User talk:Inthar/MV3

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Counterexample - XYZZY

There is a flaw in this argument somewhere because I have a specific counterexample.

Let's consider the MV3 scale XYZZY (verifying that this is indeed MV3 is a simple exercise).

If we discard the X, the result is YZZY which is not a MOS - so this is a counterexample to what you're trying to show. I believe (along with its repetitions such as XYZZYXYZZY) it's the only counterexample.

The contiguous run of Y'z and Z's, YZZY, does not have the interval 2Y in it, but if you consider the whole scale with the X removed, 2Y does appear. So the logical flaw is somewhere in the "scooting" part. —Keenan Pepper (talk) 18:34, 22 April 2021 (UTC)

Did we look at scales where you just eliminate one of the steps? I do remember looking at scales where you *equate* pairs of steps, which I think was what the Zabka paper was about. Mike Battaglia (talk) 18:49, 22 April 2021 (UTC)

Prior Results

Some prior results that are likely of interest to you:

It is somewhat simpler, at first, to look at only those scales with *exactly* three sizes, rather than "at most" three sizes.

In general there are three criteria which are almost all the same: 1. Having all intervals be exactly three sizes 2. Being a "pairwise well formed scale" (which tempers to "strict MOS" in 3 different ways) 3. Being generated from alternating generators

All instances of #1 are instances of #2 except for aabcb, and all instances of #2 are instances of #3 except for abacaba, thus all instances of #1 are instances of #3 except for aabcb and abacaba. Keenan Pepper talked extensively about the first theorem on the tuning list (and I think the Zabka paper shows this); Jon Wild claimed on the tuning list to have proven the second theorem but I am not sure how the proof goes.

So we may as well look at "3-MOS" or "alternating-generator" scales. The prototypical example of such a scale is the rank-3 JI major scale of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1, or LmsLmLs. You get this from stacking alternating generators of 5/4 and 6/5. It is instructive to look at what the next "3-MOS" in the sequence is after that, as well as what you get for different choices of third. You can think of this as "stacking triads" but alternating generators is nicer, since sometimes you get one left over at the end.

Similarly, for any such set of two generators, there is a "3-MOS sequence" of the number of notes for which the scale has exactly 3 step sizes. However, unlike regular MOS's, there seems to be a "maximum" 3-MOS. I am not sure how to prove this, but it is very easy to see visually:

For this, I calculated all scales formed by iterating the generators N times where N is from 0..100, and I plotted the size of the largest "3-MOS" scale that you get within that range. Yellow is a larger value and blue is smaller. So, where the plot is blue, there were no 3-MOS's after a relatively small value. I am not sure how to prove this but you can also extend N to very large values and get the same thing; as you get to the diagonal line where the two generators are equal the max 3-MOS seems to increase until it is infinite on the diagonal line, and as you get further from it the max 3-MOS seems to be gotten pretty early on.

To extend to arbitrary max-variety-3 scales, you of course can just repeat the scales multiple times (like abcabacabcabacabcabac...), but there are also those scales who have some intervals in 2 sizes. A useful scale to look at is abacbabc. One useful view is to look at "pairwise-DE" or PDE scales, which temper to DE in three ways (if any pair of steps are equated); the last scale is PDE. I am sure there are probably some additional "sporadic" examples of max-variety-3 scales that aren't PDE, similar to aabcb and abacaba (of course you can just repeat these to get aabcbaabcbaabcb... but also there are probably others).

I remember some results involving iterating generators within "approximately equal subperiods" (like 10/7 and 7/5 rather than an exact 1/2-octave), or perhaps iterating generators within an MOS (like having the "period" be LLsLs), but I can't seem to locate them...

Some other random points

  • Another useful view is that if you take the sequence of "stacked thirds" or "stacked fourths" for any MOS you get another MOS. So for LLsLLLs let's say C D E F G A B C, the thirds are C E G B D F A C, which is MmMmmMm (starting at the tonic and going up by third), which is another MOS. You can do the same with pairwise well-formed scales, and for each transformed PWF or PDE scale that you get this way, it must also temper to a strict MOS or DE scale. PDE scales can temper to "multi-MOS" that aren't strict, e.g. they have some interval which comes in only 1 size.
  • There is a subtlety in the notation here. The harmonic minor scale, which I'll write LsLLsAs (where A is augmented second), is a max-variety-3 scale, but the generic word abaabcb is *not* max-variety-3. The thing is, "almost all" tunings of abaabcb are not max-variety-3, but this one is because you have the "LL" third tempered equal to the "sA" third, which is not reflected in the notation of the steps. The problem is that the step notation is not adequate to reflect all the properties of the scale that are relevant to max-variety-3, because there can be "hidden temperings" in the thirds or larger intervals which aren't in that notation. If you do the above "MOS transformation" or use Rothenberg's rank-order matrices you will see these things. I suggest, though, looking at properties that hold for "generic" scale words like abaabcb; even though there are some special tunings for which that is max-variety-3, this isn't true for that word in general.
  • MODMOS's in general are also often max-variety-3, like the melodic minor scale. I am not sure if the melodic minor scale is always max-variety-3 (unlike the harmonic minor scale), but clearly it doesn't seem to fit nicely into the above picture of alternating generators. I would suggest looking at the alternating generator picture first and then seeing how things like this fit into it.
  • If you take a rank-2 generator, like the meantone fifth, and you look at some generated scale that is *NOT* an MOS - like "meantone[6]", I guess - it will be a max-variety-3 scale. Often these seem to be "rank-3 tempered versions" of generic rank-3 max-variety scales. For instance, with meantone[6], you get LLsLLm for C D E F G A C. In this case we have that "Ls" = m, but the generic word aabaac is also max-variety-3 anyway.
  • If you do look at "3-MOS's" with alternating generators, you will see that each scale word comes in two pairs: one that you get from stacking generator "a" before "b," and the one that you get by stacking generator "b" before "a". For instance, if you stack 6/5 before 5/4, you get the rank-3 JI minor scale of 1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1, or LsmLsLm. This is not a mode or rotation of the LmsLmLs scale that you get by stacking 5/4 before 6/5 - it is almost a rotation, but one of the notes is 81/80 off. But, in some generalized sense, all of these scales are "modes" of the abstract system that you get from stacking 5/4's and 6/5's. We were calling these pairs "domes" of one another, so that there are 2 domes and 7 modes of each dome for 14 total modes you can get from this generator system in general.