# TAMNAMS/Appendix

(Redirected from Hypersoft)

## 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, summarised and detailed above, just for the regions affected to avoid repetition.

### Extended spectrum

Extended spectrum of step ratio ranges and specific step ratios
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.

## Reasoning for mos 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.

### Deriving 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 (which is the reverse pattern for the mos's 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.

Given the mos xL ys, the following algorithm is used to find the brightest mode for that mos.

1. If either x or y is equal to 1 (base cases):
1. If both x and y are equal to 1, then the final scale is "Ls".
2. If only x is equal to 1, then the final scale is L followed by y s's.
3. If only y is equal to 1, then the final scale is x L's followed by s.
2. If neither x nor y is equal to 1 (recursive cases):
1. Let k be the greatest common factor of x and y.
2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the scale for (x/k)L (y/k)s; the final scale will be (x/k)L (y/k)s duplicated k times.
3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
1. Let m1 = min(x, y) and m2 = max(x, y).
2. Let z = m2 mod m1 and w = m1 - z.
3. Let prescale be the mos string for zL ws. Recursively call this algorithm to find the scale for zL ws; the final scale will be based on this.
4. If x < y, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.
5. To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
1. If x > y, every instance of an L in prescale is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale in its brightest mode.
2. If y > x, every instance of an L in prescale is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale in its brightest mode.

Using 3L 4s as an example, 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.

Interval Large size (LsLsLss) 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 LsLsL 3L+2s ssLsL 2L+3s
6-mosstep LsLsLs 3L+3s ssLsLs 2L+4s
7-mosstep (octave) LsLsLss 3L+4s ssLsLsL 3L+4s

Given the mos xL ys, the following algorithm is used to find the bright generator and its complement.

1. If either x or y is equal to 1 (base cases):
1. If both x and y are equal to 1, then the generator is "L" and its complement is "s".
2. If only x is equal to 1, then the generator is "L" followed by y-1 s's, and the complement is "s".
3. If only y is equal to 1, then the generator is "L" and the complement is x-1 L's followed by "s".
2. If neither x nor y is equal to 1 (recursive cases):
1. Let k be the greatest common factor of x and y.
2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generator and complement for (x/k)L (y/k)s; the intervals returned this way will apply to the period rather than the octave.
3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
1. Let m1 = min(x, y) and m2 = max(x, y).
2. Let z = m2 mod m1 and w = m1 - z.
3. Let gen be the scale's generator and comp be the generator's octave complement for the mos zL ws. Recursively call this algorithm to find these intervals for zL ws; the final scale's generator and complement will be based on this.
4. If x < y, reverse the order of characters in gen and comp, then swap gen and comp. This is only needed if there are more L's than s's in the scale.
5. To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
1. If x > y, every instance of an L in both intervals is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generator and complement.
2. If y > x, every instance of an L in both intervals is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generator and complement.

The length of gen is the number of mossteps for the bright generator, and the length of comp is the number of mossteps in the dark generator. For our example of 3L 4s, the algorithm returns the step pattern Ls as the bright generator and LsLss as its complement, which are 2 and 5 mossteps wide, respectively. Since the large size of a bright generator is perfect and its small size diminished, and the small size of a dark generator is perfect and its small size perfect, the scale's generators can be identified as shown in the table.

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

## Expanding names for smaller mosses

TAMANMS additionally provides optional names for mosses with fewer than six steps. These mosses require that some small integer multiple of the period is equal to an octave, under the reasoning that such step patterns are common and broad in tuning that their names can be validly reused in non-octave contexts. As a result, these names are chosen to be as general as possible, so as to avoid any bias or flavor towards anything other than their step counts or step patterns.

The exception to this are the names monowood and biwood, which must refer to an octave-equivalent mos pattern of 1L 1s or 2L 2s, respectively. Additionally, the name monowood is advised over trivial to refer to an octave-equivalent 1L 1s scale.

