Techniques of Polytempic Polymicrotonal Music Creation: Microchromatic Collections Using the Span of a Tetrachord, part XV

 Controlled Organization of Microchroma within Polymicrotonal Systems

For me, pitch organization is either atonal, tonal, or centric. One can use any approach one wishes in polytempic polymicrotonality. I prefer to be somewhere in between. Although I have not used twelve-tone procedures for my compositions, I have used cells, tetrachords, centric quasi-tonal areas, and even jazz harmony. 

Again, I refrain from "laying down" a law or system, and I am only trying to cover an array of topics related to all things poly. I know that Joseph Straus's book Introduction to Post-Tonal Theory is NOT everyone's favorite textbook, as it resembles a math class, but it has crystallized some basic atonal concepts for me in my undergrad and graduate years. 

I know I rail against the new simplicity of microtonal music adopting minimalism and drones, but there really needs to be an alternative to this. It is good that a simpler approach is pertinent today, as this period of microtonality ought to be a period of inculcation and ear-friendliness. There are even jazz microtonalists and progressive rock groups, such as King Gizzard and the Lizard Wizard, and other groups out there doing great things. Check out Ron Sword's group and his microtonal guitars! But atonal pitch and interval cells still seem a bit too unfriendly in these neo-tonal times. Jazz, while wonderful and beautiful and chromatic, still adheres to the laws of tonality and ii-V-I. 

One reason why 12-tone panchromatic music and dodecaphony works is that the chromatic pitches and intervals are still ear-friendly. People know them. But, if we apply some of these techniques to microtonal scales, will these techniques hold up? Will the trichord set (014) be heard as such? No. It won't. But this is the opportunity to explore the microchromatic personality of each tuning and microtonal system. For me, rather than relying on ii-V-I and standard diatonic chord progressions, I am trying to get to know the microtonal systems right down to their smallest intervals. 

In Figure 1, I begin with 13tet, as I indicated that 12tet has already been explored. We know what (014) and (0257) sound like. The intervals are indicated by their chromatic step relations. But what does (014) mean in 13tet? The chromatic half step of 100 cents is reduced to 92.3 cents. The 400-cent major third is reduced to 367 cents, just under the 386-cent major third. The minor third of 300 cents is reduced to 277 cents. This is already a foreign-sounding structure, and this is only 13tet! There are still 5000 other microtonal systems in the Scala file. 

Figure 1. > 12 microtonal systems from 13 to 16tet

Instead of taking the tetrachord as a four-note structure, as the Greeks did with their genera, I am filling up the space of the tetrachord, up to 498, or 500 cents, with the microchroma of each particular system. These tones, 6 for 13tet, inherently contain the classic major third interval, the CI, as well as the pyknon remaining, according to the Greek enharmonic genus. Therefore, for interested others, the intervals you seek are still there, but I chose to fill the span of the tetrachordal space with all the constituent small intervals of that particular system. I do this with all succeeding systems, as shown in the following figures. 

Of course, a tetrachord is not the only tool for pitch organization for microtonal systems. Just use the gamut. But paring these systems down to a manageable size, like the 6 chromatic tones of the 13tet tetrachord, makes the system more approachable. 

The following Figures 1-3 show the cents of the micro-interval, the number of pitches contained in the tetrachord, the cents value for each successive interval, and potential note names. Albeit, nominal ordering of these can be very confusing, as I have crossed out many times note names in deference to their higher or lower neighbor. Since the 12tet 12th root of 2, is still the overriding system of today, I use that as the basis for all other systems. Some may use the overtone series unequal divisions as their personal rudder. But, again, even in overtone tuning, we still wind up comparing those pitches to the 12th root of two for cents values and deviations. 

Figure 2. Systems from 17 to 22tet

One will begin to notice that as the microintervals compress, the tetrachord will a fortiori begin to expand. For example, in Figure 1, the 6-note tetrachords (Why not call these hexachords?) begin to accommodate 7 notes at the 15tet system, and then begin to house 8 notes at 17tet tetrachords. (shown in Figure 2) I suppose in answer to the parenthetical query in this paragraph, that these sets are not true hexachords, as a hexachord is defined by occupying the space of exactly half the octave, which is 600 cents, or the interval of the augmented fourth (tetrachord) based on the chromatic octave of the 12tet system. On the other hand, there really is no reason not to call it a hexachord, except that in my "system," the tetrachord is the "defined" space of 498-500 cents, and that space is reserved for the span of the tetrachord. One could call these other-numbered systems a hexachordal tetrachord, but that gets a little mouthy and bombastic. 

The 20tet system increases the pitch content to 9 notes in Figure 2. Figure 3, below, shows the 9 and 10 note tetrachords of 21tet and 22tet, respectively. 

Figure 3. Systems above 23, including quartertones and eighth tones

Figure 3 displays an 11-tone tetrachord of quartertones and the 21-tone tetrachord of eighth tones, which itself needs further subdivisions of organization, leading to some potential problems. But Ben Johnston has made use of 22-tone ultrachromatic intervals by scalar ordering as well as his own 53-tone system, rising scale-wise the full octave of his system. 

In my case, I used a tetrachord span of 9 notes chosen out of the 53-tone gamut for Hypercube, in the cello part. The tetrachord spanned C to F. In my tetrachord, pitch class C was divided into four different shades of microtones, the point being to reveal the multifaceted microtonal shades of one pitch class.

Below, Figures 4-7 show Straus's set classes, originally intended for pantonal 12-tone chromatic atonal music, but in my process of this blog, I am trying to reinvent forgotten systems to present-day polymicrotonality pitch organization and set strategies for multiple systems. 

As discussed earlier, the trichord set (014) varies as microtonal systems progress, compressing the intervallic sizes down to the characteristic microintervals of any system used. As familiarity with each microtonal system is gained, the otherworldly-sounding sets shown in these figures will begin to sound "normal" over time. 

Figure 4. Straus set classes: trichords and complements

Figure 5. Straus tetrachords and complement

Figure 6. Pentachords and complement

Figure 7. Hexachords with P R I RI and complement

The "complement" is the remaining notes after the indicated groupings; therefore, trichords' complement will be nonochords, the other remaining notes of that set. This principle can be used for all microtonal systems, contingent on the number of degrees per system. 

As a matter of fact, Straus's entire book can be reconfigured for each microtonal system, including unequal just intoned systems and the stretched Bohlen-Pierce, with the exception that mod x will not work, as this is a non-octave system. But other aspects of Straus's work may apply. 

Use his book for your own system of atonal, tonal, or centric techniques for your own microtonal systems. If you are not poly, then that is perfectly fine. Well. Not really, but I can live with it. If I have to. I guess. 

to be continued in the future. I really have to get back to composing again.