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'''TAMNAMS''' (read "tame names"; from '''''T'''emperament-'''A'''gnostic '''M'''os '''NAM'''ing '''S'''ystem''), devised by the XA Discord, is a system of temperament-agnostic names for scales (designed especially with small octave-equivalent [[mos]]ses in mind), their intervals and their associated generator ranges, taking into account the relative sizes of large and small steps.
{{Mbox|text=The content of this page is maintained by '''members of the Xenharmonic Alliance Discord'''. If you have any questions, spot any errors, or have any suggestions, be sure to ask there!}}
 
'''TAMNAMS''' (from '''''T'''emperament-'''A'''gnostic '''M'''os '''NAM'''ing '''S'''ystem'', read as /ˈteɪmneɪmz/ or /ˈtæmnæmz/), devised by the XA Discord in 2021, is a system of temperament-agnostic names for scales—primarily [[Octave equivalence|octave-equivalent]] [[moment of symmetry]] scales—as well as 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 ==
== Step ratio spectrum ==
=== Simple step ratios ===
{{Main| Step ratio }}TAMNAMS names nine specific simple [[Blackwood's R|L:s ratios]] tabulated below, which correspond to the simplest edos that have the mos scale. 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 ({{nowrap|L {{=}} s}}), so the mos pattern is no longer apparent. A collapsed mos has small steps shrunken down to zero ({{nowrap|s {{=}} 0}}), merging adjacent tones s apart into a single tone. In both cases, the mos structure is no longer valid.
The TAMNAMS system names nine specific simple [[Blackwood's R|L:s ratios]]. These correspond to the simplest edos that have the mos scale.


{| class=wikitable
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.
|-
 
|+ Step ratio names
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''.
|-
 
! TAMNAMS Name
In some cases it can be clearer to name step ratio ranges by their ranges in hardness (for example, 1-1.33 for ultrasoft) or by their boundary step ratios (for example, equalized-to-supersoft for ultrasoft) than by the step ratio ranges tabulated here.
! Ratio
 
! Diatonic example
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Spectrum of step ratio ranges and specific step ratios
|-
|-
| Equalized
! colspan="3" | Step ratio ranges
| L:s = 1:1
! Specific<br />step ratios
| [[7edo]]
! Hardness
! Notes
|-
|-
| Supersoft
|  
| L:s = 4:3
|  
| [[26edo]]
|
| '''1:1<br />(equalized)'''
| 1
| Trivial/pathological
|-
|-
| Soft (or monosoft)
| rowspan="7" | 1:1 to 2:1<br />(soft-of-basic)
| L:s = 3:2
| colspan="2" | 1:1 to 4:3<br />(ultrasoft)
| [[19edo]]
|
|
| Step ratios especially close to 1:1 may be called pseudoequalized
|-
|-
| Semisoft
|  
| L:s = 5:3
|  
| [[31edo]]
| '''4:3<br />(supersoft)'''
| 1.33
|  
|-
|-
| Basic (or quintessential)
| colspan="2" | 4:3 to 3:2<br />(parasoft)
| L:s = 2:1
|  
| [[12edo]]
|  
|
|-
|-
| Semihard
|  
| L:s = 5:2
|  
| [[29edo]]
| '''3:2<br />(soft)'''
| 1.5
| Also called monosoft
|-
|-
| Hard (or monohard)
| rowspan="3" | 3:2 to 2:1<br />(hyposoft)
| L:s = 3:1
| 3:2 to 5:3<br />(quasisoft)
| [[17edo]]
|
|
|  
|-
|-
| Superhard
|  
| L:s = 4:1
| '''5:3<br />(semisoft)'''
| [[22edo]]
| 1.67
|-
|  
| Collapsed
| L:s = 1:0
| [[5edo]]
|}
 
For example, the 5L2s (diatonic) scale of 19edo has a step ratio of 3:2, which is ''soft''. We call the 19edo diatonic scale ''soft diatonic''. Tunings of a mos with L:s larger are ''harder'', and tunings with L:s smaller are ''softer''.
 
The two extremes, equalized and collapsed, are degenerate cases. An equalized mos has L equal to s, so the mos pattern is no longer apparent. A collapsed mos has 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 ranges of ratios. Each range has a name. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. ''Hypohard'' could be used for tunings that are harder than basic but not as hard as the 3:1 tuning; similarly, ''hyposoft'' can be used for the range between soft and basic. Note that the soft-of-basic range is always strictly proper while the hard-of-basic range is often improper but is always proper in the case that there is 1 small step per period in the mos pattern.
 
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''.
{| class="wikitable"
|+Intermediate ranges
!TAMNAMS Name
!Range
|-
|-
|Hyposoft
| 5:3 to 2:1<br />(minisoft)
|3:2 ≤ L:s ≤ 2:1
|
|
|
|-
|-
|Ultrasoft
|  
|1:1 ≤ L:s ≤ 4:3
|  
|
| '''2:1<br />(basic)'''
| 2
|
|-
|-
|Parasoft
| rowspan="7" | 2:1 to 1:0<br />(hard-of-basic)
|4:3 ≤ L:s ≤ 3:2
| rowspan="3" | 2:1 to 3:1<br />(hypohard)
| 2:1 to 5:2<br />(minihard)
|
|
|
|-
|-
|Quasisoft
|  
|3:2 ≤ L:s ≤ 5:3
| '''5:2<br />(semihard)'''
| 2.5
|
|-
|-
|Minisoft
| 5:2 to 3:1<br />(quasihard)
|5:3 ≤ L:s ≤ 2:1
|
|
|
|-
|-
|Minihard
|  
|2:1 ≤ L:s ≤ 5:2
|  
| '''3:1<br />(hard)'''
| 3
| Also called monohard
|-
|-
|Quasihard
| colspan="2" | 3:1 to 4:1<br />(parahard)
|5:2 ≤ L:s ≤ 3:1
|
|
|
|-
|-
|Parahard
|  
|3:1 ≤ L:s ≤ 4:1
|  
| '''4:1<br />(superhard)'''
| 4
|
|-
|-
|Ultrahard
| colspan="2" | 4:1 to 1:0<br />(ultrahard)
|4:1 ≤ L:s ≤ 1:0
|
|
| Step ratios especially close to 1:0 may be called pseudocollapsed
|-
|-
|Hypohard
|  
|2:1 ≤ L:s ≤ 3:1
|  
|
| '''1:0<br />(collapsed)'''
| infinity
| Trivial/pathological
|}
|}


=== Central spectrum ===
=== Extended spectrum ===
'''Equalized''': L/s = 1/1 (trivial/pathological)
{{Main|TAMNAMS/Appendix#Extended spectrum}}
 
See [[TAMNAMS/Appendix#Extended spectrum]] which details a more complete glossary that this set of terms is a subset of.
::: ('''Ultrasoft''' range here, may also be called ''pseudoequalized'' if especially close to equalized.)
 
