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→Name-specific reasonings: Consolidated the sections on the -aric mosses into one; felt they were too small left separated
Ganaram inukshuk (talk | contribs)
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Deployed changes I was working on in my sandbox sections, namely having separate subsections for: reasoning for 0-index intervals; naming specific intervals, naming alterations; adding a somewhat detailed walkthrough/example for 3L 4s; separate mos chords section. Some additional rewording may be necessary (the diatonic interval examples are gone), so feel free to edit.
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== Naming mos intervals ==
== Naming mos intervals ==
To denote interval classes within the mos, TAMNAMS uses the generic prefix ''mos-'', or the specific prefixes and abbreviations listed under ''mos pattern names''. One might be tempted to
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'' (abbreviated as ''k''ms). A mos's intervals are a 0-mosstep or ''mosunison'', followed by a 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.
generalize diatonic 1-indexed ordinal names: ''In 31edo's ultrasoft [[mosh]] scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth.''


The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named ''(n+1)th''. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.
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. Additionally, for [[non-octave]] scales, the term ''mosoctave'' is replaced with the term ''mosequave''.


Thus TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals: First, use the term ''mosstep'' for steps of the mos, large or small. From there, an interval which is k mossteps wide is a ''k-mosstep'', short for ''k-mosstep interval''. Major, minor, perfect, etc would apply as established. The names ''mosoctave'' (or ''mosequave'' for nonoctave mosses) and ''mosunison'' could still be used, interchangeably with ''n-mosstep'' (for an n-tone mos) and ''0-mosstep'' respectively. This change makes the arithmetic needed to understand mos intervals much smoother.
This section's running example will be 3L 4s.
===Reasoning for 0-indexed intervals ===
Note that a mosunison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. One might be tempted to generalize diatonic 1-indexed ordinal names: ''In 31edo's ultrasoft [[mosh]] scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth.'' The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named ''(n+1)th''. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.


In contexts where it doesn't cause ambiguity, ''k-mosstep'' can be shortened to ''k-step''. ''k-step'' is also generalizable to non-mos scale types such as 3-step-size scales; see below for naming in scales with 3 step sizes.
To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.


(The ordinal names could still be suggestive for e.g. (tunings of) heptatonic mosses where the ordinal names tend to match up well with diatonic ordinal categories.)
The ordinal names could still be suggestive for e.g. (tunings of) heptatonic mosses where the ordinal names tend to match up well with diatonic ordinal categories.
===Naming specific mos intervals===
The phrase ''k-mosstep'' by itself does not specify the exact size of an interval. 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 (except for the mosunison and mosoctave) will be in no more than two sizes.
 
