Techniques of Polytempic Polymicrotonal Music Creation: Divisions of the Whole Note and Time Signatures, part V.i

 "Irrational" Time Signatures

What Is a Whole Note?

Technically, it is assumed that a whole note lasts four beats. The whole note, lasting four beats, has been a law of music since mensuration was gradually developed and codified throughout the Middle Ages, up to the present day. We simply accept that the value of a whole note is equal to four counts, with one single beat lasting those four counts. This, then, becomes the entirety of the measure (bar) and is called Common Time, or 4/4 time. But the truth is that a whole note really does not mean anything at all. One could argue that a whole note lasts a full measure. But what does that mean? If we stipulate that a whole note, at 100 BPM, is the set tempo, then how do we really quantify the whole note? Each 1/100th part of this whole note lasts approximately .6 seconds. 

Theoretically, the whole note could last forever. Ostensibly, a time signature is needed to give a quantifiable value to the whole note. Therefore, the whole note should, but it does not, equal the value of an entire measure, regardless of time signature. This argument may or may not be an "aside" or non-sequitur, but for my music, it is key. Initially, the original master time signature was based on the number three, because European classical music began in the Catholic Church, where three represents the Father, the Son, and the holy spirit. Three was the de facto standard organizer for the whole note (and time signatures), not four. This proves that musical law is very much a matter of ideological and cultural preference. Choice. Druthers. This is not a basis for lawmaking, but it was. Particularly because our Western Occidental music came from the Catholic Church, including influence from Pope Gregory and Guido d'Arezzo.

What am I getting at? I am getting at this: that the whole note is malleable and redefinable. Henry Cowell also realized this. For practicality, it is easiest to simply apply duple metrics for easily quantifiable maths for music. A whole note is four beats. Common time is no longer 3/4; it is now 4/4, because the maths of number 3 gets tricky, while number 4-based arithmetic is simple. 

Division of the whole note has opened up, not only polyrhythms, but also polymeter and time signatures to a new perspective and reevaluation. The rigid 2's give way to prime numbers, such as 3, 5, 7, etc. These numbers become the VALUE, in addition to the quantity, of new time signatures that do not neatly divide. Although a whole note in 4/4 and a whole note in 5/5 may be the same "length" of time, they are not the same in value or tempo. 

Figure 1. Note division acceleration percentages

Figure 1 chart shows the relative percentage of speed acceleration per division of the whole note values from the whole note, down to the 16th note, with percentages of increase or decrease, if applied to these time signatures. These changes affect surface speed, but also tempo, by default. Let me explain: if the underlying tempo is 60 bpm, and the surface speed of a prime valued time signature, such as 5/5 is applied, the beats and frenetic activity increase to 72 bpm, where we take the new prime number value of fifth notes, at 20% over quarternotes, and multiply that with the fifth-note value, therefore, 4/4 at 60 BPM, becomes 5/5 at 72 BPM. If the new time signature remains for a few measures, this new time signature becomes a "regional" change, which eventually becomes an established tempo change after a period of time. The global tempo is 60, but the odd-numbered prime time signature has "modulated" the tempo to 72 from 60. Should this new time signature revert back to the prime tempo, then it would be a simple meter change or a local tempo change. This technique could also be considered an analog to Carter's metric modulation, but as a different methodology. 

The beat "value" of the Cowell-based time signatures must be non-2-based numbers; otherwise, the tempo remains already related, by doubling or halving the beat numbers, as in cut time, for example. But prime number values in the time signatures will alter the tempo, albeit briefly, in a local change. 

These same temporal effects happen with respect to the superimposition of tuplets. Composers since Chopin, particularly, have been using tuplets to break out of the tight, rigid duple-based rhythms, which all stem from dance forms. Chopin's use of polyrhythms is, for some unknown reason, totally ignored by composers and others. Chopin, more than many other composers, made use of a huge array of prime-numbered, odd, polyrhythmic structures that deserve their own analysis, and may have influenced Cowell to write what he did in his New Musical Resources. 

Figure 2.  Seventh note-based time signature from my string quartet #9,

                Fractured Consciousness

This snippet features seventh notes and their concomitant speed acceleration at 14.3% from the previous materials. The ratios of 2-4-8 never truly violate tempic or beat scansion, as the mere doubling or halving of numbers does not offset rhythmic placement, while prime numbers and other values make quite a dent in the flow of the music. 

So, just as note value changes make a more subtle difference in tempo acceleration, so, then, do these prime number divisions of the whole note make a difference, functioning as time signatures locally affecting the speed of the rhythms. 

I believe the first composer to make use of these divisions of the whole note in time signatures is Brian Ferneyhough. He told me to read Cowell's book a few years ago (2002), and it seems to have made an impression on both of us. So, a measure of 7/5, ultimately, is really not irrational at all. 

Figure 3. Working on a measure of 7/5, collapsing a nested tuplet for an unfinished piece

Figure 3 shows a bar in 7/5, seven notes at 20% faster than previous materials, adjusting a five-layer nested rhythm. Although I am not a New-Complexist and do not use an algorithmic system, I, as a drummer, have always enjoyed rhythmic puzzles. 

to be continued...