User:Ganaram inukshuk/Notes/TAMNAMS: Difference between revisions

Mos intervals are denoted as a ''quantity'' of mossteps, large or small. An interval that is k mossteps wide is referred to as a ''k-mosstep interval'' or simply ''k-mosstep'', producing a 0-mosstep or ''mosunison'', 1-mosstep, and so on, until an n-mosstep or ''mosoctave'' is reached, where n is the number of pitches in the mos. The prefix of mos- in the terms mosstep, mosunison, and mosoctave may be replaced with the mos's prefix, specified in the section mos pattern names.

In contexts where it doesn't cause ambiguity, the term ''k-mosstep'' can be shortened to ''k-step'', which allows for generalizing terminology described here to non-mos scales, such as strict-variety-3 scales, scales with three specific interval sizes rather than two. Additionally, for [[non-octave]] scales, the term ''mosoctave'' is replaced with the term ''mosequave''.

=== Reasoning for 0-indexed intervals ===

The phrase ''k-mosstep'' by itself does not specify whether an interval is major or minor. To refer to specific intervals, the familiar modifiers of ''major'', ''minor'', ''augment'', ''perfect'', and ''diminished'' are used. As mosses are [[Distributional evenness|distributionally even]], every interval will be in no more than two sizes, except for the mosoctave and mosunison, which only has one size.

To find what mos intervals are found in a mos xL ys, start with the pattern of large and small steps that represents the mos in its brightest mode (the following subsection explains how to do this). This section's running example will be 3L 4s, with the pattern (or string) LsLsLss as its brightest mode. Since a k-mosstep is reached by going up k mossteps up from the root, to find every mos interval, we consider the first k steps of the mos pattern to find each interval's large size. To find the intervals' small size, we repeat the same process of finding mos intervals using the step pattern in the mos's darkest mode, which is the pattern of steps in the brightest mode reversed. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps iL+js, where i and j are the number of L's and s's in the interval's pattern. Note that the size difference between a large interval and small interval corresponds with replacing an L with an s.

|}The modifiers of major, minor, augmented, perfect, and diminished are assigned in the following manner:

* The generators are referred to as '''perfect''' by default. However, the generators have two interval sizes, and all mosses actually have two generators: a bright and dark generator (the following subsection explains how to find these). For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. Specifically:

** Multiples of the period are '''perfect''', as are multiples of a mosoctave.

** Both the bright and dark generators appear in every period, not just every octave. Generators that are raised some multiple of the mos's period are also '''perfect''', as are generators raised by some multiple of the mosoctave.

!Specific mos interval

|Minor mosstep (or small mosstep)

|Major mosstep (or large mosstep)

==== How to find a mos's brightest mode and its generators ====

 

* To find the generators for a mos, follow the algorithm described here: [[Recursive structure of MOS scales#Finding a generator]]. Be sure to follow the additional instructions to produce the generators as some quantity of mossteps. Alternatively, produce an interval matrix using the instructions here ([[Interval matrix#Using step sizes]]) for making an interval matrix out of a mos pattern. The generators are the intervals that appear as one size in all but one mode. The interval that appears in its large size in all but one mode is the perfect bright generator, and the interval that appears in its small size in all but one mode is the perfect dark generator.

TAMNAMS also uses the designations of ''augmented'' and ''diminished'' to refer to ''alterations'' of a mos interval, much like with using sharps and flats in standard notation. However, mos intervals are altered by raising or lowering it by a ''moschroma'' (or simply ''chroma'', if context allows), a generalized sharp/flat that is the difference between a large step and a small step. Raising a minor mos interval by a chroma makes it major, and lowering a major mos interval makes it minor. A major or perfect mos interval can be raised by a chroma repeatedly to produce an augmented, doubly-augmented, and (uncommonly) a triply-augmented mos interval. Likewise, a minor or perfect mos interval can be lowered by a chroma repeatedly to produce a diminished, doubly-diminished, and (uncommonly) a triply-diminished mos interval.  

A mosunison or mosoctave that is itself augmented or diminished may also be referred to a ''mosaugmented'' or ''mosdiminished'' unison or octave. Here, the meaning of unison and octave does not change depending on the mos pattern, but the meanings of augmented and diminished do.

Other intervals include the following:

Individual mos degrees are based on the modifiers assigned to intervals using the process for naming mos intervals and alterations. Mos degrees are enumerated starting at the 0-mosdegree, the tonic. For example, if you go up a major k-mosstep up from the root, then the mos degree reached this way is a major k-mosdegree. Much like "k-mossteps" being shortened to "k-steps" if context allows, k-mosdegrees may also be shortened to "k-degrees". The modifiers of major/minor or augmented/perfect/diminished may also be omitted when clear from context.

** In the spirit of TAMNAMS being temperament-agnostic, a proper solution may be to not use and shoehorn temperament-suggestive prefixes, but rather use the names for step ratios. This lines up with Frostburn's proposal, but applies to the first three generations, not just the third. (Frostburn's proposed abbreviations may also work.) Under this system, all prefixes can work for all three generations, so soft-chromatic, hyposoft-chromatic, and minisoft-chromatic is allowed, just as soft-subchromatic, hyposoft-subchromatic, and minisoft-subchromatic. The absence of prefixes is also allowed.