Nabokov, Nicolas [Nikolay]

Numbers and music.

Numbers appear in musical notation in a variety of ways, sharing, for instance, tempos, pulses, figured basses and bar numbers; they are also used to describe intervals (e.g. 3rds, 4ths, 5ths) and tunings of the scale in temperaments. This article, however, is concerned with numbers that can be neither heard nor seen but are used in the process of constructing a composition. For other uses of numbers in music see Cryptography, musical, and Rhetoric and music.

1. Introduction.

2. Up to 1600.

3. 1600 to 1750.

4. 1750 to 1900.

5. From 1900.

BIBLIOGRAPHY

RUTH TATLOW (1–4), PAUL GRIFFITHS (5)

Numbers and music

1. Introduction.

In the past 50 years some startling claims have been made about how Renaissance and Baroque composers might have used numbers in their music. The numbers are generated from the score by counting, for example, how many breves, bars, pulses or notes there are, or how many times a word is repeated, in a phrase, section, complete voice or movement. Characteristic of this so-called ‘numerological’ approach is the swift move from counting to interpreting. When a number recurs or is deemed significant it is treated as symbolic and interpreted either by traditional symbols or through the use of a number alphabet.

The number alphabet was first introduced to musicology by Friedrich Smend. Smend and his colleague Martin Jansen spent many years trying to discover the meanings of numbers that recur in the works of J.S. Bach. They began from the premise that every numerical relationship in the score was consciously placed there by the composer and could therefore be considered symbolic. In 1943 Smend added the use of the natural-order number alphabet (A = 1, B = 2, C = 3 etc.; IJ = 9, UV = 20, Z = 24) to their growing list of interpretative methods. Jansen was alarmed by this, arguing that the method could make any number mean many things, but his early death in 1944 cut short his moderating influence.

In 1947 Smend wrote publicly about the number alphabet in the third and fourth of six booklets of programme notes for a series of performances of Bach’s church cantatas in Berlin. He unwittingly introduced several factual errors, however, which in turn misled him as he interpreted the numbers: he gave examples of only two number alphabets, whereas over 40 may be found in various printed German sources of the 17th and 18th centuries, and he confused two distinct number alphabet traditions, the cabalistic gematria, which is a means of interpreting the Bible, and the poetical paragram (see §3 below), which is a means of generating ideas used by poets. Since 1947 other musicologists have experimented with Smend’s theory, with varying degrees of credibility. Some fundamental errors have been perpetuated, including the inaccurate use of the term ‘cabalistic gematria’, indiscriminate use of the natural-order number alphabet and historically incongruous interpretations of numbers.

There is value in Smend’s work, however. It has recently been shown that, since at least the 1630s, the poetical paragram had been a widely known technique for generating ideas before writing a poem (Tatlow, 1991). In Bach’s day it was one of the techniques listed among the loci topici (see Rhetoric and music, §I, 2, and §3 below) in poetry textbooks. Although the paragram technique does not appear in books on music theory, it is possible that musicians may have applied it to music.

Musicology is left with a dilemma. Counting notes and pulses frequently reveals a numerical correlation between the sections of a musical work. This could imply that the composition was organized numerically at an early stage, and the temptation for the modern analyst is to assert that the numerical relationships were devised by the composer. Yet there is slender historical evidence to support this: little is known from music theory or surviving sketchbooks about the pre-compositional processes of composers before Beethoven. Without a firm historical basis it is both premature and irresponsible to draw conclusions about compositional procedure from numbers in the score. A separation must be maintained between numerical analysis, comment upon the compositional process and speculative interpretation of the numbers. There is also a need to consider whether there is any historical justification for the analytical techniques used to generate the numbers; and if so, whether the numbers in the score were created consciously by the composer and whether the numbers are wholly structural or have some further significance.

Numbers and music

2. Up to 1600.

Number was of fundamental importance to both the Judaeo-Christian and the Greek view of the Creator and Creation, and therefore also to that of Renaissance man. Plato wrote: ‘The body of the universe was created to be at unity owing to proportion; in consequence it acquired concord’, (Timaeus, trans. H.D.P. Lee, 1965, p.44); and the anonymous verse from the Apocrypha frequently quoted in music treatises reads: ‘Thou hast arranged all things by measure and number and weight’, (Wisdom of Solomon xi.21).

