Mathematics of MOS: Difference between revisions

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1. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)

3. No more than two sizes of scale steps (Large and small, often written "L" and "s")

These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.

Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. Suppose the fractions a/b and c/d are a [[Wikipedia:Farey_sequence#Farey_neighbours|Farey pair]], meaning that a/b < c/d and bc - ad = 1. If g = (1-t)(a/b) + t(c/d) for 0 ? t ? 1, then when t = 0, the scale generated by g will consist of an equal division of 1 (representing P) into steps of size 1/b, and when t = 1 into steps of size 1/d. In between, when t = b/(b + d), we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] (a + c)/(b + d) and which will divide the period into b+d equal steps. For all other values a/b < g < c/d we obtain two different sizes of steps, the small steps s, and the large steps L, with the total number of steps b+d, and these scales are the MOS associated to the Farey pair. When g is between a/b and (a + c)/(b + d) there will be b large steps and d small steps, and when it is between (a + c)/(b + d) and c/d, d large steps and b small ones.

Given a generator g, we can find an MOS for g with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to g. A pair of successive semiconvergents have the property that they define a Farey pair, and when g is contained in the pair, that is, a/b < g < c/d, we have defined a MOS for g with b+d as the number of notes in the MOS, with b notes of one size and d of the other.

For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be log2(5)/4, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is 29/50 < x < 25/43. Since g is in that range and not equal to 18/31, we will get a strictly proper MOS.

As MOS Scales are generated by repeated iterations of a single interval, the generator, it is useful to visualize a contiguous "generator chain" that organizes the scale. For example, if the generator is some kind of perfect fifth, then the generator chain is a chain of fifths: FCGDAEB. We know that adjacent tones in the chain are a perfect fifth apart (possibly with octave-displacement), eg. F to C, and that tones two spaces away are some kind of second or ninth apart, eg. F to G. It is clear from the chain that B to F is *not* a perfect fifth, but must be something else (unless the chain closes to form a circle, as would be the case in [[7edo]]). The generator chain shows us that every interval of the MOS scale represents a move on the generator chain some number of generators up or down.

For more images showing the hierarchical arrangement of MOS scales, see [[MOS scales of 17edo]], [[31edo MOS scales]], [[MOS Scales of 37edo]], and [[MOS Scales of 127edo]].

Since MOS scales always consist of some number of large steps and some number of small steps, they can be classified simply by the number of large steps and the number of small steps, in the form #L#s--e.g., the diatonic scale can be described as 5L2s (5 large steps and 2 small steps) or simply [5, 2]. It is typical to ignore the period when specifying MOS scales and instead use the number of large and small steps that make up the interval of equivalence (typically assumed to be the octave--a frequency ratio of 2/1--unless otherwise specified). For instance, the diminished scale in 12-TET is typically classified as 4L4s rather than 1L1s, since there are 4 large and 4 small steps that make up an octave.

If the period is assumed to be 2^(1/n) for some integer n, we can give instead the total number of large and small steps in the octave, instead of just the period, and this is commonly done. In this case, GCD(L, s) gives the number of periods in an octave.

Yet another way of classifying MOS is via [[Wikipedia:Minkowski's question mark function|Minkowski's ? function]]. Here ?(x) is a continuous increasing function from the real numbers to the real numbers which has some peculiar properties, one being that it sends rational numbers to [[Wikipedia:dyadic rational|dyadic rational]]s. Hence if q is a rational number 0 < q < 1 in use in the mediant system of classifying MOS, r = ?(q) = A/2^n will be a dyadic rational number which can also be used. Note that the ? function is invertible, and it and its inverse function, the Box function, have code given for them in the algorithms section at the bottom of the article.

The integer n in the denominator of r (with A assumed to be odd) is the order (or n+1 is, according to some sources) of q in the [[Wikipedia:Stern-Brocot tree|Stern-Brocot tree]]. The two neighboring numbers of order n+1, which define the range of propriety, can also be expressed in terms of the ? and Box functions as Box(r - 2^(-n-1) and Box(r + 2^(-n-1)). If r represents a MOS, the range of possible values for a generator of the MOS will be Box(r) < g < Box(r + 2^(-n)), and the proper generators will be Box(r) < g < Box(r + 2^(-n-1)). So, for example, the MOS denoted by 3/2048 will be between Box(3/2048) and Box(4/2048), which means that 2/21 < g < 1/10, and will be proper if 2/21 < g < 3/31. Hence 7/72, a generator for miracle temperament, will define a MOS but it will not be proper since 7/72 > 3/31 = Box(3/2048 + 1/4096)).

Lp + sq = N.