arXiv:1407.5757v1 [math.HO] 22 Jul 2014

MATHEMATICS AND GROUP THEORY IN MUSIC

ATHANASE PAPADOPOULOS

Abstract. The purpose of this paper is to show through particular examples

how group theory is used in music. The examples are chosen from the theoret-

ical work and from the compositions of Olivier Messiaen (1908-1992), one of

the most inﬂuential twentieth century composers and pedagogues. Messiaen

consciously used mathematical concepts derived from symmetry and groups,

in his teaching and in his compositions. Before dwelling on this, I will give a

quick overview of the relation between mathematics and music. This will put

the discussion on symmetry and group theory in music in a broader context

and it will provide the reader of this handbook some background and some

motivation for the subject. The relation between mathematics and music, dur-

ing more than two millennia, was lively, widespread, and extremely enriching

for both domains.

2000 Mathematics Subject Classiﬁcation: 00A65

Keywords and Phrases: Group theory, mathematics and music, Greek music,

non-retrogradable rhythm, symmetrical permutation, mode of limited trans-

posi tion, Pythagoras, Olivier Messiaen.

This paper will appear in the H andbook of Group actions, vol. II (ed. L. Ji, A.

Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press.

The author acknowledges support from the Erwin Schr¨odinger International

Institute for Mathematical Physics (Vienna). The work was also funded by

GREAM (Groupe de Recherches Exp´erimentales sur l’Acte Musical ; Labex

de l’Universit´e de Strasbourg), 5, all´ee du G´en´eral Rouvillois CS 50008 67083

Strasbourg Cedex. Email: papadop@math.unistra.fr

1. introduction

Mathematics is the sister as well as the servant of the arts.

(Marston Morse in [81])

2. introduction

Music is a privileged ground for an alliance between arts and sciences, and in

this alliance, mathematics plays a central role . In the ﬁrst part of this paper, I will

highlight some elements o f this relation a nd I will also point out some important

works done in this ar e a which are due to mathematicians and which are spread over

several centuries. In the second part (§3 to 6), I will discuss in some detail the group

theory that is involved in the compositions a nd in the theoretical work of Olivier

Messiaen, one of the major twentieth-century composers and music teachers.

Let me mention right away that beside s group theory, there are many other ﬁelds

of mathematics that are involved in music theory, in composition and in musical

analysis: geometry, probability, category theory, combinatorics, graph theory, etc.,

Date: July 23, 2014.

2 ATHANASE PAPADOPOULOS

but I do not develop any of these ideas here because they do not really belong to

the subject of this handbook. In fact, any mathematical theory or idea may have

its counterpart in music. Let me also mention that there are presently some very

active music research groups in which mathematics plays a major role, like the

IRCAM group in Paris,

1

whose members include Moreno Andreatta, Emmanuel

Amiot, G´erard Assayag, Chantal Buteau, Marc C hemillier, Jan Haluska, Franck

Jedrzejewski and Fran¸cois Nicolas. Andreatta is one of the leaders of the group;

several of his works are related to group theory; see his habilitation document [2]

and the references there ; e.g. [3] [4] [5] [8]. The writings of Jedrze jewski are also

related to symmetry and groups; s e e [58] [59] [60] [61] [62] [63] [64] [65] [66]. There

are many other mode rn writings on mathematics and music, done by researchers

in France and outside France; see e.g. [13], [25], [39], [43], [44], [73], [4 9], [68]

[56], [71], [105]; see also the shor t introduction [85] which is more intended for

mathematicians.

I would like to thank Moreno Andreatta, Ma ttia Bergomi, Pierre Jehel, Thomas

Flore, and Franck Jedr zejewski for their detailed comments on an early versio n of

this paper. Andreatta and Jedrzejewski gave me (among other things) valuable

biographical references.

3. A brief overview of the interaction between

mathematics and mus ic

Historically, mathematics and music are intricately linked. Pythagoras , who is

considered as the founder of the ﬁrst school of mathematics as a purely deduc-

tive science, is also the founder of a school of theoretical music (may be also the

ﬁrst one).

2

Besides being a mathematician, Pythagoras was a music theorist and a

composer, and his biographers des c ribe him as playing several instruments (see for

instance [90] and [57]). We owe him the discovery of the fundamental correspon-

dence between musical intervals (that is, pairs of pitches, or of musical notes) and

numerical ratios. The quickest way to desc ribe this correspondence is by saying

that to a musical interval, we associate the ratio of the frequency of the higher note

to that of the lower-pitched note. Although the Pythagoreans did not talk about

the frequency of a note, they were aware of this correspondence between musical

interva ls and fra ctions. The Greek mathematicians were aware of the fac t that

sound is produced from a periodic vibration in the air, and that a sharper note

corres ponds to a more rapid vibration. These views were known for instance to the

author of the Division of the canon [33], which is presumably Euclid, based on an

earlier version due the Pythagorean mathematician Archytas (428-347 b.c.).

We also owe to Pythagoras the ﬁrst classiﬁcation of conso nant intervals . We

recall that consonance results from playing together two (or more) diﬀer ent sounds,

and the main question in this domain is when does such a combination give a

harmonious (or “consona nt”) sound a nd what is the reason for that. This question

occupied several mathematicians and scientists, and a mong those who wrote on

1

Institut de Recherche et Coor dination Acoustique/Musique.

2

It is fair to add right away that the culture of Greek antiquity is, in its turn, indebted to

other cultures. Several of the major Greek philosophers and scientists travelled widely, and they

acquired an important part of their knowledge from older Eastern civilizations. For instance,

the compiler Plutarch (c. 40-120 a.d.) writes in his essay On Isis and O si ris ([88] Chap. 10,

354B p. 130): “Solon, Thales, Plato, Eudoxus and Pythagoras, and some say, Lycurgus [...]

came to Egypt and were in touch with the priests there”. Plutarch even gives the names of the

Egyptian priests from whom these scholars received their teaching. In the same book, Plutarch

(who was a Delphic priest), provides an explanation of some of Pythagoras’ aphorisms by making

a comparison between these sayings and Egyptian (hieroglyphic) writings. There are many other

sources of information on the inﬂuence of Eastern civilizations on Greek culture.

MATHEMATICS AND GROUP THEORY IN MUSIC 3

this subject we mention Aristotle, Euclid, Ptolemy, Desc artes, Huygens, Galileo,

Kepler, Me rsenne, d’Alembert and Euler.

The two major Py thagorean discoveries, namely, the correspondence between

musical intervals and numerical ratios, and the c lassiﬁcation of consonances together

with the questions related to this classiﬁcation, are at the basis of all the subsequent

theories of harmony.

Pythagoras did not leave anything written – or at least, no writing of him sur-

vives.

3

But several treatises on harmony, written by later mathematicia ns and based

on Pythagorean ideas, survive at least in part; we shall mention a few of them be-

low. The disc overies of Pythagoras are described in the Handbook of Harmonics

of Nicomachus

4

(see [17], [84] [27]) and in a biography written by Iamblichus [57].

5

Aristotle, who is a reliable source, reports in his Metaphysics ([11] A5, 986a16) that

Pythagoras used to say that “everything is number”.

The works of the Pythagore ans reached us in the form of quotations, in relatively

small numbe r, but very r ich in content, see e.g. the volumes [91], [17], and [92].

Let us also recall that, in principle, every mathematical treatise of cla ssical

Greece contained a chapter on music. In fa ct, such a treatise usua lly consisted

of four parts: Number theory, Music, Geometry, and Astronomy, in that order,

because the part o n music was based on the r e sults of number theory, and the

part on as tronomy was based on the r e sults of geometry. To give the reader an

idea of the important connections between number theory and music, we reca ll

that the theory of proportions and the theory of means were developed precisely

for their use in music theory. The division of a musical interval into two or more

subintervals is deﬁned in terms of prop ortions that depend on the pitches of the

musical notes involved, and this division was formulated in terms of ratios of lengths

of the subinterva ls to the length of the whole interval. This mathematical theor y

of division of musical intervals was made possible by the Pythagorean discovery of

the correspondence between musical intervals and fractions that we mentioned, and

of the logarithmic law that governs this correspondence (concatenation of musical

interva ls corresponds to a multiplication at the level of the numerical values).

6

To introduce things more precise ly, let us give a concrete example of how music

theory acted as a mo tivation for number-theoretic research. The example concerns

the arithmetic of the so-called superpa rticular ratios. These are the numerical ratios

of the form (n + 1)/n, where n is a positive integer. Superparticular ratios are

important for several reasons, one of them being that the corresponding musical

interva ls appear as the successive intervals in the decomposition of a sound into

3

According to Pythagoras’ biographers, it was part of his s trict rules – which he applied to

himself and to his followers – that the discoveries and the results in mathematics and music theory

obtained by the members of his school should not be written up but only taught orally to the

small circle of devotees which constituted that community; see e.g. [57].

4

Nicomachus of Gerasa (c. 60-120 A.D) was a neo-Pythagorean mathematician, well known for

his Introduction to Arit hmetic and his Enchiridion (or Handbook of Harmonics). His reverence

for number is expressed in another work called Theologumena Arithmetica (Theology of number).

5

Iamblichus (c. 245-325 a.d.) was a neo-Platonist philosopher, known for his cosmologica

system based on mathematical formalism. Both Nicomachus and Iamblichus were Syrian. (Syria,

at that time, was a Roman province). Onl y a small portion of the works of Nichomachus and of

Iamblichus survives.

6

This logarithmic law was known in Greek antiquity, and it was used long before logarithms

were formalized by mathematicians. For instance, Theon of Smyrna in his M athematical exposition

writes in [103] (p. 103): “Since the ratio of the consonance of octave is 2/1 and the one of the

consonance of a fourth is 4/3, the ratio of their sum is 8/3.” That is, he knew that the ratio of

the sum of two musical intervals is the product of the corresponding ratios. Likewise, Iamblichus,

in [57] §115, reports that Pythagoras noticed that “that by what the ﬁfth surpasses a fourth is

precisely the ratio of 9/8”; in other words, he saw that the diﬀerence between the two ratios 3/2

and 4/3 is 9/8 (which corresponds to the f act that 9/8 is 3/2 divided by 4/3).

4 ATHANASE PAPADOPOULOS

harmonics, and thus we constantly hear them in any sound that is produced around

us. Therefore, the ear is familiar with them, and this makes them important.

Another reason for which these ratios are consequential is that in the anc ient Greek

classiﬁcation, the so-called consonant intervals are either superparticular or of the

form n/1 (see e.g. [103], Chapter II). We already mentioned (see Footnote 6) the

values of the octave (2/1), the ﬁfth (3/2 ), the fourth (4/3) a nd the major tone (9/8).

Some other important superparticular ratios which are useful in music appear in

the tables that follow. The major and minor thirds, deﬁned respectively by (5/4)

and (6/5), started to be considered as “impe rfect consonances” at the thirteenth

century and they played, after that period an important role in composition. The

“Didymus comma” (also called the syntonic comma), whose value is (81/8 0), is the

diﬀerence between the major tone (9/8) and the minor tone (10/9), and it played

a signiﬁcant r ole in Greek theoretical music.

