# TAMNAMS

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TAMNAMS (read "tame names"; from Temperament-Agnostic Mos NAMing System), devised by the XA Discord in 2021, is a system of temperament-agnostic names for scales – primarily octave-equivalent moment of symmetry scales – as well as their their intervals, their associated generator ranges, and the ratios describing the proportions of large and small steps.

The goal of TAMNAMS is to allow musicians and theorists to discuss moment-of-symmetry scales, or mosses, independent of the language of regular temperament theory. For example, the names flattone[7], meantone[7], pythagorean[7], and superpyth[7] all describe the same step pattern of 5L 2s, with different proportions of large and small steps. Under TAMNAMS parlance, these names can be described broadly as soft 5L 2s (for flattone and meantone) and hard 5L 2s (for pythagorean and superpyth). For discussions of the step pattern itself, the name 5L 2s or, in this example, diatonic, is used.

## Credits

This page and its associated pages were mainly written by User:Godtone, User:SupahstarSaga, User:Inthar, and User:Ganaram inukshuk.

## Step ratio spectrum

### Simple step ratios

TAMNAMS names nine specific simple L:s ratios. These correspond to the simplest edos that have the mos scale.

Step ratio names
TAMNAMS Name Ratio Hardness Diatonic example
Equalized L:s = 1:1 1.000 7edo
Supersoft L:s = 4:3 1.333 26edo
Soft (or monosoft) L:s = 3:2 1.500 19edo
Semisoft L:s = 5:3 1.667 31edo
Basic L:s = 2:1 2.000 12edo
Semihard L:s = 5:2 2.500 29edo
Hard (or monohard) L:s = 3:1 3.000 17edo
Superhard L:s = 4:1 4.000 22edo
Collapsed L:s = 1:0 ∞ (infinity) 5edo

For example, the 5L 2s (diatonic) scale of 19edo has a step ratio of 3:2, which is soft, and is thus called soft diatonic. Tunings of a mos with L:s larger than that ratio are harder, and tunings with L:s smaller than that are softer.

The two extremes, equalized and collapsed, are degenerate cases and define the boundaries for valid tuning ranges. An equalized mos has large and small steps be the same size (L=s), so the mos pattern is no longer apparent. A collapsed mos has small steps shrunken down to zero (s=0), merging adjacent tones s apart into a single tone. In both cases, the mos structure is no longer valid.

### Step ratio ranges

In between the nine specific ratios there are eight named intermediate ranges of step ratios. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. There are also two additional terms for broader ranges: the term hyposoft describes step ratios that are soft-of-basic but not as soft as 3:2; similarly, the term hypohard describes step ratios that are hard-of-basic but not as hard as 3:1.

By default, all ranges include their endpoints. For example, a hard tuning is considered a quasihard tuning. To exclude endpoints, the modifier strict can be used, for example strict hyposoft.

Note that mosses with soft-of-basic step ratios always exhibit Rothenberg propriety, or are proper, whereas mosses with hard-of-basic step ratios do not, or are not proper, with one exception: mosses with only one small step per period are always proper, regardless of the step ratio. It has been argued, however, that this is not a particularly important property, both because "improper" MOSSes still admit an ordering if you allow "off-by-one" errors and because larger moses tend to sound more distinct when L/s > 1, which is in some sense the more vast/varied side of the tuning spectrum, because as L/s becomes larger, the scale becomes increasingly close to the equalized tuning, which is usually radically different from most "proper" tunings while softer tunings don't have much room to be different compared to the basic tuning. (This is explained in more detail in TAMNAMS/Appendix#Extending the spectrum's edges.)

