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  and , 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.
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  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 , 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
A la recherche du rythme” in . 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
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.