:: '''Supersoft''': L/s = 4/3
 
::: ('''Parasoft''' range here.)
 
: '''Soft''': L/s = 3/2
 
::: (Beginning of '''hyposoft''' range here.)
 
::: ('''Quasisoft''' range here.)
 
:: '''Semisoft''': L/s = 5/3
 
::: ('''Minisoft''' range here.)
 
::: (End of '''hyposoft''' range here.)
 
'''Quintesssential''': L/s = 2/1
 
::: (Beginning of '''hypohard''' range here.)
 
::: ('''Minihard''' range here.)
 
:: '''Semihard''': L/s = 5/2
 
::: ('''Quasihard''' range here.)
 
::: (End of '''hypohard''' range here.)
 
: '''Hard''': L/s = 3/1
 
::: ('''Parahard''' range here.)
 
:: '''Superhard''': L/s = 4/1
 
::: ('''Ultrahard''' range here, may also be called ''pseudocollapsed'' if especially close to collapsed.)
 
'''Collapsed''': L/s = 1/0 = infinity (trivial/pathological)


== Naming mos intervals ==
== 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 ''k''ms). A mos's intervals are a 0-mosstep or [[1/1|''unison'']], followed by a 1-mosstep or ''mossecond'', then a 2-mosstep or ''mosthird'', 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.
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 [[1/1|''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.
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.
This section's running example will be [[3L&nbsp;4s]].
===Reasoning for 0-indexed intervals ===
Note that a unison 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.


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.
=== 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 [[1/1|unison]] and multiples of the [[period]] which is usually the [[2/1|octave]]) will be in no more than two sizes.


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 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&nbsp;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 ''n''L&nbsp;''n''s 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.


===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 are [[Distributional evenness|distributionally even]], every interval (except for the [[1/1|unison]] and [[2/1|octave]]) will be in no more than two sizes.
To find what mos interval sizes are found in a mos, 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"
|+Specific interval sizes for 3L 4s
|+ style="font-size: 105%;" | Names for mos intervals for 3L&nbsp;4s
! rowspan="2" | Interval
! colspan="2" |Large size (LsLsLss)
! colspan="2" |Small size (ssLsLsL)
|-
! Step pattern
!Sum
!Step pattern
!Sum
|-
|0-mosstep (unison)
|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
! Interval classes
|LsLsL
! Specific intervals
|'''3L+2s'''
|ssLsL
|'''2L+3s'''
|-
|6-mosstep
| LsLsLs
|'''3L+3s'''
|ssLsLs
|'''2L+4s'''
|-
|7-mosstep (octave)
|LsLsLss
|'''3L+4s'''
|ssLsLsL
|'''3L+4s'''
|}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 ''n''L ''n''s 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.
{| class="wikitable"
|+Names for mos intervals for 3L 4s
! Interval
! Specific mos interval
!Abbreviation
! Interval size
! Interval size
!Gens up
! Abbreviation
! Gens up
|-
|-
|0-mosstep (unison)
| 0-mosstep (unison)
|Perfect unison
| Perfect unison
|P0ms
| 0
| P0ms
| 0
| 0
|0
|-
|-
| rowspan="2" |1-mosstep
| rowspan="2" | 1-mosstep
|Minor mosstep (or small mosstep)
| Minor mosstep (or small mosstep)
| s
| m1ms
| m1ms
|s
| −3
| -3
|-
|-
|Major mosstep (or large mosstep)
| Major mosstep (or large mosstep)
|M1ms
| L
|L
| M1ms
|4
| 4
|-
|-
| rowspan="2" |2-mosstep
| rowspan="2" | '''2-mosstep'''
|Diminished 2-mosstep
| Diminished 2-mosstep
|d2ms
| 2s
|2s
| d2ms
| -6
| −6
|-
|-
| Perfect 2-mosstep
| '''Perfect 2-mosstep'''
|P2ms
| L + s
|L+s
| P2ms
|1
| 1
|-
|-
| rowspan="2" |3-mosstep
| rowspan="2" | 3-mosstep
|Minor 3-mosstep
| Minor 3-mosstep
|m3ms
| 1L + 2s
|1L+2s
| m3ms
| -2
| −2
|-
|-
|Major 3-mosstep
| Major 3-mosstep
|M3ms
| 2L + s
|2L+s
| M3ms
|5
| 5
|-
|-
| rowspan="2" |4-mosstep
| rowspan="2" | 4-mosstep
|Minor 4-mosstep
| Minor 4-mosstep
|m4ms
| 1L + 3s
|1L+3s
| m4ms
| -5
| −5
|-
|-
| Major 4-mosstep
| Major 4-mosstep
|M4ms
| 2L + 2s
|2L+2s
| M4ms
|2
| 2
|-
|-
| rowspan="2" | 5-mosstep
| rowspan="2" | '''5-mosstep'''
|Perfect 5-mosstep
| '''Perfect 5-mosstep'''
|P5ms
| 2L + 3s
| 2L+3s
| P5ms
| -1
| −1
|-
|-
|Augmented 5-mosstep
| Augmented 5-mosstep
|A5ms
| 3L + 2s
|3L+2s
| A5ms
|6
| 6
|-
|-
| rowspan="2" |6-mosstep
| rowspan="2" | 6-mosstep
|Minor 6-mosstep
| Minor 6-mosstep
|m6ms
| 2L + 4s
|2L+4s
| m6ms
| -4
| −4
|-
|-
|Major 6-mosstep
| Major 6-mosstep
|M6ms
| 3L + 3s
|3L+3s
| M6ms
|3
| 3
|-
|-
|7-mosstep (octave)
| 7-mosstep (octave)
| Perfect octave
| Perfect octave
|P7ms
| 3L + 4s
|3L+4s
| P7ms
|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===
=== 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.
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.
f
 