Intermediate ranges
TAMNAMS Name Ratio range Hardness
Hyposoft 3:2 ≤ L:s ≤ 2:1 1.500 ≤ L/s ≤ 2.000
Ultrasoft 1:1 ≤ L:s ≤ 4:3 1.000 ≤ L/s ≤ 1.333
Parasoft 4:3 ≤ L:s ≤ 3:2 1.333 ≤ L/s ≤ 1.500
Quasisoft 3:2 ≤ L:s ≤ 5:3 1.500 ≤ L/s ≤ 1.667
Minisoft 5:3 ≤ L:s ≤ 2:1 1.667 ≤ L/s ≤ 2.000
Minihard 2:1 ≤ L:s ≤ 5:2 2.000 ≤ L/s ≤ 2.500
Quasihard 5:2 ≤ L:s ≤ 3:1 2.500 ≤ L/s ≤ 3.000
Parahard 3:1 ≤ L:s ≤ 4:1 3.000 ≤ L/s ≤ 4.000
Ultrahard 4:1 ≤ L:s ≤ 1:0 4.000 ≤ L/s ≤ ∞
Hypohard 2:1 ≤ L:s ≤ 3:1 2.000 ≤ L/s ≤ 3.000

One may ask "what about hypersoft and hyperhard, given you have hyposoft and hypohard?" and they would be right: see TAMNAMS/Appendix#Extended spectrum which details a more complete glossary that this set of terms is a subset of.

### Central spectrum

Central spectrum of step ratio ranges and specific step ratios
Step ratio ranges Specific step ratios Notes
1:1 (equalized) Trivial/pathological
1:1 to 2:1 (soft-of-basic) 1:1 to 4:3 (ultrasoft) Step ratios especially close to 1:1 may be called pseudoequalized
4:3 (supersoft)
4:3 to 3:2 (parasoft)
3:2 (soft) Also called monosoft
3:2 to 2:1 (hyposoft) 3:2 to 5:3 (quasisoft)
5:3 (semisoft)
5:3 to 2:1 (minisoft)
2:1 (basic)
2:1 to 1:0 (hard-of-basic) 2:1 to 3:1 (hypohard) 2:1 to 5:2 (minihard)
5:2 (semihard)
5:2 to 3:1 (quasihard)
3:1 (hard) Also called monohard
3:1 to 4:1 (parahard)
4:1 (superhard)
4:1 to 1:0 (ultrahard) Step ratios especially close to 1:0 may be called pseudocollapsed
1:0 (collapsed) Trivial/pathological

### Extended spectrum

See TAMNAMS/Appendix#Extended spectrum which details a more complete glossary that this set of terms is a subset of.

## Naming mos intervals

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 kms). A mos's intervals are a 0-mosstep or unison, followed by a 1-mosstep, then a 2-mosstep, and so on, until an n-mosstep interval equal to the period is reached, where n is thus the number of pitches in the mos per period. If a positive integer multiple of the period equals an octave (or some close approximation thereof), that interval can be referred to plainly as an octave if one prefers, but mosoctave should not be used unless there is exactly 7 notes per octave. The prefix of mos- in the term mosstep 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. Additionally, for non-octave scales that assume some generalisation of octave equivalence, the term octave is replaced with the term equave. Note this also means that if an n-mosstep interval is an octave, this can be referred to as the mosequave unambiguously and unconfusingly, regardless of what positive integer n is.

This section's running example will be 3L 4s.

### 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, augmented, diminished and perfect are used. As mosses have maximum variety 2, every interval (except for the unison and multiples of the period which is usually the octave) will be in no more than two sizes.

The modifiers of major, minor, augmented, perfect, and diminished (abbreviated as M, m, A, P, and d respectively) are given as such:

• Integer multiples of the period, such as the unison and (often but not always) the octave, 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, specifically, the only time the less common size appears is at the end of the generator chain. 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 (a.k.a the common 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 the octave are perfect, as are generators raised by some multiple of the octave.

