Pythagoras and the Music of the Future Part III (contd.)


time sigsAs was noted during the preceding discussion of melody & harmony, the rise of tonality eventually gave way to its own dissolution, brought about by ever-increasing ambiguity of tonal centres which, for purposes of extra-musical expression, were eventually excised from musical discourse. The development of temporal structures shows a similar trajectory: moving towards clarity of rhythm, metre and tempo, and then away from it.

As in the case of harmony, we find striking examples in the work of Beethoven. In the first movement of his Symphony No. 3 in E flat (Eroica) of 1803, for example, we find several parallels between harmony and rhythm: in the 8th bar (roughly a mere five seconds into a work of some fifty minutes duration), the ambiguity of harmonic direction brought about by the C sharp in the ‘cellos is echoed by syncopation, at the interval of the tritone, in the violins. Throughout the movement, which is in triple time (a first movement in triple time was an innovation in itself),  the sense of metre is frequently disturbed by an emphasis on the second beat of successive bars, or by placing accents on every other beat, over a period of several bars. This is taken to extremes in the development section where a deliberate sense of duple time, in direct conflict with the underlying triple pulse, is generated. Those familiar with the work will know that the excerpt below occurs four times, treated sequentially, with the metrical ambiguity heightened by the downbeat which sounds once in every six bars.

Beethoven – Symphony No 3 in E flat (Eroica)

It is also notable that the passage, with which the above example begins, culminates in a brief episode that deploys one of the most daring dissonances to have been used up to that point. When the harmony resolves onto the key of E minor (bar 288), only then does the metre also resolve and take on the clear aspect of simple triple time once more.

Later composers such as Chopin, Liszt and of course, Wagner and Debussy, further blurred tonal centres while revealing parallels between harmony and musical time in various ways. As in the case of harmony, the need to give expression to intense emotions, moods or other extra-musical ideas was an imperative. Several of Chopin’s piano works, as well as some of those of Brahms and Schumann, contain layers of rhythm that are ostensibly so complex as to be almost unrealisable with any degree of clarity – if, indeed, that was as important a criterion for performance as it is today. In fact, some of the rhythms, such as the ‘quintuplets’ against semiquavers in the example below, are incorrectly aligned, or simply don’t ‘add up’, leading to questions as to what the composer’s rhythmic intentions actually were. However, even if realised only approximately, they succeed in conveying the sense of drama that was, no doubt, intended.

 Chopin, Nocturne in F-sharp Major, Op. 15, No. 2

Chopin_Nocturne example

In the example from the opening of Wagner’s Tristan & Isolde below, we note not only the absence of a clear tonal centre but also a lack of any obvious rhythm or metrical foundation; ambiguities which are only intensified by the degree of rubato that the performance of this passage invites.

Wagner – Prelude to Tristan & Isolde

Tristan Opening

A similar correlation between harmony and metre, with increasing degrees of ambiguity in both, can also be found in the work of Claude Debussy (1862-1918) and in the music of the Second Viennese School. Along with the phenomenon that Arnold Schoenberg (1874-1951) termed the emancipation of the dissonance, came a similar emancipation from the tyranny of the bar line.


Stravinsky & his music depicted by Jean Cocteau

The composer most readily acknowledged as having accomplished this was Igor Stravinsky (1882-1970), but the methods by which he achieved such metrical liberation were markedly dissimilar to those of the composers of the late Romantic period and the Second Viennese School. In their work, the characteristic indistinctness of rhythm and metre was produced by a number of means, among them the juxtaposition of duplet and triplet divisions of the beat often coupled to notes tied across bar lines; polyrhythm (e.g. duplet divisions of the beat in one voice and triplet divisions in another); irregular or frequently changing time signatures; frequent fluctuations of tempo by way of accelerando, ritenuto and rubato together with an absence of regular rhythmic accompaniment. In Stravinsky’s case, however, the innovation was in the bold complexity and irregularity of metre. The best-known example is provided by his ballet, Le Sacre du Printemps (The Rite of Spring), in which he took rhythmic and metrical complexity to unprecedented lengths: immediately following the work’s introduction, the first section proper, Les Augures Printaniers, features a principal theme which is purely rhythmic.  It consists of a single repeated chord, punctuated by accents which, indeed, augur the metrical complexities to come. Initially, there is a complete absence of melodic movement, and even the chord played by the strings, punctuated by sforzando horns, is percussive rather than harmonic in its effect, thus emphasising the rhythmic aspect;[1]