Besides the q ue stion of classiﬁcation, which was motivated by music, there are

purely mathematical developements. Indeed, several natural questions concerning

sup erparticular ratios were formulated a nd studied by the mathematically-oriented

Pythagoreans. I shall mention a few of them as examples; so me of these questions

are e asy, and others are diﬃcult.

(1) Ca n the square root of a superparticular ra tio be superparticular ? Or at

least, ca n it be rational?

A musical naturally related ques tion is the following: can we divide a

consonant interval into two equal consonant intervals?

(2) Given a superparticular ratio, can we enumerate all the vario us ways of

expressing it as a product of superparticular ratios? Is this number ﬁnite

or inﬁnite? Is the number of possibilities ﬁnite if we ﬁx a bound on the

number of fac tors?

This question is related to the question of dividing a c onsonant interval

into a c ertain number of consonant intervals and the problem of construct-

ing sca les whose all intervals are consonant.

(3) Given a ﬁnite set of primes, e.g. {2, 3}, {2, 3, 5} or {2, 3, 5, 7}, is the set of

sup erparticular ratios whose prime factors (of the numerator as well as the

denominator) belong to that set always ﬁnite? Can we enumerate all the

elements of this set?

This question is related to the construction of scales out of a ﬁnite set

of prime numbers. As an example, the reader can no tice that the numer-

ical values of the ratios tha t appear in the following table extracted from

Descartes’ Compendium are all multiples of 2, 3 and 5.

7

2/1 8ve

3/1 12th 3/2 5th

4/1 15th 4/2 8ve 4/3 4th

5/1 17th 5/2 10th maj. 5/3 6th maj. 5/4 2nd

6/1 19th 6/2 12th 6/3 8ve 6/4 5th 6/5 3rd min.

Table 1. A table of musical intervals, ordered according to the

denominator; e xtracted from Descartes’ Compendium musicae, [29]

Tome X p. 98.

7

Descartes writes in [28], p. 122: “All the variety of sounds, for what concerns pitch, stems

only from the numbers 2, 3 and 5; and all the numbers that deﬁne the [musical] degrees as well

as dissonance are multiples of these three sole numbers”.

MATHEMATICS AND GROUP THEORY IN MUSIC 5

Let me make a few more comments on these questions.

The response to Question 1 is known since antiquity. A proof of the fact that

there is no rational fraction whose square is equal to a superpar ticular ratio, at-

tributed to Archytas, a Pythagorean from the ﬁrst half of the fourth century b.c.,

is contained in Boethius’ Musical Institution, Book III. A more gener al result is a

consequence of Propos ition 3 of the Euclidean Section of the Canon which says the

following: For any pair of integers B, C whose quotient is equal to a superparticu-

lar ratio, there is no sequence of integers D, E, F, . . . N between B and C satisfying

B/D = D/E = E/F = . . . = N/C. (cf. [17], vol. II , p. 195). Note that in this

statement, the fraction B/C is no t necessarily in reduced form.

In Boethius’ Musical Institution III.5 d [19], the author mentions that in order

to circumvent the impossibility of dividing the tone (9/8) into two equal parts ,

Philolaos divided it into two unequa l parts , the one being “less than a semitone”,

which he called the diesis or lemma, and which is also called the minor semitone,

and the other one being “greater than a semitone”, called apotome. The comma is

the diﬀerence between these two intervals. (See also [91] p. 500 ).

Regarding Question 2, one can note that any s uperparticular ratio can be written

as a product of two others, using the following:

p + 1

p

=

2p + 2

2p

=

2p + 2

2p + 1

×

2p + 1

2p

.

We can then apply the same trick to the fraction

2p + 2

2p + 1

, or to

2p + 1

2p

(or to both),

and therefore the process goes on indeﬁnitely. In particular, this shows that every

sup erparticular ratio can be written as a product of superparticular ratios in an

inﬁnite number of ways.

Aristides Quintilianus, in Book III of his De Musica [10], used this method to

describe the division of the tone as 17/16 × 18/17 = 9/8, and then the following

division of semitones and of quarter tones: 33/32 × 34/33 = 17/16 and 35/3 4 ×

36/35 = 18/17.

There is another general method for obtaining products of superparticular ratios

which is based on the following equalities:

p + 1

p

=

3p + 3

3p

=

3p + 3

3p + 2

×

3p + 2

3p + 1

×

3p + 1

3p

.

This gives the following well known division of the fourth:

4

3

=

12

11

×

11

10

×

10

9

.

Question 3 was solved in the aﬃrmative by the mathematician Carl Størmer

who proved in [102] (1897) that for any ﬁnite set of primes {p

1

, . . . , p

n

}, there are

only ﬁnitely many superparticular ratios whose numerator and denominator are

products of elements in this ﬁnite s et. He also described a procedure to ﬁnd such

fractions. As a consequence, Table 3 gives the list of all superparticular ratios

whose pr ime factors of the numerator and denominato r belo ng to the set {2, 3, 5}.

Let us note that all the fractions in Table 3 were used in music, since Antiquity.

This is another example where music theorists were far ahead of mathematicians.

For more on superparticular ratios in music, see [47]. See also [97] and [99] for an

account of some combinatorial problems related to music theory.

Our next example is extracted from Ptolemy’s Harmonics (see [1 7] p. 203 and

[91] p. 5 28), where the author comments on the following divisions of the fourth

6 ATHANASE PAPADOPOULOS

(4/3) (such divisions are traditionally called tetrachords), which are due to Archy-

tas, and which are calle d respec tively enharmonic, chromatic and diatonic:

5

4

×

36

35

×

28

27

=

32

27

×

243

224

×

28

27

=

9

8

×

8

7

×

28

27

=

4

3

.

Several tables of ancient Gree k musical tetrachords are contained in Reinach’s

Musique Grecque, [98] see e.g. Table 2. Mos t (but not all) of the numerical values

in these tables are superparticular ratios.

Archytas 9/8 8/7 28/27

Eratos thenes 9/8 9/8 256/243

Didymus 9 /8 10/9 16/15

Ptolemy 10/9 11/10 12/11

Table 2. Diatonic genus (after Reinach)

Some more questions on superparticular ratio s in music are discussed in [47]

The next table of interva ls is extra c ted from Euler’s book on music, the Ten-

tamen [34], which we shall mention again in what follows. In the tradition of the

Greek musicologists, Euler made a systematic classiﬁcation of the useful musical

interva ls according to their numerical value s and he developed a theory of the mu-

sical signiﬁcance of the ordering in these lists. His tables, in the Tentamen, involve

the pr ime numbers 2, 3 and 5. But Euler also used the set {2, 3, 5, 7}, for instance

in his memoir Conjecture sur la raison de quelques dissonances g´en´eralement re¸cues

dans la musique [35], and this was considered as a novelty, compared to the smaller

set {2, 3, 5} which was used by his predecessors in the post-Renaissance Western

world.

In the twentieth century, Hindemith, in his famous treatise [5 1], also uses the

integer 7. In Greek antiquity not only the number 7 was used, but in principle

no number was excluded; we alr eady mentioned a few examples. Aristoxenus of

Tarentum, the great Greek music theorist of the fourth century b.c., tr ie d to make

exhaustive lists of scales where a larg e number of pr imes appear, see [18].

2/1 Octave

3/2 Fifth

4/3 Fourth

5/4 Major Third

6/5 Minor Third

9/8 Major Tone

10/9 Minor Tone

16/15 Diatonic Semitone

25/24 Chromatic Semitone

81/80 Didymus Comma

Table 3 . The list of superparticula r ra tios whose prime factors

belong to the set {2, 3, 5}.

Some relations b etween music theory and number theory are also manifested by

the ter minology. The Greek word “diastema” mea ns at the same time “ratio” and

MATHEMATICS AND GROUP THEORY IN MUSIC 7

“interval”. The same is tr ue for the word “log os”.

8

The theory of mea ns, in Ancient

Greece, found its main applicatio ns in music. Deﬁning the va rious means between

two given integers a and b (a < b) was seen practically as inserting various notes

in the musical interval who se numerical value is the quotient b/a. For instance,

the harmonic mean of the interval [6, 12] (which is 8) corresponds to the note that

divides an octave into a fourth followed by a ﬁfth. Thus, it is not surprising that

the olde st expositions of the theories of proportions and of means a re contained

in musical textbook s, and the examples, in these writing s, that illustrate these

mathematical theories are often bor rowed from music theory.

9

The discovery of

irrational numbers was motivated in part by the mathematical diﬃculty of dividing

a tone into two equal parts. The distinction continuous vs. discontinuous arose

from the attempt of splitting up the musical continuum into the smallest audible

interva ls. We note by the way that not all the intervals useful in music were rational.

Aristoxenus made a distinction between r ational and irrational musical intervals.

There a re also important repercussions of musical theories in geometrical prob-

lems, e.g. on the geometric divisions of the musical intervals and on the g eometric

constructions on means. Ptolemy (c. 90-168 a.d.), in his Harmonics (Book II,

ch. 2) describes a geometric instrument, called helicon, which was used to measure

consonances. There were also impacts on famous problems like the duplication

of the cube and on several questions on constructions with compass and straight-

edge. This came very naturally, since the same people who worked on music theory

worked on these geometrical problems.

The division of the teaching of mathematics into four parts, which was given

later on the name Quadrivium

10

(the “four ways”) lasted until the middle ages, and

the status of theoretical music as part of mathematics persisted in Western Europe

until the beginning of the Renaissance (c. 1550). A textbook on the quadrivium

available in French translation [103] is the one written by Theon of Smyrna (c7 0-135

a.d.) which we already mentioned.

11

One of the oldest Pythagor ean texts that sur vives describes geometry, arithmetic,

astronomy (referred to as spherics), and music as “sister sciences”. This text is a

fragment fr om a book titled On mathematics by Archytas, a Pythagorean from

the ﬁrst half of the fourth century b.c. and it is known through a quotation by

the philosopher Porphyry

12

in his Commentary on Ptolemy’s Harmonics (part of

which is attributed to the mathematicia n Pappus) [89]. The text of Porphyry

was later on edited, with a Latin translation accompanying the Greek original, by

the mathematician John Wallis (1616-1703), and it was published as part of his

collected works (Opera Mathematica [111] Vol. III).

13

Porphyry writes:

8

Theon’s treatise [103] contains a section on the various meanings of the word “logos”.

9

Examples of computations illustrating mathematical theories that have a musical signiﬁcance

may also be found in later works. For instance, in his famous Introductio in analysin inﬁnitorum

(Intr oduction to the analysis of the inﬁnite, published in 1748), Euler, whi le presenting his m ethods

of computation using logarithmes, explains how one can ﬁnd the twelfth root of 2, which in fact is

the value of the unit in the chromatic tempered scale. In Chapter VI of the same treatise, Euler

works out an approximate value of 2

7/12

, which of course corresponds to the ﬁfth. There are other

examples of this sort.

10

The Latin word quadrivium was introduced by Boethius (5th century a.d.).

11

The book, in the form it survives, contains three parts; the part on geometry is missing.

12

Porphyry (c. 233-309 a.d.) was a H el lenized Phoenician, born in Tyre (presently in

Lebanon). In 262 he went to Rome, where he stayed six years, during which he studied un-

der Plotinus, one of the main founders of neo-Platonism. He is known for his Commentary on

Ptolemy’s Harmonics and for several books on philosophy and a book on the history of philosophy.

