A History of
'Consonance'
and
'Dissonance'
by
James Tenney
I Excelsior Music Publishing Company, New York

A HISTORY OF: 'Consonance' and 'Dissonance'
published by Excelsior Music Publishing Company
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Library of Congress Cataloging-in-Publication Data
Tcnncy, James. 1934 -
A history of consonance & dissonance
Includes bibliographies.
I . Ham~ony-History. 2. Music-Theory. 3. Music-Philosophy and aesthetics.
I . Title. 11. Title: History of consonancc and dissonance.
ML444.T45 1987 781.3 86-82304
ISBN 0-935016-99-6
Manufactured in the United States of America
Contents
Introduction / 1.
Part One:
From Antiquity through the "Ars Antiqua"
Section I. The pre-polyphonic era (CDC-1) / 9
Section 11. The early-polyphonic period, ca. 900-1300
(CDC-2) / 17.
Notes: Section I and I1 / 32.
Part Two:
From the "Ars Nova" through the
''Seconda Pratica' '
Section 111. The contrapuntril and figured-bass
periods, ca. 1300-1 700 (CDC-3) / 39.
Notes: Section III / 59.
Part Three:
From Rameau to the Present
Section IV. Rrinteau and his successors (CDC-4) / 65
Section V . Helntholtz and the theory of beats
(CDC-5) / 87.
Section VI. Suntmary and Conclusions: Toward a
New Terminology / 95.
Notes: Section IV, V , and V l / 105
Appendix / 109.

INTRODUCTION
There is surely nothing in the language of discourse about music that is more
burdened with purely semantic problems than are the terms consonance and
dissonance. A comparison of some of the definitions of these words to be found
in current dictionaries, harmony textbooks, and books on musical acoustics
indicates that there is considerable confusion and disagreement as to their
meaning-if indeed there is any meaning still to be attributed to them.' Con-
sider, for example, the following:
CONSONANCE.. .agreement of sounds; pleasing combina-
tion of sounds. . .
DISSONANCE.. .an inharmonious or harsh sound or com-
bination of sounds.. .
(lhe Oxford English Dictionary, 196
CONSONANCE.. .a combination of musical tones felt as
satisfying and restful; specif: an interval included in a major
or minor triad and its inversions.. .
DISSONANCE.. .an unresloved musical note or chord;
specif: an interval not included in a major or minor triad or
its inversions.. .
(Webster 's lhird New International Dictionary, 1971)3
A combination of two or more tones of different frequen-
cies that is generally agreed to have a pleasing sound is call-
ed a consonance.
(Backus, lhe Acoustical Foundations of Music, 1969)4
Stable intervals are defined as consonant; unstable inter-
vals.. .asdissonant...the term dissonant dces not mean discor-
dant or unpleasant. On the contrary, the most interesting and
beautiful sounds in music are usually the dissonant ones.. .
w a f t , Gradus, 1976)'
CONSONANCE, DISSONANCE. The terms are used to
describe the agreeable effect produced by certain inter-
vals ...as against the disagreeable effect produced by
others.. .or similar effects produced by chords.. .
(lhe Harvard Dictionary of Music. 1953).6
Note that some of these definitions have "functional" implications, others do
not, and some equate consonant with "pleasant," dissonant with "unplea-
sant," while others do not-and one of them actually reverses the equation.'

2. Introduction
A historical context for the semantic problems associated with consonance
and dissonance is suggested (though not pursued very far) by Paul Hindemith
in the Craji of Musical Composition, when he says:
The two concepts have never been completely explained, and
for a thousand years the definitions have varied. At first thirds
were dissonant; later they became consonant. A distinction
was made between perfect and imperfect consonances. The
wide use of seventh-chords has made the major second and
the minor seventh almost consonant to our ears. The situa-
tion of the fourth has never been cleared up. Theorists, bas-
ing their reasoning on acoustical phenomena, have repeatedly
come to conclusions wholly at variance with those of prac-
tical musician^.^
And yet-in addition to being an eminently "practical" musician-Hindemith
himself was certainly one of the most prominent of modem theorists "basing
their reasoning on acoustical phenomena," and he does in fact imply a con-
ception of consonance and dissonance which is quite different from the purely
functional definition found in most contemporary textbooks on traditional har-
mony when he says, in the very next paragraph:
Between the octave as the most perfect and the major seventh
as the least perfect intemals, there is a series of interval-pairs
which decrease in euphony in proportion as their distance
from the octave and their proximity to the major seventh in-
creases. The tritone belongs neither to the realm of euphony
nor to that of cacophony.. .9
Arnold Schoenberg had been more careful to avoid such correlations of con-
sonance and dissonance with "euphony" and "cacophony" when he wrote
(in "Problems of Harmony," 1934):
Dissonances, even the simplest, are more difficult to com-
prehend than consonances. And therefore the battle about
them goes on throughout the length of music history.. .The
criterion for the acceptance or rejection of dissonances is not
that of their beauty, but rather only their perceptibility.1°
A few years earlier (in "Opinion or Insight?", 1926), he had written:
'The emancipation of the dissonance'. That is to say, it came
to be placed on an equal footing with the sounds regarded
as consonances (in my Hannonielehre the explanation of this
lies in the insight that consonance and dissonance differ not
as opposites do, but only in point of degree...consonances
are the sounds closer to the fundamental, dissonances those
farther away...their comprehensibility os graduated accor-
dingly, since the nearer ones are easier to comprehend than
those farther off). l 1
Introduction 3.
Still another interpretation of consonance and dissonance is suggested by
the following passage from Igor Stravinsky's Poetics of Music:
...the concepts of consonance and dissonance have given rise
to tendentious interpretations that should definitely be set
aright.. . Consonance, says the dictionary, is the combina-
tion of several tones into an harmonic unit. Dissonance results
from the deranging of his harmony by the addition of tones
foreign to it. One must admit that all this is not clear. Ever
since it appeared in our vocabulary, the word dissonance has
carried with it a certain odor of sinfulness... Let us light our
lantern: in textbook language, dissonance is an element of
transition, a complex or interval of tones which is not com-
plete in itself and which must be resolved to the ear's satisfac-
tion into a perfect consonance... But nothing forces us to be
looking constantly for satisfaction that resides only in repose.
And for over a century music has provided repeated examples
of a style in which dissonance has emancipated itself.. .I2
Although this is a plausible description of certain aspects or late 19th- and
early 20th-century developments leading toward what Schoenberg had called
"the emancipation of the dissonance," our "lantern" does not seem much
brighter than before with regard to the more basic question of the meaning
of the words consonance and dissonance. Are we to interpret dissonance, for
example, as meaning "not complete in itself," as implied here by Stravinsky,
or as less "comprehensible" (Schonberg), less "euphonious" (Hindemith),
less "agreeable" or "pleasant" (The Oxford Dictionary, Backus, et al), more
"beautiful' ' (Kraft), more "active' ' or "unstable' ' (Kraft and others), etc.-
or as some combination of some or all of these meanings?
It seems obvious that our first problem is indeed a semantic one, and that-
among many other difficulties which ensue from this-until this semantic pro-
blem has been solved any speculative theory that might be developed in an
effort to explain the nature of consonance and dissonance in musical percep-
tion is doomed to failure from the very start, since there is no common
understanding about what it is that such a theory ought to "explain." What
is perhaps not so obvious is that the semantic problems associated with con-
sonance and dissonance are rooted in the complex historical development of
what I will call the "consonance/dissonance-concept" (or CDC) in western
musical culture, and that a careful analysis of that historical development is
the only hope we have of unraveling the tangled network of meanings and
interpretations which so confuse the issue today. In The Style of Palestrina
and the Dissonance, Knud Jeppesen said:
When we encounter a.. .difficulty of linguistic-psychologic
nature, it generally repays the trouble to delve into history
and, seeking here anterior forms of the linguistic feature in
which we are especially interested, to work our way through
its genetic course.. .l 3
Accordingly-and in the spirit of this observation by Jeppesen-this book

4. Introduction
will examine the historical development of the CDC as it can be deduced
(primarily) from theoretical writings from the 3rd century B.C. through the
19th and early 20th centuries. It will shown that the words consonance and
dissonance (andlor their Greek, Latin, and modern-language equivalents) have
been used, historically, in at leastjve dzfferent ways-expressing five distinctly
different forms of the CDC. Before the rise of polyphonic practice they were
used in an essentially melodic sense, to distinguish degrees of affinity, agree-
ment, similarity, or relatedness between pitches sounding successively. Dur-
ing the first four centuries of the development of polyphony they were used
to describe an aspect of the sonorous character of simultaneous dyads, relatively
independent of any musical context in which they might occur. In the 14th
century the CDC began to change (again) in conjunction with the newly
developing rules of counterpoint, and a new system of interval-classification
emerged which involved the perceptual clarity of the lower voice in a
polyphonic texture )and of the text which it camed). In the early 18th cen-
tury, 'consonance' and 'dissonance' came to be applied to individual tones
in a chord, giving rise to a new interpretation of these terms which would
eventually yield results in diametric opposition to all of the earlier forms of
the CDC. Finally-in the mid-19th century-a conception f consonance and
dissonance arose in which 'dissonance' was equated with "roughness," and
this had implications quite different from those of earlier forms of the CDC.
These five different conceptions of consonance and dissonance will here be
called CDC- 1, CDC-2.. .(etc.). ..through CDC-5, in the order of their historical
emergence, and the development of each of these forms of the CDC will be
analyzed in their natural chronological sequence. In order to further clarify
the intentions and scope of this book, certain additional distinctions will have
to be made, as follows:
First, it is absolutely essential that we distinguish between conceptions of
consonance and dissonance, on the one hand, and on the other, explanatory
theories of, aesthetic attitudes toward, and practical uses of consonance and
dissonance. In spite of the obvious and intimate interrelations between these
various aspects of the larger problem of consonance and dissonance, they have
each followed a relatively independent course of historical development. Thus,
for example, the debate which raged in the early 17th century between Artusi
and the brothers Monteverdi involved disagreements regarding the proper use
of dissonance-and thus also aesthetic attitudes toward consonance and
dissonance-but no essential disagreement regarding the meaning of these
terms-and thus of the conception of consonance and dissonance. This book
is not intended to be a history of consonance/dissonance"treatment" as such,
or a history of theories of consonance and dissonance, but rather a history
of the underlying conceptions of consonance and dissonance, and these other
aspects of the problem will be dealt with only to the extent that they may be
helpful in clarifying the nature of these conceptions in a given historical period.
Second, the words consonance and dissonance seem to have been used-in
every historical period-in two different grammatical senses, which I will
distinguish as qualitive vs. entitive, corresponding to their use as "abstract"
vs. "concrete" nouns. The first of these refers to the property, attribute, or
quality associated with a sound or aggregate of sounds, while the second refers
Introduction 5.
to the sound or aggregate itself which manifests that quality. Which of these
two senses is intended is generally made clear by the context andlor by the
presence or absence of certain grammatical "markers" (e.g. the use of an
article, and the possibility of pluralization of the concrete noun expressing the
entitive sense). Several important words frequently used in musical discourse
have this dual nature-e.g. form, stnrcture, texture, etc.-and we generally
have no difficulty in distinguishing the two meanings. But since the historical
developments of the CDC sometimes involved a change of meaning or usage
of one of these forms without a corresponding change in the other, the descrip-
tive terms defined above will be found useful in tracing those historical
developments.
Finally, in order to clarify my own uses of the words consonance and
dissonance in this book, the following typographical procedures will be adopted:
(1) italicization-when not obviously intended merely for emphasis of a word
or phrase-will be used in the conventional way to mean the word itself (e.g.
consonance, concord, consonantia, symphonia, etc.); ( 2 ) when the reference
is to the semantic "cluster" composed of the written word plus any or all of
its cognates and equilvalent forms, single quotation marks will be used (e.g.
'consonance' = consonance/concord/consonantia/symphonia,etc.). But note
here that-since such semantic clusters hardly have any real existence separable
from the meanings they carry in common-a phrase like "'consonance' and
'dissonance"' (as used in the title of this book) becomes indistinguishable from
the term "consonanceldissonance-concept"-generally abbreviated here to
"CDC"; (3) double quotation marks will be used when the reference is to
some writer's (or group of writers') actual or imagined use of a word or its
equivalents (thus "consonance" might stand for symphonia in Aristoxenus,
concordantia in Odington, consonanza in Zarlino, etc,); and (4) no special
markers will be used when the words are intended to refer to the
acoustical/musical/perceptualphenomenon itself-whatever that may have been
during the period under consideration-or (in the entitive sense) to the sounds
in which that phenomenon was manifested.

