TAMNAMS

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TAMNAMS (read "tame names"; from Temperament-Agnostic Mos NAMing System), devised by the XA Discord, is a system of temperament-agnostic names for scales (designed especially with small octave-equivalent mosses in mind), their intervals and their associated generator ranges, taking into account the relative sizes of large and small steps.

Step ratio spectrum

Simple step ratios

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

Step ratio names
TAMNAMS Name Ratio Diatonic example
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 (or quintessential) 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
Paucitonic (from few tones) 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 paucitonic, are degenerate cases. An equalized mos has L equal to s, so the mos pattern is no longer apparent. A paucitonic 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 In both cases, the mos structure is no longer valid.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.

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

Equalized: L/s = 1/1 (trivial/pathological)

(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 pseudopaucitonic if especially close to paucitonic.)

Paucitonic: L/s = 1/0 = infinity (trivial/pathological)

Naming mos intervals

To denote interval classes within the mos, TAMNAMS uses the generic prefix mos-, or the specific prefixes and abbreviations listed under mos pattern names. 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.

Thus TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals: First, use the term mosstep for steps of the mos, large or small. From there, an interval which is k mossteps wide is a k-mosstep, short for k-mosstep interval. Major, minor, perfect, etc would apply as established. The names mosoctave (or mosequave for nonoctave mosses) and mosunison could still be used, interchangeably with n-mosstep (for an n-tone mos) and 0-mosstep respectively. This change makes the arithmetic needed to understand mos intervals much smoother.

In contexts where it doesn't cause ambiguity, k-mosstep can be shortened to k-step. k-step is also generalizable to non-mos scale types such as 3-step-size scales; see below for naming in scales with 3 step 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.)

Example: 3L4s
Interval name Abbreviation 10edo Size Gens up
Perfect mosunison P0ms 0\10 0
Minor mosstep (or small mosstep) m1ms 1\10 -3
Major mosstep (or large mosstep) M1ms 2\10 4
Diminished 2-mosstep d2ms 2\10 -6
Perfect 2-mosstep P2ms 3\10 1
Minor 3-mosstep m3ms 4\10 -2
Major 3-mosstep M3ms 5\10 5
Minor 4-mosstep m4ms 5\10 -5
Major 4-mosstep M4ms 6\10 2
Perfect 5-mosstep P5ms 7\10 -1
Augmented 5-mosstep A5ms 8\10 6
Minor 6-mosstep m6ms 8\10 -4
Major 6-mosstep M6ms 9\10 3
Perfect mosoctave P7ms 10\10 0

TAMNAMS uses the following modifiers to denote different interval sizes within a mos interval class:

  • For multiples of the period plus or minus 0 or 1 generators: perfect. (Diatonic examples: perfect mos4th (Pmos4th), perfect mos5th (Pmos5th), perfect mos8th (Pmos8th), perfect mos12th (Pmos12th), etc.)
  • For generic interval classes with 2 specific sizes of intervals therein (which are therefore separated by a chroma of c = L - s), major and minor are used to distinguish the larger (L) and smaller (s) intervals. Note that the generator, its period-equivalents, and the generator's period-complement and its period-equivalents are the only intervals excluded from this rule due to their inclusion in the previous rule. Diatonic examples: major mos2nd (abbreviated Lmos2nd), minor mos3rd (abbreviated smos3rd), major mos3rd (Lmos3rd), etc.)
  • For nL ns scales, there's an exception to the above two rules. Only multiples of the period (1\n) are called perfect. Other intervals are called major or minor, despite being period-equivalent to a generator. The reason for this exception is that otherwise all intervals would be called perfect, leading to ambiguity.
  • If you subtract a chroma from a perfect (Pmos) or minor (smos) interval, it becomes diminished (d; dmos). If you subtract two chromas instead, it becomes doubly diminished (dd; ddmos). (Diatonic examples: diminished mos3rd (dmos3rd), diminished mos4th (dmos4th), doubly diminished mos5th (ddmos5th), etc.)
    • When modifying unisons or octave multiples, mosdiminished and mosaugmented could be used (e.g. mosdiminished octave instead of diminished mosoctave), because the unison and the octave don't change depending on the mos pattern, but the meanings of augmented and diminished.
  • If you add a chroma to a perfect (Pmos) or major (Lmos) interval, it becomes augmented (A; Amos). If you add two chromas instead, it becomes doubly augmented (AA; AAmos). (Diatonic examples: augmented mos2nd (Amos2nd), augmented mos4th (Amos4th), doubly augmented mos5th (AAmos5th).)
  • The pattern continues, ddd for triply diminished and AAA for triply augmented. Note that applying this operation more than 3 times is an unlikely usecase, and a shorthand notaton of d^3 and A^3 or an alternative notation or terminology entirely would likely be preferable in such circumstances, hence repetition of the corresponding letter is a sufficient system.

Other interval names:

  • moschroma or chroma: L − s
  • mosdiesis: |chroma − s| = |L − 2s|

Naming mos degrees

To denote degrees in a given mos (or altered mos) mode, we use the term k-mosdegree, abbreviated as kmd (or k-degree and kd if context allows). Just like for mos interval names, we use 0-indexing, so the perfect 0-mosdegree is the tonic. The modifiers perfect, minor, major, augmented and diminished can be used just like for mos interval names: for example, minor 3-mosdegree (m3md) denotes the degree that lies a minor 3-mosstep above the tonic. Modifiers can be omitted when clear from context.

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 out the mode in UDP notation or in L and s steps. 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|0 (LLsLLsLs) mode on the 2-mosdegree which is a minor 2-mosstep in LsLLsLLs.

