Maximum variety

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The maximum variety (MV) of a scale is a way of quantifying how many different "flavors" of intervals there are in it. Scales with high maximum variety have many different intervals of similar size occuring at different places in the scale. Scales with low maximum variety may be easier for composers and listeners to understand, because there is more uniformity and consistence between different parts of the scale. In a low max-variety scale, simply knowing how many scale steps an interval spans gives you a lot of information about the interval (by narrowing it down to a small set of choices).

An "interval class" is the set of all intervals of a scale which span the same number of steps. For example, in the diatonic scale, the 2-step class consists of the minor third and the major third, because those are the only intervals that are divided into 2 steps of the scale. The "variety" of an interval class is the number of different intervals in it. The maximum variety of a scale is simply the maximum variety of any interval class.

Any scale with all equal steps (such as an EDO) has maximum variety 1. All MOS and distributionally even scales have maximum variety 2 (in fact this can be taken as the definition of distributional evenness). An example of a scale with high max variety is the harmonic series, because the steps get gradually smaller as you go up the scale, and none of them are equal.

Max-variety-3 scales

Max-variety-3 scales are an attempt to generalize distributional evenness (closely related to the MOS property) to scales with three different step sizes rather than two (for example, those related to rank-3 regular temperaments). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.

When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between unconditionally MV3 scale patterns or abstractly MV3 ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and conditionally MV3 patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.

MV3 Structure Theorem (conjectured)

Consider a(n unconditionally, so independently of tuning) max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, with only one exception, at least two of the three steps must occur the same number of times. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.)

If in addition the scale has odd size and does not have the same number of every step size, there always exists some "generator" interval for any max-variety-3 scale (other than the one exception) such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the generator-offset property (GO). (Proof?)

Generating MV3 scales

Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).

To illustrate this process, let us use the simplest and most familiar rank-3 system: 5-limit JI, and let us use 3/2 as a generator. Because 3/2 is the generator, these max-variety-3 scales will be related (albeit not in a simple 1-to-1 way) with MOSes of Pythagorean, the rank-2 temperament with 3/2 as a generator.

Numbers of notes in Pythagorean MOSes follow the pattern 1+1=2, 2+1=3, 2+3=5, 5+2=7, 5+7=12... (related to the continued fraction for log2(3/2)). The numbers of notes in the 5-limit max-variety-3 scales we're constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are:

1+1+1=3, 2+1+1=4, 2+2+1=5, 2+2+3=7, 2+3+3=8, 5+2+2=9, 5+5+2=12, 5+5+7=17, 5+7+7=19...

If the number of notes is even, the max-variety-3 scale consists of two chains of 3/2 of equal length, each of which contains half the notes. If the number of notes is odd, the two 3/2-chains differ by 1 in length.

Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{1/1,10/9,3/2,5/3}, {1/1,5/4,3/2,5/3}, {1/1,5/4,3/2,15/8}, {1/1,45/32,3/2,125/64}... Only the ones where the two chains are lined up closely have good potential for harmony.

The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the same pattern as {1/1, 9/8, 3/2}, and in order for it to make an actual max-variety-3 scale, the other 2 notes must fall in between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.

1/1 9/8 40/27 3/2 160/81 ... max variety 3

1/1 10/9 9/8 40/27 3/2 ... fails, max variety 4

1/1 10/9 9/8 3/2 5/3 ... fails, max variety 4

1/1 9/8 5/4 3/2 5/3 ... max variety 3

1/1 9/8 5/4 3/2 15/8 ... max variety 3

1/1 9/8 45/32 3/2 15/8 ... max variety 3

1/1 135/128 9/8 45/32 3/2 ... fails, max variety 4

1/1 135/128 9/8 3/2 405/256 ... fails, max variety 4

1/1 9/8 1215/1024 3/2 405/256 ... max variety 3

Examples testing for MV

MV2

Positive

MV2: For each generic interval class, we find a maximum of 2 different sizes.

Consider the 5L 2s diatonic scale: LLsLLLs. For each generic interval class, we must confirm that there are only 2 specific intervals:

  1. L, s
  2. LL, Ls
  3. LLL, LLs

Actually we only have to check up to halfway, because all of the generic interval classes beyond this are period complements of the ones we already checked. And so it's confirmed: this is an MV2.

Negative

Not MV2: For the 3rd generic interval class, we find 3 different sizes.

How about a counterexample, LsLLLLs:

  1. L, s
  2. LL, Ls
  3. LLL, LLs, Lss — stop!

We've found that for the generic interval class 3, this scale has three different specific intervals, so it is not MV2.

MV3

Positive

MV3: For each generic interval class, we find a maximum of 3 sizes.

Consider the 2L 2M 3s scale with pattern LsMLsMs.

  1. L, M, s
  2. Ls, Ms, LM
  3. LMs, Mss, Lss

Great, this is MV3.

Negative

Not MV3: For the 2nd generic interval class, we find 4 different sizes.

Consider the 2L 2M 3s scale with pattern LssMLMs.

  1. L, M, s
  2. Ls, ss, Ms, LM — stop!

This scale has more than three different specific intervals for a generic interval class, so it is not MV3.

Conditional

Conditionally MV3: For the 2nd and 3rd generic interval classes, if the condition that MM=Ls is met, then this scale has a maximum of 3 different sizes; otherwise it has 4 and is therefore not MV3.

How about the 2L 3M 2s scale with pattern LMMsMLs.

  1. L, M, s
  2. LM, MM, Ms, Ls — stop! ...but wait. What if MM=Ls? Then actually this would still be only 3 specific intervals. So let's go with that, and continue.
  3. LMM=LLs, MMs, LMs

So this scale is Conditionally MV3 (MM=Ls).