Numbers as proportions feature prominently in the music theory of this period, reflecting the classical Greek ideas of the schools of Pythagoras and Aristoxenus. Although it is possible to imagine ways of introducing proportions at every stage of the compositional process, Renaissance treatises discuss proportions solely in their capacity of describing intervals and duration. Following Pythagoras, intervals were expressed as ratios, based on the division of the monochord: the octave as 2:1, the 5th as 3:2, the 4th as 4:3, the major 3rd as 5:4, major 6th as 5:3, minor 3rd as 6:5 and the minor 6th as 8:5. During the early 17th century these were rejected in favour of more complicated arithmetical expressions of each interval. Numbers as proportions were also used to express duration in different mensural notation systems. In some compositions there is an exact proportional relationship between individual sections. In Du Fay’s motet Nuper rosarum flores, for example, the mensuration within the four sections is in the ratio 6:4:2:3 (Warren, 1973), and in Leonel Power’s Mass Alma Redemptoris mater the relationship between the cantus-firmus and non cantus-firmus sections is 48:12:48:24, or 4:1:2:1 (Sandresky, 1979).

These two works also illustrate another aspect of the interpretation of numbers and proportions in studies of Renaissance music, the claim that proportions in certain compositions are identical to those of specific buildings. Alberti’s design for the upper façade of S Maria Novella, Florence, supposedly has the proportions 4:1:2:1, corresponding with the proportional design of Power’s mass (Sandresky, 1979), and the proportions 6:4:2:3 of Brunelleschi’s dome for Florence Cathedral have been claimed to correspond with those of Du Fay’s motet (Warren, 1973). It has been argued that Warren’s architectural figures are faulty, and that the proportions 6:4:2:3 were used by Du Fay in imitation of those specified by God (1 Kings vi.1–20) for building Solomon’s temple (Wright, 1994). Interestingly, it is the interpretation of the proportions that has received criticism rather than the method of generating the proportions from the music, which could be considered conjectural.

Similarly, some 20th-century analysts have sought to demonstrate the use of Fibonacci numbers as a conscious compositional device in Renaissance music. The ratio between the successive terms of the Fibonacci series (in which each is the sum of the previous two, thus: 0, 1, 1, 2, 3, 5 etc.) is an arithmetical expression of Euclid’s golden ratio. Although the Fibonacci series was first described in 1202, it was not widely known to be connected to Euclid’s formula until the mid-19th century. It is thus extremely unlikely that works composed before 1600 were deliberately constructed to express the golden ratio, and any attempt to prove otherwise is little more than an interesting 20th-century analytical exercise.

Many biblical numbers had significance for the Renaissance musician. In his De mystica numerorum significatione (Bergamo, 1583), Petrus Bongus discussed the symbolic significance of specific numbers, showing that there is frequently more than one interpretation of a number. More importantly, though, Bongus did not describe how, if at all, the numbers are incorporated into a musical composition.

Following the work of Friedrich Smend there has been an awakening of interest in number symbolism in Renaissance music. The use of the misleading term ‘cabalistic gematria’ by some scholars has perpetuated the confusion over the use and implications of number alphabets. The natural-order number alphabet is invariably used to decode recurring numbers, while others, such as the Hebrew milesian alphabet (ℵ=1, = 2, … = 10, = 20 … = 100, =400) associated with Jewish or Christian cabalism, may be overlooked. Number alphabets have been used to reveal the name of a person closely associated with the composition hidden in the music. Some scholars have applied this technique to works by composers such as Jacob Obrecht (9+1+3+14+2 14+2+17+5+3+8+19 = 97), Ockeghem (14+3+10+5+7+8+5+12 = 64), Du Fay (4+20+6+1+23 = 54), Josquin des Prez (9+14+18+16+20+9+13 = 99, 4+5+18+17+5+24 = 88) and Tinctoris (19+9+13+3+19+14+17+9+18 = 121) (Elders, 1985).

