MOS substitution - Revision history

Inthar: /* Formal definition */

2026-03-20T20:40:42Z

Formal definition

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	A ternary scale word ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale word if there exists a permutation <math>\pi \in S_3</math> such that the following holds:		A ternary scale word ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale word if there exists a permutation <math>\pi \in S_3</math> such that the following holds:
	* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and		* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and
	* deleting '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').		* deleting all instances of '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').

	== Original derivation ==		== Original derivation ==

Inthar: /* Formal definition */

2026-03-20T20:40:08Z

Formal definition

← Older revision		Revision as of 20:40, 20 March 2026
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	== Formal definition ==		== Formal definition ==
	A ternary ~~necklace~~ ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale if there exists a permutation <math>\pi \in S_3</math> such that the following holds:		A ternary scale word ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale word if there exists a permutation <math>\pi \in S_3</math> such that the following holds:
	* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and		* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and
	* deleting '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').		* deleting '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').

Inthar: /* Formal definition */ Replace with shorter and more functional definition

2026-03-20T20:39:45Z

Formal definition: Replace with shorter and more functional definition

← Older revision		Revision as of 20:39, 20 March 2026
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	== Formal definition ==		== Formal definition ==
	~~Suppose~~ ''a'', ''b''~~, and~~ ''~~c'' are positive integers. Suppose that the noncircular binary word ''t''(~~'''x''', '''X''') is ~~the lexicographically first mode of the~~ MOS ~~scale word with step signature~~ ''a'''''x'''(~~''b'' + ''c''~~)~~'''X''' under the ordering~~ '''x''' < ~~'''X''', and that the noncircular binary word ''f''~~(~~'''y''', '''z'''~~) ~~is one rotation of the MOS scale word with step signature ''b'''''y'''''c'''''z'''. Then the result of replacing the ''i''th occurrence of '''X'''~~ in ~~''t'' replaced with the ''i''th letter of ''f'' for 0 ≤ ''i'' < \|''f''\|, treated as a circular word, is said to be~~ a ''MOS ~~substitution scale word'' with~~ ''template MOS'' ~~''a''~~'''x'''(''~~b'' + ''c~~'')~~'''X''' and ''filling MOS'' ''b'''''y'''''c'''''z''', or~~ in ~~shorthand,~~ a ~~"subst ''a'''''x'''~~(''~~b'''''y'''''c'''''z'~~'')".		A ternary necklace ''w''('''x'''<sub>1</sub>, '''x'''<sub>2</sub>, '''x'''<sub>3</sub>) is a ''MOS substitution'' scale if there exists a permutation <math>\pi \in S_3</math> such that the following holds:
			* identifying '''x'''<sub>π(1)</sub> and '''x'''<sub>π(2)</sub> results in a MOS (called the ''template MOS'') and
			* deleting '''x'''<sub>π(3)</sub> (called the ''slot letter'') results in a MOS (called the ''filling MOS'').

	== Original derivation ==		== Original derivation ==

Inthar at 03:21, 15 March 2026

2026-03-15T03:21:26Z

← Older revision		Revision as of 03:21, 15 March 2026
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	'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the filling MOS), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the template MOS). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.		'''MOS substitution''' is a procedure for obtaining a ternary (3 step sizes) scale from two [[MOS]] patterns. It consists of substituting the step pattern of one MOS pattern (called the ''filling MOS''), scale step for scale step, for all occurrences of a chosen step size of another MOS pattern (called the ''template MOS''). Unlike MV3 scales, a MOS substitution scale may have any combination of step sizes.

	[[Aberrismic theory]] uses MOS substitution. In fact, groundfault reports having come up with a similar concept but not following up on it.		[[Aberrismic theory]] uses MOS substitution. In fact, groundfault reports having come up with a similar concept but not following up on it.

Inthar: /* Formal definition */

2026-03-14T19:43:28Z

Formal definition

← Older revision		Revision as of 19:43, 14 March 2026
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	== Formal definition ==		== Formal definition ==
	Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose that the noncircular binary word ''t''('''x''', '''X''') is the ~~brightest~~ mode of the MOS scale word with step signature ''a'''''x'''(''b'' + ''c'')'''X''' ~~when treating~~ '''x''' ~~as larger than~~ '''X''', and that the noncircular binary word ''f''('''y''', '''z''') is one rotation of the MOS scale word with step signature ''b'''''y'''''c'''''z'''. Then the result of replacing the ''i''th occurrence of '''X''' in ''t'' replaced with the ''i''th letter of ''f'' for 0 ≤ ''i'' < \|''f''\|, treated as a circular word, is said to be a ''MOS substitution scale word'' with ''template MOS'' ''a'''''x'''(''b'' + ''c'')'''X''' and ''filling MOS'' ''b'''''y'''''c'''''z''', or in shorthand, a "subst ''a'''''x'''(''b'''''y'''''c'''''z''')".		Suppose ''a'', ''b'', and ''c'' are positive integers. Suppose that the noncircular binary word ''t''('''x''', '''X''') is the lexicographically first mode of the MOS scale word with step signature ''a'''''x'''(''b'' + ''c'')'''X''' under the ordering '''x''' < '''X''', and that the noncircular binary word ''f''('''y''', '''z''') is one rotation of the MOS scale word with step signature ''b'''''y'''''c'''''z'''. Then the result of replacing the ''i''th occurrence of '''X''' in ''t'' replaced with the ''i''th letter of ''f'' for 0 ≤ ''i'' < \|''f''\|, treated as a circular word, is said to be a ''MOS substitution scale word'' with ''template MOS'' ''a'''''x'''(''b'' + ''c'')'''X''' and ''filling MOS'' ''b'''''y'''''c'''''z''', or in shorthand, a "subst ''a'''''x'''(''b'''''y'''''c'''''z''')".

