Pythagoras and the Music of the Future Part III – Musical Time

metronome_by_All_Dressed_Up2Pitch is rhythm. Let us start with relatively recent events. In the 1950s, the German composer Karlheinz Stockhausen (1928-2007) wrote an article for the new music journal Die Rheihe [1], entitled How Time Passes…  Before him, the American composer and theorist Henry Cowell (1897-1965) had produced his New Musical Resources [2] which we shall refer to later. Even before him, Johannes Brahms (1833-1897) had commented on the unity that exists between metrical structures and those which underlie harmony and tonality. The phenomenon they were all discussing in their individual ways is widely known as the pitch-rhythm continuum.

In his article, Stockhausen states, somewhat inaccurately in fact [3], that if you route the output of an oscillator producing a sinusoidal tone (a tone consisting of a single frequency with no overtones) at a frequency of less than 16-20 Hz. via loudspeakers, then you would hear just a series of clicks since the frequency in question is below the human audio range [4]. For example, if the frequency were 8 Hz, then there would be 8 of these clicks every second. He goes on to state, rightly, that if you were gradually to increase the frequency of the oscillator’s output until the signal was above the lower threshold of the audio range, then the individual clicks would become a continuous, low pitched, tone.  If you were to increase the frequency further, the pitch of the tone would rise.  It follows that if we were to start with a continuous sinusoidal tone at a frequency lying within the human audio range and gradually reduce the frequency, then we would reach a point where we ceased to hear a single tone but, according to Stockhausen, a series of individual impulses. One second of this, at a frequency of 16 Hz., could be represented in musical notation like this;

one beat pulses

Reducing the frequency still further, to the point where there was one cycle every two seconds (0.5 Hz), would create a series of impulses that could be notated as follows;

2 beat pulses

Now, if we were to imagine starting off not with a sine tone, but with a slightly more complex timbre comprising the first four partials of the harmonic series, reducing the fundamental to the same frequency (0.5 Hz), would result, not in a regular series of clicks (to retain the terminology) but, in the following rhythmic pattern;

Four Harmonics Rhythm Result

Since each of the four partials would now be making its own ‘clicks’ in whole number ratio to those of the fundamental (in this case, 1:2:3:4) [5], the above composite rhythm would, in effect, arise out of the juxtaposition of the following, again expressed in musical notation with the minim (half note) representing the fundamental;

From this, we can see clearly that the frequency ratios of the harmonic series, at frequencies below the audio range, correspond to definite rhythmic patterns. Previously, we saw how combinations of frequencies in specific ratios within the audio range give rise to distinct timbres (Pythagoras and the Music of the Future – Part I) and how such combinations play a part in our perception of the qualities of intervals and indeed in the development of harmony from the middle ages.   Let us undertake a related survey with regard to musical time.


Illumination purportedly of Leonin

The plainchant of medieval monastic music (generally accepted as the starting point for the development of western ‘classical’ music) featured melismatic, modal melody in unmeasured rhythm. This, accentuated by the absence of audible or even implied pulse, lent it a meditative, prayer-like quality. As we noted in the previous article, plainchant developed into organum by the addition of parallel upper parts at the interval of the octave, fifth or fourth. The performance of this would have involved a degree of coordination so that the voices did indeed stay parallel. Perhaps this contained the seeds of metre, early signs of which can be discerned in the florid organum typified by the Notre Dame School of composers, the best-known representatives being Leonin (c. 1150-1201) and Perotin (c. 1160-1230). Typically, this consisted of a part utilising the notes of an established chant (cantus firmus), but with values held for so long as to render the original chant unrecognisable. This part, known as the tenor, by virtue of the long-held notes of the chant (tenor being derived from the Latin tenere which means ‘to hold’), was combined with one or more faster-moving parts, based on religious texts or secular poetry, in measured rhythm. These are now known as duplum, triplum or quadruplum depending on the numbers of such parts, which increased over time. Their rhythm was dictated by that of the poetry; the strong/weak syllabic patterns of the poetry becoming long/short rhythmic patterns of the upper parts of the music. For example, iambic rhythm could be represented thus;


And trochaic, thus;


The weak syllable preceded by a strong syllable of iambic rhythm, in a musical context, becomes long preceded by short, whereas the strong followed by weak of trochaic rhythm becomes short-long. These patterns came to be known as the first and second rhythmic modes, of which six are generally recognised. In addition to those mentioned above, there are further correspondences to the  dactylic (3rd mode), anapaestic (4th mode), spondaic (5th mode), and tribrachic (6th mode) rhythms of poetry;

Rhythmic Modes2

Melodic movement consisting of repetitions, or successions, of the rhythmic modes, would inevitably give rise to the feeling of a regular ‘beat’ with each being divided into three (as is clearly discernible in this rendition of Leonin’s Viderunt Omnes). This unit of pulse eventually became formally known as the tempus.

While the rhythms of poetry, and indeed the reverence for the Holy Trinity which was symbolised by a division of the tempus into three, played their part in the development of the modes, there is another important observation to be made. Earlier in this discussion, we saw the example of the ‘rhythm’, which would be produced by lowering the frequency of a signal with four harmonics below the audio threshold;

Four Harmonics Rhythm Result

If we look closely, we see that it also contains the basis of five of the medieval rhythmic modes. In the following example, the time values are doubled in keeping with the commonly used notations of the rhythmic modes, but the relationships remain the same;

Rhythmic Modes Illustrated

In addition to the poetic and religious factors discussed above, we find a third factor namely the intuitive awareness of the rhythmic relationships that seemed most natural; those which we have identified as corresponding to the frequency ratios of the most audible partials [7] of the harmonic series produced by the male voice [8];

Male Voice Spectrum A3

In the above graph, we see clearly that the strongest partials are those whose frequencies are, in order of intensity; 2, 3 & 4 times that of the fundamental. If we regard the fourth partial as the octave above the second, then the ratios ‘cancel down’ into 1:2 and 2:3.