There are certain problems in the application of these techniques. Methods of number counting can appear more subjective than scientific; and ambiguities in the music can lead to inconsistencies in choices such as whether or not to count a repeat, whether a long is worth two or three breves, whether or not a corona adds to the duration, and where to divide a section or group of notes. Additional problems for music of this period are the lack of autograph sources and the issue of errors introduced by a scribe. Until there is more historical or documentary evidence to support number counting techniques, the naive musicologist could easily find himself spinning neat sequences of numbers to no scholarly purpose. Musicology of this nature must for the present be treated as analytical exploration.

Numbers and music

3. 1600 to 1750.

Numbers expressed as ratios or fractions appear in many music treatises published between 1600 and 1750, as a result of their authors’ preoccupation with tuning systems. Numbers linked to letters, pitches and rhythms also appear in descriptions of cryptographic techniques (see Cryptography, musical). Numbers as ratios expressing the consonant intervals of the octave were observed by Kepler to be similar to those between the speeds of the planets in their orbits (Harmonices mundi, bk 3, 1619). The medieval view that musical ratios and harmony reflect the nature of God was endorsed by Kircher in his widely distributed treatise Musurgia universalis (1650). Numbers with symbolic values are rarely referred to, although Werckmeister believed that the triad represented the Trinity (1 = God the Father, 2 = God the Son, 3 = God the Holy Spirit). As Werckmeister gave no practical examples of how or when the triad should be interpreted symbolically in a specific composition, however, this seems to be a perpetuation of the general medieval view set out in Kircher’s work.

The lack of historical evidence of the use of compositional numbers seems extraordinary in view of the quantity of writing on Bach and number symbolism. A series of number techniques based on Smend’s work has evolved and become accepted by dint of repetition. Yet in treatises of this period there is virtually no discussion of the use of numbers in the construction of a composition, either proportionally or symbolically, nor any description of numbers used as a pre-compositional aid to invention.

A popular way of generating ideas at this time was through the ars combinatoria. In his dissertation De arte combinatoria (1666) the mathematician and philosopher G.W. Leibniz described the principle of a universal language, and 12 years later he produced a fully developed artificial language which he believed could be translated into music by using intervals instead of consonants and vowels.

The loci topici or loci dialectici (which despite their classical-sounding title, were not known to the ancient Greeks) became popular as devices for generating philosophical and rhetorical arguments in the early 16th century, and were also applied to poetry and music. The classification of loci species varied from author to author. In a lecture given at Leipzig University in 1695 the poet Erdmann Neumeister described 15 species: locus (i) notationis, (ii) definitionis, (iii) generis & specierum, (iv) totius & partium, (v) causa efficientis, (vi) causae materialis, (vii) causae formalis, (viii) causae finalis, (ix) effectorum, (x) adjunctorum, (xi) circumstantiarum, (xii) comparatorum, (xiii) oppositorum, (xiv) exemplorum and (xv) testimoniorum. The locus notationis itself was subdivided into (i) derivation, ii) aequivocation, iii) synonyma, iv) anagramma and v) artificium cabalae. Under locus notationis (v) artificium cabalae five different number alphabets are listed, each of which could be used in several ways to generate ideas in poetry. Among these is the poetical paragram, a technique adapted from cabalism simply to stimulate the imagination. For example: Table 1. As the words ‘Margaretha’ and ‘Meine Seele’ have the same numerical value using the natural-order number alphabet, the poet could use them as the starting point of his poem. Neumeister’s work was published in 1707 by the poet and librettist ‘Menantes’ (C.F. Hunold). Both men were known to J.S. Bach, who was probably familiar with this publication but there is no proof that he adapted the paragram technique to musical invention.

Several German theorists, from Burmeister (Musica poetica, 1606) to Spiess (Tractatus musicus compositorio-practicus, 1745), used rhetorical models and the loci topici, in their discussions of music (see Rhetoric and music, §I). Mattheson used the terms inventio, dispositio, elaboratio and decoratio to structure his discussion of compositional procedure in Der vollkommene Capellmeister (1739). In the section on Inventio he applied Neumeister’s 15 loci to musical composition, but included neither number alphabets nor any adaptation of the poetical paragram to music in his illustration of the locus notationis.

