltd.

18 GREAT MARLBOROUGH STREET,

CONDUIT STREET (Regent Street Corner) & 57 HIGH STREET, MARYLEBONE,
LONDON, W. 1.

MT Frout, Ebenezer

55 Double counterpoint and

P76 canon

Maitf

PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET

UNIVERSITY OF TORONTO LIBRARY

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Double counterpoint and canon"

See other formats