	== Original derivation ==		== Original derivation ==

Inthar: /* Ternary parallelogram scales are MOS substitution scales */

2026-03-14T04:27:37Z

Ternary parallelogram scales are MOS substitution scales

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	=== Ternary parallelogram scales are MOS substitution scales ===		=== Ternary parallelogram scales are MOS substitution scales ===
	:''Main article: [[~~Parallelogram substring scale~~]]''		:''Main article: [[Ternary parallelogram scales are MOS substitution]]''

	== MOS substitution scales and RTT ==		== MOS substitution scales and RTT ==

Inthar: /* If the template is a primitive MOS, and for some perfect generators p_T, p_F, \ r := \left|p_T\right|_\mathbf{X} = \left|p_F\right|, then MOS substitution yields a parallelogram substring in the lattice */

2026-03-14T04:21:00Z

If the template is a primitive MOS, and for some perfect generators p_T, p_F, \ r := \left|p_T\right|_\mathbf{X} = \left|p_F\right|, then MOS substitution yields a parallelogram substring in the lattice

← Older revision		Revision as of 04:21, 14 March 2026
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	* the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' − 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.		* the imperfect generator of the filling MOS corresponds to looping back to ''C''<sub>1</sub> but on the next note of ''C''<sub>1</sub>, so it and the ''q'' − 1 notes thereafter are advanced by 1 note from any predecessor notes in the chains.

	Hence these particular MOS substitution scales satisfy a property that we call ''parallelogram substring''. An '''e'''-equivalent scale is a ''parallelogram substring'' if there exist integers ''m'' > 0, ''n'' > 0, 0 ≤ ''a'' < ''n'', 0 ≤ ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is		Hence these particular MOS substitution scales satisfy a property that we call ''[[parallelogram substring scale\|parallelogram substring]]''. An '''e'''-equivalent scale is a ''parallelogram substring'' if there exist integers ''m'' > 0, ''n'' > 0, 0 ≤ ''a'' < ''n'', 0 ≤ ''b'' < ''n'', a vector '''a''', and two linearly independent vectors '''v''' and '''w''' such that the set of notes in the scale as a subset of the lattice of '''e'''-equivalent pitches is

	<math>		<math>

Inthar: /* MOS substitution scales have block balance at most 2 */

2026-03-14T04:19:09Z

MOS substitution scales have block balance at most 2

← Older revision		Revision as of 04:19, 14 March 2026
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	Case 2: Neither of ''k'' and {{nowrap\|''k'' + 1}} equals ({{nowrap\|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap\|''j'' + 1}} times in a window of size ''k'', then '''Y''' occurs {{nowrap\|''j'' + 1}} or {{nowrap\|''j'' + 2}} times in a window of size {{nowrap\|''k'' + 2}}.		Case 2: Neither of ''k'' and {{nowrap\|''k'' + 1}} equals ({{nowrap\|''b'' + ''c''}}). Here, if '''Y''' occurs ''j'' or {{nowrap\|''j'' + 1}} times in a window of size ''k'', then '''Y''' occurs {{nowrap\|''j'' + 1}} or {{nowrap\|''j'' + 2}} times in a window of size {{nowrap\|''k'' + 2}}.

			=== Ternary parallelogram scales are MOS substitution scales ===
			:''Main article: [[Parallelogram substring scale]]''

	== MOS substitution scales and RTT ==		== MOS substitution scales and RTT ==

Inthar: /* If a ternary scale satisfies all three possible MOS-substitution types, then it is pairwise-MOS and deletion-MOS */

2026-01-05T20:54:29Z

If a ternary scale satisfies all three possible MOS-substitution types, then it is pairwise-MOS and deletion-MOS

← Older revision		Revision as of 20:54, 5 January 2026
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	This fact is immediate. (See [[pairwise-MOS]] and [[deletion-MOS]].)		This fact is immediate. (See [[pairwise-MOS]] and [[deletion-MOS]].)

	Corollary (by [[Ternary scale theorems]]): Such a scale is Fraenkel, odd-regular, or even-regular.		Corollary (by [[Ternary scale theorems]]): Such a scale is [[Fraenkel word\|Fraenkel]], [[odd-regular]], or [[even-regular]].

	=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===		=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===

Inthar: /* If a ternary scale satisfies all three possible MOS-substitution types, then it is pairwise-MOS and deletion-MOS */

2026-01-05T20:51:17Z

If a ternary scale satisfies all three possible MOS-substitution types, then it is pairwise-MOS and deletion-MOS

← Older revision		Revision as of 20:51, 5 January 2026
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	This fact is immediate. (See [[pairwise-MOS]] and [[deletion-MOS]].)		This fact is immediate. (See [[pairwise-MOS]] and [[deletion-MOS]].)

	Corollary (by [[Ternary scale theorems]]]: Such a scale is Fraenkel, odd-regular, or even-regular.		Corollary (by [[Ternary scale theorems]]): Such a scale is Fraenkel, odd-regular, or even-regular.

	=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===		=== If the template MOS is primitive, MOS substitution yields binary well-formed generator sequences ===