Following the many experiments in using the ars combinatoria for musical invention (notably by Leibniz, Euler, Riepel, Christian Wolff and Gottsched), Lorenz Mizler von Kolof produced his own theoretical explanation of music. An important debate between Mizler and Mattheson about numbers in music is documented in Mizler’s journal the Neu eröffnete musikalische Bibliothek (founded in 1737) and Mattheson’s treatise Plus ultra (1754–6). In response to Mattheson’s assertion that mathematics is not the basis of music (Harriss, 1981, p.46), Mizler wrote:

Mathematics is the heart and soul of music … Without question the bar, the rhythm, the proportion of the parts of a musical work and so on must all be measured … Notes and other signs are only tools in music, the heart and soul is the good proportion of melody and harmony. It is ridiculous to say that mathematics is not the heart and soul of music [Neu eröffnete musikalische Bibliothek, ii, 1743, p.54]

It is highly likely that Bach was aware of these discussions, as he knew both men and in 1747 became the 14th member of the society founded by Mizler in 1738 in order to stimulate discussion about music among composers (numerologists have made great play of this since BACH = 2+1+3+8 = 14 in the natural-order number alphabet). But the analyst must be cautious: documentary evidence that Bach’s sympathies lay with Mizler rather than Mattheson may not necessarily be a sufficiently firm foundation on which to build a theory of Bach’s pre-compositional numerical method.

The evidence that comes closest to implying the use of number in the pre-compositional organisation of a work comes not from Mattheson’s section on Inventio, but from his section entitled Dispositio. Again combining artistic forms, he likened compositional construction to architecture:

DISPOSITIO is a neat ordering of all the parts and details in the melody,

or in an entire musical work, almost in the manner in which one arranges or
draws a building, makes a plan or sketch, a ground plan, to show where
e.g. an assembly room, an apartment, a bedroom etc. should be situated.
[Harriss, 469]

Although numbers are not specified, one could argue that Mattheson strongly implied their use since architectural plans at that time were ordered numerically. Mattheson’s clearest articulation of pre-compositional planning can be read in paragraph 30:

§30 [the composer] should outline his complete project on a sheet, sketch it roughly and arrange it in an orderly manner before he proceeds to the elaboration. In my humble opinion this is the best way of all through which a work obtains its proper fitness, and each part thus can be measured to determine if it would demonstrate a certain relationship, similarity, and concurrence with the rest: in as much as nothing in the world is more pleasing to the hearing than that. [Harriss, 478]

Again Mattheson does not specify numbers, but a recommendation that could easily be a practical demonstration of Mattheson’s principles appears in volume iv (1754) of Mizler’s Bibliothek (pt 1, p.108). In a section that directly follows the announcement of Bach’s presentation to the society of canon bwv1076, the anonymous author writes:

In the winter the cantata should be somewhat shorter than in summer … From experience one can specify the duration, namely that a cantata 350 bars long of varying mensuration takes roughly 25 minutes to perform, which in winter is long enough, whereas in summer it can be 8 to 10 minutes longer and so give a cantata of roughly 400 bars.

Although Bach may not have devised these guidelines, he would, as a society member, have been involved in the discussions and endorsed the recommendations. In 1619 Michael Praetorius had made a similar recommendation for measuring the duration of a composition:

80 tempora take half of a quarter of an hour, 160 tempora take a quarter of an hour, 320 tempora half an hour, 640 tempora an hour. In this way one can so much better judge how long the song or work is so that the sermon may begin at the correct time and the other church ceremonies adapted accordingly.

In both of these examples numbers are used as a tool to measure the length of a church cantata in bars and in minutes. It is an indication that in this period there was an increasingly pragmatic approach towards composition.

Numbers and music

4. 1750 to 1900.

