User:Ganaram inukshuk/TAMNAMS

This page is a work-in-progress, proposed rewrite of the following page: TAMNAMS

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

As opposed to referring to scales by temperament names followed by a note count, such as with meantone[7], names for mosses are referred to by a step pattern and a step ratio.

Step ratio spectrum

Simple step ratios

TAMNAMS names nine specific simple step ratios, corresponding to the simplest edos that contain the mos scale.

Step ratio names

TAMNAMS Name	Ratio	Example with 5L 2s
Equalized	L:s = 1:1	7edo
Supersoft	L:s = 4:3	26edo
Soft (or monosoft)	L:s = 3:2	19edo
Semisoft	L:s = 5:3	31edo
Basic	L:s = 2:1	12edo
Semihard	L:s = 5:2	29edo
Hard (or monohard)	L:s = 3:1	17edo
Superhard	L:s = 4:1	22edo
Collapsed	L:s = 1:0	5edo

For example, the 5L2s (diatonic) scale of 19edo has a step ratio of 3:2, which is soft, and is thus called soft diatonic. Tunings of a mos with L:s larger 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 In both cases, the mos structure is no longer valid.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 named ranges of step ratios. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. The term hyposoft describes step ratios that are soft-of-basic but not as soft as 3:2; similarly, the term hypohard describes step ratios that are hard-of-basic but not as hard as 3:1.

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

Note that mosses with soft-of-basic step ratios always exhibit Rothenberg propriety, or proper, whereas mosses with hard-of-basic step ratios do not, with one exception: mosses with only one small step per period are always proper, regardless of the step ratio.

Intermediate ranges

TAMNAMS Name	Range
Hyposoft	3:2 ≤ L:s ≤ 2:1
Ultrasoft	1:1 ≤ L:s ≤ 4:3
Parasoft	4:3 ≤ L:s ≤ 3:2
Quasisoft	3:2 ≤ L:s ≤ 5:3
Minisoft	5:3 ≤ L:s ≤ 2:1
Minihard	2:1 ≤ L:s ≤ 5:2
Quasihard	5:2 ≤ L:s ≤ 3:1
Parahard	3:1 ≤ L:s ≤ 4:1
Ultrahard	4:1 ≤ L:s ≤ 1:0
Hypohard	2:1 ≤ L:s ≤ 3:1

Central spectrum

Central spectrum of step ratios and step ratio ranges

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

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 are distributionally even, every interval (except for the unison and 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. 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.

Intervals of 3L 4s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-moshstep	Perfect 0-moshstep	P0moshs	0	0.0 ¢
1-moshstep	Minor 1-moshstep	m1moshs	s	0.0 ¢ to 171.4 ¢
	Major 1-moshstep	M1moshs	L	171.4 ¢ to 400.0 ¢
2-moshstep	Diminished 2-moshstep	d2moshs	2s	0.0 ¢ to 342.9 ¢
	Perfect 2-moshstep	P2moshs	L + s	342.9 ¢ to 400.0 ¢
3-moshstep	Minor 3-moshstep	m3moshs	L + 2s	400.0 ¢ to 514.3 ¢
	Major 3-moshstep	M3moshs	2L + s	514.3 ¢ to 800.0 ¢
4-moshstep	Minor 4-moshstep	m4moshs	L + 3s	400.0 ¢ to 685.7 ¢
	Major 4-moshstep	M4moshs	2L + 2s	685.7 ¢ to 800.0 ¢
5-moshstep	Perfect 5-moshstep	P5moshs	2L + 3s	800.0 ¢ to 857.1 ¢
	Augmented 5-moshstep	A5moshs	3L + 2s	857.1 ¢ to 1200.0 ¢
6-moshstep	Minor 6-moshstep	m6moshs	2L + 4s	800.0 ¢ to 1028.6 ¢
	Major 6-moshstep	M6moshs	3L + 3s	1028.6 ¢ to 1200.0 ¢
7-moshstep	Perfect 7-moshstep	P7moshs	3L + 4s	1200.0 ¢

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)

Naming neutral and interordinal intervals

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

Naming mos degrees

Individual mos degrees, 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. 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. 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.

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.

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

TAMNAMS moss names

6-note mosses
Pattern	Name	Prefix	Abbr.	Etymology
1L 5s	selenite, moonstone	sel-	sel	indirect reference to luna temperament
2L 4s	malic	mal-	mal	apples have two concave ends, lemons have two pointy ends.
3L 3s	triwood	triwd-	trw	from 3-wood
4L 2s	citric	citro-	cit	parent mos of lemon and lime
5L 1s	machinoid	mech-	mech	from machine temperament
7-note mosses
Pattern	Name	Prefix	Abbr.	Etymology
1L 6s	onyx	on-	on	from a lot of naming puns
2L 5s	antidiatonic	pel-	pel	pel- is from pelog
3L 4s	mosh	mosh-	mosh	Graham Breed's name; from "mohajira-ish"
4L 3s	smitonic	smi-	smi	from "sharp minor third"
5L 2s	diatonic	dia-	dia
6L 1s	arch(a)eotonic	arch-	arch	originally a name for 13edo's 6L 1s
8-note mosses
Pattern	Name	Prefix	Abbr.	Etymology
1L 7s	spinel	spin-	sp	contains the string "pine", referencing its sister mos
2L 6s	subaric	subar-	subar	largest subset mos of jaric and taric
3L 5s	checkertonic	check-	chk	from the Kite guitar checkerboard scale
4L 4s	tetrawood; diminished	tetrawd-	ttw	from 4-wood
5L 3s	oneirotonic	oneiro-	onei	originally a name for 13edo's 5L 3s
6L 2s	ekic	ek-	ek	from temperaments echidna and hedgehog
7L 1s	pine	pine-	pine	from porcupine temperament
9-note mosses
Pattern	Name	Prefix	Abbr.	Etymology
1L 8s	agate	ag-	ag	rhymes with "eight", depending on one's pronunciation
2L 7s	balzano	bal- /bæl/	bal	from Balzano scale in 20edo which is 2L 7s
3L 6s	tcherepnin	cher-	ch	common name
4L 5s	gramitonic	gram-	gram	from "grave minor third"
5L 4s	semiquartal	cthon-	cth	from "half fourth" and "chthonic"
6L 3s	hyrulic	hyru-	hyru	allusion to triforce temperament
7L 2s	superdiatonic; armotonic	arm-	arm	superdiatonic is a common name; arm- and armotonic references Armodue
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	olivnie	oli-	oli	rhymes with "nine", depending on one's pronunciation
2L 8s	jaric	jara-	jar	from temperaments pajara, injera and diaschismic
3L 7s	sephiroid	seph-	seph	from sephiroth temperament
4L 6s	lime	lime-	lime	limes/4L 6s's steps tend to be smaller than lemons/6L 4s's steps
5L 5s	pentawood	pentawd-	pw	from 5-wood
6L 4s	lemon	lem-	lem	from lemba temperament
7L 3s	dicoid /'daɪˌkɔɪd/	dico-	dico	from exotemperaments dichotic and dicot (dicoid)
8L 2s	taric	tara-	tar	from Hindi aṭhārah '18'
9L 1s	sinatonic	sina-	si	from sinaic

