Rank-3 scale theorems: Difference between revisions
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== MV3 proofs == | == MV3 proofs == | ||
Under construction | Under construction | ||
== Definitions and theorems == | === Definitions and theorems === | ||
Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used). | Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used). | ||
=== Definition: PMOS === | ==== Definition: PMOS ==== | ||
''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS. | ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS. | ||
=== Definition: AG === | ==== Definition: AG ==== | ||
''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties: | ''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties: | ||
# ''S'' can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | # ''S'' can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | ||
# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''. | # ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''. | ||
=== Definition: LQ === | ==== Definition: LQ ==== | ||
A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if S, when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k. (Closest approx in what sense?) | A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if S, when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k. (Closest approx in what sense?) | ||
=== MV3 Theorem 1 === | ==== MV3 Theorem 1 ==== | ||
''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:'' | ''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:'' | ||
# ''S is MV3.'' | # ''S is MV3.'' | ||
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# ''S is AG, or S is of the form x'y'z'y'x' or its repetitions, or x'y'x'z'x'y'x' or its repetitions.'' | # ''S is AG, or S is of the form x'y'z'y'x' or its repetitions, or x'y'x'z'x'y'x' or its repetitions.'' | ||
==== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ==== | ====== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ====== | ||
TODO: account for case xyzyx. | TODO: account for case xyzyx. | ||
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''To be continued...'' | ''To be continued...'' | ||
==== PMOS implies AG (except in the case xyxzxyx) ==== | ====== PMOS implies AG (except in the case xyxzxyx) ====== | ||
We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG. | We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG. | ||
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A gen chain g1...g1g1' [assuming this word is not multiperiod] detempers to G1i(1)G1i(2)...G1i(n-1)G13, where i(t) is in {1,2} and n = len(S). This word must be MV3, since otherwise the original scale wouldn't be MV3. Similarly, g2...g2g2' detempers to G2j(1)...G2j(n-1)G23, and g3...g3g3' detempers to G3k(1)...G3k(n-1)G33.--> | A gen chain g1...g1g1' [assuming this word is not multiperiod] detempers to G1i(1)G1i(2)...G1i(n-1)G13, where i(t) is in {1,2} and n = len(S). This word must be MV3, since otherwise the original scale wouldn't be MV3. Similarly, g2...g2g2' detempers to G2j(1)...G2j(n-1)G23, and g3...g3g3' detempers to G3k(1)...G3k(n-1)G33.--> | ||
==== AG implies "ax by bz" ==== | ====== AG implies "ax by bz" ====== | ||
'''Assuming the alternating generator property''', we have two chains of generator g0 (going right). The two cases are: | '''Assuming the alternating generator property''', we have two chains of generator g0 (going right). The two cases are: | ||
O-O-...-O (m notes) | O-O-...-O (m notes) | ||
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if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED. | if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED. | ||
=== 3-DE implies MV3 === | ==== 3-DE implies MV3 ==== | ||
We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3. | We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3. | ||
=== MV3 Theorem 2 === | ==== MV3 Theorem 2 ==== | ||
''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).'' | ''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).'' | ||