MODMOS scale
Introduction
A scale is considered to be a MOS scale if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS.
MODMOS scales, also known as altered MOS scales, generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS.
The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS, they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.
A chromatic alteration means changing the size of an interval by increments of the MOS's chroma, where the chroma is the difference between any pair of intervals sharing the same interval class.
Alteration by increments of some other interval is possible, but they lack the useful properties of MODMOS scales, most importantly epimorphism, so they are inflected MOS scales, rather than true MODMOS scales.
In the exposition below, we give a formal treatment of MODMOS scales.
Definitions
An MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating periodic scale by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the "chroma". A MODMOS, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS.
Ways of Looking at MODMOS Scales
Any number of alterations are permitted; it is up to the judgment of the composer which of the resulting scales are most musically useful. However, clearly, some MODMOS's will be more useful than others, and it is good to talk about some of the ways this can be the case.
For starters, certain alterations will cause the notes of the scale to no longer be "monotonic" (in ascending order). Typically we are most interested in those MODMOS's which are. In fact, for any MOS, only finitely many MODMOS's will be monotonic in this way (up to transpositional equivalence). To see this, note that there is a smallest possible type of step any MODMOS of the original MOS can have, which has been chroma-flattened as much as possible; thus there is a flattest MODMOS which is made up of N-1 of these minimal seconds in a row, followed by one huge "maximal second" to make up the difference with the octave. Similarly, there will be a sharpest MODMOS which starts with one huge second, and then the N-1 minimal seconds. Every monotonic MODMOS will be intermediate to these two, formed from various intermediate seconds (of which there are only finitely many type).
Another important note is that the more alterations are made, the less the resulting scale will resemble the original MOS. Thus, it can be very useful, when trying to "organize" the universe of MODMOS's generated by an MOS, to sort them by the total number of alterations that have been made. Thus one can look at single-alteration MODMOS's, double-alteration MODMOS's, and so on, each of which gets further from the character of the core MOS. Similarly, one can look at the maximum number of chroma-alterations that has been made to any particular note at a time: are all notes formed by one chroma alteration, or do we have any notes which have been doubly adjusted? Or triply adjusted? etc.
It is also important to look at, for some MODMOS, how many generators the entire thing will span, which is called the generator span or coverage of the MODMOS. For instance, the diatonic scale requires 7 contiguous generators, whereas the melodic minor requires 9, the harmonic minor and major scales require 10, and the double harmonic scale requires 11. It can be quite useful to look at the "coverage" of a MODMOS on the generator chain, particularly if one want the MODMOS to fit into a single larger "chromatic" or "enharmonic" sized MOS.
There are doubtless many other useful ways in which one can analyze the MODMOS universe associated to an MOS. As a baseline definition, however, all of these scales are still MODMOS scales.
Gene's Terminology
The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave.
If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS; these have all steps the same as Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = |s-c|; these we may call enharmonic MODMOS. If we were to limit ourselves to note adjustments of one chroma, then in no case can the complexity be more than 2N.
Melisse Series
One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 < k < N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.
Examples
Consider the MOS series of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone "c", equal to 193.157 - 117.108 = 78.049 cents. This interval is the chroma for meantone[7], and the adjustment of any note up or down by this interval is represented by the sharp # or flat b accidentals.
The diatonic scale has steps LLsLLLs, which in the key of C can be written C D E F G A B C'. From the definition of a MODMOS, if we add sharps and flats to this, and do not get another diatonic scale, then we have a MODMOS. For example, if we flatten the third, we obtain C D Eb F G A B C', the melodic minor scale, or LsLLLLs. Since this scale contains three types of fourths (C-F, "perfect", Eb-A, "augmented", B-Eb, "diminished"), it is no longer an MOS and is therefore a MODMOS. If we apply a further alteration and flatten the sixth as well, we obtain the harmonic minor scale of C D Eb F G Ab B C', which now has three sizes of second and fourth and is therefore also a MODMOS. However, if we apply one more alteration and flatten the seventh, we're left with the natural minor scale of C D Eb F G Ab Bb C' - this is a mode of the diatonic scale, and hence is a MOS rather than a MODMOS.
If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in 50et, for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called "smithgw_modmos12a.scl" in the Scala Scale Archive. Another MODMOS of Meantone[12] in the archive is wreckpop, "smithgw_wreckpop.scl". This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)
Of course, MODMOS are not confined to scales of meantone. If we take the hobbit scale prodigy11 and tune it in a miracle tuning such as 72et, we obtain a MODMOS of Miracle[11]. In general, if we choose a rank three temperament with an optimal tuning very close to an optimal tuning for a rank two temperament and then tune a hobbit for it in that optimal rank two temperament tuning, we are very likely to construct an interesting MODMOS scale. It is particularly useful in connection with MODMOS of temperaments where the basic MOS doesn't contain a lot of consonant chords, such as Miracle[11].
Another quite interesting result of this is that if the rank-3 JI 5-limit major scale of (9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1) is tuned to any rank-2 5-limit temperament with a 7-note MOS, this scale will typically be a MODMOS of that MOS in that temperament. So for instance, this major scale will be a MODMOS of porcupine[7] (as Lssssss b4 #7), of tetracot[7] (as LLLLLLs #2 #3 b4 #7), etc.