For multi-period mosses, note that both the bright and dark generators appear in every period, not just every octave, as what it means for a mos to be multi-period is that there is multiple periods per octave so that some number of periods is (intended to be interpreted to) equal the octave. Therefore, generators that are raised or lowered by some integer multiple of the mos's period are also perfect. There is an important exception in interval naming for nL ns mosses, in which the generators are major and minor (for the bright and dark generator respectively) rather than augmented, perfect and diminished, and all other intervals (the octave, unison and multiples of the period) are perfect as would be expected. This is to prevent ambiguity over calling every interval present perfect.

Names for mos intervals for 3L 4s
Interval classes Specific intervals Interval size Abbreviation Gens up
0-mosstep (unison) Perfect unison 0 P0ms 0
1-mosstep Minor mosstep (or small mosstep) s m1ms −3
Major mosstep (or large mosstep) L M1ms 4
2-mosstep Diminished 2-mosstep 2s d2ms −6
Perfect 2-mosstep L + s P2ms 1
3-mosstep Minor 3-mosstep 1L + 2s m3ms −2
Major 3-mosstep 2L + s M3ms 5
4-mosstep Minor 4-mosstep 1L + 3s m4ms −5
Major 4-mosstep 2L + 2s M4ms 2
5-mosstep Perfect 5-mosstep 2L + 3s P5ms −1
Augmented 5-mosstep 3L + 2s A5ms 6
6-mosstep Minor 6-mosstep 2L + 4s m6ms −4
Major 6-mosstep 3L + 3s M6ms 3
7-mosstep (octave) Perfect octave 3L + 4s P7ms 0

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

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.

Table of alterations, with abbreviations
Number of chromas Perfect intervals Major/minor intervals
+3 chromas Triply-augmented (AAA, A³, or A^3) Triply-augmented (AAA, A³, or A^3)
+2 chromas Doubly-augmented (AA) Doubly-augmented (AA)
+1 chroma Augmented (A) Augmented (A)
0 chromas (unaltered) 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:

• A generalized diesis, or mosdiesis: |L - 2s|
• A generalized kleisma, or more specifically:
• m-moskleisma: |mosdiesis - s|
• p-moskleisma: |mosdiesis - (L-s)|

### Naming neutral and interordinal intervals

For a discussion of semi-moschroma-altered versions of mos intervals, see Neutral and interordinal k-mossteps.

### Other terminology

The tonic (unison), the period, the generator and the period-complement of the generator make up all the intervals in any given mos scale that might be labelled "perfect". With the exception of the tonic and the period, they may also be "imperfect". Therefore, the degrees of a mos scale which come in a "perfect" variety are called perfectable degrees and the degrees of a mos scale which do not come in a "perfect" variety are called non-perfectable degrees.

## Naming mos degrees

Individual mos degrees, (that is, specific notes of a mos scale,) or k-mosdegrees (abbreviated kmd), 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/root of the scale. 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-mosdegree may also be shortened to k-degree to allow generalization to non-mos scales. When the modifiers major/minor or augmented/perfect/diminished are omitted, they are assumed to be the unmodified degrees of the current mode.

### Naming mos chords

To denote a chord or a mode on a given degree, write the notes of the chord separated by spaces or commas, 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|0), meaning that we have the 7| (LLsLLsLs) mode on the 2-mosdegree which is a minor 2-mosstep in LsLLsLLs (see below for the convention we have used to name the mode).

To analyze a chord as an inversion of another chord (i.e. when the bass is not seen as the root), the following strategies can be used:

1. One can write the root degree first: (6s, 0s, 2s, 7s). The first degree is assumed to be the tonic unless the following method is used:
2. One can write "T" to the left of the tonic: (0s, 2s, T6s, 7s).
3. One can use 0 for the root, using negative numbers for notes below the root. For example, to analyze (0s, 2s, 6s, 7s) on the 7-degree of the LsLLsLLs mode as being rooted on its 6s (thus on the 5-degree of LsLLsLLs), we write 5d(0s, -6s, -4s, 1s). The "5d" here is essential for avoiding confusion with the previous notation.
4. If clarity is desired as to what the root position chord is, slash notation can be used as in conventional notation. Thus the above chord can be written 5d(0s 1s 2s 4s)/7d.

