User:Unque/On Voice Leading

In discussions of theory, I'm often asked why I have such a focus on the divisibility of important intervals; 29edo's perfect fourth being at a highly divisible 12 steps is a property that I've praised a lot, though rarely have I managed to successfully articulate what makes this property so useful for composition.

In this dissertation, I plan to discuss my thoughts on voice leading in single-voice and multiple-voice harmony to provide my reasoning for this approach to divisibility.

On Mediators

The simplest type of splitting is what I'll call mediation, where one interval is cloven into some number of equal parts to move smoothly from one melodic target to another. These are an extremely common example of voice leading in action, as runs of equal steps can mitigate the strict sense of "pull" towards any one note in the middle of that stack, and the melodic targets no longer feel like "peaks" or "valleys" being pointed to by the voice leading.

Take a simple chord progression like the one below:

It's fine and functional, but it's also rather boring.

We can create more movement by placing mediators in the bass:

Hear how much better that sounds?

Notice that the last mediator splits the space unevenly, and as such has a much clearer distinct pull; meanwhile, the precise half-way splits don't produce such a pull, preventing a sense of settling onto any of the chords in the middle of the progression.

Mediation doesn't always have to cleave intervals into two; we could just as easily use divisions of three:

Note that this is written using ups and downs notation for 36edo, with each arrow accidental altering the pitch up or down by a sixth-tone.

Greater numbers of divisions like four or five are often desirable for longer melodic runs, leading to entire scales like 6L 1s that can be conceptualized by splitting the perfect fifth into four equal parts.

On Mediation by Contrary Motion

Another common usage of mediation is to create a form of resolution by contrary motion wherein the two voices move by the same interval in opposite directions.

Consider two pitches separated by a tension such as 13/10. When moving by contrary motion, the harmonic target would be a consonance such as 3/2. However, if the two pitches are to move by unequal distances to the target, such as 15/14 and 14/13, the stronger pull created by the shorter of the two steps makes one of the voices necessarily more prominent. The closer in size the two steps are, the more this effect is mitigated, and the closer the two voices come to standing on equal ground with one another; thus, it follows that the optimal step size for this voice leading is 15/13 split into two equal parts. If desired, the voices may require further subdivisions to reach the consonance; for instance, if an intermediary is desired between the 13/10 and 3/2 dyads, then 15/13 may be divided into four parts, or further into six, etc.

Now consider the tension that is precisely half of 4/3, and assume that one wants to resolve to a perfect 4/3 via contrary motion. Using the above method, one can see that the optimal step for voice leading by contrary motion is half of that half, or one quarter of 4/3. This sense of dividing divisions may occur as many times as one pleases; if one wants the precise semi-tritave to be used as the main consonance and the semi-semi-tritave or quarter tritave to be the main tension, then the step size which follows from it is the semi-semi-semi tritave, or one eighth of a tritave.

On Common Denominators

When writing melodies or counterpoint using this method of mediation, one often encounters the issue that mediators sound awkward; for example, if one voice drones a tonic while another voice walks up from the tonic to the perfect fourth, the mediator notes are likely going to be discordant against the tonic. The most intuitive way to remedy this is to temper the melodic target such that the mediator intervals reach some consonance; in the previous example, this means to ensure that some other melodic target is divisible by the same type of mediator as the fourth. One solution is to split the fourth into three parts, such that two of them are used as a type of minor third; this roughly creates a Porcupine structure, assuming that the minor third is analyzed as a sharp tempering of 6/5.

I'm told that this approach of common denominators has something to do with Don Page theory, but I can't for the life of me understand a thing that page says.