Among the music treatises of the late 18th and early 19th centuries is a corpus of material that recommends using numbers for composition. At least 20 different methods of composing music by numbers were published between 1757 and 1812, the first being Der allezeit fertige Polonoisen- und Menuettencomponist (1757) by J.P. Kirnberger, a pupil of J.S. Bach. According to Kirnberger:

Anyone who is familiar only with dice and numbers and can write down notes is capable of composing as many of the aforesaid little pieces as he desires.

Kirnberger’s method was repeated and adapted several times, including in two publications attributed to Haydn (1793) and Mozart (1793). A different method, based on the use of a nine-sided top, was first published in H.-F. Delange’s Le toton harmonique (1768) and in an anonymous Ludus melothedicus (1760s). C.P.E. Bach published a variant of this second method as ‘A Method for Making Six Bars of Double Counterpoint at the Octave Without Knowing the Rules’ in the first part of Marpurg’s Historisch-kritische Beyträge zur Aufnahme der Musik (1754). A publication that uses neither system but allows random selection of any number between eight and 48 is Piere Hoegi’s A Tabular System whereby the Art of Composing Minuets is Made so Easy that any Person, without the Least Knowledge of Musick, may Compose Ten Thousand, all Different, and in the most Pleasing and Correct Manner (London, ?1770). This is the epitome of instant composition: it shows a healthy playfulness with compositional method, at variance with the received view of the closed master–pupil apprenticeship of earlier generations and the later cult of the inspired genius. Analyses of surviving polonaises by Kirnberger or of C.P.E. Bach’s double counterpoint have yet to be made to assess whether they used these methods in their own compositions.

Based on principles taken from Smend, the natural-order number alphabet, traditional symbolic numbers (such as 3 = the Trinity, 33 = number of years of Jesus’s life) and cabalistic techniques (such as triangular numbers, or the cubing, squaring or doubling of a number to increase its potency) have infiltrated musicological studies of this period. It has been claimed, for example, that certain works by Mozart, with his Masonic associations, and Beethoven demonstrate a well worked-out numerological plan.

The study of numbers in the background compositional design of works written before 1900 has great potential for musicology; but until a historically consistent theory can be formulated that resolves the problems described in the introduction, number studies will remain at best interesting speculation.

Numbers and music

5. From 1900.

That the fate of numbers in music greatly improved around 1900 and thereafter can be attributed to various changes in how music was made and perceived: the arrival of recording technology, which facilitated numerical measurements (of duration, frequencies etc.); intensified interest in folk music and in non-European cultures, bringing an awareness of other scales and, perhaps most significantly, other ways of handling rhythm, not as a hierarchy of nested elements (beat, bar, phrase) but as pulse, which might invite composers to count up to more than four; and an increased systematization of composition. These phenomena were linked. For example, the study of ethnic music was greatly assisted by recording, at a time when mechanization in other forms (in agriculture, for instance) was placing such music under threat. And the triumph of the machine in the 20th century – including most particularly the computer revolution that began in the late 1940s – must have had a part in changing ideas about the brain and therefore about creativity, which could now be seen as dependent on processes of selection and arrangement, and so able to profit from systematic methods.

One of the earliest and most influential of such methods, serialism, was not meant by Schoenberg to be any more (or any less) systematic than tonal composition, and even the works of Webern, who used serialism highly systematically, do not appear to have recourse to numbers in any but the most traditional ways (such as rhythmic augmentation and diminution). Schoenberg was undoubtedly no stranger to numerology – he changed the name ‘Aaron’ to ‘Aron’ so that the title of his opera would have 12 and not 13 letters – but there is no evidence that he used numbers in the substance of his music. With Berg, though, there is abundant proof, and not only in his serial works. Indeed, a pre-serial composition, his Kammerkonzert, is one of the most conspicuously number-infested pieces he achieved, being constructed in units of 30 bars, on three themes, with three basic colours (piano, violin, wind): three and its multiples everywhere to celebrate the triumvirate of the Second Viennese School. Berg's conception of a ‘Hauptrhythmus’, in this work and others, could also have sprung from an arithmetical turn of mind. In his Lyrische Suite the guiding numbers, affecting both lengths in bars and metronome markings, are 23 and 50: the former, which has an important role too in Lulu and the Violin Concerto, he felt to be his personal number, for the reason, as he told Willi Reich, that he had suffered his first asthma attack on the 23rd of the month.