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.
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.
{| class="wikitable"
{| class="wikitable"
|+Table of alterations, with abbreviations
|+ style="font-size: 105%;" | Table of alterations, with abbreviations
|-
|-
!Number of chromas
! Number of chromas
!Perfect intervals
! Perfect intervals
!Major/minor intervals
! Major/minor intervals
|-
|-
| +3 chromas
| +3 chromas
|Triply-augmented (AAA, A³, or A^3)
| Triply-augmented (AAA, A³, or A^3)
|Triply-augmented (AAA, A³, or A^3)
| Triply-augmented (AAA, A³, or A^3)
|-
|-
| +2 chromas
| +2 chromas
|Doubly-augmented (AA)
| Doubly-augmented (AA)
|Doubly-augmented (AA)
| Doubly-augmented (AA)
|-
|-
| +1 chroma
| +1 chroma
|Augmented (A)
| Augmented (A)
|Augmented (A)
| Augmented (A)
|-
|-
| rowspan="2" |0 chromas (unaltered)
| rowspan="2" | 0 chromas (unaltered)
| rowspan="2" |Perfect (P)
| rowspan="2" | Perfect (P)
|Major (M)
| Major (M)
|-
|-
|Minor (m)
| Minor (m)
|-
|-
| -1 chroma
| −1 chroma
|Diminished (d)
| Diminished (d)
|Diminished (d)
| Diminished (d)
|-
|-
| -2 chromas
| −2 chromas
|Doubly-diminished (dd)
| Doubly-diminished (dd)
|Doubly-diminished (dd)
| Doubly-diminished (dd)
|-
|-
| -3 chromas
| −3 chromas
|Triply-diminished (ddd, d³, or d^3)
| Triply-diminished (ddd, d³, or d^3)
|Triply-diminished (ddd, d³, or d^3)
| Triply-diminished (ddd, d³, or d^3)
|}Other intervals include the following:
|}
*A generalized [[Diesis (scale theory)|diesis]], or ''mosdiesis'': |L - 2s|
*A generalized [[kleisma]], or more specifically:
**''m-moskleisma'': |mosdiesis - s|
**''p-moskleisma'': |mosdiesis - (L-s)|


==Naming mos degrees==
=== Smaller intervals ===
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.
{| class="wikitable"
===Naming mos chords===
|+ style="font-size: 105%;" | Mos intervals smaller than a moschroma
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).
|-
! Interval name
! Absolute value of a...
|-
| Moschroma (generalized [[chroma]], provided for reference)
| Large step minus a small step
|-
| Mosdiesis (generalized [[Diesis (scale theory)|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]].


To analyze a chord as an inversion of another chord (i.e. when the bass is not seen as the root), one way is to write out the chord as usual in mos steps, edo steps or cents, except we put an "r" before the root to indicate that that note is whatever degree number we have written. 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), we write 5d(0s-2s-r6s-7s). 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.
=== 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.


== Mos pattern names ==
== Naming mos degrees ==
NOTE: Names are finalized, prefixes and abbreviations are still open to change.
Individual mos degrees, (that is, specific notes of a mos scale,) 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/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.


TAMNAMS suggests 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 mosthird is a neutral third of size 342.9 cents.''
=== 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&nbsp;3s]], the ({{nowrap|0 369 646}}) chord can be written ({{nowrap|0 4 7}})\13, ({{nowrap|P0ms M2ms M4ms}}), or {{nowrap|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&nbsp;3s, we have m2md(0 369 646), or the chord ({{nowrap|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|&nbsp;(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).
 
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:
# 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:
# One can write "T" to the left of the tonic: (0s, 2s, T6s, 7s).
# 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.
# 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&nbsp;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 [[Graham Breed's MOS naming scheme|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.
Some of the names come from older temperament-agnostic mos names, such as names (such as ''mosh'') from [[Graham Breed]]'s [[Graham Breed's MOS naming scheme|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.


1L ns names are named with the an- prefix if they are generalised names and anti- prefix if the name for the corresponding nL1s scale assumed a period of an octave.
=== Names for mosses with 6-10 steps ===
This list is maintained by [[User:Inthar]] and [[User:Godtone]].