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 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 3-step-size 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.

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.

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

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 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 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 basic is minisoft and between semihard and basic 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 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, summarized and detailed above, just for the regions affected to avoid repetition.

Expanded spectrum of step ratios and step ratio ranges

Central ranges			Extended ranges		Specific step ratios	Notes
					1:1 (equalized)	Trivial/pathological
1:1 to 2:1 (soft-of-basic)	1:1 to 4:3 (ultrasoft)		1:1 to 6:5 (pseudoequalized)
					6:5 (semiequalized)
			6:5 to 4:3 (ultrasoft)
					4:3 (supersoft)	Nonextreme range, as detailed by central spectrum
	4:3 to 3:2 (parasoft)		4:3 to 3:2 (parasoft)
					3:2 (soft)
	3:2 to 2:1 (hyposoft)	3:2 to 5:3 (quasisoft)	3:2 to 5:3 (quasisoft)
					5:3 (semisoft)
		5:3 to 2:1 (minisoft)	5:3 to 2:1 (minisoft)
					2:1 (basic)
2:1 to 1:0 (hard-of-basic)	2:1 to 3:1 (hypohard)	2:1 to 5:2 (minihard)	2:1 to 5:2 (minihard)
					5:2 (semihard)
		5:2 to 3:1 (quasihard)	5:2 to 3:1 (quasihard)
					3:1 (hard)
	3:1 to 4:1 (parahard)		3:1 to 4:1 (parahard)
					4:1 (superhard)
	4:1 to 1:0 (ultrahard)		4:1 to 10:1 (ultrahard)	4:1 to 6:1 (hyperhard)
					6:1 (extrahard)
				6:1 to 10:1 (clustered)
					10:1 (semicollapsed)
			10:1 to 1:0 (pseudocollapsed)
					1:0 (collapsed)	Trivial/pathological

Terminology and final notes

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.

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.

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.

Appendix: reasoning for interval names

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.

Determining the names for mossteps

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 and its darkest mode, the reverse of the brightest mode. The idea of mos recursion may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and 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#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.

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.

Appendix: reasoning for mos names

The goal of TAMNAMS mos names is to choose memorable but aesthetically neutral names. This section describes historic changes to the scheme that have occurred since the inception of TAMNAMS.

Names that end with the -oid suffix

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.

Former names for mosses

Most notable are the names of m-chromatic, p-chromatic, kleistonic, and suprasmitonic. Following discussions in August 2022, the maximum step count for mosses that are eligible for names is 10, meaning that these names were dropped.

Abandoned names

Former names like "orwelloid" and "sensoid" were abandoned because the names were too temperament-specific in the sense that even considering extreme tunings did not 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".

Names for multi-period mosses

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).

Names for mosses that bear the anti- prefix

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

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.

This temperament is supported by 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 equal divisions, with non-patent val tunings including 5+5=10e, 5+10e+12=21be, 5+5+5+5+6=26qe, which are mentioned here 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).

Selenite/moonstone (1L 5s), spinel (1L 7s), agate (1L 8s), and olivine (1L 9s)

The names "agate" and "olivine" follow in the spirit as the name for 1L 6s, onyx. For "selenite", "moonstone", and "spinel", the theme of using gemstone names is expanded further. In the case of selenite (a mineral) and moonstone, both names reference the moon in some way, which is an indirect reference to luna temperament. For spinel, the name contains the word "pine", referencing its sister mos of pine (7L 1s).

Subaric (2L 6s), jaric (2L 8s), and taric (8L 2s)

The name "subaric" alludes to the fact that 2L 6s is the largest proper subset mos of both jaric (2L 8s) and taric (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, diaschismic and injera (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 basic tuning of 18edo (because 7L 4s 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 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.

This temperament is supported by 3, 10, 13, 16, 23 and 26 equal divisions, 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 (7L 3s)

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:

This temperament is supported by 7, 10 and 17 equal divisions, with non-patent val tunings including 14cd(=7+7), 20e(=10+10), 24cd(=17+7), 27ce(=17+10).

Note there are many more warted tunings than this with even more extreme tunings, which makes it reasonable to loosely 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 detempered (if the philosophy is RTT-based) or deregularized/detuned (RTT-agnostic) diatonic scales. The adjectives diatonic-like or quasi-diatonic 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.)