A more concealed use of numbers to control formal proportions (i.e. to do what Berg did openly) has been proposed in other music of roughly the same time, notably that of Debussy and Bartók. In both cases the golden section (see Golden number) is a favoured analytical goal: Howat (1983) found this in Reflets dans l'eau, La merand other works of Debussy; Lendvai (1971) has it in Bartók's Sonata for Two Pianos and Percussion. However, neither composer said much about his compositional technique, nor left sketches to indicate that forms (or, to follow Lendvai, rhythms and scales) were numerically derived. The appearance of the Fibonacci series in a rhythmic pattern at the start of the slow movement of Bartók's Music for Strings, Percussion and Celesta (1–2–3–5–8–5–3–2–1) is suggestive, but no more, and Somfai (1996) discounted the notion that Bartók used numbers in any non-traditional way.

The new importance of pulse in Stravinsky's music, particularly during the decade from Petrushka (1910–11) onwards, led to rhythms based on numbers. For example, the first of the Three Pieces for string quartet (1914) has overlaid ostinatos repeating at different intervals through time, and the tempos of The Wedding are geared in the ratios 2:3:4. But number is less an issue here than mechanical regularity, as it is in Satie's assemblage of music by time-lengths in his music for the film episode in Relâche(1924). Working for the cinema encourages a composer to think in terms of absolute duration, and the arrival of film provided at least a parallel for, if not a stimulus to, the new sense of time (pulsed, chronometrical) and form (edited, cross-cut) in Stravinsky and Satie. The latter in particular, together with information brought back from Bali by McPhee, prompted Cage to base most of his works of the late 1930s and the 1940s on durational frames based on numbers, and now musical arithmetic is often to the forefront. In Cage's First Construction (in Metal) for percussion ensemble (1939), for instance, the sequence 4–3–2–3–4 governs the grouping of the work's 16 sections and also the grouping of bars within each section.

At the same time Messiaen was beginning to use numbers consciously in formal and rhythmic construction, influenced perhaps by Stravinsky and certainly by the importance of number in Christian symbolism. Again, pulsed rhythm was part of the game. In his work with rhythmic cells, Messiaen preferred ‘non-retrogradable’ (i.e. palindromic) patterns as well as figures whose lengths were determined by prime numbers. To these in the late 1940s he added, with a nod to serialism, the use of durations embodying the arithmetical series from one to 12 (e.g. demisemiquaver to dotted crotchet) and a system of rhythmic ‘interversion’, by which sequences of numbers expressed as durations would be changed by rule. If, to give a simplified example, the sequence 1–2–3–4 were followed by 2–3–4–1, the next sequence would have to make the same changes (1 to 2, 2 to 3, etc.) and so would be 3–4–1–2. Such procedures were stimulated by a reverence for number as part of the divine order; as for numerical symbolism, that tends to determine more often the number of movements in a work: seven (the perfect number) in Les corps glorieux, Visions de l'amen, Chronochromie, Sept haïkaïand each part of La Transfiguration, for example; eight (perfection plus) in Quatuor pour la fin du temps and Saint François d'Assise; or 12 in La nativité du Seigneur.

Messiaen's contemporary Babbitt looked not at all to the extra-humanity or symbolic significance of numbers but rather to how mathematics might help composers to explore serialism more cogently. Combinatoriality, being a branch of set theory (see Set), provided an understanding of how Schoenberg and Webern had worked with like or unlike hexachords, and so suggested extensions to their procedures. The differences from Messiaen are most conspicuous at the rhythmic level, for Babbitt's inventions – the use of small durational sets amenable to inversion and retrograding, and the ‘time point’ system, by which the key phenomenon is the point at which an event occurs in the bar – are consonant both with Schoenbergian pitch serialism and a traditional (if highly sophisticated) understanding of rhythm as metrical.