To find what mos interval sizes are found in a mos xL ys, start with the patterns of large and small steps that represents the mos in its brightest mode (the following subsection explains how to do this) and its darkest mode (which is the reverse pattern for the mos's brightest mode). For our running example of 3L 4s, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each k-mosstep, consider the first k mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each k-mosstep's small size. 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 step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.
{| class="wikitable"
{| class="wikitable"
|+Example: 3L4s
|+Specific interval sizes for 3L 4s
!Interval name
! rowspan="2" | Interval
! colspan="2" |Large size (LsLsLss)
! colspan="2" |Small size (ssLsLsL)
|-
! Step pattern
!Sum
!Step pattern
!Sum
|-
|0-mosstep (mosunison)
|none
|'''0'''
|none
|'''0'''
|-
|1-mosstep
|L
|'''L'''
|s
|'''s'''
|-
|2-mosstep
|Ls
|'''L+s'''
|ss
|'''2s'''
|-
| 3-mosstep
|LsL
|'''2L+s'''
|ssL
|'''1L+2s'''
|-
|4-mosstep
|LsLs
|'''2L+2s'''
| ssLs
|'''1L+3s'''
|-
|5-mosstep
|LsLsL
|'''3L+2s'''
|ssLsL
|'''2L+3s'''
|-
|6-mosstep
| LsLsLs
|'''3L+3s'''
|ssLsLs
|'''2L+4s'''
|-
|7-mosstep (mosoctave)
|LsLsLss
|'''3L+4s'''
|ssLsLsL
|'''3L+4s'''
|}The modifiers of ''major'', ''minor'', ''augment'', ''perfect'', and ''diminished'' (abbreviated as M, m, A, P, and d respectively) are given as such:
*The mosunison and mosoctave are '''perfect''' because they only have one size each.
*The generating intervals, or generators, are referred to as '''perfect'''. Note that a mos actually has two generators - a bright and dark generator - and both generators have two sizes each. For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep (the following subsection explains how to find these). Referring to a mos's generating intervals usually implies its perfect form; specifically:
**The large size of the bright generator is '''perfect''', and the small size is '''diminished'''.
**The large size of the dark generator is '''augmented''', and the small size is '''perfect'''.
*For all other intervals, the large size is '''major''' and the small size is '''minor'''.
*For k-mossteps where k is greater than the number of pitches in the mos, those intervals have the same modifiers as an octave-reduced interval. Similarly, multiples of a mosoctave are perfect, as are generators raised by some multiple of the mosoctave.
For multi-period mosses, the additional rules apply:
*For multi-period mosses not of the form nL ns, there is an additional interval that occurs periodically that only appears as one size. This interval, the mos's period, is '''perfect'''. Additionally:
**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.
*For multi-period mosses that are of the form nL ns, the generators are '''major''' and '''minor''', rather than augmented, perfect, and diminished. This is to prevent ambiguity over calling every interval perfect.
{| class="wikitable"
|+Names for mos intervals for 3L 4s
! Interval
! Specific mos interval
!Abbreviation
!Abbreviation
!10edo Size
! Interval size
!Gens up
!Gens up
|-
|-
|0-mosstep (mosunison)
|Perfect mosunison
|Perfect mosunison
|P0ms
|P0ms
|0\10
| 0
|0
|0
|-
|-
| rowspan="2" |1-mosstep
|Minor mosstep (or small mosstep)
|Minor mosstep (or small mosstep)
|m1ms
| m1ms
|1\10
|s
| -3
| -3
|-
|-
|Major mosstep (or large mosstep)
|Major mosstep (or large mosstep)
|M1ms
|M1ms
|2\10
|L
|4
|4
|-
|-
| rowspan="2" |2-mosstep
|Diminished 2-mosstep
|Diminished 2-mosstep
|d2ms
|d2ms
|2\10
|2s
| -6
| -6
|-
|-
|Perfect 2-mosstep
| Perfect 2-mosstep
|P2ms
|P2ms
|3\10
|L+s
|1
|1
|-
|-
| rowspan="2" |3-mosstep
|Minor 3-mosstep
|Minor 3-mosstep
|m3ms
|m3ms
|4\10
|1L+2s
| -2
| -2
|-
|-
|Major 3-mosstep
|Major 3-mosstep
|M3ms
|M3ms
|5\10
|2L+s
|5
|5
|-
|-
| rowspan="2" |4-mosstep
|Minor 4-mosstep
|Minor 4-mosstep
|m4ms
|m4ms
|5\10
|1L+3s
| -5
| -5
|-
|-
|Major 4-mosstep
| Major 4-mosstep
|M4ms
|M4ms
|6\10
|2L+2s
|2
|2
|-
|-
| rowspan="2" | 5-mosstep
|Perfect 5-mosstep
|Perfect 5-mosstep
|P5ms
|P5ms
|7\10
| 2L+3s
| -1
| -1
|-
|-
|Augmented 5-mosstep
|Augmented 5-mosstep
|A5ms
|A5ms
|8\10
|3L+2s
|6
|6
|-
|-
| rowspan="2" |6-mosstep
|Minor 6-mosstep
|Minor 6-mosstep
|m6ms
|m6ms
|8\10
|2L+4s
| -4
| -4
|-
|-
|Major 6-mosstep
|Major 6-mosstep
|M6ms
|M6ms
|9\10
|3L+3s
|3
|3
|-
|-
|Perfect mosoctave
|7-mosstep (mosoctave)
| Perfect mosoctave
|P7ms
|P7ms
|10\10
|3L+4s
|0
|0
|}
|}
====How to find a mos's brightest mode and its generators====
The idea of [[Recursive structure of MOS scales|mos recursion]] may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and [[Modal UDP notation|UDP]] may be of help for a mos's modes.
*To find the mos whose order of steps represent the mos's brightest mode, follow the algorithm described here: [[Recursive structure of MOS scales|Recursive structure of MOS scales#Finding the MOS pattern from xL ys]].
*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.
===Naming alterations by a chroma===
TAMNAMS also uses the modifiers of ''augmented'' and ''diminished'' to refer to ''alterations'' of a mos interval, much like with using sharps and flats in standard notation. 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; the reverse is true. Raising a major or perfect mos interval repeatedly makes an augmented, doubly-augmented, and a triply-augmented mos interval. Likewise, lowering a minor or perfect mos interval repeatedly makes a diminished, doubly-diminished, and 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.


