MOS substitution - Revision history

Inthar: /* Aberrismic scales */

2026-04-13T13:20:07Z

Aberrismic scales

← Older revision		Revision as of 13:20, 13 April 2026
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	\|}		\|}

	For more examples, see [[Aberrisma#~~Aberrismic~~ scales]].		For more examples, see [[Aberrisma#List of aberrismic scales]].

	=== Some classes of MOS substitutions ===		=== Some classes of MOS substitutions ===

Inthar: /* Aberrismic scales */

2026-04-13T13:19:47Z

Aberrismic scales

← Older revision		Revision as of 13:19, 13 April 2026
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	\| colspan="2" \| GS(L+s, L+s, L+m)		\| colspan="2" \| GS(L+s, L+s, L+m)
	\|}		\|}

			For more examples, see [[Aberrisma#Aberrismic scales]].

	=== Some classes of MOS substitutions ===		=== Some classes of MOS substitutions ===

Inthar: /* Parallelogram substring scales */

2026-03-14T04:09:45Z

Parallelogram substring scales

← Older revision		Revision as of 04:09, 14 March 2026
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	# the first row is not required to be full but must be a suffix of a full row		# the first row is not required to be full but must be a suffix of a full row
	# the last row is not required to be full but must be a prefix of a full row.		# the last row is not required to be full but must be a prefix of a full row.
	We call such scales "parallelogram substring scales". Not all ternary parallelogram substring scales are MOS substitution scales.		We call such scales "parallelogram substring scales". Not all ternary parallelogram substring scales are MOS substitution scales. However, fully parallelogram (PS scales with full first and last rows) ternary scales are always MOS substitution scales.

	{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated~~. Fully parallelogram (PS scales with full first and last rows) ternary scales are always MOS substitution scales~~.}}		{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}
	{{cat\|		{{cat\|
	Ternary scales		Ternary scales
	Scale construction		Scale construction
	}}		}}

Inthar: /* Parallelogram substring scales */

2026-03-14T04:07:33Z

Parallelogram substring scales

← Older revision		Revision as of 04:07, 14 March 2026
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	We call such scales "parallelogram substring scales". Not all ternary parallelogram substring scales are MOS substitution scales.		We call such scales "parallelogram substring scales". Not all ternary parallelogram substring scales are MOS substitution scales.

	{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}		{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated. Fully parallelogram (PS scales with full first and last rows) ternary scales are always MOS substitution scales.}}

	~~{{adv\|It is conjectured that fully~~ parallelogram (PS scales with full first and last rows) ternary scales are ~~all~~ MOS substitution scales.}}
	{{cat\|		{{cat\|
	Ternary scales		Ternary scales
	Scale construction		Scale construction
	}}		}}

Inthar: /* Some classes of MOS substitutions */

2026-01-28T10:05:08Z

Some classes of MOS substitutions

← Older revision		Revision as of 10:05, 28 January 2026
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	** {{adv\|''Odd-regular'' MV3 scales are subst ax(ky kz){{angbr\|something}} scales where a is odd. Such scales come in a chiral pair.}}		** {{adv\|''Odd-regular'' MV3 scales are subst ax(ky kz){{angbr\|something}} scales where a is odd. Such scales come in a chiral pair.}}
	*** {{adv\|Example: Diasem (5L(2m2s)) has chiral variants LmLsLmLsL and LsLmLsLmL.}}		*** {{adv\|Example: Diasem (5L(2m2s)) has chiral variants LmLsLmLsL and LsLmLsLmL.}}
	** {{adv\|''Even-regular'' MV3 scales (such as 5sC diachrome) are subst ax(ky kz){{angbr\|something}} scales where a is even and gcd(a, k) {{=}} 1. In this case, "ax(ky kz)" specifies the scale uniquely.}}		** {{adv\|''Even-regular'' MV3 scales (such as 5sC diachrome) are subst ax(ky kz){{angbr\|something}} scales where a is even and gcd(a, k) {{=}} 1. These scales are achiral. In this case, "ax(ky kz)" specifies the scale uniquely.}}
	*** {{adv\|Example: 5sC diachrome is a subst 2m(5L5s): LsLsLmsLsLsm.}}		*** {{adv\|Example: 5sC diachrome is a subst 2m(5L5s): LsLsLmsLsLsm.}}
	* {{adv\|A ternary scale is always a MOS substitution scale if the L {{=}} m and s {{=}} 0 [[monotone-MOS property\|monotone-MOS]] properties are satisfied.}}		* {{adv\|A ternary scale is always a MOS substitution scale if the L {{=}} m and s {{=}} 0 [[monotone-MOS property\|monotone-MOS]] properties are satisfied.}}

Inthar: /* Some kinds of MOS substitutions */

2026-01-28T10:03:56Z

Some kinds of MOS substitutions

← Older revision		Revision as of 10:03, 28 January 2026
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	\|}		\|}

	=== Some ~~kinds~~ of MOS substitutions ===		=== Some classes of MOS substitutions ===
	* Ternary [[Interleaving\|interleaved]]-MOS scales such as blackdye (5L2m3s, sLmLsLmLsL) and whitedye (5L2m7s, sLsLsLsmsLsLsm) are MOS substitution scales. In those particular examples, the template MOS is of the form ky kz.		* Ternary [[Interleaving\|interleaved]]-MOS scales such as blackdye (5L2m3s, sLmLsLmLsL) and whitedye (5L2m7s, sLsLsLsmsLsLsm) are MOS substitution scales. In those particular examples, the template MOS is of the form ky kz.
	* {{adv\|MOS substitution scales where the ''filling'' MOS is of the form ky kz are of the following two types:}}		* {{adv\|MOS substitution scales where the ''filling'' MOS is of the form ky kz are of the following two types:}}