The period immediately after World War II, when Babbitt produced his first acknowledged works, also saw a growth in number music as a result of Messiaen's influence on the younger composers who were his pupils (Boulez, later Stockhausen and Xenakis) and of new means of sound programming – not only electronic music but also, in the case of Nancarrow, the player piano. Nancarrow used this instrument from the late 1940s onwards in order to realize patterns of arithmetical durations (invented, it would seem, independently of Messiaen) and, most usually and variously, overlays of different tempos and/or metres executed by different canonic lines. In his Study no.1, for example, there are five simultaneous tempos related as 2:3:5:8:14; Study no.33 uses the tempo relationship √2:2; and Study no.27 has, around one voice in constant tempo, others in accelerandos or ritardandos defined in terms of percentages, the change being by 5%, 6%, 8% or 11% from one sound to the next.

Like the player-piano roll, magnetic tape was a handicraft medium (in the 1950s and 60s works could only be created by splicing together lengths of tape), and so invited the planning of rhythms and forms in measured durations. If no composer of electronic music equalled Nancarrow's rhythmic virtuosity, many used number systems, often by analogy with serialism. For instance, the sequence of whole numbers from one to six is important in Stockhausen's early pieces, both electronic and instrumental; later he used the Fibonacci series, most conspicuously in his Klavierstück IX(1961). The frequencies of electronic music also had to be determined, and here too Stockhausen used arithmetic to derive new tuning systems. Xenakis at the same time was using numerical calculations, especially from the mathematics of probability (Poisson distributions, Markov chains), to determine pitches, durations and timbres in events conceived globally; the tasks were eminently suited to computers, which he began to use in 1956.

To the extent that computers are number machines, all computer music is number music – but then any violinist playing in just intonation is also spinning integers. Numbers, in short, may be essential but not evident, and they are perhaps only likely to be evident when, as often in Messiaen and Nancarrow, and sometimes in Cage and Stockhausen, they are simple integers controlling durations, tempos or structural proportions. Nevertheless, the ubiquity of computers since the early 1980s may have encouraged composers, both in and outside the field of computer music, to think of music as a play of numbers: the later works of Ligeti, often based on numerical rules and systems that gradually produce results of high complexity, offer many examples. Ligeti is also, with John Adams, among the composers who have been spurred by the graphic results of fractal mathematics to create similar self-similar musical constructions. Minimalist music, too, may be number music: Glass's Einstein on the Beach begins with the chanting of numbers. And Lendvai's study, however dubious its application to its ostensible subject, surely helped promote interest in musical numbers among composers two and three generations younger.

Numbers and music

BIBLIOGRAPHY

up to 1900

W. Werker: Studien über die Symmetrie im Bau der Fugen … des Wohltemperierten Klaviers von J.S. Bach (Leipzig, 1922)

M. Jansen: ‘Bach Zahlensymbolik, an seinen Passionen untersucht’, BJb 1937, 98–117

F. Smend: Johann Sebastian Bachs Kirchen-Kantaten (Berlin, 1947–9, 2/1966)

F. Feldmann: ‘Numerorum mysteria’, AMw, xiv (1957), 102–29

L.G. Ratner: ‘Ars combinatoria: Chance and Choice in Eighteenth-Century Music’, Studies in Eighteenth-Century Music: a Tribute to Karl Geiringer, ed. H.C.R. Landon and R.E. Chapman (New York and London, 1970), 343–63

R.L. Marshall: The Compositional Process of J.S. Bach (Princeton, NJ, 1972)

C. Warren: ‘Brunelleschi's Dome and Dufay's Motet’, MQ, lix (1973), 92–105

U. Siegele: Bachs theologische Formbegriff und das Duett F-Dur (Stuttgart, 1978; Eng. trans. in MAn, xi (1992), 245–78)

B. Trowell: ‘Proportion in the Music of Dunstable’, PRMA, cv (1978–9), 100–41

M. Sandresky: ‘The Continuing Concept of the Platonic Pythagorean System and its Application to the Analysis of Fifteenth-Century Music’, Music Theory Spectrum, i (1979), 107–20