2-note mosses
Pattern Name Prefix Abbr. Must be 2/1-equivalent? Etymology
1L 1s trivial triv- trv No The simplest valid mos pattern.
monowood monowd- w Yes Blackwood[10] and whitewood[14] generalized to 1 period.
3-note mosses
Pattern Name Prefix Abbr. Must be 2/1-equivalent? Etymology
1L 2s antrial atri- at No Opposite pattern of 2L 1s, with broader range. Shortening of anti-trial.
2L 1s trial tri- t No From tri- for 3.
4-note mosses
Pattern Name Prefix Abbr. Must be 2/1-equivalent? Etymology
1L 3s antetric atetra- att No Opposite pattern of 3L 1s, with broader range. Shortening of anti-tetric.
2L 2s biwood biwd- bw Yes Blackwood[10] and whitewood[14] generalized to 2 periods.
3L 1s tetric tetra- tt No From tetra- for 4.
5-note mosses
Pattern Name Prefix Abbr. Must be 2/1-equivalent? Etymology
1L 4s pedal ped- pd No From Latin ped, for foot; one big toe and four small toes.
2L 3s pentic pent- pt No Common pentatonic; from penta- for 5.
3L 2s antipentic apent- apt No Opposite pattern of 2L 3s.
4L 1s manual manu- mnu No From Latin manus, for hand; one thumb and four longer fingers.

## Reasoning for mos pattern names

The goal of TAMNAMS mos names is to choose memorable names for the most common octave-equivalent mosses. Generally, names should befit the mos they're describing no matter what temperaments support it, allowing them to be discussed agnostically of any RTT-related contexts.

Names are given to mosses that are the most likely to be used by musicians. As such, TAMNAMS primarily provides names for mosses within the range of 6 to 10 steps (or 2 to 10 steps, when including the names for smaller mosses). This range is chosen to avoid naming large mosses for the sake of naming. Additionally, some of these reasonings also serve as justifications for changing earlier names. As such, this section not only provides reasonings for their names but also a record of how those reasonings were developed.

### General reasonings

The following reasonings cover most TAMNAMS names and should be considered the minimum criteria for naming mosses.

Notable non-temperament names are incorporated into TAMNAMS if they do not cause confusion, or are given names that reference notable things. Such names include mosh, tcherepnin, archaeotonic, oneirotonic, balzano, armotonic, checkertonic, and diatonic.

The name of an interval or a diatonic interval quality can be incorporated into the name of a mos. Such names include smitonic, gramitonic, semiquartal, subneutralic, and sinatonic, from "sharp minor third", "grave minor third", "half-fourth", "between supraminor and neutral", and the interval sinaic, respectively.

Temperament-based names ending in the suffix -oid refer to exotemperaments (low-accuracy temperametns) whose tuning ranges, when including extreme tunings, cover the entirety of their corresponding mosses. Therefore, edos with simple step ratios (2:1, 3:1, 3:2, etc) for that mos will correspond to valid tunings for that temperament (if not by patent val, then with a small number of warts). Such names include machinoid, dicoid, and sephiroid, in reference to machine, dichotic/dicot, and sephiroth temperaments, respectively; for information regarding these temperaments' tunings, see their specific reasonings below.

Temperament-based names that don't refer to exotemperaments are used as a last resort, and if used should be based on a notable temperament. Most of these names are abstractions of their original temperament names insofar that they refer to a temperament. Such names include pine, hyrulic, jaric, ekic and lemon; these reference the temperaments of porcupine, triforce, pajara (along with diaschismic and injera), echidna, and lemba, respectively, with jaric and lemon having additional reasonings of their own.

#### Reasonings for nL ns mosses

Mosses of the form nL ns are given names based on a Greek numeral prefix added to the base name wood, in reference to the temperaments blackwood and whitewood. These mosses are special in that all mosses with the same number of periods n can be traced back to an nL ns mos, representing a mos consisting of only its generators and periods. In other words, these mosses are a 1L 1s pattern repeated n times in one octave. Coincidentally, all mosses with n periods form a binary tree whose root is nL ns (and wood is generally known to come from trees), lending credence to the wood-based name.

#### Monolarge mosses

Monolarge mosses (mosses of the form 1L ns) are given names based on their sister mos (nL 1s), with the anti- prefix added. The exception to this is 1L 6s, given the name onyx for the following reasonings:

"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).

Monolarge mosses were originally left unnamed due to the tuning ranges for these mosses being so large that they were unhelpful with knowing how they sound. This position was later amended as it's useful for describing structure in situations where one does not want to use the mathematical name, and especially in such contexts, a specific tuning will likely be specified.