This list is maintained by [[User:Inthar]] and [[User:Godtone]].
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ TAMNAMS moss names
|+ style="font-size: 105%;" | TAMNAMS moss names
!colspan=6| 2-note mosses
|-
! Pattern !! Name !! Prefix<ref name=prefix>used in interval, degree and mode names, e.g. ''perfect 3-oneirostep, perfect 3-oneirodegree, oneiro-3-up''</ref> !! Abbreviation<ref name="abbr">written abbreviations of prefixes, e.g. ''P3oneis, P3oneid, onei-3|4''</ref> !! Allows non-octave tunings?<ref name="general">whether the name can be used for mosses with no octaves; lightly tempered octaves are allowed;<br/>names for mosses with more than 5 notes do not admit nonoctave tunings because the names are specific to the corresponding valid tuning range</ref> !! 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
|-
!colspan=6| 3-note mosses (non-octave<ref name=general/>)
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! (Non-octave periods allowed)<ref name=general/> !! 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
|-
!colspan=6| 4-note mosses
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! Allows non-octave tunings?<ref name=general/> !! 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
|-
!colspan=6| 5-note mosses (non-octave<ref name=general/>)
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! (Non-octave periods allowed)<ref name=general/> !! Etymology
|-
| [[1L 4s]] || pedal || ped- || ped || || [[#Pedal (1L 4s)|one big toe and four small toes]]
|-
|-
| [[2L 3s]] || pentic || pent- || pt || || common pentatonic; from penta- for 5
! colspan="5" | 6-note mosses
|-
|-
| [[3L 2s]] || antipentic || apent- || apt || || opposite pattern of common pentatonic mos
! Pattern !! Name !! Prefix<ref group="note" name="prefix">used in interval, degree and mode names, e.g. ''perfect 3-oneirostep, perfect 3-oneirodegree, oneiro-3-up''</ref> !! Abbr.<ref group="note" name="abbr">written abbreviations of prefixes, e.g. ''P3oneis, P3oneid, onei-3|4''</ref> !! Etymology
|-
|-
| [[4L 1s]] || manual || manu- || manu || || one thumb and four longer fingers
| [[1L&nbsp;5s]] || antimachinoid || amech- || amk || Opposite pattern of machinoid.
|-
|-
!colspan=6| 6-note mosses
| [[2L&nbsp;4s]] || malic || mal- || mal || Sister mos of 4L&nbsp;2s; apples have concave ends, whereas lemons/limes have convex ends.
|-
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! Notes on tuning<ref name=general/> !! Etymology
| [[3L&nbsp;3s]] || triwood || triwd- || tw || [[Blackwood]][10] and [[whitewood]][14] generalized to 3 periods.
|-
|-
| [[1L 5s]] || antimachinoid || amech- || amech || || opposite pattern of machinoid
| [[4L&nbsp;2s]] || citric || citro- || cit || Parent (or subset) mos of 4L&nbsp;6s and 6L&nbsp;4s.
|-
|-
| [[2L 4s]] || malic || mal- || mal || antrial mos with 2 periods per octave || apples have two concave ends, lemons have two pointy ends.
| [[5L&nbsp;1s]] || machinoid || mech- || mk || From [[machine]] temperament.
|-
|-
| [[3L 3s]] || triwood || triwud- || trw || trivial mos with 3 periods per octave || from 3-wood
! colspan="5" | 7-note mosses
|-
|-
| [[4L 2s]] || citric || citro- || cit || trial mos with 2 periods per octave || parent mos of lemon and lime
! Pattern !! Name !! Prefix !! Abbr. !! Etymology
|-
|-
| [[5L 1s]] || machinoid || mech- || mech || || from [[machine]] temperament
| [[1L&nbsp;6s]] || onyx || on- || on || Sounds like "one-six" depending on one's pronunciation; also called ''anti-archeotonic<ref group="note" name="anti-name">Alternate name based on the name of its sister mos, with anti- prefix added.</ref>''.
|-
|-
!colspan=6| 7-note mosses
| [[2L&nbsp;5s]] || antidiatonic || pel- || pel || Opposite pattern of diatonic; pel- is from pelog.
|-
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! (Octave periods only.)<ref name=general/> !! Etymology
| [[3L&nbsp;4s]] || mosh || mosh- || mosh || From "mohajira-ish", a name from [[Graham Breed's MOS naming scheme|Graham Breed's naming scheme]].
|-
|-
| [[1L 6s]] || onyx || on- || on || || [[#Onyx (1L 6s)|from a ''lot'' of naming puns]]
| [[4L&nbsp;3s]] || smitonic || smi- || smi || From "sharp minor third".
|-
|-
| [[2L 5s]] || antidiatonic || pel- || pel || || pel- is from pelog
| [[5L&nbsp;2s]] || diatonic || dia- || dia ||  
|-
|-
| [[3L 4s]] || mosh || mosh- || mosh || || Graham Breed's name; from "mohajira-ish"
| [[6L&nbsp;1s]] || archaeotonic || arch- || arc || Originally a name for 13edo's 6L&nbsp;1s scale; also called ''archæotonic/archeotonic<ref group="note" name="spelling">Spelling variant.</ref>''.
|-
|-
| [[4L 3s]] || smitonic || smi- || smi || || from "sharp minor third"
! colspan="5" | 8-note mosses
|-
|-
| [[5L 2s]] || diatonic || ''none'' || ''none'' || ||
! Pattern !! Name !! Prefix !! Abbr. !! Etymology
|-
|-
| [[6L 1s]] || arch(a)eotonic || arch- || arch || || originally a name for 13edo's 6L 1s
| [[1L&nbsp;7s]] || antipine || apine- || ap || Opposite pattern of pine.
|-
|-
!colspan=6| 8-note mosses
| [[2L&nbsp;6s]] || subaric || subar- || sb || Parent (or subset) mos of 2L&nbsp;8s and 8L&nbsp;2s.
|-
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! Notes on tuning<ref name=general/> !! Etymology
| [[3L&nbsp;5s]] || checkertonic || check- || chk || From the [[Kite Giedraitis's Categorizations of 41edo Scales|Kite guitar checkerboard scale]].
|-
|-
| [[1L 7s]] || antipine || apine- || apine || || opposite pattern of pine
| [[4L&nbsp;4s]] || tetrawood || tetrawd- || ttw || Blackwood[10] and whitewood[14] generalized to 4 periods; also called ''diminished<ref group="note" name="unofficial2">Common name no longer recommend by TAMNAMS due to risk of ambiguity. Provided for reference.</ref>.''
|-
|-
| [[2L 6s]] || subaric || subar- || subar || antetric mos with 2 periods per octave || largest subset mos of jaric and taric
| [[5L&nbsp;3s]] || oneirotonic || oneiro- || onei || Originally a name for 13edo's 5L&nbsp;3s scale; also called ''oneiro''<ref group="note">Shortened form of name.</ref>.
|-
|-
| [[3L 5s]] || checkertonic || check- || chk || || from the [[Kite Giedraitis's Categorizations of 41edo Scales|Kite guitar checkerboard scale]]
| [[6L&nbsp;2s]] || ekic || ek- || ek || From [[echidna]] and [[hedgehog]] temperaments.
|-
|-
| [[4L 4s]] || tetrawood; diminished || tetwud- || ttw || trivial mos with 4 periods per octave || from 4-wood
| [[7L&nbsp;1s]] || pine || pine- || p || From [[porcupine]] temperament.
|-
|-
| [[5L 3s]] || oneirotonic || oneiro- || onei || || originally a name for 13edo's 5L 3s
! colspan="5" | 9-note mosses
|-
|-
| [[6L 2s]] || ekic || ek- || ek || tetric mos with 2 periods per octave || from temperaments [[echidna]] and [[hedgehog]]
! Pattern !! Name !! Prefix !! Abbr. !! Etymology
|-
|-
| [[7L 1s]] || pine || pine- || pine || || from [[porcupine]] temperament
| [[1L&nbsp;8s]] || antisubneutralic || ablu- || ablu || Opposite pattern of subneutralic.
|-
|-
!colspan=6| 9-note mosses
| [[2L&nbsp;7s]] || balzano || bal- || bz || Originally a name for 20edo's 2L&nbsp;7s (and 2L 11) scales; bal- is pronounced /bæl/.
|-
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! Notes on tuning<ref name=general/> !! Etymology
| [[3L&nbsp;6s]] || tcherepnin || cher- || ch || In reference to Tcherepnin's 9-note scale in 12edo.
|-
|-
| [[1L 8s]] || antisubneutralic || ablu- || ablu || || opposite pattern of subneutralic
| [[4L&nbsp;5s]] || gramitonic || gram- || gm || From "grave minor third".
|-
|-
| [[2L 7s]] || balzano || bal- /bæl/ || bal || || from Balzano scale in 20edo which is 2L 7s
| [[5L&nbsp;4s]] || semiquartal || cthon- || ct || From "half fourth"; cthon- is from "chthonic".
|-
|-
| [[3L 6s]] || tcherepnin || cher- || ch || antrial mos with 3 periods per octave || common name
| [[6L&nbsp;3s]] || hyrulic || hyru- || hy || References [[triforce]] temperament.
|-
|-
| [[4L 5s]] || gramitonic || gram- || gram || ||from "grave minor third"
| [[7L&nbsp;2s]] || armotonic || arm- || arm || From [[Armodue]] theory; also called ''superdiatonic<ref group="note" name="unofficial2" />.''
|-
|-
| [[5L 4s]] || semiquartal || cthon- || cth || || from "half fourth" and "chthonic"
| [[8L&nbsp;1s]] || subneutralic || blu- || blu || Derived from the generator being between supraminor and neutral quality; blu- is from [[bleu]] temperament.
|-
|-
| [[6L 3s]] || hyrulic || hyru- || hyru || trial mos with 3 periods per octave || allusion to [[triforce]] temperament
! colspan="5" | 10-note mosses
|-
|-
| [[7L 2s]] || superdiatonic; armotonic || arm- || arm || || superdiatonic is a common name; arm- and armotonic reference [[Armodue]]
! Pattern !! Name !! Prefix !! Abbr. !! Etymology
|-
|-
| [[8L 1s]] || subneutralic || blu- || blu || || blu- is from [[bleu]] temperament
| [[1L&nbsp;9s]] || antisinatonic || asina- || asi || Opposite pattern of sinatonic.
|-
|-
!colspan=6| 10-note mosses
| [[2L&nbsp;8s]] || jaric || jara- || ja || From [[pajara]], [[injera]], and [[diaschismic]] temperaments.
|-
|-
! Pattern !! Name !! Prefix<ref name=prefix/> !! Abbreviation<ref name=abbr/> !! Notes on tuning<ref name=general/> !! Etymology
| [[3L&nbsp;7s]] || sephiroid || seph- || sp || From [[sephiroth]] temperament.
|-
|-
| [[1L 9s]] || antisinatonic || asina- || asi || || opposite pattern of sinatonic
| [[4L&nbsp;6s]] || lime || lime- || lm || Sister mos of 6L&nbsp;4s; limes are smaller than lemons, as are 4L&nbsp;6s's step sizes compared to 6L&nbsp;4s.
|-
|-
| [[2L 8s]] || jaric || jara- || jar || pedal mos with 2 periods per octave || from temperaments [[pajara]], [[injera]] and [[diaschismic]]
| [[5L&nbsp;5s]] || pentawood || pentawd- || pw || Blackwood[10] and whitewood[14] generalized to 5 periods.
|-
|-
| [[3L 7s]] || sephiroid || seph- || seph || || from [[sephiroth]] temperament
| [[6L&nbsp;4s]] || lemon || lem- || le || From [[lemba]] temperament. Also sister mos of 4L&nbsp;6s.
|-
|-
| [[4L 6s]] || lime || lime- || lime || pentic mos with 2 periods per octave || limes/4L 6s's steps tend to be smaller than lemons/6L 4s's steps
| [[7L&nbsp;3s]] || dicoid || dico- || di || From [[Dicot family#Dichotic|dichotic]] and [[dicot]] (dicoid) exotemperaments; pronounced /'daɪˌkɔɪd/.
|-
|-
| [[5L 5s]] || pentawood || penwud- || pw || trivial mos with 5 periods per octave || from 5-wood
| [[8L&nbsp;2s]] || taric || tara- || ta || Sister mos of 2L&nbsp;8s; based off of the [[wikipedia:Hindustani_numerals|Hindi]] word for 18 (aṭhārah), since 18edo contains basic 8L&nbsp;2s.
|-
|-
| [[6L 4s]] || lemon || lem- || lem || anpentic mos with 2 periods per octave || from [[lemba]] temperament
| [[9L&nbsp;1s]] || sinatonic || sina- || si || Derived from the generator being within the range of a [[sinaic]].
|-
| [[7L 3s]] || dicoid /'daɪˌkɔɪd/; zaltertic || dico- or zal- || dico or zal || || from exotemperaments [[Dicot family#Dichotic|dichotic]] and [[dicot]] (dicoid), from "central Zalzalian third" for neutral third (zaltertic)
|-
| [[8L 2s]] || taric || tara- || tar || manual mos with 2 periods per octave || from Hindi ''aṭhārah'' '[[#Taric (8L 2s)|18]]'
|-
| [[9L 1s]] || sinatonic || sina- || si || || from [[sinaic]]
|}
|}
<references/>
<references group="note" />
=== Reasoning for the names ===
The goal of TAMNAMS mos names is to choose memorable but aesthetically neutral names.