His Pythagorean life is part of the latter. He wrote a Life of his master Plotinus, and he edited

his works under the name of Enneads.

13

Wallis also worked on critical editions of Ptolemy’s Harmonics and of the Harmonics of

Manuel Bryennius, the fourteenth century Byzantine music theorist. These two editions, together

8 ATHANASE PAPADOPOULOS

Let us now cite the words of Archytas the Pyt hagorean, whose writings

are said to be mainly authentic. In his book On Mathematics, right at

the beginning of the argument, he writes: “The mathematicians seem to

me to have arrived at true knowledge, and it is not surprising that they

rightly conceive th e nature of each individual thing; for, having reached

true knowledge about the nature of the universe as a whole, they were

bound to see in its true light the nature of the parts as well. Thus,

they have handed down to us clear knowledge about the speed of the

stars, and their risings and settings, and about geometry, arithmetic,

and spherics, an d, not least, about music; for these studies appear to be

sisters”.

14

The use of the word “sister” in the preceding quote is similar to the one in the

quote by Marston Morse (whom we shall mention again below) which is at the

beginning of the pres ent paper .

Euclid wrote several treatises on music. Among them is the Division of the canon

which we already mentioned, in which he gives an account of the Pythagorean the-

ory of music, and which contains in particular a careful exposition of the mathemat-

ical theory of proportions applied to musical harmony. Proclus

15

in his Commentary

to Euclid’s First Book of Elements attributes to Euclid another treatise titled Ele-

ments of Music, which unfortunately did not survive into our time. Eratosthenes

(c. 276-194 b.c.) also had important impacts on both ﬁelds, mathematics and mu-

sic. His work, the Platonicus, contains a section on music theory which is referred

to several times by Theon of Smyr na in [103].

Several of the mathematicians-musicians we mentioned were equally erudite in

other domains of knowledge. For instance, Er atosthenes, who was the administra-

tor of the fa mous library of Alexandria, was considered as the most learned person

of his time, and for this reaso n he was known under the name “β”, the second letter

of the alphabet, which was a manner of indicating that he was “second” in every

domain of knowledge. Ptolemy, whom we already mentioned, was a mathemati-

cian, geographer, astr onomer, poet and expert in oriental mysticism, and he was

probably the greatest music theorist of the Greco-Roman period. His major work,

the Mathematiki Syntaxis (Mathematical collection), a treatise on astronomy in 13

books, reached us through the Arabs with the title Almagest (a corrupt form of the

Greek superlative Megistos, meaning “the greatest”). Ptolemy is also the author

of an impor tant musical treatise, the Harmonics, in which he exposes and de velops

Pythagorean musical theories. This treatise was also translated into Arabic in the

ninth century and into Latin in the sixteenth century. From the later Greek period,

we can mention the mathematician Pappus (third century a.d.) who, like Euclid,

Eratos thenes, and Ptolemy, lived in Egypt, a nd who also was an excellent music

theorist. He wrote an impressive exposition of all of what was known in geometry at

the time, with the title Synagoge (or Collection). He also wrote a Commentary on

Euclid’s Elements and a Commentary on Ptolemy’s work, including his Harmonics.

Proclus, whom we already mentioned several times and the author of the famous

with teh one of Porphyry’s Commentary to Ptolemy’s Harmonics, with a Latin translation ac-

companying the Greek text, together with editions of works by Archimedes and Aristarchus of

Samos, constitute Volume III of Wallis’ collected works, published in three volumes in Oxford, in

1699.

14

This English translation is taken from the Selections illustrating the history of Greek Math-

ematics, edited by Ivor Thomas, s ee [104], Vol. 1, p. 5. The text is also quoted in French in the

volume [91], p. 533.

15

Proclus (412-485 a.d.) ﬁrst studied m athematics in Alexandria under Her on, and then

philosophy in Athens under Plutarch. He became the head of the neo-Platonic school of Athens,

after Plutarch and Syrianus.

MATHEMATICS AND GROUP THEORY IN MUSIC 9

Commentary to Euclid’s First Book of Elements, also wrote co mmentaries on sev-

eral of Plato’s dialogues, including the Timaeus, a dialogue which is essentially a

treatise on mathematics and music. The subject of this dialogue is the creation of

the universe, desc ribed alleg orically as a long musical scale.

The belief in a stro ng connectio n between the four ﬁelds of the quadrivium is

also part of a broader deep feeling of order and harmony in nature and in human

kind. This is also the origin of the word “Harmonics”, which is used in many places

instead of the word music. This word has the ﬂavour of order, of structure and of

measure. This feeling of harmony which was shared by most of the major thinkers

of Greek antiquity was a vehicle for an extraordinar y ﬂourishing of arts and sci-

ences which included the development of an abstract and high- level mathematics

and the construction of co he rent systems governing the sciences of music, astron-

omy, physics, metaphysics, history and theatre. It is generally accepted that these

systems had a real and probably irreversible impact on all human thought and in

any event, they continued to dominate most branches of knowledge in Europe until

the end of the middle ages.

The belief in an intima te relatio n between mathematics and music, which was

stressed primarily by the Greek thinkers, sometimes took the form of a belief in the

fact tha t music – not only its theory, but also the emotion that it produces – is in

many ways identical to the emotion that mathematical pure thought c an produce.

Such a feeling was also formulated in modern thought. Let us quote for instance

Marston Morse, from his paper Mathematics and the Arts [81]:

Most convincing to me of the spiritual relations between mathematics

and music, is my own very personal experience. Composing a little in

an amateurish way, I get exactly the same elevation from a prelude that

has come to me at t he piano, as I d o from a new idea that has come to

me in mathematics.

Although, by the end of the sixteenth century, the antique tradition considering

music as part of mathematics progre ssively disappeared, the development of mu-

sic theory and practice continued to be accompanied with a fruitful alliance with

mathematics.

Among the seventeenth century mathematicians involved in this alliance, we ﬁrst

mention Newton, one of the principa l founders of modern science.

Newton was interested in every k ind of intellectual activity, and of course he

was naturally led to music theory.

16

A notebook left from his early college days

(c. 1665) concerns this subject, and it contains, in the old Greek tradition, a

theory and computations of the division of musical intervals. Newton is als o known

for the use of the logarithms in his musical computations. He discuss ed several

points of music theor y in his correspondence, in particular with John Collins [82].

He made relations between some divisions of musical intervals , and in particular

of symmetrical divisions (palindromes), and questions in optics on the divisio n

of the color spectrum. In 1666, Newton discovered that sunlight is a mixture

of several colors, and this was one of the starting points for his theory on the

corres pondence between the color spectrum and the musical scale, which b e came

later one of his favorite subjects. This topic is also discussed in his correspondence

[82], in particular with Henry Oldenburg in 1675 and with William Briggs in 1685

and in his popula r work Opticks (1704). Newton’s theory of sound is also discussed

in his famous pap er New Theory about Light and Colors. (16 72). Let us note right

16

Pythagoras, for w hom Newton had a great respect, is mentioned several times in the Prin-

cipia. In the Scholia on Prop. VIII Book III on universal gravitation, Newton declares that

Pythagoras was aware of several physical laws, for instance the fact that square of the distance of

the planets to the sun is inversely proportional to the weights of these planets, but that because of

the nature of hi s teaching (which was essentially esoterical), nothing written by him could survive.

10 ATHANASE PAPADOPOULOS

away that the co rrespondence between colors and pitch is a lso one of the main

themes in the theoretica l work of Olivier Messiaen that we shall discuss later in

this pape r.

One should emphasize here that Newton’s ideas about the relation between the

spectrum of colors and the musical diatonic scale, and more precisely the fact that

the two spectr a ar e gove rned by the same numerical r atios, was part of his ﬁrm

convec tion that the same universa l laws rule all asp ects of nature. Voltaire was one

of several theorists on the continent who were eager to adopt and promote Newton’s

ideas, in particular his theory concerning the r elation between the seven-scale color

spectrum and the seven-scale diatonic scale, see [108] and [109].

After Newton, it is natural to mention Leibniz, with his famous sentence: Mu -

sica est exercitium arithmeticae occultum nescientis se numerare animi (Music is

a secret arithmetic exercise of the mind which is unaware of this count), that is,

music consists in a mathematics count, even though his listener is unconscious of

that. The sentence is extracted from the correspondence of Le ibniz with Christian

Godlbach [69] [70], the famous number theorist and friend of Euler.

Without going into any detail, we now mention some seventeenth century works

on music written by scientists. Kepler’s famous Harmony of the world (1616) [55]

contains se veral sections on music theory, written in the Pythagorean tradition.

(We note by the way that Kepler described himself as a “neo-Pythagorean”). See

also [87] on the relation between mathematics and music in Kepler’s Harmony of

the world. The ﬁrst book that Descartes wrote is a b ook on music, Compendium

Musicae (1618) [28]. Mersenne, the well known number theorist, wrote a music

treatise called Trait´e de l’harmonie universelle (1627) [7 8]. In this treatise, he

states, on p. 35: “Music is part of mathematics”, on 39: “Music is a science; it has

its real proofs which are based on its proper principles”, and on p. 47: “The music

I consider is subordinate to arithmetic, geometry, and physics”. Galileo Galilei’s

Discourses and dialogues concerning the two sciences (1638) [46] , which was his

last writing, contains sections on theoretical music. Christiaan Huygens also wrote

important treatises on music, e.g. his Lett er concern ing the harmonic cycle [52]

and his works on multi-divisions of the octave [53]. There are several other works

of Huygens on music theory. There are also several sets of letters on music theory

in the correspo ndenc e of several mathematicians, including Descartes, Huygens and

Leibniz,

17

and we already mentioned the correspondence of Newton [82].

Among the eighteenth centur y mathematicians who worked on music, we men-

tion Euler, who wrote a book, Tentamen novae theoriae musicae ex certissimis

harmoniae principiis dilucide expositae (Essay on a new musical theory exposed

in all clearness according to the most well-founded principles of harmony), already

mentioned, and several memoirs on music theory,

18

see [34], [35], [36] [37]. See also

the forthcoming boo ks [50] and [23] on Euler’s musical works. Euler formulates as

follows the basic principle on which he builds his music theories:

What makes music pleasant to our ears depends neither on will nor on

habits. [...] Aristoxenus denied the fact that one has to search for the

pleasant eﬀect of music in the proportions established by Pythagoras ...

Led by reasoning and by experiments, we have solved this problem and

we have established that two or more sounds produce a pleasant eﬀ ect

17

We recall that in that period there were still very few scientiﬁc journals, and that scientists

used to communicate their results by correspondence. The letters of major mathematicians were

collected and published, usually after their author’s death, but sometimes even during their life-

time. The correspondence [30] of Descartes occupies Volume 1 to 5 of his twelve-set Collected

Works [29]. The correspondence of Euler occupies several volumes of his Collected Works [38],

and up to now only part of it has been published.

18

There are also several papers of Euler on acoustics, but this is another subject.