## Mos pattern names

TAMNAMS uses the following names for selected small mosses. These names are optional; interval size names and step ratio names can be combined with conventional xL ys names. For example: 21edo is the soft 5L 3s tuning and its major 2-step is a neutral third of size 342.9 cents.

Some of the names come from older temperament-agnostic mos names, such as names (such as mosh) from Graham Breed's mos names. These names have been coined so that mosses can be discussed more independently of RTT temperaments. Sometimes the prefix has a different source than the scale name for euphonic reasons.

### Names for mosses with 2-10 steps

This list is maintained by User:Inthar and User:Godtone.

TAMNAMS moss names
2-note mosses
Pattern Name Prefix[1] Abbr.[2] Allows non-octave tunings?[3] Etymology
1L 1s trivial triv- trv Yes; can have any period The simplest valid mos pattern
1L 1s monowood monowd- wood No; must have octave period blackwood[10] & whitewood[14] generalized to n-wood for nL ns
3-note mosses (non-octave[3])
Pattern Name Prefix[1] Abbr.[2] (Non-octave periods allowed)[3] Etymology
1L 2s antrial atri- atri Yes; can have any period broader range than trial so named w.r.t. it (anti-trial; antial; antrial)
2L 1s trial tri- tri Yes; can have any period from tri- for 3
4-note mosses
Pattern Name Prefix[1] Abbr.[2] Allows non-octave tunings?[3] Etymology
1L 3s antetric atetra- att Yes; can have any period broader range than tetric so named w.r.t. it (anti-tetric; antetric)
2L 2s biwood biwd- bw No; two periods must be an octave from 2-wood
3L 1s tetric tetra- tt Yes; can have any period from tetra- for 4
5-note mosses (non-octave[3])
Pattern Name Prefix[1] Abbr.[2] (Non-octave periods allowed)[3] Etymology
1L 4s pedal ped- ped one big toe and four small toes
2L 3s pentic pent- pt common pentatonic; from penta- for 5
3L 2s antipentic apent- apt opposite pattern of common pentatonic mos
4L 1s manual manu- manu one thumb and four longer fingers
6-note mosses
Pattern Name Prefix[1] Abbr.[2] See notes on tuning[3] Etymology
1L 5s antimachinoid amech- amech opposite pattern of machinoid
2L 4s malic mal- mal antrial mos w/ 2 periods per octave apples have two concave ends, lemons have two pointy ends.
3L 3s triwood triwd- trw trivial mos w/ 3 periods per octave from 3-wood
4L 2s citric citro- cit trial mos w/ 2 periods per octave parent mos of lemon and lime
5L 1s machinoid mech- mech from machine temperament
7-note mosses
Pattern Name Prefix[1] Abbr.[2] See notes on tuning[3] Etymology
1L 6s onyx on- on from a lot of naming puns
2L 5s antidiatonic pel- pel pel- is from pelog
3L 4s mosh mosh- mosh Graham Breed's name; from "mohajira-ish"
4L 3s smitonic smi- smi from "sharp minor third"
5L 2s diatonic dia- dia
6L 1s arch(a)eotonic arch- arch originally a name for 13edo's 6L 1s
8-note mosses
Pattern Name Prefix[1] Abbr.[2] See notes on tuning[3] Etymology
1L 7s antipine apine- apine opposite pattern of pine
2L 6s subaric subar- subar antetric mos w/ 2 periods per octave largest subset mos of jaric and taric
3L 5s checkertonic check- chk from the Kite guitar checkerboard scale
4L 4s tetrawood (aka diminished[4]) tetrawd- ttw trivial mos w/ 4 periods per octave from 4-wood
5L 3s oneirotonic oneiro- onei originally a name for 13edo's 5L 3s
6L 2s ekic ek- ek tetric mos w/ 2 periods per octave from temperaments echidna and hedgehog
7L 1s pine pine- pine from porcupine temperament
9-note mosses
Pattern Name Prefix[1] Abbr.[2] See notes on tuning[3] Etymology
1L 8s antisubneutralic ablu- ablu opposite pattern of subneutralic
2L 7s balzano bal- /bæl/ bal from Balzano scale in 20edo which is 2L 7s
3L 6s tcherepnin cher- ch antrial mos w/ 3 periods per octave common name
4L 5s gramitonic gram- gram from "grave minor third"
5L 4s semiquartal cthon- cth from "half fourth" and "chthonic"
6L 3s hyrulic hyru- hyru trial mos w/ 3 periods per octave allusion to triforce temperament
7L 2s armotonic (aka superdiatonic[4]) arm- arm arm-(otonic) references Armodue
8L 1s subneutralic blu- blu from the gen's flat neutral quality. blu- is from bleu temperament
10-note mosses
Pattern Name Prefix[1] Abbr.[2] See notes on tuning[3] Etymology
1L 9s antisinatonic asina- asi opposite pattern of sinatonic
2L 8s jaric jara- jar pedal mos w/ 2 periods per octave from temperaments pajara, injera and diaschismic
3L 7s sephiroid seph- seph from sephiroth temperament
4L 6s lime lime- lime pentic mos w/ 2 periods per octave limes/4L 6s's steps tend to be smaller than lemons/6L 4s's steps
5L 5s pentawood pentawd- pw trivial mos w/ 5 periods per octave from 5-wood
6L 4s lemon lem- lem anpentic mos w/ 2 periods per octave from lemba temperament
7L 3s dicoid /'daɪˌkɔɪd/ dico- dico from exotemperaments dichotic and dicot (dicoid)
8L 2s taric tara- tar manual mos w/ 2 periods per octave from Hindi aṭhārah '18'
9L 1s sinatonic sina- si from sinaic
1. used in interval, degree and mode names, e.g. perfect 3-oneirostep, perfect 3-oneirodegree, oneiro-3-up
2. written abbreviations of prefixes, e.g. P3oneis, P3oneid, onei-3|4
3. whether the name can be used for mosses with no octaves; lightly tempered octaves are allowed;
names for mosses with more than 5 notes do not admit nonoctave tunings because the names are specific to the corresponding valid tuning range
4. This is a common name but is no longer the recommended TAMNAMS name due to ambiguity; we provide it here for reference.