#### Malic (2L 4s), citric (4L 2s), lime (4L 6s), and lemon (6L 4s)

The names for 2L 4s and 4L 2s come from Latin malus and citrus, meaning 'apple' and 'citrus', respectively. Apples have concave ends, whereas lemons and limes – both types of citrus fruits – have convex ends. Both are ubiquitous foods, justifying their use for these fairly small mosses.

The name citric is given to 4L 2s, as it is the parent mos of 6L 4s and 4L 6s, named after the citrus fruits lemon and lime, respectively, under the reasoning that lemons are larger than limes, as are the step sizes of 6L 4s compared to that of 4L 6s.

Originally, the names for 4L 6s and 6L 4s were based on the duplication of the 2L 3s mos and were called dipentic and antidipentic, respectively. These were changed to their current names as, at the time, the 5-note mosses required an octave period, thus these names required an equivalence interval of 4/1. Although the name pentic can currently apply to a 2L 3s pattern with any size period, the current names were given for completeness, which warranted renaming the related mosses of 2L 4s (renamed from antilemon to malic) and 4L 2s (renamed from lemon to citric).

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

The name jaric alludes to a few highly notable temperaments that exist in the tuning range of 8L 2s, which are alluded to through the spelling and pronunciation of jaric: pajara, injera, and diaschismic. These temperaments, except for diaschismic, have generally inaccurate tunings.

The name taric was named based on it being the only named-range mos with a basic tuning (L:s = 2:1) of 18edo and, as it and 2L 8s share the same parent of 2L 6s, was made to rhyme with jaric.

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).

Originally, the names for 2L 8s and 8L 2s were based on the duplication of the 3L 2s mos and were called called antidimanic and dimanic, respectively (note that manic was since changed to manual). These were changed for the same reasons as with 4L 6s and 6L 4s, and similarly warranted renaming the related mosses of 2L 6s (renamed from antiechidnoid to subaric) and 6L 2s (renamed from echidnoid to ekic).

Two-period mosses and name changes
Pattern Name Pattern Name Pattern Name Pattern Name
2L 2s biwood

(formerly unnamed)

4L 2s citric

(formerly lemon)

4L 6s lime

(formerly dipentic)

6L 4s lemon

(formerly antidipentic)

2L 4s malic

(formerly antilemon)

6L 2s ekic

(formerly echidnoid)

2L 6s subaric

(formerly antiechidnoid)

8L 2s taric

(formerly antidimanic)

2L 8s jaric

(formerly dimanic)

### Reasonings for specific names

#### 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, many of which correspond to both simple tunings (L:s = 2:1, 3:1, 3:2, etc) and degenerate tunings (L:s = 1:1 or 1:0) for 5L 1s. Non-patent val tunings include 5+5=10e, 5+10e+12=21be, 5+5+5+5+6=26qe; these are mentioned here for demonstrating virtual completeness of the tuning range, as is 33edo to show 11edo's strength as a tuning.

#### 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. Like with that of 5L 1s, these represent both simple and degenerate tunings for 3L 7s. Extreme tunings, such as 7e, may lie outside the mos's step ratio spectrum, although such tunings are generally not considered good tunings.

#### Dicoid (7L 3s)

Dichotic is the 7&10 temperament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63. This 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.

This temperament is supported by 7, 10 and 17 equal divisions, with non-patent val tunings including (but not limited to) 7+7=14cd, 10+10=20e, 17+7=24cd, and 17+10=27ce.

#### Armotonic (7L 2s)

Originally, the name superdiatonic was used for 7L 2s, as it has seen some precedent of use on the wiki to refer to an octave-equivalent 7L 2s pattern, although it has had earlier use to refer to the expansion of a smaller mos to a larger one. Due to these concerns, the name armotonic is normally advised over superdiatonic as the former is unambiguous as to what it refers to, and the name superdiatonic is only allowed in situations where it's truly unambiguous if the writer prefers it.

#### On the term diatonic

Although the term diatonic has accrued a variety of exact meanings over time, both within and outside the contexts of xenharmonic music theory, in the context of TAMNAMS and moment-of-symmetry scales, the term diatonic exclusively refers to 5L 2s.