User:Inthar/List of ternary parallelogram scales - Revision history

Inthar at 04:14, 14 March 2026

2026-03-14T04:14:03Z

← Older revision		Revision as of 04:14, 14 March 2026
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	Parallelogram-shaped (parallelogram substring with full first and last rows) ~~[[MOS substitution\|MOS-substitution]]~~ scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.		Parallelogram-shaped (parallelogram substring with full first and last rows) scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.

	~~It is conjectured that fully~~ parallelogram scales are always MOS-substitution. ~~This has been checked for sizes ≤ 15.~~		Ternary parallelogram scales are always MOS substitution. (Proof forthcoming)
	== 6 notes ==		== 6 notes ==
	* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]		* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]

Inthar: Inthar moved page User:Inthar/List of parallelogram MOS-substitution scales to User:Inthar/List of ternary parallelogram scales

2026-03-14T04:13:19Z

Inthar moved page User:Inthar/List of parallelogram MOS-substitution scales to User:Inthar/List of ternary parallelogram scales

← Older revision	Revision as of 04:13, 14 March 2026
(No difference)

Inthar at 00:47, 16 January 2026

2026-01-16T00:47:50Z

← Older revision		Revision as of 00:47, 16 January 2026
Line 1:		Line 1:
	Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.		Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.

	It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes ~~<= 14~~.		It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes ≤ 15.
	== 6 notes ==		== 6 notes ==
	* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]		* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]

Inthar at 19:38, 2 January 2026

2026-01-02T19:38:38Z

← Older revision		Revision as of 19:38, 2 January 2026
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	Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which [a, b, c~~] is a partition of the scale size~~ are shown.		Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.

	It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 14.		It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 14.