Mos pattern names

The following names are suggested 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.

Some of the names come from older temperament-agnostic mos names, such as names (such as mosh) from Graham Breed's mos names. Some are named by taking an arbitrary temperament that generates the scale (preferably in the mos's proper range) and suffixing -oid. These names have been coined so that mosses can be discussed more independently of RTT temperaments (while drawing on an established RTT tradition in the xen community which may help make the names more meaningful to more people).

1L ns names are not given because the generator can be anywhere from the octave to to 1\(n+1) and those scales can better be viewed as subsets of larger mosses, for example 1L 6s as a subset of 7L 1s.

TAMNAMS mos names
5-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 3s pentic pent- pent Shortening of pentatonic.
3L 2s antipentic apent- apent Anti of pentic.
4L 1s manic man- man From Latin manus "hand". A hand has 4 fingers and one thumb.
6-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 4s antilemon alem- alem
3L 3s triwood triwood- trw tri- + -wood for nL ns.
4L 2s lemon lem- lem Named after the 2.5.7.13 5&16 temperament lemba.
5L 1s machinoid mech- mech Named after the 2.9.7.11 5&6 temperament machine.
7-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 5s antidiatonic pel- pel Established name. pel comes 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 none none
6L 1s arch(a)eotonic arch(a)eo- arch A name originally given to 13edo's 6L 1s.
8-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 6s antiechinoid anech- anech
3L 5s sensoid sen- sen From sensi temperament.
4L 4s tetrawood; diminished tetwood- ttw tetra- + -wood for nL ns.
5L 3s oneirotonic oneiro- on A name originally given to 13edo's 5L 3s.
6L 2s echinoid ech- ech From hedgehog and echidna temperaments.
7L 1s pine pine- pine Named after the 11-limit 7&8 temperament porcupine.
9-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 7s joanatonic jo- jo From joan temperament.
3L 6s tcherepnin tcher- tch Common name.
4L 5s orwelloid or- or From orwell temperament.
5L 4s semiquartal sequar- seq From half-fourth.
6L 3s hyrulic hyru- hy From triforce temperament.
7L 2s superdiatonic arm- arm Established name. arm- comes from armodue theory.
8L 1s subneutralic blu- blu From subneutral 2nd generator. blu comes from bleu temperament.
10-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
2L 8s antidimanic adiman- adman
3L 7s sephiroid sephi- seph Named after the 2.5.11.13.17 3&10 temperament sephiroth.
4L 6s dipentic dipen- dpen di- + pentic (2L 3s).
5L 5s pentawood penwood- pw penta- + -wood for nL ns.
6L 4s antidipentic adipen- adpen
7L 3s dicotonic dico- dico Named after the 11-limit 7&10 temperament dichotic.
8L 2s dimanic diman- dman di- + manic (4L 1s).
9L 1s sinatonic sina- si Named after the sinaic that generates the pattern, which in turn is named after Ibn Sina.
11-note mosses
4L 7s kleistonic klei- klei- Named after kleismic and its extensions.
7L 4s suprasmitonic ssmi- ssmi- Generated by (sharp) supraminor thirds.
12-note mosses
Pattern Name Interval prefix[1] Abbreviation[2] Notes
5L 7s p-chromatic p- is for "pure or sharp (para-/super-)pyth(agorean).
7L 5s m-chromatic m- is for "(maybe-mellow) meantone chromatic".
  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 used in interval, degree and mode names, e.g. perfect 3-oneirostep, perfect 3-oneirodegree, oneiro-3-up
  2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 written abbreviations of prefixes, e.g. P3ons, P3ond, on-3|

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 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.)

Proposal: Naming mos modes

For any mos pattern, TAMNAMS allows one to use a modified version of UDP notation to name a given mos's mode. We use "xL ys k|" to denote the mode of the mos xL ys that has k generators above the tonic and n − 1 − k generators below the tonic. For example, "5L 3s 5|" for LsLLsLLs, read "5 ell 3 ess 5 pipe". We skip the number of chroma-positive generators down and the number of periods, which are redundant though these are required by standard UDP notation. The | symbol is read "pipe" instead of "up", because "up" here may be interpreted as part of the accidental alteration modifier (see below).

For an altered mode we can use any accidental alteration whose meaning is clear. For non-diatonic mosses, the degree modified is indicated using TAMNAMS's 0-indexing convention. For example, LsLsLLLs can be written "5L 3s 5| @4d" (read "5L 3s 5 pipe at-4-degree"), using the @ accidental from diamond-mos notation. Using this convention, any mode of any mos can be referred to without introducing new mode names to memorize.

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 called "oneiro-5-pipe" and written "on-5|".

Non-mos scales

Intervals in arbitrary scales

Zero-indexed interval names are also used for arbitrary scales, so a k-step interval can still be called a k-step. But instead of k-mosstep, we use k-scalestep for arbitrary scales.

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

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 paucitonic, meaning few tones, as the resulting scale is also equalized but with fewer tones per period than expected.
  • 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 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 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 pseudopaucitonic by adding a step ratio corresponding to semipaucitonic. Thus:

Ultrasoft is between supersoft and semiequalized and pseudoequalized is between semiequalized and equalized.

Ultrahard is between superhard and semipaucitonic, and pseudopaucitonic is between semipaucitonic and paucitonic.

Then all that's left is to decide what the step ratios for semipaucitonic 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 semipaucitonic 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 semipaucitonic 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 semipaucitonic 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 semipaucitonic (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.)
Semipaucitonic: L/s = 10/1
(Pseudopaucitonic range here.)

Paucitonic: L/s = 1/0 = infinity (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.