User talk:Inthar/MV3: Difference between revisions

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3. Being generated from alternating generators

All instances of #2 are instances of #1 except for aabcb and abacaba, ~~and~~ all instances of #3 are instances of #2 except for 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.

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.

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 . Being generated from alternating generators
-All instances of #2 are instances of #1 except for aabcb and abacaba, and all instances of #3 are instances of #2 except for 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.
+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.