Inthar at 18:51, 2 January 2026

2026-01-02T18:51:52Z

← Older revision		Revision as of 18:51, 2 January 2026
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	It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 14.		It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 14.
	== 6 ~~note scales~~ ==		== 6 notes ==
	* ~~xxyxxz~~ row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]		* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
	== 8 ~~note scales~~ ==		* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
	* xyxyxyxz row_count: 2 full_row_len: 4, lattice basis: gx: [1, 0, 0], gy: [3, 2, 1]		== 8 notes ==
	* xxxyxxxz row_count: 2 full_row_len: 4, lattice basis: gx: [3, 0, 1], gy: [5, 1, 1]		* xyxzyxyz row_count: 2 full_row_len: 4, lattice basis: gx: [4, 0, 2], gy: [4, 1, 2]
	== 9 ~~note scales~~ ==		* xyxyxyxz row_count: 2 full_row_len: 4, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
	* ~~xxyxxyxxz~~ row_count: 3 full_row_len: 3, lattice basis: gx: [5, 2, 1], gy: [4, 1, 1]		* xxxyxxxz row_count: 2 full_row_len: 4, lattice basis: gx: [2, 1, 3], gy: [4, 2, 3]
	== 10 ~~note scales~~ ==		== 9 notes ==
	* xyxzyxzxyz row_count: 2 full_row_len: 5, lattice basis: gx: [2, 2, 1], gy: [1, 1, 1]		* xxyzxyxzy row_count: 3 full_row_len: 3, lattice basis: gx: [1, 1, 1], gy: [2, 1, 1]
	* ~~xyxyxyxyxz~~ row_count: 5 full_row_len: 2, lattice basis: gx: [4, 3, 1], gy: [1, 0, 0]		* xxyxxyxxz row_count: 3 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
	* xxxxyxxxxz row_count: 2 full_row_len: 5, lattice basis: gx: [4, 0, 1], gy: [7, 1, 1]		== 10 notes ==
	== 12 ~~note scales~~ ==		* xyxzyxzxyz row_count: 2 full_row_len: 5, lattice basis: gx: [3, 1, 1], gy: [1, 1, 1]
	* ~~xyxyxyxyxyxz~~ row_count: 6 full_row_len: 2, lattice basis: gx: [5, 4, 1], gy: [1, 0, 0]		* xyxyxzxyxz row_count: 2 full_row_len: 5, lattice basis: gx: [4, 2, 3], gy: [3, 1, 2]
	* xxyxxyxxyxxz row_count: 4 full_row_len: 3, lattice basis: gx: [6, 2, 1], gy: [7, 3~~, 1~~]		* xyxyxyxyxz row_count: 2 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
	* xxxyxxxyxxxz row_count: 4 full_row_len: 3, lattice basis: gx: [8, 2, 1], gy: [6, 1, 1]		* xxxxyxxxxz row_count: 2 full_row_len: 5, lattice basis: gx: [1, 1, 3], gy: [4, 2, 3]
	* xxxxxyxxxxxz row_count: 2 full_row_len: 6, lattice basis: gx: [5, 0, 1], gy: [9, 1, 1]		== 12 notes ==
	== 14 ~~note scales~~ ==		* xyxyxzyxyxyz row_count: 2 full_row_len: 6, lattice basis: gx: [3, -1, 2], gy: [3, 0, 2]
	* xyxzyxzyxyzxyz row_count: 7 full_row_len: 2, lattice basis: gx: [4, 4, 3], gy: [2, 3, 2]		* xyxyxyxyxyxz row_count: 2 full_row_len: 6, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
	* ~~xyxyxzxyxzxyxz~~ row_count: 7 full_row_len: 2, lattice basis: gx: [2, ~~1, 1~~], gy: [6, 4, 3]		* xxyxxyxxyxxz row_count: 4 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
	* xyxyxyxyxyxyxz row_count: 7 full_row_len: 2, lattice basis: gx: [6, 5, 1], gy: [1, ~~0, 0~~]		* xxxyxxxyxxxz row_count: 3 full_row_len: 4, lattice basis: gx: [2, 1, 3], gy: [4, 2, 3]
			* xxxxxyxxxxxz row_count: 2 full_row_len: 6, lattice basis: gx: [0, 1, 3], gy: [4, 2, 3]
			== 14 notes ==
			* xyxzyxzyxyzxyz row_count: 2 full_row_len: 7, lattice basis: gx: [3, -1, 1], gy: [4, 1, 2]
			* xyxyxzxyxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 3], gy: [3, 1, 2]
			* xyxyxyxzxyxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 3], gy: [2, 0, 2]
			* xyxyxyxyxyxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
	* xxyxzxyxxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 1], gy: [3, 1, 1]		* xxyxzxyxxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 1], gy: [3, 1, 1]
	* xxxxxxyxxxxxxz row_count: 2 full_row_len: 7, lattice basis: gx: [6, 0, 1], gy: [11, 1, 1]		* xxxxxxyxxxxxxz row_count: 2 full_row_len: 7, lattice basis: gx: [-1, 1, 3], gy: [4, 2, 3]
	~~== 15 note scales ==~~		== 15 notes ==
	* xxyxyxxzxyxxyxz row_count: 3 full_row_len: 5, lattice basis: gx: [6, 3, 1], gy: [4, 2, 1]		* xxyzxyzxyxzyxzy row_count: 3 full_row_len: 5, lattice basis: gx: [3, 2, 2], gy: [1, 1, 1]
	* xxyxxyxxzxxyxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [6, 2, 1]		* xxyxzxyxxzyxxyz row_count: 3 full_row_len: 5, lattice basis: gx: [3, 1, 1], gy: [2, 1, 1]
	* xxyxxyxxyxxyxxz row_count: 5 full_row_len: 3, lattice basis: gx: [8, 3, 1], gy: [9, 4, 1]		* xxyxxyzxyxxyxzy row_count: 3 full_row_len: 5, lattice basis: gx: [4, 2, 1], gy: [3, 2, 1]
	* xxxxyxxxxyxxxxz row_count: 5 full_row_len: 3, lattice basis: gx: [11, 2, 1], gy: [8, 1, 1]		* xxyxyxxzxyxxyxz row_count: 3 full_row_len: 5, lattice basis: gx: [2, 1, 2], gy: [4, 2, 1]
	~~== 16 note scales ==~~		* xxyxxyxxzxxyxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [1, 1, 2]
	* xyxzyxzyxzxyzxyz row_count: 8 full_row_len: 2, lattice		* xxyxxyxxyxxyxxz row_count: 5 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
			* xxxxyxxxxyxxxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 1,

Inthar at 02:10, 2 January 2026

2026-01-02T02:10:48Z

← Older revision		Revision as of 02:10, 2 January 2026
Line 1:		Line 1:
	Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which [a, b, c] is a partition of the scale size are shown.		Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which [a, b, c] is a partition of the scale size are shown.