All names ending in -oid refer to an exotemperament which, when including extreme tunings, covers the entire range of the corresponding octave-period mos, such that many edos with simple step ratios for that mos will correspond to valid tunings, if not by patent val, then with a small number of warts.
=== Expansion to smaller mosses ===
For names for mosses with fewer than 6 steps, see [[TAMNAMS/Appendix#Expanding names for smaller mosses|here]].


All names for mosses with five or less notes - excluding (mono)wood and biwood (which like all n-wood mosses are specific to octave tuning) - require that some small integer multiple of the period is equal to an octave, under the reasoning that mosses with five or less notes are common and broad in tuning enough that they are much more likely to find interest in non-octave contexts. Because of this, their names were chosen to be extremely general, both to avoid bias/being too flavorful and (correspondingly) so that the terms could validly be reused for any mos for which the period is not equal to a (potentially tempered) octave.
=== Expansion to larger mosses ===
 
{{see also| TAMNAMS Extension}}
Any multiperiod mos with more than five notes was given a name that wasn't reliant on the name of a mos with five or less notes as such names were based on those mos names formerly requiring an octave tuning (which is to say some small integer multiple of their period must be equal to a (potentially slightly tempered) octave).
Various users have proposed names for mosses with more than 10 steps, commonly referred to as "TAMNAMS extensions". Chief among these are the following:
 
Former names like "orwelloid" and "sensoid" were abandoned because the names were too temperament-specific in the sense that even considering extreme tunings didnt cover the whole range of the mos. The remaining temperament-based names have been abstracted or altered heavily, namely "pine", "hyrulic", "jaric", "ekic" and "lemon".
 