Stravinsky – Le Sacre du Printemps; Les augures printaniers

Augurs Spring eg

As is well known, Stravinsky’s ballet culminates, in the Danse Sacrale, with the chosen maiden dancing until death. The accompanying music is brutal and convulsive in its effect, due to irregular metrical patterns, coupled to stringent, percussive harmony, this time more discernible as harmony (although not functional harmony) due to its being place in mid-register with explosive forte-fortissimo interjections on muted brass.

StravinskyLe Sacre du Printemps; Danse Sacrale

Danse SacraleIn 1913, when ‘Le Sacre‘ was first performed, many might have thought they had indeed caught a glimpse of the music of the future. Nothing quite like it – especially in terms of rhythm and metre – had ever been heard before. Despite this, there was, in fact, an important element that was, in a sense, not at all new: it is readily observable that the metrical ratios deployed here do not venture beyond 2:3. The semiquaver (16th note) groupings in the example above are 3, 2+3, 3, 2+2, 2+3, simple and compound beats having been placed in succession, thus creating beats of unequal length and, therefore, irregular metre.

Even patterns such as those in example a) below, often encountered in the music of Stravinsky, Bartok (1881-1945) and composers such as those who comprised Les Six, are not, as it might at first appear, in the ratio 4:5, but, in fact, in the ratio 2:3 – the metrical pattern is 2+2+2+3 or possibly, 2+2+3+2.[2]


4 then 5

This emphasises the aforementioned fact that, although there is a clear correlation between the development of harmony and of rhythm, and that these developments are closely bound up with the frequency relationships inherent in the naturally occurring harmonic series, rhythmic development did not, for several reasons, ‘keep up’ with intervallic development: even the complexities of Stravinsky et al (save for the instances of 6:7 ratio in the first tableau of Petrushka) amount to no more than juxtapositions of metrical divisions of 2+3 or 3+2; horizontally, to create irregular metres or time signatures such as in the Danse Sacrale, or vertically to create polyrhythm.[3] Patterns such as b) & c) below, which incorporate the true ratio 4:5, although found in certain folk music, did not become commonplace in Western music until the mid-twentieth century.[4]


4 v 5 horizontal


4 v 5 vertical

The composer who, perhaps, made the most significant contribution to rhythmic thinking was the Frenchman, Olivier Messiaen (1908-1992). In much of his music, he embraced the concept of additive rhythm, which consists of successive durations comprising numbers of a basic unit rather than divisions of a larger unit. In the following example of an additive rhythm (not from Messiaen), each duration is measured in semiquavers (16th notes) as indicated;[5]

additive rhythm ex

Compare this to a typical divisive rhythm, such as the following, which incorporates successive divisions of the semibreve (whole note);

divisive rhythm

In considering the additive example above once more, it also becomes clear that the rhythm is palindromic i.e. the same forwards as backwards. Here, yet another link with the harmonic series is revealed: In the earlier example, in which we saw a composite rhythm resulting from the first four partials of a simple timbre, the fact that it too is palindromic (this time, the illustration consists of a single ‘cycle’) is obvious;

Four Harmonics Rhythm Result

When working with higher ratios of the harmonic series, however, we would find less straightforward rhythms but they too, like any composite rhythm derived from the interplay of frequency ratios of the harmonic series are palindromic. The following example is the result of superimposing values in the ratio 8:9, the frequency ratio of the major 2nd. We see that the pattern of durations (measured in semiquavers or 16th notes) up to the dotted line is reversed thereafter.

8 9 Composite Rhythm

See how this rhythm is derived

Moreover, it can be noted that the rhythm in this example is additive, rather than divisive, in nature. Therefore, we can deduce that additive, palindromic (termed by Messiaen non-retrogradeable) rhythms, can be derived from ratios to be found in the same areas of the harmonic series as dissonant intervals and inharmonic timbres.