Part One
From Antiquity through the
"Ars Antiqua"

Section I
The pre-polyphonic era (CDC-1)
In most pre-9th-century theoretical sources, the cognates of consonance and
dissonance-or of related words like concord and discord, symphony and
diaphony, and even our more general term harmony-refer neither to the
sonorous qualities of simultaneous tones nor to their functional characteristics
in a musical context but rather to some more abstract (and yet perhaps more
basic) sense of relatedness between sounds which-though it might determine
in certain ways their effects in a piece of music-is logically antecedent to
these effects. Among the Pythagoreans, as Arthur Koestler reminds us (in Zhe
Sleepwalkers, 1959):
..the concept armonia did not have quite the same meaning
that we lend to "harmony." It is not the pleasing effect of
simultaneously-sounded concordant strings-"harmony" in
that sense was absent from classical Greek music-but
something more austere: armonia is simply the attunement
of the strings to the intervals in the scale, and the pattern
of the scale itself. It means that balance and order, not sweet
pleasure, are the law of the world.14
Edmond de Coussemaker had said essentially the same thing (if somewhat less
colorfully) a hundred years earlier, when he wrote (in his Histoire de 1'Har-
monie au Moyen Age, 1852):
The word "harmony". ..signified to the Greeks the arrange-
ment or ordering of sounds considered with respect to the
melodic relationship between their pitches. It was not at all
(concerned with) the mixture of several sounds striking the
ear at the same time.. . We do not mean to say by this that
music (involving) simultaneous sounds was excluded from
Greek treatises on music, or that it is only a question of
melody; one finds there in fact more than one passage where
(the word) is used in the sense we call "harmony ." We on-
ly wish to demonstrate that the word "harmony" did not
have for the Greeks the restrictive meaning that it has to-
day, and that one would be in error if one took it in this
sense.I5
In fact, a m n i a (or harmonia) had an earlier meaning quite unrelated to music.
In Pythagoras and the Pythagoreans (1966), J.A. Philip says of this word:
Its primary meaning is not musical concord but a "fitting
together" produced by a craftsman such as to result in a

10. The pre-polyphonic era (CDC-1)
unified object, or "perfect fit" ...It is in this sense that the
word is often used in other than Pythagorean contexts.. . Thus
the primary sense is that of a principle producing a unified
complex, and, deriving from this sense, that of a complex
in musical sound and, finally, such a complex as a succes-
sion of sounds, a type of scale or mode expressible in
numerical ratios which state how the sounds "fit
together. ' ' ...In the Presocratics we find the word used first
by Heraclitus largely in a musical sense of an accord or
fitting-together of sounds in a melody, each note "fitting"
that which precedes it (and not as what we call harmony,
part-singing or p l y phonic music). ..Though we find that the
Pythagorean usage is predominantly musical. ..the earlier
sense of a fitting-together-into-one imposed by a craftsman
is often also present. '6
The opening sentence of l'he Harmonics of Aristoxenus (3rd century, B.C.)
confirms this purely melodic connotation of the term "harmonic"
(hannoniken):
The branch of study which bears the name of Harmonic is
to be regarded as one of the several divisions or special
sciences embraced by the general science that concerns itself
with Melody.''
and in a later passage, he says:
We shall now proceed to the consideration of Harmonic and
its parts. It is to be observed that in general the subject of
our study is the question, In melody of every kind, what are
the natural laws according to which the voice in ascending
or descending places the intervals?lB
In "Ancient Greek Music," Isobel Henderson says that the discipline of
"harmonics" :
...meant tuning, or acoustic theory. Greek postulates were
melodic and heterophonic, and ignored 'harmony' in our
sense.. .The term 'consonant' (sympho11os) refers to melodic
progressions. Music had nothing nearer to 'harmony' than
choirs doubling at the octave.. .'9
and concerning this word 'symphony' and its cognates, Gustave Reese has
written:
The original meaning of symphonia was "a concord of
sounds," and the fourth, fifth, and octave were, in fact, wn-
sonances for both the Greeks and the medieval "sym-
phonists" 6.e. the singers/composers/theorists of early
organum]-but from different points of view. Basically, the
intervals constituting the consonances were melodic (i.e.
The pre-polyphonic era (CDC-1) 11.
successively sounded) with the Greeks, harmonic (i.e.
simultaneously sounded) with the "symphonists. "lo
The purely melodic implication-to Aristoxenus-of the word symphonia
(translated by Macran as "concord") can be seen quite clearly in the follow-
ing passage from the Harmonics:
Whatever be the genus, from whatever note one starts, if
the melody moves in continuous progression either upwards
or downwards, the fourth note in order from any note must
form with it the concord [symphonon] of the fourth, or the
fifth note in order from it the concord of the Fifth. Any note
that answers neither of these tests must be regarded as out
of tune [&ynphonoi] in relation to those notes with which
it fails to form the above-mentioned concords.21
For Aristoxenus, the fourth was the first consonance, but simply because
it was the smallest. The others were the fifth, the octave, and several of the
composite intervals formed by octave-expansion of the fourth, fifth, and oc-
tave. Intervals smaller than the fourth, and those lying between the fourth and
fifth, and between the fifth and the octave (and the composite forms of these)
were all dissonant (diaphonos, ''discordant"). After thus classifying the various
intervals used in melody, Aristoxenus says:
So far we have been stating what we have learned from our
predecessors; henceforth we must amve at our conclusions
unaided. 22
The most prominent and influential of his predecessors, of course, were the
Pythagoreans. Aristoxenus-like his more celebrated teacher, Aristotle-was
highly critical of the Pythagoreans' numerological mysticism, and has therefore
come to represent to us an anti-Pythagorean point of view, as expressed, for
example, in the following passage:
...we hold that the voice follows a natural law in its motion,
ad does not place the intervals at random. And of our answers
we endeavour to supply proofs that will be in agreement with
the phenomena-in this unlike our predecessors. For some
of these introduced extraneous reasoning, and rejecting the
senses as inaccurate fabricated rational principles, asserting
that height and depth of pitch consist in cetain numerical ratios
and relative rates of vibration-a theory utterly extraneous
to the subject and quite at variance with the phenomena.. .23
And yet Aristoxenus does not offer an alternative explanation as to why cer-
tain intervals were judged to be 'consonant' and others 'dissonant', and his
classification of the fourth, fifth, and octave (and their compounds) as "con-
cords1'-and of all other intervals as "discords"-is perfectly consistent with
the results of m a g o r e a n doctrine, if not with its philosophical rationale. The
universality of the 'consonance' and 'dissonance' categories during this period

12. The pre-polyphonic era (CDC-1)
is exemplified again in the following passage from the same work:
If.. .a certain note be given, and it be required to find a cer-
tain discord [diaphonon] below it, such as the ditone.. .one
should take the Fourth above the given note, then descend
a Fifth, then ascend a Fourth again, and finally descend
another Fifth.. .If it be required to ascertain the discord in
the other direction, the concords must be taken in the other
direction. 24
This constitutes a set of instructions for tuning what has come to be called
a "Pythagorean scale," whose pitches are all derived from a sequence of tun-
i n g ~in perfect fourths, fifths, and octaves. The association of these three basic
"concords" with simple integer ratios between string-lengths was an essen-
tial element in the Pythagorean tradition, generally considered to have been
discovered by Pythagoras himself some three centuries earlier.
In its earliest manifestation, then, CDC-I involved relations between pit-
ches in a purely melodic context. Intervals that were precisely and directly
tunable were considered consonant, while d others-thos which were tunable
only indirectly-were dissonant, and there were no degrees of relative con-
sonance or dissonance in between these two extremes. Since it will be of
relevance to our discussion of later developments of the CDC, we must con-
sider the question: why was the class of "concords" thus limited to just these
three basic intervals (and their octave-compounds)-the fourth, fifth, and oc-
tave? The answer to this question involves what the Pythagoreans called the
"tetraktys (or quaternary) of the decadm--the geometric or "figurate"
representation of the number 10 as the sum of the first four integers (i.e.
1 2 3 4 =lo), arranged in triangular fashion as follows:
The purely musical significance of this "tetraktys of the decad" resides in
the fact that the string-length ratios for the fourth, fifth, octave, twelfth (but
not the eleventh), and double octave involve only these first four integers (i.e.
413, 312, 211, 311, 411, but not 813).25 But the decad had significance for the
Pythagoreans that went far beyond this musical application. According to a
later Pythagorean, Theon of Smyrna (2nd century, A.D.):
The importance of the quaternary.. .is great in music because
all of the consonances are found in it. But it is not only for
this reason that all Pythagoreans hold it in highest esteem:
it is also because it seems to outline the entire nature of the
The pre-polyphonic era (CDC-1) 13.
universe. It is for this reason that the formula of their oath
was: "I swear by the one who has bestowed the tetraktys
to the coming generations, source of eternal nature, into our
souls." The one who bestowed it was Pythagoras, and it has
been said that the tetraktys appears indeed to have been
discovered by him.26
This aspect of Pythagorean philosophy had been summarized much earlier
by Aristotle (in the Metaphysics), as follows:
...the Pythagoreans.. .devoted themselves to mathematics;
they were the first to advance this study, and having been
brought up in it they thought its principles were the prin-
ciples of all things. Since of these principles numbers are
by nature the first, and in numbers they seemed to see many
resemblances to the things that exist and come into be-
ing.. .since, again, they saw that the attributes and the ratios
of the musical scales were expressible in numbers; since,
then, all other things seemed in their whole nature to be
modeled after numbers, and numbers seemed to be the
elements of all things, and the whole heaven to be a musical
scale and a number. And all the properties of numbers and
scales which they could show to agree with the attributes and
parts and the whole arrangement of the heavens, they col-
lected and fitted into their scheme; and if there was a gap
anywhere, they readily made additions so as to make their
whole theory coherent. E.g. as the number 10 is thought to
be perfect and to comprise the whole nature of numbers, they
say that the bodies which move through the heavens are ten,
but as the visible bodies are only nine, to meet this they in-
vent a tenth-the 'counter-earth'. We have discussed these
matters elsewhere.. .27
In spite of the disdain which Aristotle (and Aristoxenus) evidently felt for
such notions, the Pythagorean viewpoint exerted a very powerful influence
on later Medieval philosophy in general, and music theory in particular-an
ibfluence that continued even into the Renaissance and beyond. As will be
seen later, the limit or boundary imposed on the conception of 'consonance'
by the tetraktys of the decad was finally broken only by an appeal to
cosmological arguments which were themselves fundamentally Pythagorean
in spirit.
The same two-fold classification of intervals is to be found in the writings
of nearly every subsequent theorist (whether "Pythagorean" or "Aristoxe-
nian") until about the 9th century.28Thus, for example, Cleonides, in the Har-
monic Introduction (1st c. A.D.)-obviously borrowing from
Aristoxenus-say s:
Of intervals the differences are five, in that they differ from
one another in magnitude, and in genus, and as the symphonic
from the diaphonic, and as the composite from the incom-
posite, and as the rational from the irrational.. .The symphonic

