User:Inthar/MV3 - Revision history

Inthar: Redirected page to Rank-3 scale theorems#MV3 proofs

2021-06-23T21:48:38Z

Redirected page to Rank-3 scale theorems#MV3 proofs

Inthar: /* Definition: LQ */

2021-06-21T05:05:59Z

Definition: LQ

← Older revision		Revision as of 05:05, 21 June 2021
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	=== 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 the ~~coordinates increase monotonically~~), 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 ===

Inthar: /* Definition: LQ */

2021-06-21T05:05:06Z

Definition: LQ

← Older revision		Revision as of 05:05, 21 June 2021
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	=== 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, 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 the coordinates increase monotonically), 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 ===

Inthar: /* Definition: LQ */

2021-06-21T04:51:08Z

Definition: LQ

← Older revision		Revision as of 04:51, 21 June 2021
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	=== 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, results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.		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, 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 ===

Inthar: /* Definition: LQ */

2021-06-21T04:50:49Z

Definition: LQ

← Older revision		Revision as of 04:50, 21 June 2021
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	# ''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: QL ===		=== 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 ''~~quantized~~-~~line~~'' (QL) if S, when viewed as a set of instructions tracing a path in Z^k from the origin, results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.		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, results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.

	=== MV3 Theorem 1 ===		=== MV3 Theorem 1 ===

Inthar at 04:49, 21 June 2021

2021-06-21T04:49:47Z

← Older revision		Revision as of 04:49, 21 June 2021
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	# ''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: QL ===
			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 ''quantized-line'' (QL) if S, when viewed as a set of instructions tracing a path in Z^k from the origin, results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.

	=== MV3 Theorem 1 ===		=== MV3 Theorem 1 ===

Inthar at 03:21, 10 May 2021

2021-05-10T03:21:48Z

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	== 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).
	=== PMOS ===		=== Definition: PMOS ===
	~~Definition:~~ ''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.
	=== AG ===		=== Definition: AG ===
	~~Definition:~~ ''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''.

Inthar: /* MV3 Theorem 1 */

2021-05-09T14:34:19Z

MV3 Theorem 1

← Older revision		Revision as of 14:34, 9 May 2021
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	=== MV3 Theorem 1 ===		=== MV3 Theorem 1 ===
	''The following are equivalent for a 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.''
	# ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''		# ''S is PMOS, or S is of the form x'y'z'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.''		# ''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.''

	~~Below, assume that the scale word S is not multiperiod.~~

	==== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ====		==== Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" ====

Inthar: /* 3-DE implies MV3 */

2021-05-09T14:26:16Z

3-DE implies MV3

← Older revision		Revision as of 14:26, 9 May 2021
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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).''

Inthar: /* AG */

2021-05-09T14:25:39Z

← Older revision		Revision as of 14:25, 9 May 2021
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	=== AG ===		=== AG ===
	Definition: ''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties:		Definition: ''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'' 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''.
	~~# ''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.~~

	=== MV3 Theorem 1 ===		=== MV3 Theorem 1 ===