The claim that Messiaen’s predisposition towards such structures stemmed from an acute, though probably subliminal, awareness of the inner workings of the harmonic series cannot be substantiated, but neither can it be categorically refuted.[6]  Whatever the truth of this particular matter, the relationship between patterns of the type illustrated and the harmonic series is beyond dispute; it is a physical fact that is mathematically provable. The implications of this will be discussed further in the next article.

boulez messiaaen

Olivier Messiaen (right) with Pierre Boulez

The influence of Messiaen’s use of rhythms that comprised successions of durations rather than divisions of metrical units, together with the serial approach to the organisation of pitch espoused by the Second Viennese School and by Anton Webern in particular, led to the short-lived concept of integral serialism. This compositional method involved series of durations being manipulated in ways analogous to the series of pitches that were the defining feature of Schoenberg’s twelve-tone system. Using various matrices, composers such as Karlheinz Stockhausen (q.v.), Pierre Boulez (the two most famous of Messiaen’s pupils) and the Italian, Luciano Berio (1925-2003), mapped series of durations to those of pitches, dynamics, timbres and even articulations.[7] The paradox of such extremes of organisation was that the effect was one of disorganisation or even randomness, as many of the composers involved have since acknowledged. In terms of musical time, there were no recognisable rhythmic patterns, no discernible metre (regular or irregular) and also, because of the absence of a regular pulse, no sense of tempo. This lack of perceptible coherence is due in large part to the music’s failure to obey the laws of pragnanz, but more importantly, it is proposed, it arises out of the fact that the relationships between pitch and rhythm, in integral serialism, were contrived and artificial in that were divorced from the naturally occurring pitch-rhythm continuum.[8]

One composer who indeed recognised the relevance of the pitch rhythm continuum, and who made a conscious effort to embody the properties of the harmonic series in the domain of rhythm, was the American, Henry Cowell (q.v.). Cowell’s New Musical Resources, unlike Stockhausen’s How Time Passes… (which was, in fact, a rationale for extending the principles of serialism to the domains of rhythm and tempo), proposed the derivation of rhythmic structures directly from the superposition of frequency ratios of the harmonic series, thereby creating a unity of rhythm, harmony and timbre all of which had their basis in this acoustical phenomenon. In particular, he realised that by using the higher frequency relationships of the harmonic series as a basis, rhythms of considerable complexity and interest could be produced. An idea of the complexity of such rhythms can be gained by studying the following depiction of the first 21 frequencies of the harmonic series, represented in musical notation;

Harmonic Series Rhythm

(Click to enlarge)

Cowell drew such close parallels between the ratios of the harmonic series expressed as rhythm or as harmony, that he developed a whole new musical ‘language’ which went so far as to classify rhythmic relationships as consonant or dissonant: those which were easily resolved as subdivisions (or subdivisions of subdivisions) of a basic unit being classed as consonant, while those which were not, being classed as dissonant.

To appreciate this last point fully, it is helpful to consider the harmonic series from the point of view of periodicity. The period of a wave (also known as its wavelength) is the time taken for it to complete a single cycle. The period is the reciprocal of the frequency i.e. where the frequency is 3f, the period is 1/3f. This means that, in the case of sound waves associated with sustaining instruments, each of the upper partials will complete its own fixed number of periods in the time of a single period of the fundamental. In the following diagram, it can be seen that in time of a single cycle of the fundamental (f ), the first overtone (2f ) completes two periods; the second (3f ), three and so on.

In the case of ratios to be found in the higher frequency range of the harmonic series, for example, 15:16 (in terms of pitch, the minor 2nd and in terms of timbre, an inharmonic colour), then the beginnings of the periods will coincide only after 16 cycles of the higher frequency have passed. The greater the number of cycles that occur, and therefore, the longer the time that passes, before all the periods coincide, the less stable or more dissonant will be the relationship. In Cowell’s view, therefore, the rhythm derived from a wave consisting of four partials, such as in the earlier example, would have been considered consonant, whereas the additive example derived from the ratio 7:8, illustrated above, would have been dissonant.