E. Harriss: Johann Mattheson's ‘Der vollkommene Capellmeister’: a Revised Translation with Critical Commentary (Ann Arbor, 1981)

C. Wolff: ‘“Die sonderbaren Volkommenheiten des Herrn Hofcompositeurs”: Versuch über die Eigenart der Bachscher Musik’, Bachiana et alia musicologica: Festschrift Alfred Dürr, ed. W. Rehm (Kassel, 1983), 356–62

L. Curchin and R. Herz-Fischler: ‘De quand date le premier rapprochement entre la suite de Fibonacci et la division en extrême et moyenne raison?’, Centaurus, xxviii (1985), 129–38

W. Elders: Componisten ven de Lage Landen (Utrecht, 1985; Eng. trans., 1991), 76–86

A. Dürr: Bachs Werk vom Einfall bis zur Drucklegung (Wiesbaden, 1989)

G. Butler: Bachs Clavierübung III: the Making of a Print: with a Companion Study of the Canonic Variations on ‘Vom Himmel Hoch’ BWV769 (Durham, NC,1990)

R. Tatlow: Bach and the Riddle of the Number Alphabet (Cambridge, 1991)

I. Grattan-Guinness: ‘Counting the Notes: Numerology in the Works of Mozart, especially Die Zauberflöte’,Annals of Science, xlix (1992), 201–32

I. Grattan-Guinness: ‘Why did Mozart Write Three Symphonies in the Summer of 1788?’, MR, liii (1992), 1–6

I. Grattan-Guinness: ‘Some Numerological Features of Beethoven's Output’, Annals of Science, li (1994), 103–35

R. Herz-Fischler: ‘Fibonacci and the “Fibonacci sequence”’, ‘The Golden Number, and Division in Extreme and Mean Ratio’, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, ed. I. Grattan-Guinness (London, 1994), ii, 1579–84

B. Stephenson: The Music of the Heavens: Kepler's Harmonic Astronomy (Princeton, NJ, 1994)

C. Wright: ‘Dufay's Nuper rosarum flores, King Solomon's Temple, and the Veneration of the Virgin’, JAMS, xlvii (1994), 395–441

I. Grattan-Guinness: ‘Mozart 18, Beethoven 32: Hidden Shadows of Integers in Classical Music’,The History of Mathematics: the State of the Art, ed. J.W. Dauben (Boston, 1996), 29–47

I. Grattan-Guinness: The Fontana History of the Mathematical Sciences (London, 1997)

R. Tatlow: ‘Number Symbolism’, Oxford Composer Companions: J.S. Bach, ed. M. Boyd (Oxford, 1999)

from 1900

O. Messiaen: Technique de mon langage musical (Paris, 1944; Eng. trans., 1957)

J. Cage: Silence: Lectures and Writings (London, 1961/R)

I. Xenakis: Musiques formelles (Paris, 1963/R; Eng. trans., 1971, 2/1991)

K. Stockhausen: Texte, i-iii (Cologne, 1963–71)

P. Boulez: Relevés d'apprenti (Paris, 1966; Eng. trans., 1991)

C. Samuel: Entretiens avec Olivier Messiaen (Paris, 1967; Eng. trans., 1976), enlarged asOlivier Messiaen, musique et couleur: nouveaux entretiens (Paris, 1986; Eng. trans., 1994)

E. Lendvai: Béla Bartók: an Analysis of his Music (London, 1971)

R. Maconie: The Works of Karlheinz Stockhausen (London, 1976, 2/1990)

D. Jarman: The Music of Alban Berg (Berkeley, 1979)

P. Griffiths: György Ligeti (London, 1983, 2/1996)

R. Howat: Debussy in Proportion (Cambridge, 1983)

M. Hall: Harrison Birtwistle (London, 1984)

J. Pritchett: The Music of John Cage (Cambridge, 1993)

P. Hill, ed.: The Messiaen Companion (London, 1994)

K. Gann: The Music of Conlon Nancarrow (Cambridge, 1995)

L. Somfai: Béla Bartók: Composition, Concepts, and Autograph Sources (Berkeley, 1996)