14. The pre-polyphonic em (CDC-1)
intervals are the diatessaron, diapente, diapason, and the like
[presumably the octave-compounds of these]. The diaphonic
intervals are all those smaller than the diatessaron and all
those lying between the symphonic interval^...^^
He also defines the terms 'symphony' and 'diaphony' as follows:
...symphony is a blending of two notes, a higher and a lower;
diaphony, on the contrary, is a refusal to two notes to com-
bine, with the result that they do not blend but grate harshly
on the ear.30
In the treatise by Theon of Smyrna we find the following:
The interval is defined as the relationship of sounds among
each other, such as the fourth, the fifth and the oc-
tave.. .Among the intervals, some are consonant, others disso-
nant.. ./individual tones] form a consonance with each other
when a sound which is produced by a string of an instru-
ment causes the other strings to resonate by means of a cer-
tain affinity, a kind of sympathy; and also, when two sounds
being produced at the same time result in a mixed sound
which has a sweetness and a quite particular charm.31
Similarly, in his Introduction to Harmonics (2nd c.), Gaudentius says:
Consonant tones, when they are produced simultanwusly,
either by striking or blowing on an instrument, always result
in the same musical sound, whether the motion is from the
low tones to the high, or the high tones to the low.. .Disso-
nant tones, when they are produced simultaneously.. .never
seem to be the same in any part of the musical sound.. .[and]
do not show any evidence of blending with each other.. .There
are six consonances in the perfect system [the fourth, fifth,
octave, eleventh, twelfth, and double octave]. 32
Again in the 6th century, we find essentially the same definition of 'symphony'
as that given by Cleonides four hundred years earlier, and the same set of
three basic "symphonic" intervals, in the Institutiones (ca. 550-562) by
Cassiodorus:
Symphony is the fusion [tempermn~un] of a low sound with
a high one or of a high sound with a low one ...There are
six symphonies: diatessaron, diapente, diapason, diapason
and diatessaron together, diapason and diapente together,
[and] disdiapason [sic-the double octave]. 33
Whether these categorical classifications were meant to refer to successive
or to simultaneous tones-or both-is not always altogether clear in these "post-
classical" but pre-polyphonic sources. The use of words like "combine,"
"blending," and "fusion" might tend to suggest simultaneity, and-referring
The pre-polyphonic era (CDC-1) 15.
to the passage quoted above from Gaudentius-Curt Sachs has written (in me
Rise of Music in the Ancient World, 1943):
The ancient definition of consonance had a remarkably
modem flavor [and later]. .. Consonances, Boethius says, are
"pleasant, " and the pseudo-Aristotelian Problem 19:13 states
that "any consonance is sweeter than a single note." And
are we supposed to believe that the Greeks did not use
them?34735
There are persuasive arguments, however, against an interpretation of 'con-
sonance' and 'dissonance' in these early-Medieval sources as having been in-
tended primarily as descriptions of simultaneous sounds. In "The Birth of
Polyphony, " Dom Anselm Hughes wrote:
...It seems that there was a formal, almost scholastic ter-
minology in use from the time of Cassiodorus (479-575) and
St. Isidore of Seville (565-636) down to Aurelian of Reome
and Remy of Auxerre in the ninth century, according to which
harmonia was used of melody, and symphonia of consonant
intervals.. .Borrowing.. .from Cassiodorus, [Isidore] describes
symphonia as a consonant interval, accurately sung or played,
the opposite of diaphonia (dissonance). This definition reap-
pears in Aurelian; but there is nothing to show that the defini-
tion of consonance and dissonance refers to simultaneous,
rather than successive, sounds, until we come to the specific
explanation given by Regino of Prum (d. 915) in his De har-
monica institutione. The use of diaphonia in a purely melodic
sense persisted even after polyphony was well e ~ t a b l i s h e d . ~ ~
No matter how the debate as to whether these Medieval theorists meant to
refer to the qualities of simultaneous tones is ultimately resolved, one thing
at least is clear: 'symphonic' and 'diaphonic' were terms generally used by
them-as by Aristoxenus-to describe relations between pitches, conceived
in a melodic context. The observation that two tones forming a "symphonic"
interval also "result in the same musical sound" (Gaudentius), or that one
of those tones sounded on a string "causes the other strings to resonate by
a certain affinity.. ." (Theon), merely conjnned the essential point these writers
intended to demonstrate-namely that such tones were in a concordant reltion
to each other, and that such a relation was the essential basis for melodic
organization.
In these earliest sources, then, the terms 'consonance' and 'dissonance' had
a meaning which was quite different from those which developed later. It was
certainly the prevailing (if not the only) sense of the CDC preceding the rise
of polyphony in western music. It is also important to note, however, that this
earliest sense of the CDC still exists as a musically meaningful concept, even
when expressed by different terms. It is clearly the principle behind Rameau's
rules for root-progression, as in the following (from the Treatise on Harmony,
1722):

16. The pre-polyphonic era (CDC-1)
...when we give a progression to the part representing the
undivided string fi.e. the basse-fondamentale], we can only
make it proceed by those intervals obtained from the first
divisions of the string. Each sound will consequently har-
monize with the sound preceding it [my emphasis]. 37
It is also manifested in later tonal theory in the notion of "closely related keys"
involved in modulation, but its nearest equivalent in the contemporary musical
vocabulary is perhaps simply "relations between tones7'-in the sense in which
this phrase is used by Schoenberg, for example, when he says (in "Problems
of Harmony"):
If.. .we wish to investigate what the relation of tones to each
other really is, the first question that arises is: what makes
it possible that a second tone should follow a first, a beginn-
ing tone?. ..My answer is that such a juxtaposition of tones.. .is
only possible because a relation already exists between the
tones themselves.
Finally, CDC-1 is evidently the basis for Hindemith's "Series 1" (in the
Crafr...), about which he says:
The values of the relationships established in that series will
be the basis for our understanding of the connection of tones
and chords, the ordering of harmonic progressions, and ac-
cordingly the tonal progress of compositions.39
The one essential difference between these "relations between tons" as discuss-
ed by Schoenberg and Hindemith and the earlier sense of consonance and
dissonance I am calling CDC-I is that what had originally been conceived as
a simple two-fold dichotomy is now conceived as an ordered continuum of
degrees of relatedness, within which, as Hindemith says:
We know that no point can be determined at which "con-
sonance" passes over into "di~sonance."~0
Ln every other respect, however, the musical/perceptual phenomenon to which
Schoenberg and Hindemith were addressing themselves here is equivalent to
the most ancient of all known conceptions of consonance and
dissonance-CDC-l .
Section II
The early-polyphonic period, ca. 900-1300 (CDC-2)
The second sense of the CDC-described earlier as involving an aspect of
the sonorous quality of simultaneous dyads, relatively independent of their
musical context-begins to be expressed unambiguously in the theoretical
literature only after the rise of polyphony in about the 9th century. Although
it is obviously difficult to gaive a precise date to the beginnings of polyphonic
practice, the following passage from Hucbald's early-10-century treatise De
harmonica (Melodic instruction) has been called "the earliest unmistakable
reference to harmonized music":41
"Consonance" [consonantia] is the calculated and concor-
dant blending [concordabilispennirtio]of two sounds, which
will come about only when two simultaneous sounds from
different sources combine into a single musical whole, as
happens when a man's and a boy's voices sound at once,
and indeed in what is usually called "making
organum". ..There are six of these "consonances" [con-
sonantiae], three simple and three composite ...diapason,
diapente, diatessaron, diapason-plusdiapente, diapason-plus-
diatessaron, and double diapason.42
With the advent of polyphony it had become necessary-for the first time-to
make a distinction between melodic interval and simultaneous dyad, and Huc-
bald's solution to this problem involved a subtle modification of the traditional
Latin terminology associated with plainchant. In his introduction to Warren
Babb's English translation of Hucbald's treatise, Claude Palisca explains cer-
tain differences between Hucbald's terminology and that of his 6th-century
predecessor Boethius:
At the outset Hucbald makes several distinctions among in-
tervals. First he uses the terms aequisonae and con-
some.. .terms derived from Boethius, who in turn got them
from Ptolemy , but Hucbald.. .altered their meaning. For
Boethius two notes of the same pitch are unisonae; two notes
which sound almost identical, such as the octave and double
octave, are equisonae; whereas the diapente and diatessaron
are consonae. All these together comprise the genus con-
sonantia or consonance. For Hucbald aequisonae are unisons,
consonae are simply consonances, and he transfers the con-
dition of agreeably sounding simultaneously, which Boethius
ascribed to the octaves, from these to all consonances.. .Huc-
bald distinguishes consonance.. .from melodic interval [in-
tervallum or spatium] ...Ancient theory is thus adjusted to

