MOS scale: Difference between revisions

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An '''MOS''' (can be pronounced "moss") or '''Moment Of Symmetry''' is a scale in which every interval except for the period comes in two sizes.

The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [http://anaphoria.com/mos.PDF Moments of Symmetry]. There is also an introduction by Kraig Grady here: [http://anaphoria.com/wilsonintroMOS.html Introduction to Erv Wilson's Moments of Symmetry].

 

Sometimes, scales are defined with respect to a period and an additional "equivalence interval," considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOS's'''. MOS's in which the equivalence interval is equal to the period are sometimes called '''Strict MOS's'''. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

 

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.

As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where ''s'' = 0. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. On this basis, a system has been proposed by a few users, detailed in a later section.

|+Step ratio names

|Equalized

|L:s = 1:1

|7edo

|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

|L:s = 1:0

|5edo

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.

|+Intermediate ranges

|Parasoft

|4:3 ≤ L:s ≤ 3:2

|3:2 ≤ L:s ≤ 5:3

|-

|3:1 ≤ L:s ≤ 4:1

|-

|Ultrahard

=== 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 [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the "soft" tuning, and 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". Something softer than supersoft is "ultrasoft". Something 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". 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". 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. 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 [[EPD|equal pitch division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum, detailed after.

'''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.)

: '''Hard''': L/s = 3/1

 

 

:: '''Superhard''': L/s = 4/1

 

 

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

 

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:

 

 

 

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.

 

 

 

 

 

 

 

 

::: (Beginning of '''ultrahard''' range here.)

 

 

 

 

 

 

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

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.

 

* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.

* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.

* [[MOS Diagrams]], visualizations of the MOS process.

* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]

 

* [[MODMOS Scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of L-s, the "chroma".

* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.

* [[MOS Cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.

* [[Operations on MOSes]]

== As applied to rhythms ==

* [http://anaphoria.com/horo2.pdf More on Horogram Rhythms].

[[Category:MOS scales| ]] <!-- sort order in category: this page shows above A -->

[[Category:Overview]]

[[Category:Scale theory]]