The inclusion of mos names for "multiperiod" mosses was from a desire to have all ten-note-and-under mosses named for completeness, which is also what prompted some of the reconsiderations mentioned earlier. Similarly, the inclusion of mosses of the form 1L ns using the "anti-" prefix (or an- for less-than-six-note mosses) was also for a practical consideration; although the tuning range is very unhelpful for knowing what such a mos will sound, it is nonetheless useful for describing structure in situations where one does not want to use the mathematical name, especially given that in such situations the tuning will likely be specified somewhere already. Jaric and taric specifically were chosen over bipedal and bimanual because of this, and to a lesser extent, lemon and lime were chosen over antibipentic and bipentic respectively (and for consistency with that their parent MOSS, 4L2s, is named citric).
 
The distinction between using the prefixes "anti-" vs "an-" for reversing the number of large vs. small steps is also not as trivial as it may sound. In the case of mosses with six or more notes, as the period is always an octave, there is a very large tuning range for the 1L ns mosses (hence the original reason for omitting such mosses), but the "anti-" prefix shows that what is significant is that it has the opposite structure to the corresponding nL 1s mos while pointing out the resulting ambiguity of range. In the case of mosses with five or less notes, as the period is not known and therefore could be very small, this is not as much of a concern as fuller specification is likely required anyway, especially in the case of larger periods, so the name should not be tediously long as the name refers to a very simple mos pattern, and for related reasons, the name shouldn't give as much of a sense of one 'orientation' of the structure being more 'primary' than the other, while with mosses with more than five notes, this suggestion of sense is very much intended, because it will almost always make more sense to talk about the (n+1)L 1s child mos of whatever 1L ns mos you want to speak of.
 
=== Name-specific reasonings ===
==== Pedal (1L 4s) ====
Pedals are operated with feet, which have one large toe and four small toes. Also comes from words like "bipedal", where in TAMNAMS, "bipedal" would literally mean a pedal scale with a period equal to half of some chosen interval, although such a scale would have either two right feet or two left feet depending on orientation chosen. If you think "car"/"vehicle" when you think "pedal" and don't think (or want to think) much about feet then you can think about "[[beep]]ing" (as [[beep]] is the 7-limit 4&5 exotemperament). Because this name relies so heavily and fundamentally on there being 1 large and 4 small steps per period, it is appropriate to generalise for any size of period you would want. In that regard, same goes for manual, pentic and anpentic.
==== Malic (2L 4s) and citric (4L 2s) ====
Malic derives from Latin ''malus'' 'apple'. An apple has two concave ends, and large steps in a scale with more small steps are hole-like, hence the two large steps in malic. Citric (4L 2s) is named after the child mosses of citric, namely lemon (6L 4s) and lime (4L 6s). Unlike apples, lemons have two convex pointy ends, and small steps in a scale with more large steps are pointy, hence the two small steps. Malic and citric acids are both ubiquitous in food and biology, thus justifying their use for fairly small mos scales.
 
==== Machinoid (5L 1s) ====
[[Machine]] is the 5&6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.
 
Its list of EDO tunings is {{ val list | 5, 6, (5+5=10e,) 11, 12, 16, 17, (5+10e+12=21be,) 22, 23, (5+5+5+5+6=26qe,) 27, 28, 33 }} with non-patent val EDOs in brackets but included for demonstrating virtual completeness of the tuning range and the unusually large [[33edo]] tuning being to show [[11edo]]'s strength as a tuning.
 
==== Onyx (1L 6s) ====
"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L 1s (pine), 8L 1s (subneutralic) and 9L 1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).
==== Subaric (2L 6s), jaric (2L 8s), and taric (8L 2s) ====
The name "subaric" alludes to the fact that 2L 6s is the largest proper '''sub'''set mos of both j'''aric''' (2L 8s) and t'''aric''' (8L 2s).
 
The name "jaric" alludes to a few highly notable and generally inaccurate (with the exception of diaschismic) temperaments that exist in the tuning range of this MOSS. Specifically, notice how the letters and sound of "jaric" has (or is intended to have) a lot of overlap with [[pajara|pa'''jar'''a]], [[diaschismic|diaschism'''ic''']] and [[injera|in'''jer'''a]] (listed in order of increasingly sharp fourths; note that diatonic fourths and 4-jarasteps are equated in jaric, a notable property).
 
The name "taric" was named based on it being the only octave-tuned TAMNAMS pattern with a [[#Simple step ratios|basic]] tuning of [[18edo]] (because [[5L 6s]] has more than 10 notes so is out of the scope of TAMNAMS, although not necessarily out of the scope of extensions) and it was also named based on rhyming with jaric (as they share the parent mos [[2L 6s]]).
==== Sephiroid (3L 7s) ====
[[Sephiroth]] is the 3&10 temperament in the 2.5.7.11.13.17.21 subgroup with commas including 65/64, 85/84, 221/220, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.
 
Its list of patent val EDO tunings is {{ val list | 3, 10, 13, 16, 23, 26 }} with non-patent val tunings including 6eg, 7e*, 19eg, 20e, 29g, 32egq, 33ce, 36c.
 
* Extreme tunings even occasionally go outside of this range like with 7e, but this would never be considered a good tuning.
 
(Note that ''q'' in the above is a placeholder symbol meaning that the generator 21 is warted.)
 
Note therefore how practically a full range of tunings is covered both in breadth and depth.
==== Dicoid/Zaltertic (7L 3s) ====
[[Dicot family#Dichotic|Dichotic]] is the 7&10 temerament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63 and is an extension of the 5-limit exotemperament [[dicot]] which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. To help justify using these temperament for inspiration for the name, note that:


Its list of patent val EDO tunings is {{ val list | 7, 10, 17 }} with non-patent val tunings including 14cd(=7+7), 20e(=10+10), 24cd(=17+7), 27ce(=17+10).
* [[User:Frostburn/TAMNAMS Extension]]
 
* [[User:Ganaram inukshuk/TAMNAMS Extension]]
Note there are many more warted tunings than this with even more extreme tunings, which makes it reasonable to associate the exotemperament with the range of vaguely saner tunings.
==== Superdiatonic/armotonic (7L 2s) ====
Though the term has seen some use in other contexts, the name "superdiatonic" has seen some precedent of use on the Xenwiki to refer to the mos pattern 7L 2s. This mos is part of a series of mos patterns (5+2k)L 2s, which starts with diatonic (5L 2s, k=0) and superdiatonic (7L 2s, k=1). Like 5L 2s, 7L 2s is also a fifth-generated scale and has a structure similar to diatonic in some ways, but with more large steps. In contexts where the term "superdiatonic" conveys a different meaning (other than referring to 7L 2s), the name "armotonic", in reference to Armodue theory, can be used as an alternative name.
 
