MODMOS Scales

Introduction

A scale is considered to be an MOS if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. MODMOS 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 (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.

Numerous options exist for the choice of chromatic alteration, all of which can be obtained by combining and subtracting intervals from within the MOS. The most common is alteration by chroma, where the chroma is the difference between any pair of intervals sharing the same interval class.

This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished by the sense that they are epimorphic, and hence of special musical interest. However, alterations by other intervals may also be useful.

Definitions

A 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. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.

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. 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. In no case can the complexity be more than 2N if we limit ourselves to note adjustments of one chroma.

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].