TAMNAMS uses the following modifiers to denote different interval sizes within a mos interval class:
Repetition of "A" or "d" is used to denote repeatedly augmented/diminished mos intervals, and is sufficient in most cases. It's typically uncommon to alter an interval more than three times, such as with a quadruply-augmented and quadruply-diminished interval; in such cases, it's preferable to use a shorthand such as A^n and d^n, or to use alternate notation or terminology.
* For multiples of the period plus or minus 0 or 1 generators: ''perfect''. (Diatonic examples: perfect mos4th (Pmos4th), perfect mos5th (Pmos5th), perfect mos8th (Pmos8th), perfect mos12th (Pmos12th), etc.)
{| class="wikitable"
* For generic interval classes with 2 specific sizes of intervals therein (which are therefore separated by a chroma of c = L - s), ''major'' and ''minor'' are used to distinguish the larger (L) and smaller (s) intervals. Note that the generator, its period-equivalents, and the generator's period-complement and its period-equivalents are the only intervals excluded from this rule due to their inclusion in the previous rule. Diatonic examples: major mos2nd (abbreviated Lmos2nd), minor mos3rd (abbreviated smos3rd), major mos3rd (Lmos3rd), etc.)
|+Table of alterations, with abbreviations
* For nL ns scales, there's an exception to the above two rules. Only multiples of the period (1\n) are called ''perfect''. Other intervals are called major or minor, despite being period-equivalent to a generator. The reason for this exception is that otherwise all intervals would be called ''perfect'', leading to ambiguity.
|-
* If you subtract a chroma from a perfect (Pmos) or minor (smos) interval, it becomes ''diminished'' (d; dmos). If you subtract two chromas instead, it becomes ''doubly diminished'' (dd; ddmos). (Diatonic examples: diminished mos3rd (dmos3rd), diminished mos4th (dmos4th), doubly diminished mos5th (ddmos5th), etc.)
!Number of chromas
** When modifying unisons or octave multiples, ''mosdiminished'' and ''mosaugmented'' could be used (e.g. ''mosdiminished octave'' instead of ''diminished mosoctave''), because the unison and the octave don't change depending on the mos pattern, but the meanings of ''augmented'' and ''diminished''.
!Perfect intervals
* If you add a chroma to a perfect (Pmos) or major (Lmos) interval, it becomes ''augmented'' (A; Amos). If you add two chromas instead, it becomes ''doubly augmented'' (AA; AAmos). (Diatonic examples: augmented mos2nd (Amos2nd), augmented mos4th (Amos4th), doubly augmented mos5th (AAmos5th).)
!Major/minor intervals
* The pattern continues, ddd for triply diminished and AAA for triply augmented. Note that applying this operation more than 3 times is an unlikely usecase, and a shorthand notaton of d^3 and A^3 or an alternative notation or terminology entirely would likely be preferable in such circumstances, hence repetition of the corresponding letter is a sufficient system.
|-
 
| +3 chromas
Other interval names:
|Triply-augmented (AAA, A³, or A^3)
* ''moschroma'' or ''chroma'': L − s
|Triply-augmented (AAA, A³, or A^3)
* ''mosdiesis'': |chroma − s| = |L − 2s|
|-
 