==== On the term ''diatonic'' ====
In TAMNAMS, ''diatonic'' exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized/detuned'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.
 
(The choice of how to define ''diatonic'' isn't bound by history, since many other terms have different meanings depending on the historical musical system referred to, for example the ''enharmonic scale'' may refer to a chain-of-fifths-based scale with 12edo enharmonics not equated, or to an ancient Greek genus.)


== Naming mos modes ==
== Naming mos modes ==
TAMNAMS uses [[UDP]] to name modes (i.e. the format pu|pd (p) for mosses with period 1/p of the equave, where u is the number of bright generators up and d is the number of bright generators down). For non-diatonic mosses, the diamond mos accidentals can be used to alter modes, and the degree modified is indicated using TAMNAMS's 0-indexing convention. For example, LsLsLLLs can be written "5L 3s 5|2 @4d".
By default, TAMNAMS uses a simplified version of [[Modal UDP notation]] which specifies the number of generators up and down without multiplying them by the number of periods per equave. This only affects how the modes of multi-period MOS scales are written: for example, the modes of 4L&nbsp;2s are written as 2|0, 1|1, and 0|2, instead of 4|0(2), 2|2(2), and 0|2(4). The modes for single-period MOS scales, such as 5|2 in 5L&nbsp;3s, are written the same way to that of standard UDP notation.


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".
Other mode notation schemes or mode names can be used instead, if desired.
{{MOS mode degrees|Scale Signature=5L 3s|MOS Prefix=mos|Mode Names=Default}}
For modes with altered scale degrees, the abbreviations for the scale degrees are listed after the UDP for the mode.
{{MOS mode degrees|Scale Signature=5L 3s|MOS Prefix=mos|MODMOS Step Pattern=LsLsLLLs|Mode Names=Default}}
Notation, such as [[Diamond-mos notation|diamond-mos]], can be used instead of the abbreviation of a mosdegree. For example, LsLsLLLs can be written {{nowrap|"5L 3s 5{{!}}2 m4md"}}. {{nowrap|"5L 3s 5{{!}}2 @4d"}}.


== Proposal: Extensions for Descendent MOSes ==
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&nbsp;3s mode LsLLsLLs can be written "onei-5|2".
{{see also| TAMNAMS Extension}}
{{see also| User:Frostburn/TAMNAMS Extension}}


There is currently a proposal for a series of systematic extensions to this system for naming MOSes descended from the main ones listed here, as well as a few others.  These extensions are currently being worked on mainly by [[User:Frostburn|Frostburn]].
== Generalization to non-mos scales ==
 
== Non-mos scales ==
=== Intervals in arbitrary 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.
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 3-step-size scales' step ratios ===
=== Proposal: Naming ternary scales' step ratios===
Analogously to 2-step-size scales including mosses, scales 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''.
Analogously to binary scales including mosses, ternary scales, i.e. those with three step sizes {{nowrap|L &gt; M &gt; 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&nbsp;2M&nbsp;2s, LMLSLMLSL or its inverse) has a step ratio of {{nowrap|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:
 
For step ratios where one ratio is unspecified:
* 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''.
* 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''.


=== 3-step scale pattern names ===
* 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 {{nowrap|x &ge; 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 {{nowrap|x &ge; 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 {{nowrap|x &ge; 0}} represents a free variable).


=== Naming MV3 intervals ===
=== 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''.
[[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''.
 
== Appendix ==
 
=== Derivation of the step ratio names ===
The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the [[mediant]] (aka Farey addition) to adjacent fractions.
 
* Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the ''quintessential'' (''quintess.'' or ''essential'' for short) or ''basic'' tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)
 
* As L/s = 1/1 represents L and s being equal in size, it is called ''equalized''.
 
* As L/s = 1/0 represents s = 0, it is called ''collapsed'', as the small scale steps collapse to zero cents and disappear.
 
* The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the ''soft'' tuning.
 
* Analogously, the mediant of 2/1 and 1/0, 3/1, is called the ''hard'' tuning. Thus you can say that a step ratio tuning is ''hard of'' or ''soft of'' another step ratio tuning.
 
* To get something between soft and basic we take the mediant again and get 5/3 for ''semisoft'', and analogously 5/2 for ''semihard''. To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for ''superhard'' and analogously 4/3 for ''supersoft''.
 
There are also tertiary names beyond the above:
 
* Anything softer than supersoft is ''ultrasoft,'' and anything harder than superhard is ''ultrahard''. Something between soft and supersoft is ''parasoft'', as ''para-'' means both ''beyond'' and ''next to''. Something between hard and superhard is ''parahard''.
 
* Something between soft and basic is ''hyposoft'' as it is less soft than soft. Something between hard and basic is ''hypohard'' for the same reason. Between semisoft and quintessential is ''minisoft'' and between semihard and quintessential is ''minihard''.
 
* Finally, between soft and semisoft is ''quasisoft'' as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix ''quasi-'', and between hard and semihard is ''quasihard'' for the same reason.
 
The reasoning for the ''para- super- ultra-'' progression (note that ''super-'' is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because ''parapythagorean'' is between ''pythagorean'' and ''superpythagorean''.
 
This results in the ''central spectrum'' - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an [[EPD|equal pitch division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum.
=== Extending the spectrum's edges ===
Extending the spectrum builds on the central spectrum and relies on a few key observations. Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of ''simple'' tunings for the step ratio for a given mos pattern and period, the notion of ''simple'' used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s. The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the ''hard'' end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.
 
The most obvious adjustment to the edges is to draw a distinction between ''ultrasoft'' and ''pseudoequalized'' by adding a step ratio corresponding to ''semiequalized'', and between ''ultrahard'' and ''pseudocollapsed'' by adding a step ratio corresponding to ''semicollapsed''. Thus:
 
'''Ultrasoft''' is between '''supersoft''' and '''semiequalized''' and '''pseudoequalized''' is between '''semiequalized''' and '''equalized'''.
 
'''Ultrahard''' is between '''superhard''' and '''semicollapsed''', and '''pseudocollapsed''' is between '''semicollapsed''' and '''collapsed'''.
 