18. The early-polyphonic period (CDC-2)
to the budding practice of polyphony. The Ptolemaic-Boethian
concepts are distorted in the process, to be sure, but they
are ingeniously fitted to modem use. In the Greek tradition
all consonances were essentially melodic intervals.. .43
This shift .of referent for consonantia from melodic interval to simultaneous
dyad did not become standard until much laterm, however, since it continued
to be used in the Boethian sense as melodic interval by the majority of theorists
throughout this period-and even well into the 14th century. The word most
commonly used then for consonant simultaneous dyad was concordantia (or
occasionally, concordia). Even this "most common" usage was not entirely
consistent, however, and Johames de Grocheo (writing ca. 1300) explicitly
reverses these correlation^.^^
Another solution to this problem of distinguishing between melodic interval
and simultaneous dyad involved the adaptation of the ancient Greek terms sym-
phonia and diaphonia, but here the semantic transformations were less sub-
tle. By the 10th century symphonia had come to mean a consonant simultaneous
dyad, and is used strictly in that sense by theorists as late as Walter Odington
and Jacobus of Liege (ca. 1300 and 1330, respectively). The word diaphonia,
on the other hand, entirely lost its earlier linguistic function as antonym for
symphonia, and came to mean (by the 1lth century, at least) simply singing
in separate parts simultaneously-another term for organum. In the anonymus
Musica enchiriadis, written at about the same time (ca. 895) as Hucbald's
treatise, his "consonances" are called "symphoniae," and described as
follows:
Not all tones blend together equally well, nor do they always
render harmonious effects in song in every kind of combina-
tion. Just as letters brought together at random often do not
produce either connected words or syllables, so in music only
certain fixed intervals may constitute syrnphoniae. A syrn-
phonia is a pleasant concord [dulcis concentus] of dissimilar
tones joined to one another.. .There are three simple or prime
syrnphoniae, from which the rest are compounded. Of these
one is called a fourth, another, a fifth, and the next, an oc-
tave.. . 4 5
Whereas Hucbald's treatise is primarily concerned with melody, the Musica
enchiriadis is largely devoted to early organum, and might thus be considered
the earliest known treatise on what would (much later) come to be called
"counterpoint." It includes what may be the first "rule" in the history of
that discipline, designed to avoid the tritone:
Since the sounds at the interval of a fourth do not all, without
exception, produce consonances throughout the whole series
of tones, certain intervals of the composition should not be
sung exactly. Therefore, in this kind of song the voices are
marvelously accommodated to each other by a certain
rule.. .46
The early-polyphonic period (CDC-2) 19.
Again, in a long section of the subsequent Scholia enchiriadis (ca. 900) entitl-
ed "Of Symphonies," we find the following dialogue:
(Disciple) What is a symphony? (Master) A sweet blending
of certain sounds, three of which are simple-diapason,
diapente, and diatessaron-and three composite-double
diapason, diapason plus diapente, and diapason plus
diate~saron.~'
This is followed by a detailed description of each, including the several "com-
posite" forms-this adjective now referring to various octave-doublings of one
or both of the primary tones of the dyad (in the "organal" and "principal"
voices). Note that, in all three of these treatises from the late 9th or early 10th
century, the same intervals are classified as consonant as those so designated
by Aristoxenus over twelve hundred years earlier, but the reference now is
clearly to simultaneous dyads rather than successive tones.
In its earliest manifestations then, CDC-2 is nearly indistinguishable from
CDC-1, but a growing separation between the two begins to be noticeable in
theoretical writings of the 1lth and 12th centuries, with CDC-1 implicit in
passages concerned solely with melody, CDC-2 in those describing the ef-
fects of the added voice or voices in organum. Both senses of the CDC are
to be found on the Micrologus (ca. 1026-28) by Guido d'Arezzo, as can be
seen by comparing the following definition of 'symphony' (exemplifying
CDC-2);
You should remember that these three intervals [the octave,
fifth, and fourth] are called "symphonies," that is, smooth
unions of notes [suaves vocurn copulationes], because in the
diapason the different notes sound as one [unurn sonant] and
because the diapente and the diatessaron are the basis of
diaphony, that is, organum, and produce notes similar in
every case.. .48
with another passage which occurs during his discussion of modes and melodic
organization (and thus exemplifies CDC-1):
Notice.. .that these affinities of notes [vocurn afinitates] in
the various modes are made through the diatessaron and the
diapente, for A is joined to D, and B to E, and C to F by
the lower diatessaron, but [also] by the upper diapente.
Whatever other affinities there are, they are produced
likewise by the diatessaron and the diapente.. .We have con-
fined ourselves to just a few things about the similarities bet-
ween notes, because insofar as similarity is sought out bet-
ween different things, to this extent is lessened that diversi-
ty which can prolong the labor of the confused mind.49
In a later passage, the two senses of the CDC are both referred to, and Guido
notes the close correlation between them:

20. The early-polyphonic period (CDC-2)
Diaphony sounds as a separateness of [simultaneous] sounds,
which we also call organum, in which notes distinct from
each other make dissonance harmoniously and harmonize in
the dissonance [concorditer dissonant, & dissonanter con-
cordant]. 50 Some practice diaphony in such a way that the
fourth step down always accompanies the singer, as A with
D; and if you double this organum by acute a, so that you
have A D a, then A will sound a diatessaron with D and a
diapason with a, whereas D will sound a diatessaron and a
diapente with A and a respectively, and acute a with the lower
two notes a diapente and a diapason. These three intervals
blend in organum congenially and smoothly just as it has
been shown above that they caused a resemblance of notes
[my emphasis]. Hence they are called "symphonies," that
is, compatible unions of notes, although this term symphony
is also applied to all chant. Here is an example of this
diaphony [diaphoniae]. 51
In other words, tones forming a fourth, fifth, or octave display an "affinity,"
"similarity," o r "resemblance" to each other in a melodic context (CDC-I),
and they also create "smooth,' ' "congenial,' ' or ''compatible unions' ' with
each other when sounded simultaneously (CDC-2). The same three intervals
are thus understood to satisfy two different conditions-but these conditions
a r e different.
A few generations later, Guido's "and" has become an "eitherlor" in John's
treatise De musica (ca. 1100), as we see in the following:
Among other things, one ought to know that there are just
nine intervals [modi] from which melody is put together.. .Six
of these are called "consonances" [consonantiae], either
because in singing they sound together-at the same time-
more often than the others [CDC-21; or, more likely [my em-
phasis], because they sound together in the sense that they
are related among themselves.. .[CDC-11. 52
John seems to prefer the second of the two explanations, probably because
he is primarily concerned with melody-and only secondarily with "diaphony,"
which receives a very cursory treatment in this work (one chapter out of
twenty-seven).
What eventually led to a more clear-cut distinction between CDC-1 and
CDC-2 were developments of the freer style of organum involving oblique
and contrary (as well as parallel) motion between the voices-and thus a more
frequent occurrence of simultaneous dyads other than the three classical "sym-
phonies." Guido describes one form of this freer style as follows:
..let us explain the low voice added beneath the singer of
the original line in the way that we employ. For the above
manner of diaphony [parallel organum] is hard [or
"harshM--durus], but ours is smooth [mollis]. In it we do
not admit the sernitone or the diapente[!], but we do allow
the tone, the ditone, the semiditone, and the diatessaron; and
The early-polyphonic period (CDC-2) 2 1.
of these the serniditone holds the lowest rank and the
diatessaron the chief one. With these four concords [con-
cordiis] the diaphony accompanies the chant.53
In John's treatise, parallel organum is no longer even mentioned, and he recom-
mends contrary motion as the "simplest methodm-although he seems to prefer
it for reasons other than its simplicity:
Diaphony is the sounding of different but harmonious notes,
which is carried on by at least two singers, so that while one
holds to the original melody, another may range aptly among
other tones, and at each breathing point both may come
together on the same note or at the octave.. .Different musi-
cians practice this differently. The simplest method for it is
when the various melodic progressions are borne carefully
in mind, so that wherever there is an ascent in the original
melody, there is at that point a descent in the organal part
and vice versa.54
Thus-unlike Guido-John does not limit the acceptable intervals between the
organal and principal parts to those within the compass of a fourth. Neither
does the anonymous author of Ad organum faciendum (late 1 lth century)-
even though his definition of 'diaphony' is virtually identical to Guido's. Here
we find the following:
The first note of the organum will either remain conjunct
with the cantus at the octave or unison, or disjunct at the
fifth or fourth. The middle notes, however, move at the fifth
and fourth. Then, when the cantus requires (conjunction with)
the organum, a c o p ~ l a t i o ~ ~is effected in some way.56
A companion treatise from about the same time-the anonymous Item de
o r g a n ~ ~ ~ - a d d smajor and minor thirds to the list of intervals (consonantiae)
which may be used as simultaneous dyads in organum-even at the beginning
of a phrase, and the Monipellier organum treatise58 includes sixths as well
as thirds-although Jay Huff, in his Introduction to Item de organo, says that
"None of the examples in either the present treatise or the Montpellier have
a third (or sixth) for an initial interval, as initial interval is defined in both
treatise^."^^ Even so, the relatively high incidence of thirds, at least, in music
of this period (the 1lth century) is indicated by the following statistical data
on the C hartres MS 109 (in Hughes' "The Birth of Polyphony," 195460):out
of a total of 241 intervals (i.e. simultaneous dyads), there are 67 thirds (28%),
48 fourths (20%), and only 15 fifths (6%).
By the 12th century, thirds-and to a lesser extent, sixths-were beginning
to demand recognition, but whether recognized by the theorists as consonant
or not, the important point, for our purposes, is that these intervals were now
being heard more and more often as simultaneous dyads, and this provided
an opportunity for comparing their sonorous qualities with those of the tradi-
tional "symphonies." This, in turn, led to efforts by theorists in the 13th cen-
tury to classify the various intervals with respect to their sonorous properties