MATHEMATICS AND GROUP THEORY IN MUSIC 11

when the ear recognizes t he ratio which exists among the number of

vibrations made in the same period of time; that on the contrary their

eﬀect is unpleasant when the ear do es not recognize this ratio (Extracted

from the Intro duction of [34]; see also the French translation in [23]).

In fact, the book [34] is the ﬁrst one which Euler wrote. He ﬁnished writing it

in 1731, the year he obtained his ﬁrst position, at the Saint-Petersburg Academy

of Science s, and he was 24 years old. It is most probable that several projects

in combinatoric s and in number theory occurred to E ule r while he was develop-

ing his music theory, since several natural questions regarding primes and prime

factorizations of numbers appear in that theory.

It is also interesting to hear what composers say about mathematics.

We can quote Jean-Philippe Rameau, the great eighteenth’s century French

composer and music theorist, from his famous Trait´e de l’harmonie r´eduite `a ses

principes naturels (1722) (see [94], Vol. 1, p. 3)

19

:

Music is a science which must have determined rules. These rules must

be drawn from a principle which should be evident, and t his principle

cannot be known without the help of mathematics. I must confess that

in spite of all the experience which I have acquired in music by practicing

it for a fairly long period of time, it is nevertheless only with the help of

mathematics that my ideas became d isentangled and that light succeeded

to a certain darkness of which I was not aware before.

In his D´emonstration du principe de l’harmonie, Rameau relates how, since his

childhood, he wa s aware of the role that mathema tics plays in mus ic ([94], Vol. 3,

p. 221):

Led, since my early youth, by a mathematical instinct in the study of an

art for which I found myself destined, and which occup ied me all my life

long, I wanted to know its true principle, as the only way to guide me

with certitude, regardless of the problems and accepted ideas.

Two hundred years after Rameau, Olivier Messiaen made similar statements

concerning the relation between mathematics and music, and we sha ll record them

in §4 below.

Rameau wrote a major corpus of works on music theory. The y include his

Trait´e de l’harmonie r´eduite `a ses principes naturels [94], his Nouveau Syst`eme de

Musique Th´eorique [95], his D´emonstration du principe de l’harmonie [96] and there

are many others; see the whole collection in [93]. There is also a correspondence

between Euler a nd Rameau, see [23] and [50].

In a review of Rameau’s Trait´e de l’harmonie r´eduite `a ses principes naturels

which he wrote in the famo us Journal de Tr´evoux, the Jesuit mathematician and

philosopher L.-B. Castel wrote: “Music is henceforth a vast quarry which will not

be exhausted before a long time, and it is desirable that philosophers and geometers

will want to lend themselves to the advancement of a science which is so puzzling.”

D’Alembert also became very much interested in Rameau’s theoretical writings,

and he wrote an essay explaining his theories [1]. The relation between the two men

became tense and eventually bad – d’Alembert accused Ramea u of exaggerating the

role of mathematics in his music – but this is another story.

Diderot, one of grea test ﬁgures of the French Enlightenment and one of the two

main editors (the other one being d’Alembert) of the famous Encyclop´edie, wrote

a book on the theory of sound

20

in which he writes ([31] p. 84): “The musical

19

I am translating from the French.

20

In 1784, Diderot published a collection of 5 memoirs under the general title M´emoires sur

diﬀ´erents sujets de math´ematiques; three of these memoirs concern sound and music theory.

12 ATHANASE PAPADOPOULOS

pleasure lies in the pe rception of r atios of numbers [...] Pleasure, in general, lies in

the perception of ratios”.

Finally, let us give a few examples from the modern period, by quoting a few

geometers.

In a letter to his friend G. Wolﬀ, a teacher at the conservatory of Leipzig, Bel-

trami writes ([20] p. 154, note 111):

Between music and mathematics, there is a reconciliation which has not

yet been noted [...] A mathematical reasoning is comparable to a se-

quence of chords [...] and the discovery of a new branch of mathematics

is comparable to a harmonic modulation.

In his pape r Twentieth century mathematics (1940 ) [79], Morse writes the fol-

lowing:

Mathematics is both an art and a science, and the lack of appreciation

of this fact is responsible for much misunderstanding [...] Objective ad-

vances must be revised in form to make them aesthetically acceptable

and logically comprehensible, while advances of a more subjective nature,

if complete and harmonious, will not long remain unapplied.

In a letter [80], he writes:

Mathematics is both an art and a science ... Mathematicians are the

freest and most ﬁercely individualistic artists. They are subject to n o

limitations of materials or instruments. Their direction at any time is

largely determined by their tastes and intellectual curiosity. Their studies

are really t he studies of the human mind. To me the work of Einstein is

even more important as a free and beaut iful expression of the creative

imagination of an individ ual than as a part of the science of physics.

The paper by Birkhoﬀ at the Bologna ICM conference (1928) concerns mathe-

matics and arts [21], and there are several papers by him on the relation b etween

mathematics and music, s ee e.g. his treatise Esthetic measure [22].

The compose r Milton Babbitt, whom we shall mention below and who taught

music theory and sometimes mathematics at Princeton, insisted on a rigorous and a

mathematically incline d teaching o f music. He w rites in [16] that a musical theory

should be “ statable as a c onnected set of axioms, deﬁnitions, and theore ms , the

proofs of which are derived by means of an appropriate logic”. This is in the

tradition of the great musical treatises of Mersenne [78], Ra meau [94], etc. which

we already mentioned and where the exposition is in the form of Theorem, Lemma,

etc.

Several serial composers, before Babbitt, knew that they were dealing with

groups. For instance, the Austrian composer Hanns Jelinek (1901-1969), who had

been in contact with Arnold Schoenberg and Alban Berg, wrote a book on twelve-

tone music [67], in w hich he explicitly cites the ”Gruppenpermutation” (p. 157,

vol. 2). Likewise, Herbet Eimert, who made in 1964 a compete list of the “all-

interva ll series”, knew since the 1950s that he was dealing with groups. One may

also cite Adria an Fok ker (1887-1 972), a Dutch physicist

21

and musician, who w rote

extensively on music theory, cf. for instance [40] and [41]. Fokker was very much

inﬂuenced by the works of Huygens on music theory. An impor tant article on the

systematic use of group theory in music is [48] by Halsey and Hewitt.

It is always good to see, in skimming through these papers and books, how

mathematics ca n serve music and vice versa

In the rest of this survey, I will concentrate on the work of the French composer

Olivier Messiae n, because it involves in several aspects group theory. Me ssiaen

21

Fokker obtained his PhD under Lorentz, and he also studied under Einstein, Rutherford and

Bragg. His name is attached to the Fokker-Planck equation.

MATHEMATICS AND GROUP THEORY IN MUSIC 13

stated explicitly (like Rameau, Euler and other s did before him) that what makes

the charm of a musical piece is the mathematical structures that stand behind it.

4. The music of Olivier Messiaen

We ar e concerned in the following pages with certain mathematical aspects of the

musical compositions and of the theoretical writings of Mess iaen.

22

Even though

Messiaen never considere d himself as a mathematician, he granted to mathematics

a prominent place, both in his comp ositions and in his theoretical teaching. The

titles of some of his pieces , like Le Nombre L´eger (Pr´elude No. III for pia no),

Soixante-quatre dur´ees (Piece No. VII of his Livre d’Orgue), are signiﬁcant in

this respect. The mathematical notions that are involved in his compositions are

basic notions (permutations, symmetries, prime numbers, pe riodicity, etc.), and it

may be worthwhile to stress r ight away the fact that the fact that Messiaen uses

these notions in a mathematically elementary and simple way does not reduce the

place of mathematics in his work . Questions related to prope rties of sequences

of numbers, of their transformations and of their symmetries, however elementary

they a re, ar e part of mathematics. Messiaen worked with these notions cons ciously

and s ystematically. In a book of dialogues with Claude Samuel [100], he recalls that

since he was a child, he was fascinated by certain properties of numbers, properties

which were led to play a central role in his musical langua ge. In [100], p. 118,

answering a question about de¸ci-tˆala Indian rhythms and more generally about the

reasons which guided him in his choice of certain rhythmical formulae, Messiaen

says:

I was oriented towards this kind of research, t owards asymmetrical di-

visions, and towards an element which one encounters in Greek meters

and in Indian rhythms: prime numbers. When I was a child, I already

liked p rime numbers, these numbers which, by the simple fact that they

are not divisible, emit an occult force [...]

One aspect of the music of Messiaen is a balance between reason and intuition,

between poetic creation and a rigorous formal structure. His theoretical work is in

the tradition of the Greek quadrivium, and we can quote here the composer Alain

Louvier, who was a student of Messiaen at the Conservatory of Paris and who says,

in his foreword to Messiae n’s Trait´e de rythme, de couleur et d’ornithologie [75],

that in his teaching, Messiaen placed Music at the conﬂuence of a new Septivium:

Mathematics, Physics, Cosmology, Acous tics, Physiology, Poetry, and Philosophy.

Understanding the way in which mathematical structures are present in Messiaen’s

music can at least serve the purpose o f making his music less enig matic than it

appears at ﬁrst hear ing.

Finally, beyond the description of Messiaen’s work, one of the themes which we

would like to develop in the next sections is that music (and in particular rhythm)

is a certain way of giving life to mathematical structures, and of rendering them

perceptible to our senses. More than that, music transforms these notions into

emotionally aﬀecting objects.

We have divided the re st of our exposition into three pa rts, with the following

titles:

• Rhythm.

• Counterpoint.

• Modes of limited transposition.

In each part, the reader will notice the relation with group theory.

22

I already reviewed some of these ideas in my paper [86] (2003).

14 ATHANASE PAPADOPOULOS

5. Rhythm

It is natural to start with rhythm, since in the work of Messiaen, this notion

occupies a central place. In [100], p. 101, Messiaen says: “I consider rhythm

as a fundamental element, and may be the essential element of music. It has

conceivably existed before melody and harmony, and ﬁnally, I have a preference for

this element.” His monumental theoretical work, on which he worked for more than

40 years, is titled “Trait´e de rythme, etc.”. Since the ﬁrst pages of this treatise,

Messiaen rises up against the co mmon opinion which says that music is made out

of sounds. He writes: “I say no ! No, not only with sounds ... Melody canno t

exist without Rhythm!... Music is made ﬁrst of all with durations, impulses, rests,

accents, intensities, attacks and timbres, all things which can be regrouped under a

general term: rhythm”. ([75], Tome 1 p. 40.)

The notion of rhythm is not foreign to mathema tics, at least because elements

like durations, intensities and densities are measured with numbers. Timbre, de-

composed into fundamental frequency and harmonics, can also be expressed by a

sequence of numb e rs. But of cour se, the use of number is not the main po int, a nd

there are more profound reasons for the relation between rhythm and mathematics,

and we shall discuss them below.

We shall mainly talk about rhythm as a sequence of durations. Messiaen de-

scribes such a sequence as a “chopping-up of Time”.

23

Some of his compos itions

are based on particula rly simple (but never monotonic) rhythmic for mulae. We

can mention in this r espect Piece No. VII of his Livre d’Orgue, titled Soixante

quatre dur´ees (sixty-four durations), a composition which from an abstr act point of

view resembles a game in which the composer takes diﬀerent groupings of elements

in a chr omatic

24

sequence of 64 durations and intertwines them using a geometric

process which Me ssiaen calls “symmetrical permutations”, which we shall describe

below. Let us mention also Piece No. V of his Messe de la Pentecˆote, titled Le

vent de l’Esprit, in which two chromatic sequences are superimpose d, the ﬁrst one

decreasing gradually from a value equal to 23 sixteenths notes until the value of one

sixteenth note, and the second incre asing g radually from 4 sixteenth notes until 25

sixteenth notes.