The fact that the ratios of 1 : 2 and 2 : 3 are those of the intervals of the perfect octave and the perfect 5th, the earliest intervals to be deployed in parallel organum, points to a correspondence between the development of harmony and of rhythm in medieval music. However,  as was to become the case throughout the history of western music, rhythmic developments occurred later, in this case around 300 years later, than corresponding ones in the realm of harmony: whereas parallel organum developed mostly in the 9th Century, the development of rhythmic modes did not occur until the 12th. It is also interesting to note that whereas intervals represented by higher ratios such as 15:16 (the ratio of the semitone) have been a regular feature of melody since the middle ages, and of harmony since the late classical period, in the domain of musical time (tempo, rhythm, and meter), the principal relationships hardly deviated from the ratios 1:2 or 2:3 from the time of the Notre Dame School to the 20th century – a time span of some 800 years. One reason for this lack of even slightly more complex rhythmic ratios, such as 4:5 or 5:6 etc., could well be the difficulties they present in performance. Another might be the utilitarian nature of so much music through the ages: music for dance (which formed the basis of many Baroque and Classical forms) and military music, for example, need the feet to make contact with the ground at regular intervals. Since humans are earth-bound beings with limbs in pairs, cyclic patterns such as ‘one-two’ for marching, or ‘one-and-a-two-and-a’ for dancing are inevitable.

While during the Middle Ages, the division of the tempus into three was de rigueur, by the time of the Ars Nova, its division into two had become more common. Later, during the Renaissance we divisions into two and three alternating and sometimes superimposed. Between the 15th and 17th centuries, there was increasing use of a further time division; the bar (measure). This firmly established the concept of metre in music, with two basic types emerging: simple, where each beat of the bar is divided into two, and compound, where each beat is divided into three. Henceforth, we could speak of, for example, ‘simple triple time’ (3/4) and ‘compound duple time’ (6/8), etc. After this, music tended to be based on either simple or compound metre for extended sections or for whole movements, such as in the Baroque suite. This remained the norm until the late classical and early romantic eras, although the interplay of compound and simple metres often took place by way of hemiola [9].

Frequency relationships similar to those inherent in the harmonic series are clearly also observable in relation to tempo: If we look at any metronome, we see a range of standard tempo markings, displayed in beats per minute. Some readers (at some time, at least) might have wondered why there are tempo markings of 72 and 76 but not 74 or 75. Much of the reason is to do with the fact that tempi are in fact organised in similar ratios.[10] For example, there is always ‘60 beats per minute’ and 120 which are, of course, in the ratio 1:2, the ratio of the octave and the ratio of the crotchet (quarter note) to minim (half note). However, the proportion 1:2 can be discerned across a range of tempi as can the ratios of 2:3 and 3:4. This is illustrated by the following table;

Metronome Markings Table

Placing the above values in ascending order, while eliminating duplicates, gives us the range of standard metronome markings as shown below.

metronome scale

(Click to enlarge)

Relationships between tempi give rise to the concept of tempo modulation, the simplest example of which is ‘quaver = crotchet‘ (ratio 2:1), which leads to a doubling of the speed of the music. Of course, other relationships are possible. For example, if the quaver (eighth note) in a new tempo is equal to a triplet quaver (triplet eighth note) in the previous (ratio 3:2), then the music will speed up by 1.5 times. While composers in the Baroque and Classical eras did not calculate such relationships in this way (tempi being indicated via Italian terms such as Andante, Allegro, etc.), there have been claims that relationships between markings such as Andante and Moderato, for instance, embodied similar relationships in practice. Indeed, this relationship feels entirely natural when moving between the slow introduction and the main body of the first movement of many a classical sonata symphony. Twentieth-century composers, most notably Eliot Carter, made extensive use of increasingly sophisticated tempo modulations, the impact of which will be discussed more fully in the next article.

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 [1] Universal Edition, Vienna, 1955-62

[2] Cambridge University Press, 1996

[3] It is inaccurate because if the output were a sine wave which it needs to be for the purposes of Stockhausen’s illustration, the only clicks possible would be incidental noises, possibly created by the movement of the loudspeaker cone; the sensation of sound would, in fact, disappear at such a frequency. However, it serves to illustrate the principle satisfactorily.

[4] No two individuals are the same, but the human audio range is generally accepted as being between the extremes of 16 and 20,000 cycles per second (Hz.)

[5] As was seen in Pythagoras and the Music of the Future I – Timbre, the frequency ratios of the partials (overtones) of the harmonic series are in whole number ratio to that of the fundamental.

[6] Meaning ‘held’ (from the Latin ‘tenere’ meaning ‘to hold’), not to be confused with the male voice.

[7] These vary from pitch to pitch and with changes in intensity and also with different vowel/consonant combinations. However, the graph represents a reasonable ‘average’ of the most prominent partials.

[8] It was stressed when discussing harmonic developments in the previous article, and it must be emphasised again here, that such awareness in all probability would not have been conscious.

[9] Where the beat is divided into two and three simultaneously, e.g. in 6/8 time where there can be three beats of two and two beats of three; often encountered in the music of Brahms.

[10] That Mazale, the inventor of the metronome, deliberately calculated tempo markings with this in mind is not known: it is the fact that such relationships exist in macro-temporal structures that is important.

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