Inthar: /* More general classes of MOS substitutions */

2026-01-28T10:03:35Z

More general classes of MOS substitutions

← Older revision		Revision as of 10:03, 28 January 2026
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	\|}		\|}

	=== ~~More general classes~~ of MOS substitutions ===		=== Some kinds of MOS substitutions ===
	* Ternary [[Interleaving\|interleaved]]-MOS scales such as blackdye (5L2m3s, sLmLsLmLsL) and whitedye (5L2m7s, sLsLsLsmsLsLsm) are MOS substitution scales. In those particular examples, the template MOS is of the form ky kz.		* Ternary [[Interleaving\|interleaved]]-MOS scales such as blackdye (5L2m3s, sLmLsLmLsL) and whitedye (5L2m7s, sLsLsLsmsLsLsm) are MOS substitution scales. In those particular examples, the template MOS is of the form ky kz.
	* {{adv\|MOS substitution scales where the ''filling'' MOS is of the form ky kz are of the following two types:}}		* {{adv\|MOS substitution scales where the ''filling'' MOS is of the form ky kz are of the following two types:}}

Inthar: /* Parallelogram substring scales */

2026-01-16T14:36:17Z

Parallelogram substring scales

← Older revision		Revision as of 14:36, 16 January 2026
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	=== Parallelogram substring scales ===		=== Parallelogram substring scales ===
	MOS substitution scales in general still have somewhat messy pitch-class lattices (or they may not have one at all, due to the equave being equally divided). However, some have nicer lattices in the sense that they are [[parallelogram substring scale\|substrings of a row-by-row ~~{{adv\|(lexicographical)}}~~ parallelogram traversal]]. That is, the pitch classes form a parallelogram {{adv\|(a rectangle under some basis)}}		MOS substitution scales in general still have somewhat messy pitch-class lattices (or they may not have one at all, due to the equave being equally divided). However, some have nicer lattices in the sense that they are [[parallelogram substring scale\|substrings of a row-by-row parallelogram traversal]]. That is, the pitch classes form a parallelogram {{adv\|(a rectangle under some basis)}}
	# with all rows traversed in the same direction		# with all rows traversed in the same direction
	# with some number (possibly zero) of full middle rows		# with some number (possibly zero) of full middle rows

Inthar: /* Properties */

2026-01-16T00:26:17Z

Properties

← Older revision		Revision as of 00:26, 16 January 2026
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	{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}		{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}

	{{adv\|It is conjectured that fully parallelogram (PS scales with full first and last rows) are all MOS substitution scales.}}		{{adv\|It is conjectured that fully parallelogram (PS scales with full first and last rows) ternary scales are all MOS substitution scales.}}
	{{cat\|		{{cat\|
	Ternary scales		Ternary scales
	Scale construction		Scale construction
	}}		}}

Inthar: /* Parallelogram substring scales */

2026-01-16T00:24:21Z

Parallelogram substring scales

← Older revision		Revision as of 00:24, 16 January 2026
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	{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}		{{adv\|If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.}}

			{{adv\|It is conjectured that fully parallelogram (PS scales with full first and last rows) are all MOS substitution scales.}}
	{{cat\|		{{cat\|
	Ternary scales		Ternary scales
	Scale construction		Scale construction
	}}		}}

← Older revision		Revision as of 10:05, 28 January 2026
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	** {{adv\|''Odd-regular'' MV3 scales are subst ax(ky kz){{angbr\|something}} scales where a is odd. Such scales come in a chiral pair.}}		** {{adv\|''Odd-regular'' MV3 scales are subst ax(ky kz){{angbr\|something}} scales where a is odd. Such scales come in a chiral pair.}}
	*** {{adv\|Example: Diasem (5L(2m2s)) has chiral variants LmLsLmLsL and LsLmLsLmL.}}		*** {{adv\|Example: Diasem (5L(2m2s)) has chiral variants LmLsLmLsL and LsLmLsLmL.}}
	** {{adv\|''Even-regular'' MV3 scales (such as 5sC diachrome) are subst ax(ky kz){{angbr\|something}} scales where a is even and gcd(a, k) {{=}} 1. In this case, "ax(ky kz)" specifies the scale uniquely.}}		** {{adv\|''Even-regular'' MV3 scales (such as 5sC diachrome) are subst ax(ky kz){{angbr\|something}} scales where a is even and gcd(a, k) {{=}} 1. These scales are achiral. In this case, "ax(ky kz)" specifies the scale uniquely.}}
	*** {{adv\|Example: 5sC diachrome is a subst 2m(5L5s): LsLsLmsLsLsm.}}		*** {{adv\|Example: 5sC diachrome is a subst 2m(5L5s): LsLsLmsLsLsm.}}
	* {{adv\|A ternary scale is always a MOS substitution scale if the L {{=}} m and s {{=}} 0 [[monotone-MOS property\|monotone-MOS]] properties are satisfied.}}		* {{adv\|A ternary scale is always a MOS substitution scale if the L {{=}} m and s {{=}} 0 [[monotone-MOS property\|monotone-MOS]] properties are satisfied.}}