Then all that's left is to decide what the step ratios for semicollapsed and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for '''semicollapsed''' we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as '''extrahard'''. Although L/s = 10/1 for '''semicollapsed''' may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most ''equally spaced'' continuing on from 6/1 for the same reason that L/s = 6/5 is the most ''equally spaced''. Note that while the range from '''superhard''' to '''semicollapsed''' is '''ultrahard''', the region may be split into two sub-ranges:
 
'''superhard''' (L/s=4/1) to '''extrahard''' (L/s=6/1) is '''hyperhard''' (4 < L/s < 6).
 
'''extrahard''' (L/s=6/1) to '''semicollapsed''' (L/s=10/1) is '''clustered''' (6 < L/s < 10).
 
With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.
 
 
=== Extended spectrum ===
'''Equalized''': L/s = 1/1 (trivial/pathological)
 
::: ('''Pseudoequalized''' range here.)
 
:: '''Semiequalized''': L/s = 6/5
 
::: ('''Ultrasoft''' range here.)
 
:: '''Supersoft''': L/s = 4/3
 
(4/3 < L/s < 4/1 range here, called the '''nonextreme''' range, detailed by central spectrum.)
 
:: '''Superhard''': L/s = 4/1
 
::: (Beginning of '''ultrahard''' range here.)
 
::: ('''Hyperhard''' range here.)
 
:: '''Extrahard''': L/s = 6/1
 
::: ('''Clustered''' range here.)
 
::: (End of '''ultrahard''' range here.)
 
::'''Semicollapsed''': L/s = 10/1
 
::: ('''Pseudocollapsed''' range here.)


'''Collapsed''': L/s = 1/0 = infinity (trivial/pathological)
== Appendix==
=== Terminology and final notes ===
=== Reasoning for step ratio names ===
A ratio of L/s = k/1 can be called ''k-hard'' and a ratio of L/s = k/(k-1) can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or ''pentasoft'', the simplest hyperhard tuning is 5-hard or ''pentahard'', the simplest clustered tuning is 7-hard or ''heptahard'', 8-hard is ''octahard'', 9-hard is ''nonahard'', and finally, the characteristic simple ultrahard tuning is 6-hard or ''extrahard'', as previously discussed, which can be seen to be similar to ''hexahard'' - hopefully helping with memorisation.
{{Main|TAMNAMS/Appendix#Reasoning for step ratio names}}


A perhaps useful (or otherwise mildly amusing) mnemonic is ''2-soft is too soft to be hard and 2-hard is too hard to be soft'', representing that 2-soft = 2-hard = 2/1 = '''basic'''.
=== Reasoning for mos interval names ===
{{Main|TAMNAMS/Appendix#Reasoning for mos interval names}}


Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos  pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of [[12L 17s]] represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of [[12edo]] and those of [[29edo]], which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.
=== Reasoning for mos pattern names ===
{{Main|TAMNAMS/Appendix#Reasoning for mos pattern names}}


[[Category:MOS]]
[[Category:TAMNAMS]]

Latest revision as of 03:43, 22 March 2026

The content of this page is maintained by members of the Xenharmonic Alliance Discord. If you have any questions, spot any errors, or have any suggestions, be sure to ask there!

TAMNAMS (from Temperament-Agnostic Mos NAMing System, read as /ˈteɪmneɪmz/ or /ˈtæmnæmz/), 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 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

Main article: Step ratio
TAMNAMS names nine specific simple L:s ratios tabulated below, which correspond to the simplest edos that have the mos scale. 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.

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.

In some cases it can be clearer to name step ratio ranges by their ranges in hardness (for example, 1-1.33 for ultrasoft) or by their boundary step ratios (for example, equalized-to-supersoft for ultrasoft) than by the step ratio ranges tabulated here.

Spectrum of step ratio ranges and specific step ratios
Step ratio ranges Specific
step ratios 		Hardness Notes
1:1
(equalized) 		1 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) 		1.33
4:3 to 3:2
(parasoft)
3:2
(soft) 		1.5 Also called monosoft
3:2 to 2:1
(hyposoft) 		3:2 to 5:3
(quasisoft)
5:3
(semisoft) 		1.67
5:3 to 2:1
(minisoft)
2:1
(basic) 		2
2:1 to 1:0
(hard-of-basic) 		2:1 to 3:1
(hypohard) 		2:1 to 5:2
(minihard)
5:2
(semihard) 		2.5
5:2 to 3:1
(quasihard)
3:1
(hard) 		3 Also called monohard
3:1 to 4:1
(parahard)
4:1
(superhard) 		4
4:1 to 1:0
(ultrahard) 		Step ratios especially close to 1:0 may be called pseudocollapsed
1:0
(collapsed) 		infinity Trivial/pathological

Extended spectrum

Main article: TAMNAMS/Appendix § 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.

TAMNAMS moss names
6-note mosses
Pattern Name Prefix[note 1] Abbr.[note 2] 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[note 3].
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[note 4].
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[note 5].
5L 3s oneirotonic oneiro- onei Originally a name for 13edo's 5L 3s scale; also called oneiro[note 6].
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[note 5].
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 the 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. ↑ 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. ↑ Alternate name based on the name of its sister mos, with anti- prefix added.
  4. ↑ Spelling variant.
  5. ↑ 5.0 5.1 Common name no longer recommend by TAMNAMS due to risk of ambiguity. Provided for reference.
  6. ↑ Shortened form of name.

Expansion to smaller mosses

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

Expansion to larger mosses

See also: TAMNAMS Extension

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

By default, TAMNAMS uses a simplified version of Modal UDP notation which specifies the number of generators up and down without multiplying them by the number of periods per equave. This only affects how the modes of multi-period MOS scales are written: for example, the modes of 4L 2s are written as 2|0, 1|1, and 0|2, instead of 4|0(2), 2|2(2), and 0|2(4). The modes for single-period MOS scales, such as 5|2 in 5L 3s, are written the same way to that of standard UDP notation.

Other mode notation schemes or mode names can be used instead, if desired.

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.

Appendix

Reasoning for step ratio names

Main article: TAMNAMS/Appendix § Reasoning for step ratio names

Reasoning for mos interval names

Main article: TAMNAMS/Appendix § Reasoning for mos interval names

Reasoning for mos pattern names

Main article: TAMNAMS/Appendix § Reasoning for mos pattern names
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