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

Pitch is rhythm.

Let us begin with relatively recent events. In the 1950s, the German composer Karlheinz Stockhausen (1928-2007) wrote an article for the new music journal Die Rheihe, entitled How Time Passes…[1]  Before him, the American composer and theorist Henry Cowell (1897-1965) had produced his New Musical Resources which we shall refer to later.[2] Even before him, the German music theorist, Moritz Hauptman (1792 -1868) had “…introduced to Western music theory the idea that the philosophical principles underlying metric structure are the same as those underlying the harmonic structure of tonality.”[3] 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, 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] [5]  For example, if the frequency were 8 Hz, then there would be 8 such clicks per second. He goes on to state, this time correctly, 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 hearing a single tone but a series of individual impulses. One second of this, at a frequency of 16 Hz, could be represented in musical notation as follows;

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;

Now, if we were to imagine beginning 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 Stockhausen’s terminology) but in the following rhythmic pattern;

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), 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.[6]

From this, it is clear that the ratios of the harmonic series, at frequencies below the audio range, correspond to definite rhythmic patterns.

Previously, we examined 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, later, how such combinations play a part in our assigning affective qualities to intervals. It was also found that the same frequency relationships exerted a formative influence on the development of harmony from the Middle Ages.   Here, we undertake a similar survey with regard to musical time.

The plainchant of Medieval monastic music, which is generally accepted as the origin 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 was 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 some degree of coordination so that the voices did indeed stay parallel. Perhaps, herein lay 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, florid organum consisted of a sustained part utilising the notes of an established chant, known as the cantus firmus, with values held for so long as to render the original chant unrecognisable. This part, called the tenor, (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 referred to as duplum, triplum or quadruplum depending on the number of such parts, which increased over time. Their rhythm was dictated by that of the text; the strong/weak syllabic patterns of poetry combining with long/short rhythmic patterns of the upper parts of the music. For example, iambic and trochaic rhythm could be represented musically as follows;

Iambic

Trochaic

These patterns gave rise to the first and second rhythmic modes, of which six are generally recognised. All of the rhythmic modes, however, show correspondences to the rhythms of poetry: dactylic equates to the 3rd mode, anapaestic to the 4th, spondaic to the 5th, and tribrachic to the 6th mode;

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

While the rhythms of poetry, and indeed the reverence for the Holy Trinity, which was symbolised by the division of the tempus into three, played their part in the development of the modes, another important observation can be made. To revisit the example of the hypothetical rhythm that would result from the interplay of the frequencies of a signal with four overtones;

Close inspection reveals that parts of this rhythm also correspond to no less than five of the Medieval rhythmic modes. In the following example, time values are doubled in keeping with more customary notations of the rhythmic modes, but the relationships remain the same;

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 have been identified as corresponding to the frequency ratios of the most audible partials of the harmonic series produced by the male voice:[7] [8]

From the above graph, it is clear 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’ to 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 the existence of a connection between the developments of both harmony and of rhythm in Medieval music. However, as was to become the case for the rest of western musical history, rhythmic developments occurred later than harmonic. Whereas parallel organum evolved mostly during the 9th and 10th centuries, the rhythmic modes did not become a feature of music 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 an accepted 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 rarely deviated from the ratios 1:2 or 2:3 from the time of the Notre Dame School to the late 19th century – a time span of more than 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 would have presented 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 the norm, by the time of the Ars Nova, its division into two had become more common. Later, during the Renaissance, divisions into two and three alternating, and sometimes superimposed, became more common. Between the 15th and 17th centuries, there was an increasing use of a new time division; the bar (measure). This established firmly the concept of metre in music, with two basic types emerging: simple, in which each beat of the bar is divided into two, and compound, where the beats are divided into three. Thenceforth, it was possible to refer to ‘simple triple time’ (3/4) and ‘compound duple time’ (6/8), and so forth. After this, music tended to be based on either simple or compound metre for extended sections or for whole movements, such as are to be found 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].[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. Why there are tempo markings of 72 and 76 but not 74 or 75 would appear to be open to question. However, the reasons can be traced to the fact that tempi too are organised in ratios similar to those applicable to harmony and rhythm.[10] For example, ‘60 beats per minute’ is always to be found as is 120. It need hardly be pointed out that these are 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 of 1:2 can be discerned across a range of tempi as can the ratios of 2:3 and 3:4, as is illustrated in the following table;

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

(Click to enlarge)

Such relationships between tempi give rise to the concept of tempo modulation, the simplest example of which is where a crotchet (quarter note) becomes equal to a minim (half note) in the new tempo. This change of tempo, in the ratio 2:1, 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. More recently, however, twentieth-century composers, one of the most notable being Eliot Carter (1908 – 2012), have consciously and deliberately 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]  Hauptmann, Moritz. Die Natur Der Harmonik Und Der Metrik. Zur Theorie Der Musik. MS. Notes. Leipzig, (1863) quoted in  Lewin, David “On Harmony and Meter in Brahms Op. 76, No. 8.” 19th-Century Music 4, no. 3 (1981)

[4] 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.

[5] The human audio range is generally accepted as being between the extremes of 16 and 20,000 Hz.

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

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

[8] As was stressed in the previous article, such awareness, in all probability, would not have been entirely conscious.

[9] Where, in different voices, the beat is divided into two and three simultaneously, e.g. in 6/8 time where three beats of two and two beats of three are possible. This is often encountered in the music of Brahms.

[10] Whether Maazel, 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 material to the present argument.