Creativity often implies a profound immersion in the sour ces, and Messiaen’s

sources, for what concerns his rhythmical langua ge, are India and Ancient Greece.

25

Indian and Gre ek rhythms have a dominant position, both in his comp ositions and

in his teaching. Furthermore, in his written work, Messiaen keeps ﬁxing one’s

attention on the arithmetical properties of these rhythms. We shall give some

examples.

5.1. Greek rhythms. It is well-known that in Ancient Greece , music was used

as an accompaniment to poetry and to theatre, and therefore, musical rhythm

followed the rhythm of declamation. In this setting, there are essentially two sorts

of durations for musical notes, long durations, all equal in va lue , corresponding to

long syllables, and short dura tions, also all equal in value, corresponding to short

23

The relation between rhythm and time is i mpor tant in Messiaen’s thinking. Volume 1

of hi s Treatise [75 ], which is devoted to Rhythm, starts with a long chapter on Time in all

its aspects: absolute time, relative time, biological time, cosmological time, physiological time,

psychological time, the relation of tim e to eternity, etc., with long di gressions on the concept of

time in mythologies, in the Bible, in Catholic theology, in the theories of Einstein, of Bergson and

in many other settings. See [101] for an interesting study on Time in the work of Messiaen.

24

The word c hromatic, here and below, means that the values in the sequence increase linearl y,

that is, the sequence behaves like an arithmetic sequence. The terminology chromatic is used in

music theory.

25

One has also to mention birds, but this is another story.

MATHEMATICS AND GROUP THEORY IN MUSIC 15

syllables, the va lue o f a short duration being half of the value o f a long one. A

rhythm in this sense, that is, a string of long and short durations, is called a meter.

Mathematicians know that there is a rich theory of combinatorics of strings of words

written in an alphabet of two letters.

Ancient Greek music contained a rich variety of meters, and these were classiﬁed

in particular by Aristoxenus in his impressive Harmonic Elements which we already

mentioned, see [17] and [18], Vol. II. One of the characteristics of this music is that

within the same piece, meters are of variable length, in contrast with the meters of

(pre-twentieth century) Western classical music, where a piece is divided into bars

within which the number of beats is constant. Meters of variable length existed

even in Gregorian chant, which in some sense is a heir of ancient Gr eek music, and

at some point during the Renaissance period, Greek meters were in fashio n.

26

But

then the interest in them disappeared again, altho ugh there are reminiscences of

Greek meters in Romanian folk music and in compositions by Ravel and Stravinsky.

For instance, in Stravinsky’s Rite of the Spring, at the beginning of the Introduction,

the meter switches constantly between the values 4:4, 3:4 and 2:4. Likewise, in the

last piece, Sacriﬁcial Dance, the meter changes constantly, taking values like 5:16,

3:16, 4:16, 2:8, 3:8, 3:4 , 5:4, and there are others. Messiae n revived the systematic

usage of meters of varia ble leng ths, teaching their principle in his class at the

Conservatory of Paris, and putting them into practice in his compositions. The

ﬁrst volume of his Trait´e [75] contains a 170 pages chapter on Greek meters.

These “a-metrical rhythms” were used by Messiaen since his earliest composi-

tions. It seems that he cherished this kind of freedom in rhythm, and one reason

for that is that it excludes monotony. Messiaen, who sometimes des c ribed hims e lf

as a Rhythmician, says in [100], p. 102, that “a rhythmical music is a music which

excludes repetition and equal divisions and which ﬁnds its inspiration in the move-

ments of nature, which are movements with free and non-equal durations.” On p.

103 of the same treatise, he gives examples of a non-rhythmical music: “Military

music is the negation of r hythm”, and he notes that military marches are most un-

natural. Likewise, there is no rhythm, he says, in a Concerto by Prokoﬁev, because

of the mono tonicity of the meter. On the other hand, he considers Mozart and De-

bussy as true rhythmicians. To understand this, we refer the reader to the chapter

titled “

`

A la recherche du rythme” in [100]. The reader might remember that the

word rhythm refers here to a variety of notions: sequences of durations, but also o f

attacks, intensities, timbre, etc. In the ﬁrst volume of his Trait´e, Messiaen writes

that rhythm contains p e riodicity, “but the true periodicity, the one of the waves of

the sea, which is the opposite of pure a nd simple rep etition. Each wave is diﬀerent

from the preceding one a nd fro m the fo llowing one by its volume, its height, its

duration, its slowness, the briefness of its formation, the power of its climax, the

prolongation of its fall, of its ﬂow, of its scattering...” ([7 5], Tome 1, p. 42).

Another aspect of Greek meters, which was seldom used in Western classical

music before Messiaen, is the systematic use of rhythmical patterns who se value is

a prime number (o ther than 3), for instance 5, 7, or 17. One example of a rhythm

whose total value is 5 is the Cretic rhythm, deﬁned by the sequence 2, 1, 2 (that

is, a rhythm corresponding to a long, then a short, and then a long syllable), and

its two permutations, 2, 2, 1 a nd 1, 2, 2. The rhythm 2, 1, 2 is ca lled amphimacer,

meaning (as Messiaen explains) “longs surrounding the short”. This introduces us

directly to two important notions in the rhythmical language of Messiaen. The

ﬁrst one is related to the central symmetry of the sequence 2,1,2, which makes it an

26

For instance, at the beginning of the seventeenth century, Claude le Jeune composed choral

works whose rhythm followed the principle of Greek meters, which is not based on the sole count

of syllables, but which takes into account their length or shortness.

16 ATHANASE PAPADOPOULOS

instance of a non-retrogradable rhythm, and the second one is that of a permutation

applied to a rhythm. But before dwelling on that, let us say a few words on Indian

rhythms, which also pos sess some beautiful properties.

5.2. Indian rhythms. A signiﬁcant characteristic of Indian music is the important

place that it makes for percus sion instruments like drums, cymbals, bells, ha nd-

clapping, and so on, and this makes rhythm a very important factor in that music.

Let us quote Messiaen again: “Indian music is the music which certainly went

farther than any other music in the domain of rhythm, especially in the quantitative

domain (combinations of long a nd of s hort durations). The Indian rhythms, of

unequalled reﬁnement and subtlety, leave far behind them o ur poor western rhythms

with their isochroneous bars, and their perpetual divisions and multiplications by

2 (sometimes by 3).” ([75] Tome 1, p. 258).

In the same way as do Greek rhythms, Indian rhythms a bound in Messiaen’s

compositions. In his Trait´e [7 5], the chapter concerning Indian rhythms occupies

130 pages. In this chapter, Messiaen draws up lists of the 120 de¸ci-tˆalas,

27

of the

36 rhythms of the Carnatic (that is, South-Indian) tradition and of other gr oups

of Indian rhythms, and he comments them thoroughly. Here a lso, a rhythm is a

sequence of numbers, and Messiaen expresses a real fascination for the arithmetical

properties o f these sequences, a fascination which he transmits to his reader. Let

us see a few ex amples of these pro perties. He points out, whenever this is the

case, that the sum of all the dura tions of some rhythms is a prime number. For

instance, he records several de¸ci-tˆalas whose total value are 5, 7, 11, 17 , 19, 37, and

so on. This ins istence on prime numbers may be surprising, but we have already

mentioned the importance of these numbers for Messiaen. In the ﬁrst volume of his

Trait´e, he writes that “the impossibility of dividing a prime numb er (other than by

itself and by one) grants it a sort of force which is very eﬀective in the domain of

rhythm.” ([75] Tome 1, p. 266).

Another special class of de¸ci-tˆalas which is highlighted by Messia e n is the class of

rhythms consisting of a sequence of durations which is followed by its augmentation.

For instance, the rhythm 1, 1, 1, 2, 2, 2 (de¸ci- tˆala No. 73) is made out of the

sequence 1, 1, 1 followed by its augmentation by multiplication by 2. An analogous

feature occurs in de¸ci-tˆala No. 115, which is the rhythm 4, 4, 2, 2, 1, 1, constituted

by the sequence 4, 4, followed by its diminution 2, 2, and then by the diminution of

its diminution, 1, 1. Augmentation a nd diminution are arithmetical transformations

which are important in the art of counterpoint, which is the art of transforming

and combining musical lines, and which we shall discuss below in more detail.

There are mo re complex combinations. For instance, in the rhythm 1, 3, 2, 3,

3, 3, 2, 3, 1, 3 (de¸ci-tˆala No. 27), Messiaen notes that the odd- order dura tions are

all equal, whereas the even-order durations consist in a regularly increasing and

then regularly decreasing sequence. He points out that this rhythm was used by

Stravinsky in the Rite of the Spring, and that it is at the basis of his theory of

Rhythmic characters.

28

Messiaen ma kes extensive use of rhythmic characters in his

Turangalˆıla Symphony (composed in 1946-1948).

Finally, let us mention that the de¸ci-tˆalas contain several instances of non-

retrogradable rhythms, that is, rhythms c onsisting of a sequence of durations fol-

lowed by its mirror image (with sometimes a co mmon central value). We already

27

These rhythms have been cl ass iﬁed by the 13th-century Indian musicologist Cˆarngadeva.

Messiaen explains that, in Hindi, de¸ci means rhythm, and tˆala means province. Thus, the word

de¸ci-tˆala refer the rhythms of the various provinces. There are other interpretations for tˆala; s ee

for instance the article India in the New Grove Dictionary of Music and Musicians.

28

There are three rhythmic characters here: one character stays still, another one is decreasing

and the third one is increasing.

MATHEMATICS AND GROUP THEORY IN MUSIC 17

encountered such r hythms when we talked about Greek rhythms. For instance,

de¸ci- tˆala No. 58 is the Greek amphimacer rhythm, 2, 1, 2 , which Messiaen de-

scribes as “the simplest and the most natural non-retrogradable rhythm, because

it is based on the number 5, the number of ﬁngers in the hand.” ([75], Tome 1, p.

289). Other examples of non-retro gradable de¸ci-tˆalas are 2, 2, 1, 1, 2, 2 (de¸ci-tˆala

No. 26), 1, 1 , 2, 2, 1, 1 (de¸ci- tˆala No. 80 ) a nd 2, 1, 1, 1, 2 (de¸ci-tˆala No. 111 ), a nd

there are several others . In the next se c tion, we discuss non-retrogradable rhythms

at fuller length.

5.3. Non-retrogradable rhythms. The use of non-retrogradable rhythms is an

example of the systematic use of symmetry in the music of Messiaen.