22. The early-polyphonic period (CDC-2)
as simultaneous dyads, and most of the classification systems which began
to appear in theoretical treatises now involved much finer qualitative distinc-
tions than had ever been employed in descriptions based on CDC-1. In the
D e musica libellus (Anonymous VII, ca. 1220), the class of "consonances"
is divided-apparently for the first time-into three subcategories, as follows:
Let it be observed that the unison, serniditone, ditone,
diatessaron, diapente, and diapason are more essential than
the other intervals [species], for all discant forms one of these
consonances [consonantiarum] with its tenor. It should be
noted that the unison and the diapason are pefect consonances,
the ditone and the semiditone imperfect, and the diatessaron
and the diapente
This classification of the consonances as perfect, intermediate, and imperfect
is found again in treatises by John of Garland,62Franco of C ~ l o g n e , ~ ~and
Coussemaker's Anonymi I,64,11,65and IW6-all written during the latter half
of the 13th century. In addition, some of these theorists also divided the
dissonances into similar subcategories, although here there was somewhat less
agreement among them as to the appropriate ranking of certain intervals. Some
of the many consonance/dissonance classification-systems expressed or im-
plied by theorists from the beginning of polyphonic theory in the 9th or 10th
century through the first half of the 16th century are shown in tabular form
in Figure 1. In Figure 2, a few of these are displayed in another way which
shows more clearly the changes in status of each interval during this same
period. (See Appendix, fig. 1 and 2).
John of Garland's system of interval-classification is the most elaborate of
any theorist of the 13th century, involving the largest number of subcategories
(six). In addition, his definitions of "concord'' and "discord" are fairly typical
of those given by theorists of this period (many of whom borrowed directly
from him), and are thus indicative of the qualitative connotations of 'con-
sonance' and 'dissonance' in CDC-2, as in the following (from D e mensurabili
musice, ca. 1250):
Of the consonances [consonantiarum], some are called con-
cords, some discords. Concord [concordantia] is when two
sounds are joined at the same time so that one can be heard
as compatible with the other. Discord [discordantia] is the
opposite.. .A perfect concord is when two sounds are joined
at the same time so that the ears cannot distinguish one voice
from the other on account of [this] concordance, and is call-
ed one sound, or the sounding of equals [equisonantiam],
as in the unison or diapason.. .An imperfect concord is when
two sounds appear at the same time so that the ears can wholly
distinguish one voice from the other, and I say that this is
[also] a type of concord, and there are two species, namely
the ditone and semiditone. An in-between concord is when
two voices are joined together so that they are neither perfect
nor imperfect, and there are two species, namely the diapente
and diatessaron. 67
The early-polyphonic period (CDC-2) 23.
The "discords" are similarly subdivided as follows:
When two sounds are joined at the same time so that one
sound cannot be heard as compatible with the other, it is call-
ed discord. Of the discords, some are called perfect, some
imperfect, some in-between. They are called perfect when
two sounds are combined by a certain means according to
the sympathy [compassionem] of sounds so that one voice
cannot be heard as compatible with the other, and there are
three types, namely the semitone, tritone and ditone with
diapente b.e. major seventh]. They are called imperfect when
two sounds are combined so that in a certain manner they
can be heard as compatible, but nevertheless not concordant
like the concords; and there are two species.. .the tone with
diapente and semiditone with diapente. They are called in-
between when two sounds are combined so that they are heard
partially like the perfect discords, partially like the imperfect
discords, and there are two species.. .the tone and semitone
with d i a ~ n t e . ~ ~
The definitions of "concord" and "discord" given by Franco, Anonymous
I, and Lambertus ("cujusdam Aristotelis," in Coussemaker's Scriptomm.. .I)69
are nearly identical to those of John of Garland, and the classification systems
of the first two of these writers differ from John's only with respect to the
dissonances, which they divide into two (rather than three) subcategories, rank-
ing the major second with the "imperfect," the minor sixth with the "perfect
discords." Unfortunately, the relevant portion of Coussemaker's text of the
Tractatus d e musica now attributed to Lambertus is garbled, making it im-
possible to ascertain just how he may have intended to classify each interval,
but it is clear that he distinguished the same three degrees of "discord" (as
well as of "concord") as did John of Garland.'O
The question naturally arises: to what extent did such theoretical systems
of interval-classificationreflect the actual harmonic practice of their own time,
as distinct from purely theoretical doctrines carried over from some earlier
era-or even the idiosyncracies of the individual writers? After summarizing
the statistical data cited earlier regarding the frequency of occurrence of various
simultaneous dyads in the Chartres MS 109, Dom Anselm Hughes says:
The result of this analysis shows that the actual music of the
eleventh century at Chartres at any rate was considerably
different from what we have been taught to expect from the
descriptions of the theorists, and that is, from a later point
of view, considerably in advance of it.7'
His primary reason for saying this is that fifths occur much less frequently-
and thirds more frequently-than their relative theoretical status as consonances
might lead one to expect, although-as has been suggested by Fred Blum, if
Hughes had considered the more "progressive" discussion of thirds and sixths
in the Montpellier organum treatise "he might not have found such an im-
measurable gap between theory and practice. "72 Both Hughes and Blum seem

24. The early-polyphonic period (CDC-2)
to assume, however, that there should be a simple correlation between interval-
frequency and consonant status, which is at least questionable, if not altogether
unwarranted. If this criterion were applied to the free organum style describ-
ed by Guido, for example, the resulting classifications would look very strange
indeed-the fifth would be a dissonance along with the semitone, and the thirds
and major second would be only slightly less consonant than the fourth (with
the major second more consonant than the minor third!). In an Appendix at
the end of this section some statistical data are presented regarding dyad-
frequencies in Perotin's conductus, Salvatoris Hodie, and these are compared
with corresponding data given by Hughes in the source quoted above, for music
of the 1lth-13th centuries. In nearly every case, seconds occur more often
than sixths, which suggests that dyad-frequency is determined by other fac-
tors (such as a tendency to favor smaller intervals over larger ones)-in addi-
tion to consonance and dissonance. But is it not also at least possible that
Medieval musicians actually enjoyed the sonorities of simultaneous aggregates
that even they would have called "dissonant"?-just as is clearly the case with
many 20th-century composers. It can be admitted that Hughes' suggestion that
harmonic practice was "considerably in advance o f ' theory is plausible-if
only because of the analogous discrepancies between theory and practice which
are so painfully evident in our own century. But the new musical experiences
of the 20th century have also made it possible for us to hear the magnificent
clashes of seconds and sevenths in the organa and conductus of Perotin, for
example, in a more positive way than was perhaps possible for 19th- and
early-20th-century musicians and musical scholars, for whom the music of
Perotin could only represent a "primitive" or "archaic" stage in a progressive-
evolutionary development in which "complete control" of the musical materials
was not achieved before Dufay (at the earliest), or-for some-Palestrina, or
even J.S. Bach. Such an attitude about the music (and I should add, the
theoretical writings) of the 13th century is no longer tenable, of course, but
many of the early prejudices linger on.
Harmonic practice differed from one region to another, of course, and the
English were apparently somewhat "in advance o f ' Continental musicians
in their use of thirds and sixths. In De mensuris et discantu (ca. 1275),
Anonymous IV gives the same three-fold classification of the consonances
specified by John of Garland, Franco of Cologne, and others, but then adds
the following information:
...there are excellent composers of polyphonic music in ce-
tain places, such as England.. .who consider [thirds] to be
the best possible consonances [optime concorduntie], since
they use them so much.73
Later, in reference to an example he gives of the use of the major sixth as
the penultimate dyad before a final octave at the end of a phrase, he says:
Thus, we have shown an example of that vile and loathesome
dischord [vilis discordanria sive tediosa] which is the sixth,
and which is mostly to be avoided. If, however, it is the next-
to-last note fin the duplum, above the tenor] before a perfect
The early-polyphonic period (CDC-2) 25.
consonance, which is the octave, it is the best consonance
in this arrangement of notes or sounds.. .74
Only a few years later, the English theorist, Walter Odington, in his De specula-
tione musicae (ca. 1300), classifies the fourth, fifth, octave, twelfth, and dou-
ble octave as symphoniae, while the major and minor thirds and tenths, the
major (but not the minor) sixth, and the eleventh are called concordes
discordiae- ''discordant concords"-or, in effect, "imperfect consonances."7 5
Regarding the theoretical status of thirds and sixths during this period, Hughes
has written:
The intervals of the third and, to a lesser degree, the sixth
were now /by the late 13th century] recognized by theorists.
As early as the latter half of the twelfth century Theinred
of Dover explains why the major and minor thirds are ad-
mitted in organa, in spite of the fact that they are not strictly
consonances...It is obvious that the reluctance of theorists
to admit thirds and sixths as consonances was due to the fact
that they did not fit into the acoustic theory which they had
inherited from the Greeks.. .But as Theinred himself says,
the difference between the Pythagorean and "just" forms
of these intervals] is hardly noticeable to the ear; and
Odington.. .not only mentions that many people regard the
ditonus and the semiditonus as consonant...but also observes
that intervals like this which are not mathematically conso-
nant can be made to sound so if they are skillfully and
beautifully sung.76
As Hughes suggests here, one of the primary reasons for the long delay in
accepting thirds and sixths as consonant was the persistence by theorists in
assuming the Pythagorean ratios for these intervals, in spite of what I would
consider the very great probability that what was actually being sung-and
therefore heard (in vocal music at least)-were their simpler "just" forms.
A comparison of the "just" with the Pythagorean ratios for thirds and sixths
makes it clear why the theorists (if not the practicing musicians) would have
resisted their inclusion among the consonances for so much longer than musical
practice would seem to have warranted: 514 vs. 81164 for the major third or
ditone, 615 vs. 32127 for the minor third (semiditone), 513 vs. 27116 for the
major sixth (tone plus diapente), and 815 vs. 128181 for the minor sixth
(semitone plus diapente). Sixty years after Franco's Ars cantus mensurabilis,
the ratios specified for these intervals by Philippe de Vitry in his Ars Nova77
(ca. 1320) were still those derived from Pythagorean tuning by fourths, fifths,
and octaves-and even as late as the end of the 15th century the majority of
theorists were steadfastly assigning these Pythagorean ratios to thirds and sixths.
It is no wonder then that Franco (and others) included the minor sixth (128181)
among the "pefect discords," along with the semitone, tritone, and major
seventh!
And yet-constrained as they may have been by Pythagorean doctrine-
these theorists of the 13th century were not simply rank-ordering the intervals
in some routinely mechanical way according to the relative complexities of

