User:AthiTrydhen/Abstract pergens

This is an extension of work with User:TallKite on pergens and ups and downs notation.

Pergens do not have to be based on chains of diatonic-like fifths; they could be based on other, more abstract chains of generators. The underlying principle is the same.

Color notation

When naming intervals in a system based on an abstract pergen, a TAMNAMS paradigm is selected and intervals are named according to the MOS mode of that paradigm where the tonic is at the center, if the paradigm has an odd number of notes per period, or the tonic is as close to the center as possible but the acoustically larger interval is chosen for the "half-period" in the scale, if the paradigm has an even number of notes per period.

An example of the former is diatonic, where the MOS that's used for the "natural" notes is Dorian. For oneirotonic, on the other hand, the MOS scale that's used is Ultharian, because the oneiro-fifth is the acoustically higher option.

Natural notes in this paradigm are given the color wa. Furthermore, for naming's sake, the period may be denoted P, and the brighter generator G. For instance, in oneirotonic, the generator is a wa oneiro-fourth, and the genchain is as follows:

... sw1 sw4 sw7 sw2 sw5 w8 w3 w6 w1 w4 w7 w2 w5 Lw8 Lw3 Lw6 Lw1 ...

Pergens

Extensions of MOS's like neutralization and Sabafication can be described by pergens in much the same way regular temperaments can. Neutralization adjoins a half-chroma to the underlying MOS lattice, and the pergen that results can vary wildly depending on the original lattice:

• For diatonic, the neutralization is mosh and produces a pergen of (P, G/2).
• For pentic, the neutralization is manual and also produces a pergen of (P, G/2), except that G in this case is what we'd normally call a "perfect fourth" rather than a "perfect fifth".
• For oneirotonic, the neutralization splits the period rather than the generator: (P/2, G).

Neutralization always produces a pergen which splits the original lattice into two, so it's either (P, G/2) or (P/2, G). Sabafication can produce more complex pergens -- for instance, the Sabafication of diatonic into hyrulic produces a pergen of (P/3, G/2).

Ups and downs notation, and families of neutralizations

Ups and downs notation may be used in addition to diamond-mos notation for MOS extensions. When there's a pergen there is inevitably an enharmonic unison as well, but in musical practices based on extensions of an MOS, the rank doesn't have to be preserved when ups and downs are added. An example, based on the diatonic paradigm, is Maqam Rast, which would be notated C D vE F G A vB C in ups and downs notation. In maqam practices that use binary Rast (LSSLLSS), there is an additional enharmonic unison of vvA1 which results in a pergen of (P8, P5/2). But in practices that use ternary Rast (LMSLLMS), the ups and downs would be totally independent of the chain of fifths. In extant maqam traditions that do so, the ^1 is less than half of an A1, but it doesn't have to be the case -- if otherwise, I would suggest that the larger ^1 be rewritten as a vA1, to simplify the notation. The resulting mirror-Rast scale would then be written C D ^Eb F G A ^Bb C.

The neutralization of an MOS is part of a one parameter family or a continuum of generally ternary scales (by turning Ls into Mm with L > M >= m > s), and sometimes, it may be musically worthwhile to look at points in the continuum other than the boundary where M = m. I call this process "partial neutralization". Partial neutralization can either be used for greater melodic expressivity, or to incorporate simple harmonic intervals, or both.

For diatonic, this results in various forms of ternary Rast, including a permutation of Zarlino where L = 9/8, M = 10/9, and m = 16/15. For oneirotonic, where strict neutralization would produce a half-octave, a partial neutralization can render what would be a half-octave as 7/5 or 10/7.