Techniques of Polytempic Polymicrotonal Music Creation: Prosody As Applied To Microtonal Systems, part XIIIa: two successive intervals

 Applying Prosody to a Microtonal System

What is Prosody, and what the Heck does it have to do with Microtones? 

Although Leonard Meyer created a musical analysis approach using simple prosodic elements from the analysis of poetic syllabic feet, I have always been interested in prosody and its close relationship to song and music in general. This morning, as my cat Maximus attacked my face while I was trying to sleep, I had an idea concerning the use of prosody with regard to microtonal systems. 

Fig. 1. Disyllabic permutations as applied to microtonal voice leading and pitch cell organization

In prosody, poems are written syllabically in a meter, such as in iambic pentameter. Most English classical poetry is written in iambic pentameter. Take Milton's Paradise Lost, for instance. This epic poem is entirely written in a foot resembling: ∪ — where "∪" is the unstressed syllable, and "—" is the stressed syllable. There are three categories of syllabic feet: disyllabic, trisyllabic, and tetrasyllabic. Meters are available in single monometer, all the way up to whatever you want, such as dodecameter, which is twelve feet of syllabic line. I am avoiding using the terms arsis and thesis, since they have been historically reversed, causing confusion. 

Figure 1, although sketched out today, has been subconsciously working itself out for a few years. It harkens to Webern's cell (014) system, yet I believe this prosodic alternative could be malleable enough to work for any microtonal system. Schönberg's twelve-tone system in the hands of Webern became a hallmark of intervallic cell economy while preserving the tone row order by using the basic intervallic cell structure in four divisions across the 12 pitch chromatic scale. Pitch cells are in disuse today, but I do not understand why. It seems that composers have reverted back to neotonality in the guise of microtonal advancement.

My goals for microtonal voice-leading happen to extend from the line, rather than the chord. Since polymicrotonal chords are in their infancy, I am not keen to develop a harmonic theory on polymicrotonal chords, yet. But if I compose from the line to the chord, I think this Palestrina-based linear approach would work better. For Palestrina, the line came first over the chord. For Bach, the chord came first as the line was extended from the harmony. I adore both approaches, but for my purposes, I believe the "line first, melody over harmony" works better for poly and allows for a future of undiscovered polymicrotonal harmonic structures. 

In addition to the disyllabic feet, there are trisyllabic and tetrasyllabic feet, as well. Systems of two to four intervals will be discussed. First, I will dive into disyllabic, or two-interval cells.

How it works:

Figure 1 shows examples of four types of disyllabic feet: ∪ ∪ = dibrach, ∪ — = iamb, — ∪ = trochee, and — — = spondee. 

Conjunct Motion

I would use the unstressed "∪" as a microchromatic conjunct stepwise motion, either up or down. Strictly applied, where no skips, leaps, or jumps are allowed. So, in a 13tet system, this unstressed ∪ would be the stepwise chromatic motion of a 92.3 cent interval (either up or down). This rule would apply to any microtonal system, thus exploiting the smallest cent interval of that system. For me, the "microtonalness" of the system lies inherently in its smallest micro-intervals. 

Disjunct Motion

The stressed "—" would therefore include the rest of the voice leading motion: skips (more than a chromatic interval), leaps (a skip of 4 microchromatic intervals), and jumps (any size intervallic disjunction, including register and octave displacement). Although these voice-leading motions will resemble standard, but microtonally tuned, intervals, the choice to treat them as tonal or atonal is up to the composer. For me, I would treat skips as atonal, keeping to the microtonalness of the system at hand. 

Disyllables (Di-Interval Cells)

Since these are notes, and not words, I would rather call these Di-interval cells. Although there are two successive intervals, there happen to be three pitches involved; these are relative intervals and not absolute pitch structures, due to the fact that these structures apply to all microtonal systems at play in any one given poly composition. 

The astute observer will note that there is a third interval formed by the outer pitches...As this is early in this new system, I honestly do not know what to do with those, except enjoy that they add a further dimension to this technique. Other than that, I insist that my INTENTION is that these structures are meant to be successive intervals. Intervals! Not pitches. 

Dibrach ∪∪:

Figure 2. Dibrach

The dibrach foot consists of two microchromatic, or even repeated, intervals in succession, either up or down, or static. These units are highly important for the identification of the smallest intervallic cells in the microtonal system being used. For example, quartertones would use the 50-cent intervals in succession. 

Iamb ∪ — :

Figure 3. Iamb

The iamb will move first with the smallest microchromatic interval, either up or down, followed by a skip, leap, or jump in either the same direction or opposite direction. For example, if in 13tet, the first interval is 92.3 cents, the next interval must be a multiple of that interval in any direction. Skips are small, maybe 184 cents, while leaps are at least 3 x 92.3, while jumps can be over an octave, depending on the tessitura of the voice or instrument in question. In classical counterpoint, a change in direction is not necessary here. If you are outlining a chord, fine; if not, better. 

Trochee — ∪:

Figure 4. Trochee

The trochee is the reverse of the iambic foot, where the first interval shall be a multiple of the lowest microtonal unit. So, if there are 8th-tones, then the first disjunct motion will be at least 50 cents, as the smallest interval for eighth-tones is the 25-cent "near" comma. The beginning motion can be anywhere from a skip to a leap. Be adventurous. Take the leap. Unlike Renaissance counterpoint voice-leading, I do not require the conjunct ∪ to move in the opposite direction after a leap or skip. Freedom is king. 

Spondee — —:

Figure 5. Spondee

Spondee requires two successive leaps, skips, or jumps in any direction. Just consider tessitura, range, and register of the voice or instrument. For example, if you are in Just 11-limit, and your smallest interval is @ 53 cents, then multiples of this interval can be used, as well as multiples of the largest Characteristic Interval, at @ 151 cents. Your choice. With unequal tunings, you will have a greater variety of intervals to choose from for non-chromatic skips or jumps. 

Contour Resulting from Prosodic Voice-Leading

Figure 6. Contour of Cell

There are definite benefits to this system, as highly recognizable contour lines help clarify and illustrate the melodic dynamic nature of the line, even in this small cell. Iambic cells above in Figure 6 show just one of the possible line directions the contour can take. Early sketching of contour lines on scratch paper can help give ideas as to how you want to create your melodic lines, over the entire composition, or even just a small section. Joseph Straus discusses contour relations in his book about post-tonal harmony, which may come in handy for additional reading. 

Since clarifying my own contrapuntal goals, having read Knud Jeppesen has helped a great deal; I have come to realize that my initial impetus for free lines in polytempic polymicrotonality needs an open line approach that is not dictated by chord harmony, as that has not resulted in any definitive form. 

If one defines the chords first, then the lines will be subordinated by the harmony, as Kirnberger taught Bach's methods in this way. Therefore, I will take the "two melodies sung together resulting in harmony" approach, advocated by Zarlino and ultimately Palestrina, with respect to this enterprise. 

Meter vs Time Signature

Figure 7. Meter vs Time Signature

As for meter in prosody, I have decided to use it in a "color and talea" rhythmic tension as the "meter" of the feet does not align with the time signature, adding a nice little rhythmic dissonance, in addition to the polytempo of the part, globally. 

Figure 7 shows the intervallic cell as a Trochee and its meter as trimeter; therefore, having three of these intervallic cells "over" the 7/16 time signature gives it a little rub. The Trocheeic Trimeter extends out of the time signature, freeing the rhythmic scansion, while reminiscent of color and talea of isorhythms, when parts did not align neatly, making isorhythms more interesting. 

Three and four-interval successions will be discussed next. 

to be continued...