26. The early-polyphonic period (CDC-2)
their F'ythagorean ratios-even though this factor is often invoked by them
in their discussions of consonance and dissonance. If they had been doing this,
their classifications would not have differed, one from another, and thirds and
sixths would not have been classified as consonances at all, since their
F'ythagorean ratios are more complex than those of the major second and minor
seventh, both of which were invariably classified as dissonances. This indicates
to me that the theorists of this period were making a very real effort to evaluate
the sonorous qualities of simultaneous dyads as they heard them-i.e. accor-
ding to how they actually sounded to them in the music-and this in spite of
F'ythagorean doctrine. In this respect, at least, there was an intimate connec-
tion between musical theory and practice in the 13th century-a connection
which is in no way weakened by the differences which existed between the
various systems of interval-classification formulated by individual theorists.
These differences invariably involved only certain intervals-namely those in
the middle range of the consonance/dissonance "spectrum"--whereas there
was no disagreement among them regarding the classification of intervals at
either end of that same spectrum. A comparison of the rank-orderings that
would be derived from the relative complexities of the F'ythagorean ratios,
on the one hand, and of the simpler "just" ratios, on the other, shows that
it is precisely these intermediate intervals whose relative rank would have been
most affected by any variability or ambiguity of intonation in performance
(see Figure 3). The probability that there was such variability or ambiguity-
and, more specifically-that the intonation of these intervals was tending in
the direction of the simpler "just" ratios (at least for those used most fre-
quently), is suggested in Figures 4a and 4b, where the rank-orders given by
John of Garland and Franco of Cologne (Figure 4a)-and those which would
be derived from F'ythagorean and "just" ratios (Figure 4b)-have been plot-
ted as a function of interval size (insemitones, along the horizonal axis). It
is evident that Figure 4a approximates the graph based on "just" ratios more
closely that it does the F'ythagorean, for all intervals except perhaps the two
sixths. (See Appendix, fig. 3 and 4).
Nevertheless, it was not the "just" but the F'ythagorean ratios which were
consistently used by Medieval theorists in their discussions of the objective
properties of intervals, and the fact that their consonance/dissonanceclassifica-
tions did not simply correspond to the order of complexity of these ratios is
ample evidence that their basis for classification was, in fact, some aspect of
the perceptible sonorous qualities of simultaneous dyads. The same conclu-
sion (though based perhaps on a different line of reasoning) has been express-
ed by Richard Crocker (in "Discant, Counterpoint, and Harmony," 1962)
as follows:
Medieval writers.. .consistently invoke the judgment of the
ear in discussing the degree of concord and discord.. .Clear-
ly.. .it is false to believe that the Middle Ages relied solely
on mathematics and excluded the judgment of the ear in deter-
mining the nature of c o n ~ o n a n c e . ~ ~
In support of this observation, which is, as he says, "in flat contradiction to
The early-polyphonic period (CDC-2) 27.
the opinion commonly held about medieval musicians, " he quotes statements
by several theorists from John of Garland in the mid-13th century to Tinctoris
in the late-15th-on the basis of which he further concludes:
These authors say, in sum, that the ear takes pleasure in con-
sonance, and the greater the consonance the greater the
pleasure; and that for this reason one should use chiefly con-
sonances in composing d i ~ c a n t . ~ ~
The question arises, however: was the degree of "pleasure" associated with
a given dyad the sole (or even the primary) basis for its placement within the
consonance/dissonancecontinuum? Crocker suggests that there must have been
"at least two distinct bases for judgment of consonance"-one involving the
degree to which the tones of a dyad "blend together," the other having to
do with "the function of intervals within the development of ~tyle"~O-and
regarding the first of these he says:
...the medieval musician.. .finds the simplest intervals to be
the sweetest, a judgment which one must admit to have been
reasonable in the springtime of polyphony. We, withdraw-
ing a little from sonorous reality, find the more complex,
less consonant intervals to be sweeter..
Crocker seems to equate simplicity, sweetness or pleasure, and what he calls
"the degree of sonorous blend," including them all as components of the first
of his "two distinct bases for judgment of consonance." But these are clearly
separable factors, and while sweetness and pleasure involve highly subjective
responses-and these have obviously changed considerably over the
centuries-"sonorous blend" is a rather more objective factor (though entire-
ly perceptual). By comparison with some of the definitions of 'consonance'
and 'dissonance' given by theorists ajler the 13th century, those of the 13th
century have a remarkably objective-even ascetic-character which suggests
that perhaps-for them (as Arthur Koestler said of the early Greeks)-"balance
and order, not sweet pleasure," were still "the law of the world." Consider
again the definitions of "perfect" and "imperfect concord" by John of Garland:
A perfect concord is when two sounds are joined at the same
time so that the ears cannot distinguish one voice from the
other ...and is called one sound, or the sounding of
equals.. .[whereas, with an imperfect concord]. ..the ears can
wholly distinguish one voice from the other.. .82
Taken by itself, John's definition of "perfect concord" might seem to refer
to nothing more than what we now call "octave equivalence7'--and this
phenomenon is certainly involved here, since he expressly refers to the unison
and octave as "one sound, or the sounding of equals" (equisonantium, ob-
viously related to the Boethian equisonae). But this does not explain the place-
ment of the fourth and fifth in a category immediately adjacent to these "perfect

The early-polyphonic period (CDC-2) 29.28. The early-polyphonic period (CDC-2)
concords." There is no way in which one can say that the tones forming a
fourth or a fifth are just a little less "equivalent" than those of a unison or
octave: "Octave-equivalence" may well be the reason why the octave (as a
simultaneous dyad) manifests a degree of sonorous blend nearly equal in
"perfection" to the unison, but I submit that it was the degree of sonorous
blend itself (or something very closely related to it) that formed the primary
basis for dyad classification in the 13th century.
In fact, I will carry this argument one step farther and suggest that what
13th-century theorists may have actually meant by "perfect concord" was a
condition in which a simultaneous dyad sounded like a single tone, and that
they distinguished varying degrees of consonance (and dissonance) according
to the extent to which a given dyad satisfied this condition of "singularity."
In support of this hypothesis I invoke the remarkable definition of 'discant'
given by the last of the great Medieval theorists to write extensively about
musical practices and conceptions of the 13th century-Jacobus of Liege. His
mammoth Speculum musicae, although written (ca. 1330) long after the end
of the period associated with CDC-2, is suggestive of what "might of been"
if the style he called the "ars antiqua " had not been so precipitously terminated
and replaced by the new concerns of the "ars nova. " And since Franco had
said that "every discant is governed by consonances,"83 I think we can inter-
pret this definition of discant by Jacobus-indirectly at least-as a definition
of 'consonance':
Discant is called the consonance of distinct melodies
because-just as consonance requires distinct sounds mixed
together simultaneously-so discant (requires) distinct
melodies mixed together simultaneously; and just as not all
simultaneously mixed sounds will be heard as smooth and
sweet mixtures, so not all distinct melodies mixed together
simultaneously will produce discant; but those which con-
cord with each other become, by virtue of their concord, like
one melody [quasi cantus unus], although there are many,
just as from the distinct sounds of the octave or fifth there
is brought about-by virtue of the concord-one sound, as
it were [quusi sonus unus]. Whoever therefore discords with
another does not discant. What discant is, then, (is) nothing
but two or more distinct melodies (sounding)-by virtue of
the consonance-as one melody. To discant is to make two
or more distinct melodies-through smooth concord-like one
melody; or, discant is the making of a melody above the tenor,
distinct from it, but because of the smooth mixed sound, like
one melody. To discant is to perform above the tenor or tenors
other sounds at the same time with it, sounds (which are)
concordant with it. He discants then who sings sweetly
together with another or others, so that from distinct sounds
it becomes like one sound...84
Thus-like a litany-the phrase "quasi sonus (or cantus) unus" is repeated
over and over again-six times in all-as though Jacobus felt a necessity to
display this idea in all of its possible permutations and combinations with the
other elements of discant. And-in retrospect-the definitions of 'consonance'
and 'dissonance' by the earlier polyphonic theorists appear in sharper outline.
In fact, the whole development of polyphony during these first centuries of
its history is freshly illuminated. It had originally been inspired by a desire
to glorify, amplify, or intensih the traditional plain-chant-without in any way
obscuring or distracting the listener's attention from the chant itself. In the
natural course of its development, polyphonic practice had gradually become
more and more elaborate-eventually culminating in the magnificent organa
quadrupla of Perotin and the School of Notre-Dame in the early 13th century.
But during this whole period-and until sometime after the death of Perotin
himself-this inspiration was never lost sight of, and the original desire to "in-
tensify" the chant without obscuring it continued to be an essential d e t e d -
nant of polyphonic practice. The fact that music in the second half of the 13th
century had already begun to overstep the stylistic boundaries i m p e d by these
criteria is suggested by the notorious Papal decree of 1322, which reads, in
part, as follows:
Certain disciples of the new school, much occupying
themselves with the measured dividing of the tempora,
display their prolation in notes which are new to us, prefer-
ring to devise methods of their own rather than to continue
singing in the old way.. .Moreover, they truncate the melodies
with hoquets, they deprave them with discants, sometimes
even they stuff them with upper parts made out of secular
songs. So that often they must be losing sight of the fun-
damental sources of our melodies in the Antiphoner and
Gradual, and may thus forget what that is upon which their
superstructure is raised... This state of things, hitherto the
common one, we and our brethren have regarded as stan-
ding in need of correction; and we now hasten therefore to
banish those methods, nay rather to cast them entirely away,
and to put them to flight more effectually than heretofore,
far from the house of God. Wherefore.. .we straitly command
that no one henceforward shall think himself at liberty to at-
tempt those methods, or methods like them, in the aforesaid
Offices, and especially in the canonical Hours, or in the
solemn celebrations of the Mass.. .Yet, for all this, it is not
our intention to forbid, occasionally-and especially upon
feast days or in the solemn celebrations of the Mass and in
the aforesaid divine offices-the use of some consonances,
for example the eighth, fifth, and fourth, which heighten the
beauty of the melody; such intervals therefore may be sung
above the plain cantus ecclesiasticus, yet so that the integri-
ty of the c a n m itself may remain intact, and that nothing
in the authoritative music be changed.. .'I5
By the very measures that were used to justify this condemnation of the newer
methods, the earlier polyphonic music was evidently deemed to have been of
a kind that would "heighten the beauty of the melody" while leaving "in-
tact.. .the integrity of the cantus.. .ecclesiasticus. "

30. The early-polyphonic period (CDC-2)
Music theory in the 9th through 12th centuries had been in a state of flux
which was quite without any historical precedent-and this obviously in
response to the profound changes that were taking place in musical practice.
The avalanche of new procedures, new conceptions-but above all-new
auditory experiences which must have followed one another in rapid succes-
sion during this period suggests a comparison with our own time perhaps more
than any other period in the history of western music. A new musical
''parameter" had come into existence-in addition to the parameters pitch and
time, which were the primary dimensions of monophonic music. This had
created a need to organize the perceptually distinct "values" in this new
parameter-to devise a scale of such values analogous to the scale of pitches-
just as the necessity for the rhythmic coordination of several polyphonic parts
had created a need to organize different time values, and for a r h y t h c nota-
tion to represent them. Both of these needs were finally satisfied in the 13th
century by the formulation of the system of rhythmic modes, on the one hand,
and on the other, the development of systems of classification of simultaneous
dyads with respect to consonance and dissonance.
As it turned out, the period of relative stability which might have been in-
itiated by these theoretical solutions was not destined to last very long. Already
in Franco's Ars cantus mensurabilis certain modifications of modal rhythmic
theory are evident, and later developments of the ars nova and other styles
in the 14th century eventually yielded a new system of intervalclassification
which was radically different from thoe of the 13th century.86The conception
of consonance and dissonance implicit in this new classification system will
be called CDC-3, and will be the subject of the next Section of this book. But
before moving on, it is important to note that-just as was the case with
CDC- I-CDC-2 still CDC- 1 exists as a musically meaningful concept. It is
often confused with other senses of the CDC which developed later, but it
is to be found in a relatively pure form in the concept of "tonal fusion"
(Tonverschmekung)enunciated by the 19th-century theorist Carl Stumpf, as
expressed, for example, in the following passage from his article "Konsonanz
und Dissonanz" (1898):
The combined sound of two tones approximates-now more,
now less-the impression of a single tone, and it appears that
the more this condition holds, the more consonant is the in-
terval. Even when we perceive and distinguish the tones as
two, they nevertheless form a whole in perception, and this
whole strikes us as more or less unitary. We find this pro-
perty with simple tones, just as with those with overtones.
That the octave sounds effectively like a unison, even when
we can clearly distinguish two tones in it, is always admit-
ted, although it is nothing less than self-evident, but it is a
most remarkable fact. This same property becomes weaker,
however, even with fifths and fourths, and still weaker with
thlrds and sixths...That is the rock, discarded by the builders,
which we make the c~rnerstone.~~
A comparison of this passage with the definition of discantus given by Jacobus
The earljrpolyphonic period (CDC-2) 3 1.
of ~ i e ~ ep. 28 will show that the meaning of 'consonance' implicit there (quasi
sonus unus ' 3 was virtually identical to the meaning suggested here by Stumpf.
Stumpf makes a very clear distinction between the consonance or dissonance
of successive vs. simultaneous tones (and thus between what I am calling CDC-1
and CDC-2), attributing the former (with Helmholtz) to the coincidence of
upper partials (i'Zusammenfallen von fieiltonen"), as the physical basis for
tone-relations ("Tonverhiiltniss ") or relationships (' 'Vewandtschgf?en''1. But
even in his consideration of simultaneous tones, discrepancies naturally arose
between the results of his psychological experiments on fusion and those of
other forms of the CDC which will be seen (in subsequent Sections of this
book) to have emerged later. Some years after this work on fusion, as Nor-
man Cazden tells it:
...Stumpf came to believe that as soon as combinations of
more than two tones are involved, a new and different level
of musical response arises on which operate the more com-
plex relationships among chords. Thus he regarded "Kon-
sonanz" and "Dissonanz" less asfw2damental than as merely
elementary values with little direct bearing on the art of
music, while the practice of musical harmony was seen to
involve the motion of chords, chord progressions, rather than
two-tone intervals of theoretical purity judged in isolation.
Appropriate to such a higher level of chord action, new laws
arise, which are best deduced from the observation of ac-
tual harmonic practice in music, and which cannot be ac-
counted for by the raw properties of consonant agreement.
For the sake of clarity, Stumpf proposed that the terms Con-
cordance and Discordance be applied to the qualities perceiv-
ed on this level of functional harmony
Stumpf s later distinctions between 'consonance' and 'dissonance' (and bet-
ween 'dissonance' and 'discordance') would roughly correspond to the distinc-
tions I will make later (Part m) between CDC-2 and what will be called CDC-4,
but the historical and aesthetic implications he apparently attached to these
distinctions were very different from those I will draw from them. Perhaps
if Stumpf had been prepared to limit the application of his concept of fusion
to the early-polyphonic period-during which it was, in fact, the prevailing
musical conception of consonance and dissonance-he would not have had to
relegate "fusion" to a position so unimportant as to be "merely elemen-
tary. .with little direct bearing on the art of music."

