Techniques of Polytempic Polymicrotonal Composition: Compression of Harmony while Framed in Polytempo, XIIb

 Compression of Harmony in Polymicrotonality

One technique regarding the gradual compression of micro-intervallic structures in the microchromatic line is shown below, in Figure 3, from my String Quartet 12, Big Bad Mother Fucker From Outer Space, 2019. 

Figure 1. Tuning Legend from String Quartet #12

String Quartet 12 is explicitly polymicrotonal, with microtonal systems in full, except for the last one. We start with 20tet at 60 cents, 22tet at 54.5 cents, quartertones at 50 cents, and lastly, 18 of the 26tet gamut, at 46.15 cents. Since a starting pitch needs to be chosen for further divisions of the octave of a pitch class set, and inclusive of all other pitch classes within that octave, and for all microtonal systems within the given octaves of each system, I chose C as "1/1" as indicating the "untouched," pure beginning pitch. In this case, the pitch C is from 12tet and is 261.6 Hz. Any pitch class can be used, including a starting pitch up the overtone series, say, for example, starting at the 13th overtone, but from what fundamental? See? The issue of choice rears its human-agency-head again. C 261, nevertheless, is a neutral pitch that most people feel satisfied with, as a given starting pitch for microtonal octave divisions. Note that dividing the tone would render this issue moot, since the divisions of a tone preclude any issue of its relative frequency with respect to scales, or modes, or other pitch relationships; the notion of dividing a tone precludes the tone's relative positioning to any other intervallic contingencies. In fact, when dividing the octave into equal divisions, EDO, or TET, the starting pitch is rarely discussed, even though it is highly relevant when composing. 

Choosing C for the octave equivalence does not mean the music is in the key of C, or that C is the finalis, but that the relative distances between the microtonal intervals are more important, which is to say that "absolute pitch" is irrelevant here. It does not matter that C is 261.6 Hz. What matters are the microintervals and their cent values. 

On the other hand, if one finds oneself composing for C, by adding triadic structures typical of C, or C minor, I would suggest that this becomes centric to C, and is still not tonal in the classic sense, as there are no progressions of circles of fifths, such as ii-V-I. In my case, I do choose various pitches for repetition for structural reasons and then create centric harmony in that voice, as well as responses to this in the other voices. Figure 2 at the beginning already shows the potential for tight motivic pitch-cell usage of the microtonal systems employed. 

Figure 2. Beginning of piece showing polymetric/tempic relationships between the parts to the microtonal systems.

Figure 2 shows that the polymetric beginning, if maintained (which they are), eventually leads from regional polymeters to global polytempic relationships, lending the polymeters the power to determine polytempo by assertion. The first violin is at 66 BPM, the second violin is 72, the viola is at 79, and the cello is at 86 BPM. Yes, there are fractions, or decimals to the tempo BPM, because each value is calculated through algebraic cross multiplication to solve for x, to find each tempo. I round off, for practicality, as I expect for the microtonal cents values. Computer realizations, however, are exempt from roundups. 

One may ask how I came up with 66 beats for the first violin? A: I just felt like it. BUT, there is a kinship in keeping with the nearness of tuning for the 20tet first violin tuning, as 66 is a near enough multiple of the 20tet divisions. Adagio is among my favorite tempi, not too slow, and not too fast, while an energetic string part can polyrhythmically dance all over the place at this tempo to give it life. Note the activity in the parts is not mere polytempic beat marking, but sophisticated living rhythmic organisms living in their own environments. 

Figure 3. Quasi stretto of "subject" in each different tempo and tuning.

Figure 3, as mentioned above, shows a figure in stretto across four tunings and four tempi, giving the notions of rhythmic diminution a new twist. The motive in the first violin in 20tet is being compressed microintervallically by each successive entrance of the stretto, down to the cello, at 26tet, while the figure is shortened gradually by the tempo difference in each part, getting gradually faster, revealing a percentage of decrease to the figure as it reappears, casting a new glance on diminution in counterpoint. 

Bartok is known for his use of compression regarding equally spaced chords, so, in his spirit, I am extending his vision, compressing intervals down to the micro level. 

With respect to my legend and how the notes are tuned, I prefer to keep the horizontal unfolding of music clutter-free, and the boring use of standard, but re-tuned, note names may not be as sexy as original microtonal semiotics, but I am trying to keep the integrity of my rhythmic line. If I added an army of symbols to the line, I would have to either reduce page size or else quit the rhythmic invention, which I refuse to do. 

to be continued...