MOS Step Ratio Spectrum

The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with L more similar to s sound smoother and more mellow. MOSes with L much larger than s sound jagged and dramatic. The step ratio, the ratio between the sizes of L and s, is thus important to the sound of the scale.

Relative interval sizes

Part of this perception stems from the fact that, as these L:s ratios change and pass certain critical rational values, the *next* MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "xLys" to "yLxs." As an example, with the "5L2s" diatonic MOS, if we have L/s < 2, the next MOS will be "7L5s," and if we have L/s > 2, the next MOS will be "5L7s." (At the point L/s = 2, we have that the next MOS is an equal temperament.)

Similar things happen with *all* of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is *two* steps after the current one changes. Again, as an example, with the familiar 5L2s diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.

Another way to look at this is using Rothenberg propriety: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper," all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form xL1s), at which point it is always "strictly proper." Similarly we pass the L:s 3:2 boundary, the *next* MOS changes from strictly proper to improper, and so on.

The special ratio L:s = phi is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.

Naming System

Ratio Spectrum

We in the discord have named nine specific simple L:s ratios.

Step ratio names

Name	Ratio	Diatonic example
Equalized	L:s = 1:1	7edo
Supersoft	L:s = 4:3	26edo
Soft	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	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.

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.

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

Name	Range
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

Derivation

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" below - 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, detailed after.

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)

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 rations nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed below, 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.