### Expansion to mosses with more than 10 steps

Various users have proposed names for mosses with more than 10 steps, commonly referred to as "TAMNAMS extensions". Chief among these are the following:

## Naming mos modes

TAMNAMS uses Modal UDP notation to name modes. For example, the names of modes for 5L 3s are the names of the mos followed by the UDP of that mode.

Scale degree qualities of 5L 3s modes
UDP Rotational Order Step pattern Mode names Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLsLLsLs 5L 3s 7|0 Perf. Maj. Maj. Perf. Maj. Aug. Maj. Maj. Perf.
6|1 4 LLsLsLLs 5L 3s 6|1 Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 7 LsLLsLLs 5L 3s 5|2 Perf. Maj. Min. Perf. Maj. Perf. Maj. Maj. Perf.
4|3 2 LsLLsLsL 5L 3s 4|3 Perf. Maj. Min. Perf. Maj. Perf. Maj. Min. Perf.
3|4 5 LsLsLLsL 5L 3s 3|4 Perf. Maj. Min. Perf. Min. Perf. Maj. Min. Perf.
2|5 8 sLLsLLsL 5L 3s 2|5 Perf. Min. Min. Perf. Min. Perf. Maj. Min. Perf.
1|6 3 sLLsLsLL 5L 3s 1|6 Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 6 sLsLLsLL 5L 3s 0|7 Perf. Min. Min. Dim. Min. Perf. Min. Min. Perf.