Messiaen dealt with non-retrogradable rhythms since his early compositions,

and he attached great importance to them in his ﬁrst theoretical essay, Technique

de mon Langage Musical (1944). A non-retrogr adable rhythm is a sequence of

durations which gives the same result whether it is r ead from left to r ight or from

right to left. It may be good to recall here that retrogradation is a classical device

in the art counterpoint which we shall consider mor e thoroughly later on in this

article. It transforms a certain musical motive by reading it backwards, that is,

beginning from the last no te and ending with the ﬁrst no te. The initial motive is

then called the motive in direct motion, and the transformed motive the motive

in retrograde motion. Thus, a non-retr ogradable rhythm can be regarded as the

juxtaposition of a motive in direct motion and of a motive in retrog rade motion,

with sometimes a centr al value in common.

Retrogradation, as a counterpoint operation, was use d and taught since the

beginning of this art, around the fourteenth century. But before Messiaen, it was

usually applied to a melodic motive, that is, to a sequence of pitches, whereas

with Messiaen, retrogradation acquired a more abstract character, since he applied

it systematically to rhythm, regardless of pitch. Thus, the listener of Messia en’s

music is invited to feel re trogradation at the level of dura tions only, since ther e

need not be any reg ular correspondence (transposition, symmetry, etc.) between

the pitches of the motive in direct motion and those in the motive in retrograde

motion.

For instance, in the “Danse de la fureur, pour les sept trompettes” (Part VI

of Messiaen’s Quatuor pour la ﬁn du Temps), we ﬁnd the fo llowing succession of

non-retrogradable rhythms:

3, 5, 8, 5, 3

4, 3, 7, 3, 4

2, 2, 3, 5, 3, 2, 2

1, 1, 3, 2, 2, 1, 2, 2, 3 , 1, 1

2, 1, 1, 1, 3, 1, 1, 1, 2

2, 1, 1, 1, 3, 1, 1, 1, 2

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

3, 5, 8, 5, 3

(the unit being the sixtee nth note). In the analysis of this sequence of bars that

he makes in the second volume of his Tr ait´e, Messiaen, of course, highlig hts the

fact that the total number of durations in bars 3 and 4 is 19, and that in bars 5, 6

and 7, this value is 13, pointing out again that 19 and 13 are prime numb ers ([75],

Tome 2, p. 26). It would be tedious and superﬂuous to try to draw up a long list

of such examples, since the are plenty of them in Messiaen’s compositions . But it

is natural to raise the question of why non-retrogradable r hythms are interesting.

18 ATHANASE PAPADOPOULOS

One may as well ask why is the symmetry of a face a beauty c riterion. Messiae n

answers this question on rhythm in his own way, and he gives two kinds of reasons,

one of an aesthetic nature, and the second one philosophical. In [74], he talks

about the charm which a non-retro gradable rhythm produce s on the listener of

his music . He considers that this charm is of the same order as the one which is

produced by his modes of limited transposition (which we shall discuss below), and

he calls the non-retrogradable rhythms and the modes of limited transposition as

two m athematical impossibilities. The impossibility, in the ﬁrst c ase, lies in the

fact that it is “impossible to reverse such a rhythm”, since when we reverse it, we

obtain exactly the same rhythm. Le t us quote Messiaen from the ﬁrst ﬁrst of his

Technique de mon langage musical ([74], Tome 1, p. 5):

One point will attract ﬁrst our attention: the charm of impossibilities...

This charm, at the same time voluptuous and contemplative, lies par-

ticularly in certain mathematical impossibilities in the modal and of the

rhythmic domains. The m odes which cannot be transposed beyond a

certain number of transpositions, because if one does so, he falls again

on the same notes; t he rhythms which cannot be retrograded because if

one does so, he recovers the same order of the values...

On page 13 of the same Trait´e, Messiaen describ es the impressions which these

impossibilities produce on their listener.

Let us consider now the listener of our modal and rhythmic music; there

is no time for him, at the concert, to check non-transposition and non-

retrogradation, and at that moment, these q uestions will not interest

him any more: to be seduced, this will be his unique desire. And this is

precisely what will happen: he will undergo despite his will the strange

charm of t hese impossibilities; a certain ubiquitous tonal eﬀect of the

non-transposition, a certain unity of movement (where beginning and

end merge, because they are identical) of non-retrogradation, all things

which will certainly lead him t o that sort of “theological rainbow” which

our musical language tries to be, a language which we are trying to edify

and to theorize.

29

Messiaen after this e xplains the philosophic relevance of non-retrogradable rhythms,

which justiﬁes also the term “theological rainbow” in the last sentence. A non-

retrogradable rhythm, according to him, can give the listener a feeling of inﬁn-

ity. Indeed, whether one reads it from left to right or from right to left, a non-

retrogradable rhythm stays invariably the same, and in this sense, such a rhythm

has no beginning and no end. In the second volume of his Trait´e, Messiaen says

that a non-retrogradable rhythm draws its strength from the fact that “like Time,

a non-retrogradable rhythm is irreversible. It cannot move backwards, unless it

repeats itself... The future and the past are mirr or images of each other” ([75],

Tome 2, p. 8).

In conclusion to this section on rhythm, we quote again Messiaen, who mentions

in the ﬁrst volume of his Trait´e rhythms which ar e “thought for the only intellectual

pleasure of number” ([75] Tome 1, p. 51 ). The reference to number reminds us

again of ideas that Leibniz, Euler, Diderot, and others emitted about music and

which we rec alled in §3. Numbers can a priori appear as being severe, austere, and

devoid of lyricism. Expressed as rhy thm, they are given a new life.

Messiaen notes in his Trait´e [75] that nature is full of non-retrogra dable rhythms,

starting with the human fac e, with the two ears, the two eyes and the nose at

the center, or like the teeth inside the mouth. He also makes an analogy with

several architectural ediﬁces, including the marble bridge in Beijing that leads to

29

This sentence is reminiscent of the sentence by Leibniz that we quoted in §2, saying roughly

that music is a secret exercise in arithmetic.

MATHEMATICS AND GROUP THEORY IN MUSIC 19

the Summer Palace of the Chinese emperors, together with its reﬂection in the

water.

Messiaen certainly compared the beauty of certain rhythms, which are built as

sequences of numbers with rigor ous properties, to the beauty of certain faces with

regular and s ymmetrical features, to that of French ga rdens that follow completely

symmetrical plans, to that of Romanesque cathedrals, and to that of the wings of

butterﬂies.

As a last wink to non-retrogradable rhythms, let us mention that the number of

pieces in the seven books of Messiaen’s compos ition Catalogue d’Oiseaux is respec-

tively 3, 1, 2, 1, 2, 1, 3.

5.4. Symmetrical permutations. Permutations of ﬁnite sets play an important

role in the music of Messiaen, and groups are present there. We shall see this more

precisely below.

Given a sequence of musical objects (e.g. pitches, dynamics, durations, etc.),

one obtains another seq ue nce by applying to it a permutation. For instance, ret-

rogradation is a special kind of permutation. The pr oblem is that as soon as the

number of objects is large, the total number of pe rmutations becomes too large to

play a signiﬁcant role in music. For instance, for a sequence of 5 objects, there are

120 distinct permutations, for 6 objects, there are 720 distinct permutations, for 7

objects, there are 5040 distinct permutations, a nd then the number of permutations

become huge. Thus, for the use of permutations in music, one has to make choices,

because if the order of the symmetries used is too large, the ear cannot discern

them. This leads us to the theory that Messiaen calls symmetrical permutations,

that is, permutations which have a small group of symmetries. In his compo sitions,

he applies to a musical motive the iterates of a given symmetrical permutation. For

instance, retrogradation is of order two and therefore it is symmetrical. Symmetri-

cal permutations which are more complicated than retrog radation are used already

in his early pieces, for instance in the Vingt Regards sur l’Enfant J´esus, composed

in 1944.

The piece Chronochromie (1960) starts with a chromatically increasing sequence

of 32 durations, starting with a thirty-secondth note, and ending with a 32 × thirty-

secondth note, that is, a whole note. Mess iaen applies to it the permutation 3, 28,

5, 30, 7, 32, 26, 2, 25, 1, 8, 24, 9 , 23, 16, 17, 18, 22, 21, 19, 20, 4, 31 , 6, 29, 10, 27,

11, 15, 14, 12, 13. He then applies the same permutation to the new sequence, and

so forth. After 35 steps, we recover the initial sequence, 1, 2, 3, ..., 32.

In Chronochromie, Messiaen uses, as rhythmical motives, a colle c tion of rhythms

which belong to this set of iterates.

It is natura l to ask why these permutations are impor tant in music. In his

treatise, Messiaen describes the symmetrical permutations as a third mathematical

impossibility. The listener of such permutations in a musical piece is supposed to

be dazzled by the same charm as with non-retrogradable rhythms and of mode s of

limited transposition, the two impossibilities that we already mentioned. In [1 00], p.

222, he describes the piec e Chronochromie as “durations and permutations rendered

sensible by sonorous colors; this is indeed a Color of time, a Chonochromy”.

For mathematicians, it is amusing to see a large number of explicit examples

contained in that musical treatise, and to imagine how tedious it was for Messiaen

to write them.

There is a special kind of per mutations to which Messiaen attaches more impor-

tance (and which probably are at the origin of the adjective “symmetrical”). To

obtain such a permutation, one starts at the center of a sequence of objects, and

then takes successively one object from the right and one object from the left, until

20 ATHANASE PAPADOPOULOS

one reaches the two ends of the sequence, and one of its iterates. For instance, this

process transforms the sequence 1, 2, 3, into the sequence 2, 1 , 3 . Applying the

same rule to 2, 1, 3 , we obtain 1, 2, 3, which is the sequence we started with. Thus,

the per mutation 1, 2, 3 7→ 2, 1, 3 is of order 2. Let us consider now a seq ue nc e of

four objects. The iterates are:

1, 2, 3, 4 7→ 2, 3, 1, 4 7→ 3, 1, 2, 4 7→ 1, 2, 3, 4.

and in this case, the order of the permutation is 3. This is a way of ﬁnding “by

hand” symmetrical permutation. It is interesting to see in which terms Mes siaen

describes this mathematical process. In [100], p. 119, he says:

There are durations which follow one another in a certain order, and we

read them again in the initial order. Let us take, for example, a chromatic

scale of 32 durations: we invert them according to a given order, then we

read the result according to this order, and so forth u ntil we recover the

initial chromatic scale of 32 d urations. This system produces interesting

and very strange rhythms, but above all, it has the advantage of avoiding

an absolutely fabulous number of permutations. You k now that with the

number 12, so much beloved by serialists, the number of permutations

is 479.001.600 ! One needs years to write them all. Whereas with my

procedure, one can, starting with larger numbers – for instance 32 or

64 – ob tain better permutations, suppress the secondary permutations

which lead only to repetitions, and work with a reasonable number of

permutations, not too far from the number we started with.

6. Counterpoint

Counterpoint is the c lassical art of transforming, combining, and superimposing

musical lines.

30

Symmetry is extensively applied in this art, and we shall see this in

this sec tion. Two important treatises o n counterpoint are those of Tinctoris (147 7)

[106] and Fux (1725) [45].

6.1. The use of integers mod 12. There are a few standard operations in coun-

terpoint, and it is practical to describe them using the language of integers mod 12.

We brieﬂy discuss this formalism here, and this will also serve us in the description

of Messiaen’s modes of limited transposition. We note however that Messiaen did

not use the notation of integer s modulo 12; in fact, he did not have any background

in mathematics. He had his own words to deﬁne mathematical objects and to

explain what he wanted to do with them. This usually involved a heavy language.