NOTES-Part One: Section I
1. An indication of the currently equivocal status of consonance and dissonance is the
conspicuous absence of entries for either of these terms in the otherwise very com-
prehensive Dictionary of Contemporary Music, edited by John Vinton (New York: Dut-
ton, 1971).
2. 7he Oxford English Dictionary (London: Oxford University Press, 1933), rep. 1961),
Vol. 11, p. 866 and Vol. III, p. 515.
3. Webster 's 7hird International Dictionary (Springfield, Massachusetts: G. & C. Mer-
r i m Co., 1971), pp, 484 and 657.
4. John Backus, Ihe Acoustical Foundations of Music (New York: Norton, 1969), p.
116.
5. Leo Kraft, Gradus (New York: 1976), Book I, p. 29.
6. Willi Apel, Harvard Dictionary of Music (Cambridge, Massachusetts: Harvard
University Press, 1953). D. 180.. .
7. ~ e ~ a r d i n ~these pleas~ntlunpleasantconnotations of consonance and dissonance,
Apel adds, later in the same Harvard Dictionary article: "In spite of numerous efforts
no wholly satisfactory explanation and definition of consonance and dissonance has
yet been found. The shortcoming of the explanation [quotw lies not so much in the
fact that it is based entirely on subjective impressions, but.. .in its failure to account
for the consonant quality of the fourth and fifth.. .It is chiefly for this reason that the
'pleasant-unpleasant-theory' cannot be considered satisfactory."
8. Paul Hindemith, Crafi ofMusica1 Composition, Vol. I, translated by Arthur Mendel
(New York: Schott Music, 1937), p. 85.
10. Arnold Schoenberg, Style and Idea, edited by Leonard Stein (New York: St. Mar-
tin's Press, 1975), pp. 282-3.
1I. Bid., pp. 260-6 1.
12. Igor Stravinsky, Poetics of Music, translated by Arthur Knodel and Ingolf Dahl
(Cambridge, Massachussetts: Harvard University Press, 1942), p. 34.
13. Knud Jeppesen, Tke Style of Palestrina and the Dissoruuce (London: Oxford Univer-
sity Press, 1946, and New York: Dover, 1970), p. 131.
14. Arthur Koestler, Ihe Sleepwalkers (New York: Grosset & Dunlap, 1959), p. 29,
15. E. de Coussemaker, Histoire de I'Harmonie au Moyen Age (Paris: 1852, rep.
Hildesheim: Georg Olrns, 1%6), pp. 2-3: "Le mot 'harrnonie'. ..signifiait chez les Grecs
1' arrangement ou I'enchainement des sons considCrCs sous le rapport mklodique de
leur acuiti ou de leur gravitk. Ce n'etait point le mklange de plusieurs sons frappant
I'oreille en m2me temps.. .Nous ne pdtendons pas dire par l i que la musique i sons
simultam5 soit exclue des traitis grecs sur la musique, ou qu'il n'y soit question que
de mklodie; on y trouve en effet plus d'un passage ob il est par16 de ce que nous ap-
pelons 'harmonie'. Nous voulons seulement demontrer que le mot 'harmonie' n'avait
pas chez les Grecs la signification restrictive qu'il a aujourd'hui, et qu'on serait dans
I'erreur si on le pemait dans ce sens."
16. J. A. Philip, Pythagoras and the Pythgoreans (Toronto: University of Toronto
Press, 1966), pp. 123-24.
17. Aristoxenus, Ihe Harmonics, edited with translation, notes, introduction, and in-
dex of words by Henry Stewart Macran, Oxford, 1902 (Hildesheim: Georg Olrns, 1974),
p. 165.
18. Ibid., p. 188
Notes-Part One: Section 1 33.
1 9 7he New Oxford History of Music, Vol. I , edited by Egon Wellesz (London: OX-
f&university-press, 1957), pp. 340-41.
20. Gustave Reese, Music in the Middle Ages (New York: Norton, 1940), p. 250.
21. Aristoxenus, op. cit., p. 205.
22. Aristoxenus, o p cit., p. 198.
23. Bid., pp. 188-9.
24. Bid., p. 206.
25. For an excellent and very detailed survey of Pythagorean mathematicallmusical
concepts see Richard Cracker's "Pythagorm Mathematics and Music," Journal of
Aesthetics and An Criticism, Vol. 22 (1963-64), pp. 189-198 and 325-335.
26, Theon of Smyma, Mathematics Usefulfor Understanding PInto (2nd. c., A.D.),
translated by Robert and Deborah Lawlor from the 1892 GreeWFrench edition of J.
Dupuis, edited and annotated by Christos Todis (San Diego: Wizards Bookshelf, 1979).
p. 62.
77 G.S. Kirk and J.E. Raven, Ihe Presocratic Philosophers (London: Cambridge-.. -
University Press, 1957), pp. 236-7.
28. Theon subdivides the consonant intervals into "antiphonic" (octave and double
octave) and "paraphonic" (fifthand fourth), and Gaudentius also refers to "paraphonic"
intervals (the ditone and tritone) as "intermediate between consonance and dissonance"
(see Theon of Smyma, op. cit., pp. 33-4, and Ruth Halle Rowen, Music 7hroughSources
and Documents (Englewood Cliffs, N.J. : Prentice-Hall, 1979), pp. 24-5).
29. Oliver Strunk, Source Readings in Music History (New York: Norton, 1950), p. 38.
30. Ibid.
3 1. Theon of Smyrna, op. cit., pp. 33-5.
32. Rowen, loc. cit.
33. Strunck, op. cit., p. 89.
34. Curt Sachs, Ihe Rise of Music in the Ancient World (New York: Norton, 1943),
p. 258.
35. I have not yet found access to the 16th-century De insitutione musica by Boethius,
which Gustave Reese has called "...possibly the most influential treatise in the history
of music" (Fourscore Chsics of Music Literature, 1957 (New York: Da Capo, 1970),
p. 12). This leaves a rather profound gap in my documentation of source materials
from the early-Medieval period, but I doubt that this will affect the general conclu-
sions of this book in any essential way.
36. 7he New Oxford History of Music, Vol. II, edited by Dom Anselm Hughes (Lon-
don: Oxford University Press, 1954), pp. 270-71.
37. Jean-Phili~~eRameau, Treatise on Harmony (1722), translated by Philip Grossett
(New York: Dover, 19711, p. 60.
38. Arnold Schoenberg, op. tit., P- 270.
39. Paul Hindemith, op. cit., p. 56.
40. Ibid., p. 85.
(NOTES: Section LI)
41. Hughes, op. cit., p. 276.
42. Warren Babb, translator, Hucbald, Guido and John on Music, edited, with introduc-
tions, by Claude V. Palisca (New Haven and London: Yale University Press, 19781,
i 3 . Ibid., p. 5.
4 4 . Johannes de Grocheo, Concerning Music (De musica), translated by .Albert Seay

