Recursive structure of MOS scales - Revision history

Inthar: /* Reflection of generators */

2026-03-13T18:54:36Z

Reflection of generators

← Older revision		Revision as of 18:54, 13 March 2026
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	=== Reflection of generators===		=== Reflection of generators===
			Todo: replace with a more rigorous determinant-based argument

	Suppose there is a generator after chunking. Assume also that there are more L's than s's, and that the imperfect generator is larger than the perfect generator. This means that this interval, built on the chunk boundaries after expanding, will still be "a generator" on the chunk boundaries: it will be the same size on all but one of the chunk boundaries. Take one of the "perfect" (smaller) intervals. We showed previously that, before expansion, the step to its right was an L, just like its first step; therefore we can scoot it over for the entirety of the first chunk and keep it perfect. We now come to another chunk. If a "perfect" interval can be built on it then we can repeat. If the interval is imperfect, then we know that the step to its right is an s, which will make it smaller and thus perfect, and we can keep doing this for the remainder of the chunk. Therefore, the interval is a generator after expansion as well, since it is only imperfect in one position.		Suppose there is a generator after chunking. Assume also that there are more L's than s's, and that the imperfect generator is larger than the perfect generator. This means that this interval, built on the chunk boundaries after expanding, will still be "a generator" on the chunk boundaries: it will be the same size on all but one of the chunk boundaries. Take one of the "perfect" (smaller) intervals. We showed previously that, before expansion, the step to its right was an L, just like its first step; therefore we can scoot it over for the entirety of the first chunk and keep it perfect. We now come to another chunk. If a "perfect" interval can be built on it then we can repeat. If the interval is imperfect, then we know that the step to its right is an s, which will make it smaller and thus perfect, and we can keep doing this for the remainder of the chunk. Therefore, the interval is a generator after expansion as well, since it is only imperfect in one position.

Ganaram inukshuk: Lead section (tetrachards not necessary for recursion)

2024-05-06T05:27:45Z

Lead section (tetrachards not necessary for recursion)

Show changes

Inthar at 22:31, 9 February 2024

2024-02-09T22:31:48Z

← Older revision		Revision as of 22:31, 9 February 2024
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	[[Category:MOS scale]]		[[Category:MOS scale]]
	[[Category:~~Articles~~ with proofs]]		[[Category:Pages with proofs]]

Inthar: /* See also */

2024-01-28T19:20:12Z

Inthar at 20:47, 26 December 2023

2023-12-26T20:47:09Z

← Older revision		Revision as of 20:47, 26 December 2023
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	== Finding the MOS pattern from xL ys ==		== Finding the MOS pattern from xL ys ==
	There is a linear-time algorithm for generating a MOS that works by traversing the closest approximation to ''y = b/a*x'', and the generator may be found from the integer point on this path that is closest to this line~~. This~~ is essentially the Bresenham line algorithm. The following presents the slower recursive algorithms as is topical for this article.		There is a linear-time algorithm for generating a MOS that works by traversing the closest approximation to ''y = b/a*x'', and the generator may be found from the integer point on this path that is closest to this line; this is essentially the Bresenham line algorithm. The following presents the slower recursive algorithms as is topical for this article.
	=== General algorithm ===		=== General algorithm ===
	This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.		This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.

Inthar at 20:45, 26 December 2023

2023-12-26T20:45:37Z

← Older revision		Revision as of 20:45, 26 December 2023
Line 186:		Line 186:

	== Finding the MOS pattern from xL ys ==		== Finding the MOS pattern from xL ys ==
	There is a linear-time algorithm for generating a MOS that works by traversing the closest approximation to ''y = b/a*x'', and the generator may be found from the integer point on this path that is closest to this line. The following presents the slower recursive algorithms as is topical for this article.		There is a linear-time algorithm for generating a MOS that works by traversing the closest approximation to ''y = b/a*x'', and the generator may be found from the integer point on this path that is closest to this line. This is essentially the Bresenham line algorithm. The following presents the slower recursive algorithms as is topical for this article.
	=== General algorithm ===		=== General algorithm ===
	This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.		This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.

Inthar: /* Finding the MOS pattern from xL ys */

2023-12-26T20:43:05Z

Finding the MOS pattern from xL ys

← Older revision		Revision as of 20:43, 26 December 2023
Line 186:		Line 186:

	== Finding the MOS pattern from xL ys ==		== Finding the MOS pattern from xL ys ==
			There is a linear-time algorithm for generating a MOS that works by traversing the closest approximation to ''y = b/a*x'', and the generator may be found from the integer point on this path that is closest to this line. The following presents the slower recursive algorithms as is topical for this article.
	=== General algorithm ===		=== General algorithm ===
	This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.		This is the general algorithm for generating a moment of symmetry scale of the form xL ys represented as a string of L's and s's in its brightest mode. This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.

Inthar at 17:08, 26 August 2023

2023-08-26T17:08:31Z

← Older revision		Revision as of 17:08, 26 August 2023
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	By looking at the "tetrachords" L…s of an MOS scale in word form and giving them the names "L" and "s", we get out another MOS scale. The MOS thus obtained preserves a number of important properties, such as which interval is the generator. To find properties of complex MOS word patterns, we can then just compare them to the simpler ones, whose properties we know.		By looking at the "tetrachords" L…s of an [[MOS scale]] in word form and giving them the names "L" and "s", we get out another MOS scale. The MOS thus obtained preserves a number of important properties, such as which interval is the generator. To find properties of complex MOS word patterns, we can then just compare them to the simpler ones, whose properties we know.

	== Recursive structure ==		== Recursive structure ==

Inthar: /* Preservation of the MOS property */

2023-08-26T17:03:03Z

Preservation of the MOS property

← Older revision		Revision as of 17:03, 26 August 2023
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	(Assume that the MOS has length ''n''; the notation ''w''(X, Y) for a word ''w''(L, s) in L, s means w with step sizes X substituted for L and Y substituted for s.)		(Assume that the MOS has length ''n''; the notation ''w''(X, Y) for a word ''w''(L, s) in L, s means w with step sizes X substituted for L and Y substituted for s.)

	Suppose w(L, s) had three chunks L...s with r < r+a < r+b 'L's. Then we have a length r+2 subword that's only 'L's, one that has one s at the end and one that has two 's's on either side, which means that the original scale was not MOS. Therefore the reduced word has two step sizes that differ by 1. Repeating this argument for sequences of ''k'' > 1 chunks shows that any ''k''-step in the reduced word must come in at most 2 sizes, showing that the reduced word is indeed MOS.		Suppose w(L, s) had three chunks L...s with r < r+i < r+j 'L's. Then we have a length r+2 subword that's only 'L's, one that has one s at the end and one that has two 's's on either side, which means that the original scale was not MOS. Therefore the reduced word has two step sizes that differ by 1. Repeating this argument for sequences of ''k'' > 1 chunks shows that any ''k''-step in the reduced word must come in at most 2 sizes, showing that the reduced word is indeed MOS.

	=== Preservation of generators ===		=== Preservation of generators ===

Inthar: /* Proofs */

2023-08-26T13:28:21Z

Proofs

← Older revision		Revision as of 13:28, 26 August 2023
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	== Proofs ==		== Proofs ==
			=== Definition of MOS ===
			A periodic scale is ''MOS'' if it has [[maximum variety]] 2. This in particular implies that MOS scales are binary.

	=== Preservation of the MOS property ===		=== Preservation of the MOS property ===

← Older revision		Revision as of 19:20, 28 January 2024
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	[[Category:MOS scale]]		[[Category:MOS scale]]
			[[Category:Articles with proofs]]