In two famous articles, published at the be ginning of the 19 60s (see [14] and

[15]), Babbitt applied the language and techniques of group theory in music (in

particular in twelve-tone music). He used in particular the concept of pitch-class

which became an important tool in the teaching of ce rtain musical theories; for

instance, this is an important facto r in the textbook by Allen Forte [42], which was

for many years one of the main references in the USA for twentieth-century musical

analysis techniques.

The principle of the use of integers mod 12 is the identiﬁcation of notes whose

pitches diﬀer by an octave. This is a natural identiﬁcation, beca use in practice, it is

observed that in general, the notes produced by the voices of a man and of woman

(or of a man and a child) singing the same song, diﬀer by an octave, although they

are considered to be the same notes. This octave identiﬁcation is also suggested

by the fac t that the na mes of notes that diﬀer by an octave have the same name,

30

The word counterpoint comes from the latin expression punctus contra punctum which means

“point against point”, expressing the fact that on a musical score, the dots that represent diﬀerent

pitches that are played at the same time, as the result of the superimposition of the musical lines,

appear vertically one above the other.

MATHEMATICS AND GROUP THEORY IN MUSIC 21

and therefore there is a cyclic repe tition in the names of notes that are played by a

traditional instrument. In fact, it can be diﬃcult for a non-exper t to say whether

two notes with the same na me played by diﬀerent instruments (say a ﬂute a nd a

violin) correspond to the same pitch or to pitches that diﬀer by an octave. This

fact has been pointed out and analyze d by several music theorists, in particular in

the set of problems concerning music theor y which are attributed to Ar istotle ([12],

Volume I).

31

It has also been formalize d as a principle, the “principe de l’identit´e

des octaves”, by Rameau in his Trait´e de l’harmonie r´eduite `a ses principes naturels

[94], and this principle has been thoroughly used before and after Rameau. We note

by the way that there was a correspondence between Euler and Rameau about this

principle, and some of these letters reached us, see [93], [83] and [2 3].

In any case, we are co nsidering now the so-called equally tempered chromatic

scale.

32

In other words, we are considering notes which correspond to a divisio n

of an octave into twelve equal intervals. Since the ratio of the frequencies of two

pitches separated by an octave is equal to 2/1, the ratio of two successive notes in

the equally tempered scale is equal to

12

√

2. The notes in this scale correspond to the

sequence C, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, B that appear in that order within

an octave on a piano keyboard. Re presented by the integers 0, 1, ..., 11 in that

order, these numbers are considered as elements of the cyclic group Z

12

= Z/12Z.

For our needs here, the applicability o f symmetry and group theory is easier if we

use the equally tempered scale, and these mathematical notions naturally appear

in the description of the modes of limited transposition as we shall see below.

In the language introduced by Babbitt, a nd developed by Forte and others, these

integers are called pitch-classes, and they represent e quivalence classes of pitches,

that is, pitches deﬁned up to addition of a multiple of an octave. A pitch-class set

denotes then a set of pitch-classes, and it is represented by a sequence of distinct

elements of the group Z

12

. To denote a pitch-class set, Forte uses square brackets,

for instance [0, 1, 3]. One writes, to be brief, pc and pc-s et for pitch-class and

pitch-class set.

It is fair to note here that this notion of pc set was known (without the name)

in the nineteenth century, for insta nc e by Heinrich Vincent (1819-1901) [107] and

Anatole Loq uin (1834-1903) [72]. In the twentieth century, E dmond C ost`ere [26]

made an exhaustive list of pc sets long before Babbitt and Forte, using the name

´echelonnement, but without taking into ac count the notion of inversion.

33

Let us now review the relation with counterpoint.

The three basic operations of counterpoint are transposition, inversion and ret-

rogradation, and they have been identiﬁed and highlighted early at the beginning

31

Music was not the main subject of study of Aristotle, and in fact, Aristotle had no preferred

subj ect of study; he was industrious in all sciences – and music theory was one of them – working

on them, teaching them, and writing essays on all them. He had a set of Problems that he made

avail able to the students of his school (which was called the Lyce um), which one could compare

to the lists of open problems that are known to mathematicians today, except that Aristotle’s

problems concern all sciences. The set of problems from the school of Aristotle that reached us

contains about 900 problems, classiﬁed into 38 sections, of which two are dedicated to music (one

section is more di rected towards acoustics, and the other one towards music theory). Aristotle, like

his teacher P lato, payed careful attention to music and acoustics, and Aristoxenus, who became

later on the greatest Greek musicologist, was Aristotle’s student.

32

The equally tempered scale became more or less universally adopted in the nineteenth cen-

tury for various reasons, one of them being the appearance of large orchestral ensembles, in which

several kinds of diﬀerent instruments had to be tuned in a uniform way, and the most convenient

way appeared to be the one based on equal division. Furthermore, with that scale, transposi-

tions become easy to p erform, and playing a given piece starting at any note was possible. We

note by the way that equally tempered scales were already considered in the theoretical work of

Aristoxenus.

33

I learned this from F. Jedrzejewski.

22 ATHANASE PAPADOPOULOS

of this art (that is, at the beginning of the Renaissance). Using a mathematical

language, the operations are represented respectively by a translation, a symmetr y

with respect to a horizontal line and a symmetry with respect to a vertical line

(which is outside the motive): equivalently, for the last operation, one reads the

motive backwards.

At the beginning of the twentieth century, and notably under the inﬂuence of

Arnold Sch¨onberg and his School (the s o-called Second Viennes e school), the coun-

terpoint op erations were used in a more a bstract and systematic way by composers

who became known as dodecaphonic or twelve-tone composers and later on as se-

rialists. In this setting, a series, also called a tone row, is a pitch-class set, that is,

a collection of distinct no tes, with no specia l melodic value

34

and which, together

with its transformations by transposition, retrogradation and inversion, is used as

a building block for a musical comp osition. A twelve-tone series is a tone row

in which every integer modulo 12 appears. Thus, a twelve-tone series is simply a

permutation of the seq ue nc e of integers 0, 1, 2, ..., 11. The thre e operations of

transposition, inve rsion and retrogradation can be expressed simply and elegantly

using the formalism of pitch classes, and this was done in the work of Babbitt and

Forte. Indeed, transposition cor responds to translation modulo 12 in the group Z

12

,

inversion is the map x 7→ −x mod 12, followed by a translation, and retrograda-

tion consists in replacing a certain motive n

0

, n

1

, ..., n

k

by the same motive written

backwards, n

k

, ..., n

1

, n

0

. We note that in the context of serial music, the integers

n

i

in the sequence representing a melodic motive have to be distinct as classes mod

12. It is appropriate to quote here Forte (who is not a mathematician), who on p. 8

of his es say [42], stresses the fact that this formalism is not useless: “T he notion of

mapping is more than a convenience in describing relations between pitch-class sets.

It permits the development of economical and precise descr iptions which cannot be

obtained using conventional musical terms”. Let us note that this kind of theory is

known among musicologists as set theory (and in France, it is called American set

theory, or s imply “set theory”, using the English words), although it has not much

to do with set theory as ma thematicians intend it. A recent reference is [6].

Let us now return to Messiaen.

6.2. Generalized series. Messiaen never belonged to the serial school and in fact,

he belonged to no school. However, it is true that some of his compositions contain

techniques which appertain to that school, and above all, his piece Mode de valeurs

et d’intensit´es (Piece No. II of his Quatre Etudes de Rythme for the piano, 1949).

In fact, in this piece, Mes siaen goes beyond the existing twelve-tone techniques by

using the so-called generalized series (Messiaen used the French word supers´erie),

that is, not only series of pitches, but also series of rhythms, of intensities, of attacks,

of dynamics, and of timbres. An example of a series of intensities is the ordered

sequence

ppp, pp, p, mf, f, ff, ff f.

The piece Mode de valeurs et d’intensit´es had a signiﬁcant inﬂuence on the so-

called post-serialist (o r “integral serialism”) school, to which belonged at some point

Boulez and Stockhausen, who had be en students of Messiaen. Boulez, stimulated

34

This contrasts with the usual motives of old contrapuntal writing where the building block,

called the theme, has an intrinsic musical value; it was sometimes – e.g. in the music of Bach –

a theme extracted from a popular song or of a well-known melody. It should be noted however

that Bach also wrote magniﬁcent chorals based on poor themes, in fact, themes obtained by

concatenating notes in a manner which a priori is uninteresting. The richness of the resulting

harmonies is due to the cleverness of the assembly of the theme w ith its symmetric images, obtained

through the counterpoint operations. This is another way of proving – if a proof is needed – that

it is the mathematical structures that are behind a musical piece that makes its beauty.

MATHEMATICS AND GROUP THEORY IN MUSIC 23

by this piece, used ex tensively the principle of generalized series applied to timbre,

intensity, duration, and so on. For instance, he used in his piece Polyphonie X pour

18 instruments (1951) a series of 24 quarter-tones. In his piece Structures I for

two pianos (1952), he used series of 12 durations, of 12 intensities (pppp, ppp, pp,

etc.) and of 12 attacks (staccato, legato, etc.). The position of Messiaen regarding

the serial movement was moderate, and in fact, he disliked the exc essiveness of

abstract formalism that this move ment gave rise to. But most of all, he disliked

the absence of colors in that music. He described the music of Sch¨onberg and the

Second Viennese School as black, morbid and as a “night music”. We mention also

that Boulez later on broke with the serial school, and in fact, one should also note

that the period which followed the compositio n of Mode de valeurs et d’intensit´es

was also a per iod of profound questio ning of the usefulness of the serial techniques,

by the serial composers themselves.

The new combinatorics that Messiaen introduced in his piec e Mode de valeurs et

d’intensit´es had a great inﬂuence that Messiaen himself disapproved. He declares

in [100] p. 119:

In Modes de valeurs et d’intensit´es, I used a superseries in which sounds

of the same name come past various regions m aking them change as to

octave, attack, intensity, duration. I think that this was an interesting

discovery... Everybody used to talk only about th e superserial asp ect,

and also in [10 0], p. 72:

I was very an noyed by the absolutely inordinate importance which has

been granted to that minor work, which is only three pages long an d

which is called Mode de valeurs et d’intensit´es, under the pretext that it

was supposed to be at the origin of the serial shattering in the domain of

attacks, of durations, of intensities, of timbre, in short, of all the musical

parameters. This music may have been prophetic, historically important,

but musically, it is nothing...

6.3. Rhythmical counterpoint. Messiaen developed in his teaching a theory of

counterpoint which is proper to rhythm. One of the features of this theory is that

the co unterpoint transformations are applied to a musical motive at the level of

rhythm, regardless of pitch. Counterpoint writing includes other transformations

than the three that we mentioned in §6.1, and that they can all be applied to

rhythm. In fact, we already encountered two of these transformations in the section

on Indian rhythms, augmentation and diminution. These transformations c onsist

in taking a c ertain melodic motive, keeping unchanged the seque nc e of pitches, and

multiplying the value s of all the durations by a constant fa ctor (which is > 1 in the

case of augmentation and < 1 in the case of diminution). These transforma tions

exist in classical counterpoint, where the melodic motive stays the same while

the durations are transfor med. But in the music of Messiaen, augmentation and

diminution have a more abstract character, because they aﬀect rhythm regardless

of pitch; the latter can undergo either regular or irreg ular transformations. For

example, at the beginning of Piece No . V (Regard du Fils sur le Fils) of the Vingt

Regards sur l’Enfant J´esu s, there is an augmentation of rhythm by a factor of

3/2, while the motive and the transformed one are unrelated from the point of

view of pitches. The themes are written in diﬀerent modes, whereas their various

combinations in the piece constitute a rhythmical canon. In any case, augmentation

and diminution preserve non-retrogr adability.

