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Counterexample to some Proofs?
Great page! One thing I'm curious about is that there are some scales which have MV3, but for which the "word" representing that pattern of step sizes isn't, in general, MV3. A good example is the meantone harmonic minor scale, or LsLLsAs, where A = L+L-s. This scale is MV3, but in general the pattern of step sizes "abaabcb" isn't MV3 for almost all tunings; it only happens to become MV3 in the situation where c = a+a-b.
Basically, we can look at the space of all tunings for abaabcb. Once two of the steps are determined, the remaining is also determined (if we want the result to sum to an octave), so this is a 2D space. Almost all tunings of this scale aren't MV3, but then there is a 1D subspace (or a line) for which things magically are MV3.
I think that most of the theorems on this page hold only for those scale words which are MV3 for "almost all" tunings of the scale word. For instance, I don't think there are two alternating generators which give the meantone harmonic minor scale. Maybe we should say that scale words which are always (or "almost always") MV3 are General MV3 scales, and then scale words which are generally **not** MV3 but only sometimes (like meantone harmonic minor) are Sporadic MV3 scales, or something like that.
There is another important set of MV3 scales you can get by taking just one generator and iterating it to a number of scale steps that is NOT an MOS (like meantone), but from initial inspection I do think that these tends of scales actually will be General MV3 scales.
It would be a good idea to clarify which of these theorems hold for only General MV3 scale words and which hold for every possible MV3 scale. FWIW I think this problem actually goes way way back to Keenan Pepper's original version of this page (which originated from some tuning-math and IRC discussions at least 10 years ago), so I'll ask him about it. Also kudos to Inthar for a lot of great work on this. Mike Battaglia (talk) 00:24, 24 May 2022 (UTC)
- This may be of interest to anyone interested in MV3 who hasn't seen it yet, basically I bruteforce found and listed all the MV3 up to 14 tones. May come in handy, let me know if anyone finds any mistakes. --Xenoindex (talk) 08:31, 24 May 2022 (UTC)
- Good stuff! It would be good to see if every scale is generated in this way from two alternating generators. I think if memory serves they all are except for scales of the form "aabcb" and "abacaba". Some of these aren't "pairwise well-formed" but only "pairwise DE" scales and I was quite surprised to see that even these apparently tend to be generated from two alternating generators, just in different interval classes. Mike Battaglia (talk) 00:16, 28 May 2022 (UTC)
Relationship Between Pairwise-Well-Formed scales and MV3
I wrote this elsewhere at some point, but it would also be good to clarify the relationship between
- Pairwise-Well-Formed scales - those which temper to (strict) MOS in three different ways
- Pairwise DE scales - those which temper to DE in three different ways
- "Triple Fokker PBs" or "Wakalixes" - which are a Fokker PB in three different ways (I think the current definition of "wakalix," which is from Gene, is even more general)
- Scales which have variety exactly equal to 3 for all intervals
- Scales which have variety *at most* 3 for all intervals, but for which some can be 1 or 2
- MV3 scales which also have this generator offset property
If memory serves, these are basically all the same thing, modulo the one main quirk regarding if we want *exactly 3* or *at most 3* interval sizes (and similarly with strict MOS / DE in the product words), and with some extra quirkiness with the two scales aabcb and abacaba which are exceptions otherwise. The Zabka paper had some good stuff about all of this. This is also regarding scale words which are "almost always" MV3 (see note about "Sporadic MV3" scales above.) Mike Battaglia (talk) 00:24, 24 May 2022 (UTC)
Connections to 5L 2s MODMOSes
Hey, Moremajorthanmajor, I think we need to go into the technical specifications as to how 1L 3m 3s with the pattern 4 3 2 3 2 3 2 relates to LsLLsAs and LLsLsAs. It looks like the latter two might also be Metathetic variants of the former, and thus, we need to take stock of the various properties of MV3 scales. --Aura (talk) 19:54, 12 May 2021 (UTC)