34. NOTES-Part One: Section II
(Colorado Springs: Colorado College Music Press, 1967, 2nd edition, 1974), p. 4.
The original Latin text may be found in Ernst Rohloff, Die Quellenhandschnften zum
Musiktraktat des Johannes de Grocheio (Leipzig: VEB Deutscher Verlag, 1943). The
relevant passage reads as follows: "The principles of music are normally called con-
sonances and concords [consonantiae er concorduntiae]. I say concord whenever one
sound is harmonically continued by another, just as one moment of time or a motion
is contonuous with another. I say consonance whenever two or many sounds give one
perfect harmony, united at the same moment and at the same time ...First, one must
discuss consonances, for it is through consonances that concords are found."
45. Rowen, op. cit., p. 75 (original Latin text in Martin Gerbert, Scriptores Ecclesiastici
De Musica Sacra Potissum (hereafter abbreviated GerS), 3 vols. (St. Blasien, 1784;
reprint, Hildesheim: Georg Olms, 1963), Vol. I, p. 160.
46. Rowen, op. cit., v. 77.
47. Strunk, ob. cit., 6. 126.
48. Babb, op. cit., p. 63 (see also GerS II, p. 7).
49. Ibid., pp. 63-65 (and GerS 11, pp. 8-9).
50. More literally translated by Palisca in his introduction: "notes disjoined from each
other both concordantly dissonating and dissonamly concording"-and aptly describ-
ed by him as "Guido's elegant antithesis" (ibid., p. 54).
51. Ibid., p. 77 (and GerS 11, p. 21).
52. Ibid., pp. 110-111 (and GerS 11, p. 237).
53. Ibid., p. 78 (and GerS II, p. 21).
54. Ibid., pp. 159-60.
55. The copulario was a kind of cadence formula involving the last two dyads in a
phrase, the final one always being either a unison or an octave.
56. Jay A. Huff, translator, Ad Organum Faciendwn & Item de Organo (Brooklyn:
The Institute of Mediaeval Music, (n.d.)), p. 42.
57. Ibid.. 0 . 60.. . -
58. Fred Blum, "Another Look at the Montpellier Organum Treatise," Musica
Disciplina, Vol. 13 (1959). pp. 15-24 (includes both the Latin text and an English
translation).
59. Huff, op. cit., p. 37, footnote**
60. Hughes, op. cit., pp. 282-4.
61. Janet Knapp, "TW; 13th Century Treatises on Modal Rhythm and the Discant"
(including English translations of Discantus positio vulgaris and De musica libellu).
Journal of Music Theory, Vol. 6 (1962), pp. 200-215.
62. Johannes de Garlandia, Concerning Measured Music (De rnensurabili musice),
translated by Stanley H. Birnbaum (Colorado Springs: Colorado College Music Press,
1978). pp. 15-17 (original latin text in E. de Coussemaker, Scriptorum de Musica Medii
Aevi (hereafter abbreviated CouS). 4 vols. (Paris. 1864: reprint, Hildesheim: Georg
Olms. 1963), Vol. I, vv. 104-6.
63. Franco of Cologne, Ars cantus mensurabilis, English translation in Strunk, op.
cit., pp. 139-159 (see also CouS I, pp. 117-135).
64. Anonymous I, "tractatus de consonantiis musicalibus," in CouS I, pp. 296-302.
65. Albert Seay, translator, Anonymous 11: Tractatus de Discanru (Colorado Springs:
Colorado College Music Press, 1978); (includes the Latin text).
66. Luther Dittmer, translator Anonymous N (Brooklyn: The Institute of Mediaeval
Music, 1959); original text in CouS I, pp. 327-365.
67. Johannes de Garlandia, op. cit., pp. 15-16.
68. Ibid., pp. 17-18.
69. CouS I, pp. 251-281.
70. Not only is the text in CouS I internally inconsistent; it differs from Coussemaker's
own quotations from it in his earlier Histoire ...cited above (footnote 15).
NOTES-Part One: Section 11 3 5 .
71. Hughes, op. cir., p. 285.
72. Blum, op. cit., p. 20.
73. Dittmer, op. cit., p. 63.
74. Ibid., p. 64.
75. Walter Odington, De Speculutione Musicae, edited by Frederick F. Hamrnond (Stut-
tgart: American Institute of Musicology, 1970), pp. 74-75.
76. Hughes, op. cir., pp. 339-40.
77, Leon Platinga, "Philippe de Vitry's Ars Nova: A Translation," Journal of Music
Theory, Vol. 2 (1961), pp. 204-223.
78. &chard Crocker, "Discant, Counterpoint, and Harmony ," Journal of the American
Musicological Society, Vol. 15 (1962), pp. 1-21.
79. Ibid., p. 4.
80. fbid., p. 5.
81. Ibid.
82. Vide supra, p. 28.
83. Strunk, op. cit., p. 152.
84. Jacobus Leodiensis, Speculum musicae, Book VIJ, Ch. IV ("Quid sit discantus"),
in CouS Ii, P. 387: "Dicitur discantus consonantia distinctorum cantuum, quia sicut
consonantia requirit distinctas voces sirnul mixtas, sic discantus distinctos cantus simul
mixtos; et sicut non quicunque soni simul mixti faciunt mixtionem suaviter dulciterque
auditui se facientem, sic nec omnes distincti cantus simul mixti discantum faciunt; sed
illi qui invicem concordat ut per bonam illorum concordiam ex illis fiat, quasi cantus
unus, cum sint plures sicut ex distinctis vocibus ipsius dyapason vel dyapente propter
bonam concordiam efficitur quasi sonus unus. Qui ergo cum alio discordat, non discantat.
Quid est igitur discantus nisi duorum cantuum vel plurium distinctorum propter bonam
concordiam quasi cantus unus. Discantare est de duobus vel pluribus distinctis can-
tibus propter suavem concordiam quasi cantum unum facere; vel discantus est supra
tenorem cantus factus ab iUo distinctus, sed propter suavem vocum mixtionem quasi
cantus unus. Discantare est supra tenorem vel tenores voces alias simul cum illis pro-
ferre voces illis concordantes. Discantat igitur qui simul cum uno vel pluribus dulciter
cantat, ut ex distinctis sonis quasi unus fiat.. ."
85. Both the original latin text and an English translation are to be found in H.E.
Wooldridge, 7he Oxford History of Music, Vol. I, 2nd edition, revised by Percy C.
Buck (London: Oxford University Press, 1929; reprint, New York: Cooper Square
Publishers, 1973), pp. 294-6.
86. It is interesting to note in this regard that the major theorist-spokesmen for the
ars nova-Philippe de Vitry and Jean de Muris-were so exclusively concerned with
rhythmic organization and notation that they do not deal with the question of con-
sonanceldissonanceclassificationat all (at least in the works definitely attributed to them).
87. Carl Stumpf, "Konsonanz und Dissonanz," Beitrage zur Akustik und
Musikwissenschafi, 1 (Leipzig: Johann Ambrosius Barth, 1898), p. 35: "Der Zusam-
menklang zweier Tone niihert sich bald mehr, bald weniger dem Eidruck E i e s Tones,
und es zeigt sich, dass dies urn so mehr der Fall ist, je konsonanter das Interval1 ist.
Auch damn, wenn wir die Tone als zwei erkennen und auseinanderhalten, bilden sie
doch ein Ganzes in der Empfmdung, und dieses Ganze erscheint uns bald mehr, bald
weniger einheitlich. Wir fmden diese Eigenschaft bei einfachen Tonen ebenso wie bei
K1'kngen mit Oberrijnen. Dass die Oktave dem wirklichen Unisono iihnlich klingt, auch
wenn wir deutlich zwei Tone darin unterscheiden konnen, ist allezeit anerkannt worden,
obschon es nichts weniger als selbstverstihdlich, sondern eine h k h s t merkwiirdige
Thatsache ist. Dieselbe Eigenschaft kehrt aber in angeschwachter Weise auch bei Quinten
und Ouarten. ia bei Terzen und Sexten weider. Das ist der Stein, den die Bauleute
verw;rfen ha&, den wir zurn Eckstein machen."
88. Norman Cazden, "The Definition of Consonance and Dissonance," unpublished,
1975, p. 8.

Part Two
From the "Ars Nova" through the
"Seconda Pratica"

Section I11
I The contrapuntal and figured-bass periods, ca. 1300-1700 (CDC-3).
The new system of interval-classification which emerged in theoretical
writings sometime during the 14th century differs from those of the 13th cen-
tury in several ways, but the most striking of these differences is that the number
of consonanceldissonance categories has been reduced from five or six to just
three-' 'perfect consonances, '' ''imperfect consonances, '' and "dissonances. ''
~ o t hthe major and the minor sixth (as well as the thirds) are now accepted
as consonances (albeit "imperfect" ones), the fifth has been elevated from
an intermediate to a perfect consonance whereas the fourth has become a special
kind of dissonance (or rather, a hlghly qualified consonance). All of the other
intervals-if allowed at all in the music-are simply called "dissonances."
There is a virtually unanimous consensus among theorists of the 14th through
17th centuries regarding this system of classification-and it is, in fact, essen-
tially identical to that still used in current textbooks on counterpoint and har-
mony. The efforts to distinguish and classify finer shades of relative consonance
and dissonance are now seen to have been a uniquely 13thcentury phenomenon.
How we interpret this reduction in the number of categories in the theoretical
interval-classificationsystems is crucial to an understanding of the later history
of the CDC. Obviously, it should not be taken to mean that post-13th-century
theorists' power of discrimination had become less acute than those of their
13th-century counterparts, so that the finer distinctions observed by John of
Garland or Franco of Cologne were no longer perceptible to them. Not can
it mean that their powers of discrimination had become more acute, leading
to a classification system that was in some way more "accurate" than those
of the 13th century. What had changed was not the theorists' powers of
discrimination at all, but simply their criteria for consonanceldissonance
classification, and these were now related to the newly developing rules of
counterpoint. In CDC-3, all dissonant intervals are subsumed in one undif-
ferentiated category because they are all treated the same way in these rules.
The intermediate category of consonances has been dropped, but the other two
have been retained because the rules drffer for the treatment of perfect vs. im-
perfect consonances regarding cadences and consecutive dyads in parallel mo-
tion. There is thus established a precise, one-to-one correspondence between
the rules of counterpoint and the consonance/dissonance categories referred
to by those rules-'consonance' and 'dissonance' are now defined operationally,
according to the intended functional behavior of the various dyads in the music.
This had not been the case in 13th-century theoretical writings. The finer
distinctions between varying degrees of consonance and dissonance made by
13th-century theorists were not "operational" distinctions at all,since the rules
articulated by them regarding the way different dyads were to be used in com-
position merely assumed a distinction between the two broad categories of con-
sonance and dissonance; consonances (of any kind) could be used freely, while
dissonances (of any kind) were to be used only under certain conditions- as
in the following statements by Franco:
The discant begins either in unison with the tenor.. .or at the
diapason [or one of the other "concords"]. .. , proceeding

40. The contrapuntal and figured-bas periods (CDC-3)
then by concords, so that when the tenor ascends the discant
descends, and vice versa.. .Be it also understood that in all
the [rhythmic] modes concords are always to be used at the
beginning of a perfection, whether this beginning be a long,
a breve, or a semibreve...He who shall wish to construct
a triplum ought to have the tenor and discant in mind, so
that if the triplum be discordant with the tenor, it will not
be discordant with the discant, and vice versa. And let him
proceed further by concords, ascending or descending now
with the tenor, now with the discant, so that this triplum is
not always with either one alone...He who shall wish to con-
struct a quadruplum or quintuplum ought to have in mind
the melodies already written, so that if it be discordant with
one, it will be in concord with the others. Nor ought it always
to ascend or descend with any one of these, but now with
the tenor, now with the discant, and so forth.'
By the mid-14th century there had occurred a clear shift of theoretical con-
cern from the sonorous qualities of simultaneous aggregates to the ways in
which their various qualities might be (or rather, were being) used in music-
and with that, to a strictly operational correspondence between their interval-
classification and the rules of what was now being called "counterpoint."
Regarding these rules themselves, there was somewhat less of a consensus
among theorists than there was in their interval-classifications, but during the
15th and early 16th centuries most theorists did seem to agree on the follow-
ing: (1) a piece should begin with a perfect consonance, and (2) it should also
end with a perfect consonance; (3) consecutive parallel perfect consonances
of the same kind were to be avoided, whereas (4) such consecutive imperfect
consonances might be used freely; (5) dissonances were not to be used in note-
against-note textures at all, although-in "florid" or "diminished"
counterpoint-they were allowed in unstressed rhythmic positions and shorter
note-values; and (6) stepwise and contrary motion were preferred, if not ab-
solutely required (this last rule-or pair of rules-was carried over from an
earlier period, and is often not stated explicitly, though it generally seems to
be assumed). Toward the end of the 15th century, in the writings of Tinctoris
and Gafurius, we see the first of these rules begin to break down-imperfect
consonances are allowed at the be

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