Diachrome - Revision history

Sintel: /* Pele temperament */ typo

2025-05-11T02:03:54Z

Pele temperament: typo

← Older revision		Revision as of 02:03, 11 May 2025
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	=== Pele temperament ===		=== Pele temperament ===
	In the [[pele]] ~~tempeament~~ ([41 & 46 & 58]), following interpretations are available:		In the [[pele]] temperament ([41 & 46 & 58]), following interpretations are available:
	* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80;		* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80;
	* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49;		* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49;

Xenllium at 02:30, 6 May 2025

2025-05-06T02:30:46Z

← Older revision		Revision as of 02:30, 6 May 2025
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	=== 2.3.7.11.13 parapyth ===		=== 2.3.7.11.13 parapyth ===
	Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:		Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
	* The L step becomes 12/11		* The L step becomes 12/11;
	* The m step becomes 22/21~104/99~256/243		* The m step becomes 22/21~104/99~256/243;
	* The s step becomes 28/27~33/32~1053/1024		* The s step becomes 28/27~33/32~1053/1024.

	The tempered tuning thus has the mappings		The tempered tuning thus has the mappings
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	=== 2.3.11.19.23/5.31 [17 & 24] ===		=== 2.3.11.19.23/5.31 [17 & 24] ===
	There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral \| neutral]] (2.3.11.19.23/5.31[17 & 24]):		There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral \| neutral]] (2.3.11.19.23/5.31[17 & 24]):
	* The L step is 12/11		* The L step is 12/11;
	* The m step is 256/243~128/121~93/88~19/18		* The m step is 256/243~128/121~93/88~19/18;
	* The s step is 33/32~32/31~95/92		* The s step is 33/32~32/31~95/92.

	The tempered tuning thus has the mappings		The tempered tuning thus has the mappings
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	=== 2.3.7.13.19.23 [17 & 19 & 41] ===		=== 2.3.7.13.19.23 [17 & 19 & 41] ===
	A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:		A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:
	* The L step is 13/12		* The L step is 13/12;
	* The m step is 256/243~96/91~19/18		* The m step is 256/243~96/91~19/18;
	* The s step is 28/27~27/26		* The s step is 28/27~27/26.

	The tempered tuning has the mappings		The tempered tuning has the mappings
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	=== Pele temperament ===		=== Pele temperament ===
	In the [[pele]] tempeament ([41 & 46 & 58]), following interpretations are available:		In the [[pele]] tempeament ([41 & 46 & 58]), following interpretations are available:
	* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80.		* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80;
	* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49.		* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49;
	* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63.		* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63;
			* L = 13/12, m = 21/20~22/21~104/99~256/243, s = 27/26.

	== External links ==		== External links ==

Xenllium: Added interpretations

2025-05-06T02:16:11Z

Added interpretations

← Older revision		Revision as of 02:16, 6 May 2025
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	* 5sL: LsLsLsmLsLsm		* 5sL: LsLsLsmLsLsm
	* 5sR: LmsLsLsLmsLs		* 5sR: LmsLsLsLmsLs
	* 5sC: LsLsLmsLsLsm		* 5sC: LsLsLmsLsLsm (''interchroid'' structure)

	5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have.		5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have.

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	Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243.		Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243.

	=== 7-limit[5120/5103] ===		=== 7-limit [5120/5103] ===
	In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5.8c gap between 81/80 and 64/63. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]].		In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5.8{{c}} gap between 81/80 and 64/63. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]].
	* The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80.		* The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80.
	* The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63.		* The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63.

	The tempered tuning thus has the mappings		The tempered tuning thus has the mappings
	* 3/2 = 3L + m + 3s,		* 3/2 = 3L + m + 3s,
	* 5/4 = 2L + s,		* 5/4 = 2L + s,
	* 7/4 = 4L + 2m + 3s.		* 7/4 = 4L + 2m + 3s.

	The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.		The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.

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	Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:		Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
	* The L step becomes 12/11		* The L step becomes 12/11
	* The m step becomes ~~256/243~~~22/21~104/99		* The m step becomes 22/21~104/99~256/243
	* The s step becomes 28/27~33/32~1053/1024		* The s step becomes 28/27~33/32~1053/1024

	The tempered tuning thus has the mappings		The tempered tuning thus has the mappings
	* 3/2 = 3L + m + 3s,		* 3/2 = 3L + m + 3s,
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	* 11/8 = 2L + m + 3s,		* 11/8 = 2L + m + 3s,
	* 13/8 = 3L + 2m + 4s.		* 13/8 = 3L + 2m + 4s.

	By not tempering out 144/143, parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.		By not tempering out 144/143, parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.

	The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]".		The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]".

	=== 2.3.11.19.23/5.31[17 & 24] ===		=== 2.3.11.19.23/5.31 [17 & 24] ===
	There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral \| neutral]] (2.3.11.19.23/5.31[17 & 24]):		There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral \| neutral]] (2.3.11.19.23/5.31[17 & 24]):
	* The L step is 12/11		* The L step is 12/11
	* The m step is 256/243~128/121~93/88~19/18		* The m step is 256/243~128/121~93/88~19/18
	* The s step is 33/32~32/31~95/92		* The s step is 33/32~32/31~95/92

	The tempered tuning thus has the mappings		The tempered tuning thus has the mappings
	* 3/2 = 3L + m + 3s,		* 3/2 = 3L + m + 3s,
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	* 23/20 = L + m,		* 23/20 = L + m,
	* 31/16 = 5L + 2m + 4s.		* 31/16 = 5L + 2m + 4s.

	Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety \| proper]].		Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety \| proper]].

	=== 2.3.7.13.19.23[17 & 19 & 41] ===		=== 2.3.7.13.19.23 [17 & 19 & 41] ===
	A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:		A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:
	* The L step is 13/12		* The L step is 13/12
	* The m step is 256/243~96/91~19/18		* The m step is 256/243~96/91~19/18
	* The s step is 28/27~27/26		* The s step is 28/27~27/26

	The tempered tuning has the mappings		The tempered tuning has the mappings
	* 3/2 = 3L + m + 3s,		* 3/2 = 3L + m + 3s,
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	* 19/16 = L + m + s,		* 19/16 = L + m + s,
	* 23/14 = 3L + 2m + 4s.		* 23/14 = 3L + 2m + 4s.

	Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11.		Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11.

	It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.		It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.

			=== Pele temperament ===
			In the [[pele]] tempeament ([41 & 46 & 58]), following interpretations are available:
			* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80.
			* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49.
			* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63.

	== External links ==		== External links ==

Inthar: /* External links */

2025-01-07T19:50:52Z

External links

← Older revision		Revision as of 19:50, 7 January 2025
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	== External links ==		== External links ==
	* [https://sw3.lumipakkanen.com/scale/~~Xvm32gcae~~ 5sRA Aeolian (46edo 7:3:1 chromedye)]		* [https://sw3.lumipakkanen.com/scale/XvmXDTFp0 5sRA Aeolian (46edo 7:3:1 Aberschismic chromedye)]
			* [https://sw3.lumipakkanen.com/scale/Xvmsw1KBx 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)]

	[[Category:Aberrismic theory]]		[[Category:Aberrismic theory]]
	[[Category:Rank-3 scales]]		[[Category:Rank-3 scales]]

Inthar at 19:41, 7 January 2025

2025-01-07T19:41:31Z

← Older revision		Revision as of 19:41, 7 January 2025
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	It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.		It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.

			== External links ==
			* [https://sw3.lumipakkanen.com/scale/Xvm32gcae 5sRA Aeolian (46edo 7:3:1 chromedye)]

	[[Category:Aberrismic theory]]		[[Category:Aberrismic theory]]
	[[Category:Rank-3 scales]]		[[Category:Rank-3 scales]]

Inthar: /* 7-limit[5120/5103] */

2025-01-07T19:34:03Z

7-limit[5120/5103]

← Older revision		Revision as of 19:34, 7 January 2025
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	* 5/4 = 2L + s,		* 5/4 = 2L + s,
	* 7/4 = 4L + 2m + 3s.		* 7/4 = 4L + 2m + 3s.
	The property of tempering out 5120/5103 thus lends 41edo, 46edo, 53edo, and 58edo some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.		The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.

	=== 2.3.7.11.13 parapyth ===		=== 2.3.7.11.13 parapyth ===

Inthar at 19:27, 7 January 2025

2025-01-07T19:27:15Z

← Older revision		Revision as of 19:27, 7 January 2025
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	'''Diachrome''' (~~also~~ denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s [[scale pattern]]s:		'''Diachrome''' or '''chromedye''' (denoted dia5s or 5s in [[groundfault]]'s [[aberrismic theory]] systematic naming) is a set of three 5L 2m 5s [[scale pattern]]s:
	* 5sL: LsLsLsmLsLsm		* 5sL: LsLsLsmLsLsm
	* 5sR: LmsLsLsLmsLs		* 5sR: LmsLsLsLmsLs

Inthar at 15:02, 4 January 2025

2025-01-04T15:02:15Z

← Older revision		Revision as of 15:02, 4 January 2025
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	=== Balance ===		=== Balance ===
	The 5sC pattern, LsLsLmsLsLsm, is ~~a diregular~~ scale according to the [[Ternary scale theorems#\|classification of ternary balanced scales]]. In particular, it is (as an abstract scale word) MV3 but not SV3.		The 5sC pattern, LsLsLmsLsLsm, is an even-regular scale according to the [[Ternary scale theorems#\|classification of ternary balanced scales]]. In particular, it is (as an abstract scale word) MV3 but not SV3.

	== Diachrome in edos ==		== Diachrome in edos ==

Fredg999: Update internal link

2024-08-12T16:14:36Z

Update internal link

← Older revision		Revision as of 16:14, 12 August 2024
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	'''Diachrome''' (also denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s [[~~Step pattern\|~~scale pattern]]s:		'''Diachrome''' (also denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s [[scale pattern]]s:
	* 5sL: LsLsLsmLsLsm		* 5sL: LsLsLsmLsLsm
	* 5sR: LmsLsLsLmsLs		* 5sR: LmsLsLsLmsLs

Fredg999: Internal link

2024-08-12T16:13:09Z

Internal link

← Older revision		Revision as of 16:13, 12 August 2024
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	'''Diachrome''' (also denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s scale ~~patterns~~:		'''Diachrome''' (also denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s [[Step pattern\|scale pattern]]s:
	* 5sL: LsLsLsmLsLsm		* 5sL: LsLsLsmLsLsm
	* 5sR: LmsLsLsLmsLs		* 5sR: LmsLsLsLmsLs