| +2 chromas
== Naming mos degrees ==
|Doubly-augmented (AA)
To denote degrees in a given mos (or altered mos) mode, we use the term ''k-mosdegree'', abbreviated as kmd (or ''k-degree'' and kd if context allows). Just like for mos interval names, we use 0-indexing, so the perfect 0-mosdegree is the tonic. The modifiers ''perfect'', ''minor'', ''major'', ''augmented'' and ''diminished'' can be used just like for mos interval names: for example, ''minor 3-mosdegree'' (m3md) denotes the degree that lies a minor 3-mosstep above the tonic. Modifiers can be omitted when clear from context.
|Doubly-augmented (AA)
 
|-
To denote a chord or a mode on a given degree, write the chord or the mode in parentheses after the degree symbol. The most explicit option is to write out the chord in cents, edosteps or mossteps (e.g. in [[13edo]] [[5L 3s]], the 0-369-646 chord can be written 0-4-7\13, P0ms-M2ms-M4ms or 7|0 0-2-4ms) and to write the mode. To save space, you can use whatever names or abbreviations for the chord or mode you have defined for the reader. For example, in the LsLLsLLs mode of 5L 3s, we have m2md(0-369-646), or the chord 0-369-646 on the 2-mosdegree which is a minor 2-mosstep. The LsLLsLLs mode also has m2md(7|), meaning that we have the 7| (LLsLLsLs) mode on the 2-mosdegree which is a minor 2-mosstep in LsLLsLLs (see [[TAMNAMS#Proposal: Naming mos modes|below]] for the convention we have used to name the mode).
| +1 chroma
 
|Augmented (A)
|Augmented (A)
|-
| rowspan="2" |0 chromas (unaltered)
| rowspan="2" |Perfect (P)
|Major (M)
|-
|Minor (m)
|-
|  -1 chroma
|Diminished (d)
|Diminished (d)
|-
| -2 chromas
|Doubly-diminished (dd)
|Doubly-diminished (dd)
|-
| -3 chromas
|Triply-diminished (ddd, d³, or d^3)
|Triply-diminished (ddd, d³, or d^3)
|}Other intervals include the following:
*Mosdiesis (a generalized [[Diesis (scale theory)|diesis]]): |L - 2s|
*Moskleisma (a generalized [[kleisma]]; uncommon): |L - 3s|
==Naming mos degrees==
Individual mos degrees, or ''k-mosdegrees'' (abbreviated ''k''md) are based on the modifiers given to intervals using the process for naming mos intervals and alterations. Mosdegrees are 0-indexed and 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 mossteps, the prefix of mos- may also be replaced with the mos's prefix. If context allows, ''k-mosdegrees'' may also be shortened to ''k-degrees'' to allow generalization to non-mos scales. The modifiers of major/minor or augmented/perfect/diminished may also be omitted when clear from context.
===Naming mos chords===
To denote a chord or a mode on a given degree, write the chord or the mode in parentheses after the degree symbol. The most explicit option is to write out the chord in cents, edosteps or mossteps (e.g. in [[13edo]] [[5L 3s]], the 0-369-646 chord can be written 0-4-7\13, P0ms-M2ms-M4ms or 7|0 0-2-4ms) and to write the mode. To save space, you can use whatever names or abbreviations for the chord or mode you have defined for the reader. For example, in the LsLLsLLs mode of 5L 3s, we have m2md(0-369-646), or the chord 0-369-646 on the 2-mosdegree which is a minor 2-mosstep. The LsLLsLLs mode also has m2md(7|), meaning that we have the 7| (LLsLLsLs) mode on the 2-mosdegree which is a minor 2-mosstep in LsLLsLLs (see [[TAMNAMS#Proposal:%20Naming%20mos%20modes|below]] for the convention we have used to name the mode).
== Mos pattern names ==
== Mos pattern names ==
NOTE: Names are finalized, prefixes and abbreviations are still open to change.
NOTE: Names are finalized, prefixes and abbreviations are still open to change.
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