*Figure 43 (Click to Enlarge and Play)*

If tone can be scaled down to periodic pulse, it follows that tonal consonances can be considered as rhythmic phase patterns. Figure 43 is an attempt to translate the simpler consonant ratios into rhythm harmony using standard western notation.

The score becomes a good deal less impenetrable if it is interpreted on a sampler, and the appendix provides the relevant cent values for detuning Equal Temperament pitches to the simple number ratios of natural tuning. First of all, play a metronomic pulse on any instrument - that is, a series of regular repeated notes, at about 60 bpm. Then sample and loop this pulse, assigning its original pitch to c1, so that when the key c1 is pressed on the triggering keyboard the original metronomic pulse is played continuously on the sampler. This beat is represented in figure 39 as the centre stave c, and all the cross rhythms relate to this as unity.

The figure below represents what happens if c1 and g1 are played together on the keyboard. Since the ratio between the two beats is 3/2, the speed of the sample on g1 will 3/2 faster than the sample on c, just as the pitch of the beat tone will be 3/2 higher. - in the time it takes c1 to play two beats, g1 will play three. This rhythm concord can be perceived from two angles: (a) from the perspective of the c, the patterning can be heard as a 6/8 metre, with two beats to the bar, while (b), from the point of view of g1, it can be understood as a faster 3/4 metre, with three beats to the bar. It is a question of emphasis whether we hear the rhythm in 6/8 time at 60 bpm or in 3/4 time at 90 bpm.

*Figure 44*

The dual identity of a rhythmic pattern helps us to understand harmonic ratios in tone. For this identity, signified by the under and over numbers of the ratio, informs us of the pivotal nature of ratios in general. Harmonically, 3/2 represents the convergence of two progressions: it is the third overtone term of the second undertone row at the same time as being the second undertone term of the third overtone row. It is the pivot between two different tonalities, just as between two different time feels. Harry Partch refers to these identities as Utonality (under number) and Otonality (over number) and they reveal the ratio’s potential for both harmonic and rhythmic movement along the diagonal rows of the Lambdoma, the overtone and undertone series.

Since Unity in all its ratio forms (the central vertical artery of the Lambdoma - 1/1, 2/2, 3/3 etc.) is a term in all overtone and undertone series, then all ratios relate to it in a hierarchy of rhythmic and tonal consonance. The complexity of rhythm pattern increases as the consonance diminishes, as shown in figure 39, and as it is heard on the sampler in relation to the constant pulse of c1, Unity.

However, although every tonal ratios has an implied counterpart in rhythm, I am not suggesting that every step of harmonic movement in my proposed musical system should be mirrored by its rhythmic equivalent. This would be as difficult to play as to listen to.

It’s important to remember the principle of scaling when creating this music. Generally speaking, in music ensembles, regardless of musical style, an instrument’s function and mode of playing is dictated by its pitch range. So in a military band playing a Sousa march, the piccolo weaves intricate lines in the higher pitch range, while the tuba propels the music with slower regular steps some octaves below, usually centred around the roots and fifths of the harmony. If the two instruments were to swap their roles but remain in their distinct pitch ranges, the result would be interesting, but odd. The weedy piping of a piccolo playing the tuba part is not the strongest stimulus for marching, just as the elaborate ornamentation of the piccolo part would sound murky and lugubrious when transposed down several octaves into the lower reaches of the tuba.

The way that our brains scan the musical spectrum seems to be similar to how samplers transpose tone and rhythm. The fact that the lower the sample is played on the keyboard the slower the rate of movement tallies with our preference for simple rhythm harmony movement at the lower end of the spectrum. Further, this offers insight into how we discern rhythm. For if we view rhythm as tone lowered to beats then it follows that the complexity of the rhythm harmony movement must be scaled down accordingly.

If we look back to figure 39 we see that standard musical notation soon becomes unwieldy when dealing with ratio rhythms. The under numbers in time signatures proceed geometrically (2, 4, 8 etc.) and so for grouping of 5 and 7 the notation must be customized using time signatures like 4/5, which indicates four groupings of five sub-beats. While it is a drawback of the notation system it is also testimony to our limited mental capacity for rhythmic complexity. Being able to play a cross rhythm of 5 against 7 is all very impressive but also fairly redundant if the brain cannot register it as such.

Throughout this project I have bemoaned the inability of western notation to codify Harmonicism. Now seems a good time to develop an alternative which also avoids long lists of number ratios. My musical background is in improvisation and therefore I have never viewed the notated score as sacrosanct. However, notation has its place as a guide for musical performance and as a springboard for creative development. Ideally, it should also graphically clarify the processes which define the music.

Since Harmonicism is rooted in the mathematics of simple number ratios, it would be logical to notate it with a corresponding geometry.

First of all, we must specify which elements of music we wish to represent. Let’s follow the example of standard notation and concentrate on pitch and its movement relative to time, subordinating, for the moment, timbral and dynamic features.

*Figure 45*

We can consider pitch spatially with regard to its proportionally related wavelength; the lower the pitch the larger the wavelength. If the wavelengths of the twenty-three fundamental tones of the harmonic canon are represented by proportionally sized circles (the simplest geometric figure) figure 41 emerges. These circles, wave pulses of both tone and rhythm, form a harmonic patterning when repeated and set against a horiontal axis of time (figure 46).

*Figure 46 (Click to enlarge) - or open in new window*

To borrow an idea put forward by several esoteric musicologists, figure 46 is a pleroma. That is to say, it is a fullness of all the potential vibratory patterning between the twenty-three fundamental tones of the harmonic canon. This interlacing of wave pulses represents what we would hear if we were to strike all the strings together, or, for that matter, if we were to play all the detuned keys on the sampler simultaneously. Simpler patterning can be defined out of this fullness. So figure 43 shows the harmonic and rhythmic signature of (a) the overtone series and (b) the undertone series.

*Figure 47 (Click to enlarge) - or open in new window*

How useful such notation is as a guide for playing the harmonic canon only time and practice will tell.

Trying to conceive of Harmonicism in geometric terms has implications far beyond the need for notation. I stated initially that the aim of this project was to investigate the correlation between the vibratory patterning of natural acoustics and harmonic structuring of the material world, with a view to creative cross-fertilization. Hopefully, geometry will be the bridge which links the two.