Whilst Cowell’s ideas about rhythm had great artistic potential and a firm foundation in musical acoustics, difficulties arose along with the realisation that many of the rhythmic relationships he sought to exploit were, in fact, unplayable with any degree of accuracy. Cowell’s solution was to collaborate with Theremin in the development of a new mechanical instrument called the rhythmicon.[9] The rhythmicon was able to realise the desired relationships accurately, but it was unwieldy and restricted in terms of timbre. An alternative and, it could be ventured, a more convincing, solution was proposed by the American composer, Conlan Nancarrow (1912-1997) who, taking Cowell’s ideas as a starting point, turned to the player piano. Punching the piano roll in spatial ratios corresponding to the desired temporal ratios, allowed previously unplayable rhythms and tempo relationships to be executed with accuracy, and at speeds well beyond those achievable by human performers. However, whilst Nancarrow’s works for player-piano were truly ground-breaking in their treatment of musical time, there remained limitations in terms of a lack of unity in melody, harmony, rhythm and timbre.[10] Thus, in the cases of both Cowell and Nancarrow, the solutions to the problems associated with basing rhythm on ratios of the harmonic series created further, insurmountable ones in terms of practicality and of aesthetics. Nonetheless, Cowell’s contribution in respect of rhythm was seminal in that it highlighted the physical relationships that exist between the frequency ratios constituting not only the microstructures of timbre but also the macrostructures of harmony, melodic intervals and aspects of musical time such as rhythm, metre and tempo.

In conclusion, it has, hitherto, been established that frequency ratios found in the higher reaches of the harmonic series, such as 15:16, 20:21 etc. give rise, in the micro domain, to inharmonic timbres and dissonant intervals, while in the macro domain, they generate rhythms that are additive and non-retrogradable, dubbed dissonant by Cowell. Interactions of the lowest frequency ratios, by contrast, result in harmonic timbres, consonant intervals and divisive rhythms which Cowell would have called consonant. Since conceptions of consonance and dissonance depend, to a certain extent on their context, it is proposed that a more appropriate term for such polarities – between which exists a continuum – would be harmonic and inharmonic, irrespective of whether they are describing relationships in the domains of timbre, melody, harmony, or musical time. It is similarly suggested that the concept of transposition be expanded so as to apply not only to pitch but also to timbral and temporal elements. This clears the way for a kind of musical thinking, which also seeks to establish unity in timbre, pitch and time, but which can be brought fully to realisation only by means of digital computers running dedicated software alongside traditional musical instruments. The possibilities will be explored in the forthcoming article, Pythagoras and the Music of the Future Part IV – Synthesis.



[1] The term ‘percussive harmony’ was coined to denote chords, often polytonal in nature, which are acoustically so complex that it is difficult to ascribe definite pitch to them – as is the case with unpitched percussion instruments.

[2] The French composers Francis Poulenc, Arthur Honegger, Darius Milhaud, Georges Auric, Louise Dury, Germaine Tailleferre.

[3] Where different parts or voices have different time signatures or the effect of such.  This is not quite the same as hemiola (q.v).

[4] Whilst such ratios were suggested by Romantic composers, particularly in music for piano, their notation was often inaccurate (as in the Chopin example above) and the notational context often militated against their accurate interpretation. See Hook, How to Play Impossible Rhythms’.

[5] From the author’s own Klee Connections for Piano

[6] In his treatise, The Technique of my Musical Language, Messiaen refers to the interval of the augmented 4th being perceivable by a ‘fine ear’ when a low C is played on the piano. This is clearly a reference to the audible 10th harmonic which, he proposes, can function in the harmony as an ‘added note’ and which exhibits a tendency to fall towards the principal note. He also draws a parallel between added notes in harmony and added values in rhythm e.g. the addition of a dot (lengthening the duration by half) to the penultimate chord of a cadence. This suggests, at the very least, an awareness of an acoustical correlation between harmonic and rhythmic functions.

[7] Possibly the most well-known example is Boulez’s Structure 1a from Structures Book I for two pianos (1952).

[8] A term from Gestalt psychology which states that the constituent parts of a structure such as, in this context, a melody or rhythmic pattern must exhibit similarity, continuity, proximity and closure in order to be perceived as a whole. Further discussion of gestalt psychology as applied to perception of melody is to be found the article elsewhere on this site, Let’s Make Music (forget reading it for now)!

[9] Leon Theremin, Russian inventor (1896-1993). His most well-known invention was of one of the earliest electronic musical instruments, the Theremin.

[10] It is possible to gain considerable appreciation of Conlan Nancarrow’s contribution from this performance of Study for Player Piano (Canon X).