	It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 12.		It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 14.
	== 6 note scales ==		== 6 note scales ==
	* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]		* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]

Inthar at 02:01, 2 January 2026

2026-01-02T02:01:16Z

← Older revision		Revision as of 02:01, 2 January 2026
Line 1:		Line 1:
	Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36		Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which [a, b, c] is a partition of the scale size are shown.

	~~Only axbycz for which [a, b, c]~~ is ~~a partition of the scale size~~ are ~~shown~~		It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 12.
	== 6 note scales ==		== 6 note scales ==
	* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]		* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]

Inthar at 01:54, 2 January 2026

2026-01-02T01:54:13Z

Show changes

Inthar at 01:52, 2 January 2026

2026-01-02T01:52:35Z

← Older revision		Revision as of 01:52, 2 January 2026
Line 1:		Line 1:
	Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36		Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution\|MOS-substitution]] scales for scale sizes from 6 to 36

	~~Only aLbmcs for which [a, b, c] is a partition of the scale size are shown~~		Only axbycz for which [a, b, c] is a partition of the scale size are shown
			== 6 note scales ==* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]
	~~EDIT: This list is incomplete, there was a bug that incorrectly rejected some scales~~		== 8 note scales ==* xyxyxyxz row_count: 2 full_row_len: 4, lattice basis: gx: [1, 0, 0], gy: [3, 2, 1]
	~~== 6 note scales ==~~		* xxxyxxxz row_count: 2 full_row_len: 4, lattice basis: gx: [3, 0, 1], gy: [5, 1, 1]
	* scale: LLmLLs, row_count: 2, full_row_len: 3, lattice basis: gx: [3, 1, 1], gy: [2, 0, 1]		== 9 note scales ==* xxyxxyxxz row_count: 3 full_row_len: 3, lattice basis: gx: [5, 2, 1], gy: [4, 1, 1]
	~~== 8 note scales ==~~		== 10 note scales ==* xyxzyxzxyz row_count: 2 full_row_len: 5
	* scale: LmLsmLms, row_count: 2, full_row_len: 4, lattice basis: gx: [2, 2, 1], gy: [2, 1, 1]
	* scale: LmLmLmLs, row_count: 2, full_row_len: 4, lattice basis: gx: [3, 2, 1], gy: [1, 0, 0]
	* scale: LLLmLLLs, row_count: 2, full_row_len: 4, lattice basis: gx: [5, 1, 1], gy: [3, 0, 1]
	~~== 9 note scales ==~~
	* scale: LLmsLmLsm, row_count: 3, full_row_len: 3, lattice basis: gx: [2, 1, 1], gy: [1, 1, 1]
	* scale: LLmLLmLLs, row_count: 3, full_row_len: 3, lattice

Inthar at 23:50, 1 January 2026

2026-01-01T23:50:33Z

← Older revision		Revision as of 23:50, 1 January 2026
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	Only aLbmcs for which [a, b, c] is a partition of the scale size are shown		Only aLbmcs for which [a, b, c] is a partition of the scale size are shown

			EDIT: This list is incomplete, there was a bug that incorrectly rejected some scales
	== 6 note scales ==		== 6 note scales ==
	* scale: LLmLLs, row_count: 2, full_row_len: 3, lattice basis: gx: [3, 1, 1], gy: [2, 0, 1]		* scale: LLmLLs, row_count: 2, full_row_len: 3, lattice basis: gx: [3, 1, 1], gy: [2, 0, 1]