There are other rhythm transformatio ns which preserve non-retrogradablility,

and which are discussed by Messiae n in the second volume of his Trait´e ([75], Tome

2, p. 41). These include symmetrical ampliﬁcation and symmetrical elimination of

24 ATHANASE PAPADOPOULOS

the extr e mities . A symmetrical ampliﬁcation consists in adding at the two extrem-

ities of a given rhythm another rhythm together with its retrograded form in such

a way tha t non-retrogradability is preserved. For instance, in Piece No. XX of the

Vingt Regards, the theme is exposed at bar No. 2, it a very short theme, and its

rhythmic value is 2, 1, 2 (the amphimacer non-retrogradable rhythm, with the unit

being the sixteenth note). The theme is then symmetrically ampliﬁed at bar No.

4, wher e it becomes 2, 2, 2, 1, 2, 2, 2. At bar No. 6, it is ampliﬁed in a diﬀerent

manner, where it becomes 2, 3, 2, 2, 1, 2, 2, 3, 2. We ﬁnd again this rhythm, with

two diﬀerent ampliﬁcations, la ter on in the same piece (bar 82):

2, 1, 2

2, 2, 2, 1, 2, 2, 2

2, 3/2, 2, 2, 2, 1, 2, 2, 3/2, 2,

and so on. There are other op erations that preserve non-r e trogradability, in partic-

ular the ones calle d by Messiaen enlargement and diminution of the central value

([75], Tome 2, p. 30).

It is interesting to know that the de¸ci-tˆalas contain examples of all the rhythm

transformations that we mentioned.

Classical counterpo int techniques a re usually applied in the composition of canons.

Messiaen wrote rhythmical canons, that is, pieces based on superimposition of

rhythms and their transformatio ns, following a certain regularity rule (the Greek

word “canon” means “rule” ) – for instance a periodicity pattern – governing the su-

perimposition between a motive a nd the transformed motive. A composition such as

the Vingt Regards sur l’Enfant J´esus contains several c anons of non-retrogradable

rhythms, and it is interesting that Messiaen indicates explicitly on the score, at

several places, the subjects and the counter-subjects of these canons, as well as the

various transformations which they undergo, in order to help the reader understand

the structure of the piece.

Let us note ﬁnally that Messiaen’s Turangalˆıla Symphony is consider ed in itself

as an immense counterpoint of rhythm.

In the next section, we shall se e how c ounterpoint operations were neces sary fo r

Messiaen in his transc ription of bird songs.

6.4. Counterpoint and bird singing. Birds represented one the most important

sources of inspiration for Messiaen, and probably the most important one . His

interest for bir d songs exceeded his interest for any other music. His early piece

Quatuor pour la ﬁn du Temps (1946) contains already a lot of bird song material,

and one can hea r bird songs in all the piec e s that he composed after 1950. In

fact, bird songs are even the central element in several of his pieces, including Le

r´eveil des oiseaux (1953), Oiseaux exotiques (1956), Catalogue d’oiseaux (1958), La

fauvette des jardins (1970), Petites esquisses d’oiseaux (1985), etc. In this respect,

one has to mention also his opera Saint Fran¸cois d’Assise (composed in 1975-1983)

and another major piece, D es canyons aux ´etoiles (composed in 1971-1974).

Let us see how counterpoint operations were necessary during this transcr iption

process.

Diminution, ﬁrst, was needed because some birds sing at a tempo w hich is so fast

that it is impossible to reproduce by any per former. Thus, Messiaen, in writing

these songs, was led to use rhythm diminution. Transposition was also needed

because some birds sing at a register which is too high to be played on an instrument

like the piano, for which, for instance, Catalogue d’oiseaux was written.

35

Messiaen

was led therefore to transcribe the song at a register which is one or several octaves

35

The piano is among the instruments which have the w idest register.

MATHEMATICS AND GROUP THEORY IN MUSIC 25

lower. Finally, Mes siaen appeals to an unusual operation of augmentation at the

level of pitches. Indeed, some birds use in their singing tiny interva ls (o f the order of

a comma), and to transcribe their songs for an instrument like the piano, Messiae n,

in the absence of these micro-tonal interva ls, replaced, for instance, an interval of

two commas by a semitone and the rest o f the transcription fo llows, multiplying the

interva ls by the same factor. Thus, an interval of four commas became a s econd,

and so forth.

7. Modes of limited tra n s po sitio n

The theory of modes of limited transp osition is a spectacular example where

symmetry and group theory are use d in music. It is one of the most important

characteristics of Messiaen’s compositions (saying that a certain piece o f Messiaen

is in a certain mode is like saying that Beethoven’s ﬁfth symphony is in the C minor

scale), and it is a purely mathematical notion. Before talking about this subject,

we say a few wo rds about the meaning of the word mode in music, which might be

useful for the reader . In the last subsection, we shall talk about the relation between

modes and colors as theorized by Messiaen. This subject has a long history, and

it is in the tradition of the relation between pitch and color that was studied by

Newton and others which we mentioned in §3 of this article.

7.1. Modes. The title of Messiaen’s treatis e, Trait´e de rythme, de cou leu r et

d’ornithologie [75], c ontains the word color. He describes there his perception of

the relation between mode and color, and fully understanding this req uires a cer -

tain menta l faculty which Messiaen possessed, namely, that of seeing in a precise

way the diﬀerent colors associated with modes, within harmonies or within melodic

motives. We shall say more abo ut this in the last subsection of this article, and

the interested reader can consult [100], p. 95, the chapter entitled Of sounds and

colors. In the absence of this faculty, one can resort to a more down-to-earth deﬁ-

nition, and consider a mode as a sequence of distinct notes which describe (or, in a

certain sense, characterize ) the atmosphere of a musical composition, or of the part

of the composition which is co ncerned by this mode. There also exist pieces that

are polymodal, and s ome others that are not modal. But most of the pieces that we

hear are modal. The major and minor scales which are usually as sociated to the

familiar pieces of music are approximations of this atmosphere. They classify the

so-called “tonal music”, that is, most pre-twentieth century clas sical pieces (and

practically all modern popular music piece s). Modes are described by sequences of

notes. Rameau, in his D´emonstration du principe de l’harmonie (see [94], Vol. 3),

writes: “In music, a mode is nothing else than a prescrib ed order between sounds ,

both in harmony and in melody.” T he music of Ancient Greece, that of China and

that of India possess a rich variety of modes. Byzantine chant, and to some extent

Gregor ian chant, that are considered to have their roots in Ancient Greek music,

have a lso several modes.

7.2. Modes of limited transposition. Mathematically, a mode of limited trans-

position is a sequence of distinct notes that has a nontrivial group of symmetries

by the a ction of tr anslations mod 12, that is, by transpositions mod 12. Although

Messiaen did not do so, it is convenient to describe these mo de s using the formalism

of pitch-classes which we recalled above. In this setting, a mode is a certain pc-set,

that is, a set of e lements in the group Z

12

, and a mode of limited transposition

is associated to a pc-set which is invariant by a nontrivial translation mod 12. In

this sense, a mode of limited tra nsposition is a pc-set which has some symmetry.

The order of a tr anslation in Z

12

is necessarily a divis or of 12, and we shall see

now how the various modes ar e associated to trans lations of order 2 , 3 or 6. The

26 ATHANASE PAPADOPOULOS

familiar major and minor sca le s are invariant by a translation o f order 12 and not

less (and therefore by the identity map) and in that sense they are not modes of

limited transposition. The expression of a given mode in terms of pitch-classes,

that we give below, highlights in a particula rly simple manner the symmetries of

such a mode.

Messiaen used modes of limited transposition since his ﬁrst compositions. They

are present, for instance, in his Preludes for the Piano, which he composed in 1929,

at the age o f 19. He writes in his Technique de mon langage musical [74] p. 13:

“These modes realize in the vertical sense [that is, at the level of pitch] what non-

retrogradable rhythms realize in the horizontal sense [at the level of time]”. There

is no non-re trogradability in the modes of limited transposition, and the common

feature between the two notions is the presence of symmetry.

In his classiﬁcation, which we shall r ecall below, Messiaen keeps seven modes

of limited transposition. In his compositions, he uses mostly Modes 2, 3, 4 and

6, with a preference for Mode 2, which at his ﬁrst transposition corresponds to

diﬀerent gradations of blue-violet, which was his favorite color. We also note that

this classiﬁcation in seven modes has not much to do with the previous existing

classiﬁcations (e.g. the eight modes o f Gregorian chant or of Byzantine music, or

those of Ancient Greece which are much larger in number).

As we said a bove, the theory assumes that the tuning of the instruments for

which the piece is written is the modern equally tempered tuning (equal intervals

between two c onsecutive notes).

Mode 1.— This is the sequence of notes C, D , E, F♯, G♯, B♭. Messiaen describes

this mode as being “two times transposable”, and by this he means that the cor-

responding pc-set is invariant by the translation x 7→ x + 2. The pc-set [0, 2, 4,

6, 8, 10] and it is the orbit of 0 by this translation.

36

In fa ct, as Messiaen notes,

this mode is Debussy’s whole tone scale and it was already thoro ughly used by

this composer, for instance in certain passages of his opera Pell´eas et M´elisande.

It was also used by Paul Dukas (who was Messiaen’s teacher) in his oper a Ariane

et Barbe-Bleue and by Liszt in his opera-fantasy for piano R´eminiscences de Don

Juan. This scale is called the whole tone scale because each deg ree increases by a

tone. It corresponds to the division of a n octave into six equal subintervals, and it

is represented in musical notation as follows:

Messiaen made a very moderate use of the ﬁrst mode, probably because it was not

very innovative.

37

Mode 2.— In the language of Mess iaen, this mode is “three times transposable”,

which means that, as a subset of Z

12

, it is invariant under the tra nslation x 7→ x+3.

It is given by the sequence of notes C, D♭, E♭, E♮, F♯, G, A, B♭, which corresponds

to the pc-s et [0, 1, 3, 4, 6, 7, 9, 10], tha t is, the union of the orbits of 0 and 1 by the

36

The pitch-class sets that are described here start at 0, and they correspond to modes which

are at their ﬁrst transpositions.

37

Messiaen wr ites, in citeMessiaen1944 (Vol. I p. 52): “Claude D ebussy (in Pell´eas et

M´elisande) and, after him, Paul Dukas (in Ariane et Barbe-Bleue) made such a remarkable use of

this mode that there is nothing to add to that. Therefore we shall carefully avoid using it – unless

it is hidden in a superimposition of mo des that will make it unrecognizable.” (The translations

from the French are mine.)