# TAMNAMS

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TAMNAMS (read as "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 In both cases, the mos structure is no longer valid.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)

### Smaller intervals

Mos intervals smaller than a moschroma
Interval name Absolute value of a...
Moschroma (generalized chroma, provided for reference) Large step minus a small step
Mosdiesis (generalized diesis) Large step minus two small steps
Moskleisma (generalized kleisma) Mosdiesis minus a moschroma
Mosgothma (generalized gothma) Mosdiesis minus a small step

### 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 6-10 steps

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

6-note mosses
Pattern Name Prefix Abbr. Etymology
1L 5s antimachinoid amech- amk Opposite pattern of machinoid.
2L 4s malic mal- mal Sister mos of 4L 2s; apples have concave ends, whereas lemons/limes have convex ends.
3L 3s triwood triwd- tw Blackwood[10] and whitewood[14] generalized to 3 periods.
4L 2s citric citro- cit Parent (or subset) mos of 4L 6s and 6L 4s.
5L 1s machinoid mech- mk From machine temperament.
7-note mosses
Pattern Name Prefix Abbr. Etymology
1L 6s onyx on- on Sounds like "one-six" depending on one's pronunciation; also called anti-archeotonic[1].
2L 5s antidiatonic pel- pel Opposite Pattern of diatonic;pel- is from pelog.
3L 4s mosh mosh- mosh From "mohajira-ish", a name from Graham Breed's naming scheme.
4L 3s smitonic smi- smi From "sharp minor third".
5L 2s diatonic dia- dia
6L 1s archaeotonic arch- arc Originally a name for 13edo's 6L 1s scale; also called archæotonic/archeotonic[2].
8-note mosses
Pattern Name Prefix Abbr. Etymology
1L 7s antipine apine- ap Opposite pattern of pine.
2L 6s subaric subar- sb Parent (or subset) mos of 2L 8s and 8L 2s.
3L 5s checkertonic check- chk From the Kite guitar checkerboard scale.
4L 4s tetrawood tetrawd- ttw Blackwood[10] and whitewood[14] generalized to 4 periods; also called diminished[3].
5L 3s oneirotonic oneiro- onei Originally a name for 13edo's 5L 3s scale; also called oneiro[4].
6L 2s ekic ek- ek From echidna and hedgehog temperaments.
7L 1s pine pine- p From porcupine temperament.
9-note mosses
Pattern Name Prefix Abbr. Etymology
1L 8s antisubneutralic ablu- ablu Opposite pattern of subneutralic.
2L 7s balzano bal- bz Originally a name for 20edo's 2L 7s (and 2L 11) scales; bal- is pronounced /bæl/.
3L 6s tcherepnin cher- ch In reference to Tcherepnin's 9-note scale in 12edo.
4L 5s gramitonic gram- gm From "grave minor third".
5L 4s semiquartal cthon- ct From "half fourth"; cthon- is from "chthonic".
6L 3s hyrulic hyru- hy References triforce temperament.
7L 2s armotonic arm- arm From Armodue theory; also called superdiatonic[3].
8L 1s subneutralic blu- blu Derived from the generator being between supraminor and neutral quality; blu- is from bleu temperament.
10-note mosses
Pattern Name Prefix Abbr. Etymology
1L 9s antisinatonic asina- asi Opposite pattern of sinatonic.
2L 8s jaric jara- ja From pajara, injera, and diaschismic temperaments.
3L 7s sephiroid seph- sp From sephiroth temperament.
4L 6s lime lime- lm Sister mos of 6L 4s; limes are smaller than lemons, as are 4L 6s's step sizes compared to 6L 4s.
5L 5s pentawood pentawd- pw Blackwood[10] and whitewood[14] generalized to 5 periods.
6L 4s lemon lem- le From lemba temperament. Also sister mos of 4L 6s.
7L 3s dicoid dico- di From dichotic and dicot (dicoid) exotemperaments; pronounced /'daɪˌkɔɪd/.
8L 2s taric tara- ta Sister mos of 2L 8s; based off of Hindi word for 18 (aṭhārah), since 18edo contains basic 8L 2s.
9L 1s sinatonic sina- si Derived from the generator being within the range of a sinaic.
1. Alternate name based on the name of its sister mos, with anti- prefix added.
2. Spelling variant.
3. Common name no longer recommend by TAMNAMS due to risk of ambiguity. Provided for reference.
4. Shortened form of name.

### Expansion to smaller mosses

For names for mosses with fewer than 6 steps, see here.

### Expansion to larger mosses

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 degrees of the modes of 5L 3s
UDP Cyclic
Order
Step
Pattern
Scale Degree (oneirodegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLsLLsLs Perf. Maj. Maj. Perf. Maj. Aug. Maj. Maj. Perf.
6|1 4 LLsLsLLs Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 7 LsLLsLLs Perf. Maj. Min. Perf. Maj. Perf. Maj. Maj. Perf.
4|3 2 LsLLsLsL Perf. Maj. Min. Perf. Maj. Perf. Maj. Min. Perf.
3|4 5 LsLsLLsL Perf. Maj. Min. Perf. Min. Perf. Maj. Min. Perf.
2|5 8 sLLsLLsL Perf. Min. Min. Perf. Min. Perf. Maj. Min. Perf.
1|6 3 sLLsLsLL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 6 sLsLLsLL 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 degrees of the modes of 5L 3s (LsLsLLLs)
UDP and
alterations
Cyclic
Order
Step
Pattern
Scale Degree (oneirodegree)
0 1 2 3 4 5 6 7 8
5|2 m4md
3|4 M7md
1 LsLsLLLs Perf. Maj. Min. Perf. Min. Perf. Maj. Maj. Perf.
2|5 d3md
0|7 M6md
2 sLsLLLsL Perf. Min. Min. Dim. Min. Perf. Maj. Min. Perf.
7|0 m2md
5|2 A5md
3 LsLLLsLs Perf. Maj. Min. Perf. Maj. Aug. Maj. Maj. Perf.
4|3 m1md
2|5 M4md
4 sLLLsLsL Perf. Min. Min. Perf. Maj. Perf. Maj. Min. Perf.
7|0 A3md 5 LLLsLsLs Perf. Maj. Maj. Aug. Maj. Aug. Maj. Maj. Perf.
6|1 m7md
4|3 M2md
6 LLsLsLsL Perf. Maj. Maj. Perf. Maj. Perf. Maj. Min. Perf.
3|4 m6md
1|6 M1md
7 LsLsLsLL Perf. Maj. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 d5md 8 sLsLsLLL 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.