Techniques of Polytempic Polymicrotonal Composition: Divisions of the Whole Note, part V

 More on the Divisions of the Whole Note, in a Cowellian Sense

"Decimal Rhythms"

While in Champaign back in 2007, I was given a notebook. Since I had nothing but time on my hands, I began ruminating about the divisions of the whole note. Borrowing (as the best of us do) from Cowell, I wrote out a schema for a 'newer' approach to non-duple-based rhythms. I wrote this:

         Figure 1. A figuration of Decimal Rhythms

Figure 1 is a common-time bar of decimal rhythms: one third-note, at 33.3% of a whole note; one twelfth-note, at 1/3 of a third note of 11.1%; an eleventh-note, at 1/11th of a whole note, at 9%; a seventh-note, at 14.3%; and three tenth notes, obviously at 10% x 3 = 30% of a whole note. These add up to approximately. 97.7% of a whole note, or the 4/4 measure, is indicated.

An outline of the Figure 1 decimal rhythmic measure is shown below in Figure 2.

   Figure 2. An outline of the decimal measure is hidden 

            ghost-nested tuplets

Each note represents a group of tuplets not used in the original rhythm. To understand this better, each singular note from each particular odd grouping of tuplets needs to be understood as representing the entire group of that tuplet to preserve a mental timeline in the mind. Most of the time, tuplets are superimposed on a beat, or several beats, based on tempo and time signature. In the case of decimal rhythms, extracting just one, or several, of the notes from a particular tuplet, without using the entire space of that tuplet, creates a visual and mental vacuum: What about the rest of the notes? The remainder of the tuplet is usurped by another tuplet, hence 'nested tuplets', yet, if denoting each rhythmic brand of note by "Cowellian" note heads, as Cowell did in Fabric, for piano, based on his taxonomic method for visually addressing various tuplets in the score, this same technique can be used in my decimal rhythms. 

Obviously, a "seventh note" cannot be heard as distinctly as a common 8th note. We can hear and visualize duple rhythms and meter--such that a singular 8th note by itself would not be anomalous. It would probably just be considered additive, or else a single meter of 1/8. But in viewing the breakdown of Figure 2, the visual outline of the hidden tuplets makes the decimal rhythm more understandable. So, why not simply use nested tuplets, like the New-Complexity folks? I am avoiding this precisely because I am not fulfilling the remainder of the tuplets; even Ferneyhough and Cowell, both complete the tuplets, or else use a familiar rhythmic marker on top of the staff, such as a space of music being equal to a half-note, for example. I am trying to avoid this. I want to use just ONE note of the tuplet, along with its inherent duration and oddness. This area of rhythmic development resembles microtones very strongly. It's as though decimal rhythms are the analog to microtonal pitches, where common musical structures are taken out of context, with respect to duration, pitch, or placement, and given meaning anew. 

                                                                    Figure 3. Divisions of the whole note.

                          

My system of decimal rhythms eliminates the clustered appearance of nested tuplets, which seem to make my colleagues drool. This "solution" puts a nested tuplet at five layers deep back to ONE layer. If one is writing polytempic polymicrotonal music, there will be the issue of clutter on the page. This new system eliminates clutter. But a fairly substantial understanding of minute partial prime number group elements, hearing them in context and in rhythm, is difficult. Probably as difficult as guessing the performance of numerous rhythmically vague and conflicting nested tuplets, that really is impossible to perform without guessing, which is actually part of their performance practice.

A new problem arises for this enterprise: will Finale/Dorico be able to computationally negotiate these partially nested ghost tuplets without crashing? A MIDI realization of both the precise rhythms and microtones is absolutely necessary for performing such a piece. Getting my software to cooperate is paramount. 

 Figure 4. Division of the Whole Note flow chart showing prime numbers

                                                      

Figure 4 shows the gradual breakdown of the divisions of a whole note. Each rhythmic division has its own notehead for easy identification, without cluttering the score with more signs and numbers. Isn't this just Cowell re-visited? Why didn't this stick the first time? My first guess is that Cowell was far ahead of his time. He wrote New Musical Resources at age 19. Although he did not use his own system to any large extent, he did create it for posterity. For me. Cowell, Nancarrow, and Ives all owe a debt of gratitude to Helmholtz, in any case. Without Helmholtz, none of this would have happened. 

Another potential argument is: "Why the whole note, and aren't all the whole notes going to be the same duration?" No. Polytempo can arrange different durations for each and every whole note. 

   Figure 5. Bar of Decimal Rhythms with numerical note values

Painstaking processes require a calculator. The grace note before the seventh-note is valued at 2.15% of a whole note and rounds out the bar for the full value of a whole note. This, to honor the common-time signature. 

to be continued...