For modes with altered scale degrees, the abbreviations for the scale degrees are listed after the UDP for the mode.

Scale degree qualities of 5L 3s modes (step pattern of LsLsLLLs)
UDP and alterations Rotational Order Step pattern Mode names Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8
5|2 m4md 1 LsLsLLLs 5L 3s 5|2 m4md Perf. Maj. Min. Perf. Min. Perf. Maj. Maj. Perf.
2|5 d3md 2 sLsLLLsL 5L 3s 2|5 d3md Perf. Min. Min. Dim. Min. Perf. Maj. Min. Perf.
7|0 m2md 3 LsLLLsLs 5L 3s 7|0 m2md Perf. Maj. Min. Perf. Maj. Aug. Maj. Maj. Perf.
4|3 m1md 4 sLLLsLsL 5L 3s 4|3 m1md Perf. Min. Min. Perf. Maj. Perf. Maj. Min. Perf.
7|0 A3md 5 LLLsLsLs 5L 3s 7|0 A3md Perf. Maj. Maj. Aug. Maj. Aug. Maj. Maj. Perf.
6|1 m7md 6 LLsLsLsL 5L 3s 6|1 m7md Perf. Maj. Maj. Perf. Maj. Perf. Maj. Min. Perf.
3|4 m6md 7 LsLsLsLL 5L 3s 3|4 m6md Perf. Maj. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 d5md 8 sLsLsLLL 5L 3s 0|7 d5md Perf. Min. Min. Dim. Min. Dim. Min. Min. Perf.

Notation, such as diamond-mos, can be used instead of the abbreviation of a mosdegree. For example, LsLsLLLs can be written "5L 3s 5|2 m4md". "5L 3s 5|2 @4d".

For a mos pattern given a name in TAMNAMS, there is also the option of using the prefix for the pattern instead of saying "xL ys": the 5L 3s mode LsLLsLLs can be written "onei-5|2".

## Generalization to non-mos scales

### Intervals in arbitrary scales

Zero-indexed interval names are also used for arbitrary scales, so we can still call a k-step interval a k-step and the corresponding degree the k-degree. But instead of k-mosstep and k-mosdegree, we use k-scalestep and k-scaledegree for arbitrary scales.

### Proposal: Naming ternary scales' step ratios

Analogously to binary scales including mosses, ternary scales, i.e. those with three step sizes L > M > S, including MV3 scales, can also be defined by their L:M:S ratios. Here TAMNAMS names the L/M ratio and then the M/S ratio as if these were mos step ratios: for example, 21edo diasem (5L 2M 2s, LMLSLMLSL or its inverse) has a step ratio of L:M:S = 3:2:1, so we name it soft-basic diasem. If the ratios are the same, repetition may optionally be omitted, so that 26edo diasem, 4:2:1, may optionally be called "basic diasem" instead of "basic-basic diasem". Not to be confused with step ratios where one ratio is unspecified; for that, use:

• x:y:z (where x:y is known but y:z is not) is called (hardness term for x/y)-any. x:x:1 is called equalized-any or LM-equalized (where x >= 1 represents a free variable).
• x:y:z (where y:z is known but x:y is not) is called any-(hardness term for y/z). x:1:1 is called any-equalized or MS-equalized (where x >= 1 represents a free variable).
• x:y:z (where x:z is known but x:y and y:z are not) is called outer-(hardness term for x/z)-any. x:1:x is called outer-equalized-any or LS-equalized. (where x >= 0 represents a free variable).

### Naming MV3 intervals

MV3 scales, such as diasem, have at most 3 sizes for each interval class. For every interval class that occurs in exactly 3 sizes, we use large, medium and small k-step. For every interval class that occurs in 2 sizes, we use large k-step and small k-step. If an interval class only has one size, then we call it perfect k-step.