Diaschismic - Revision history

Inthar at 16:55, 2 July 2026

2026-07-02T16:55:31Z

← Older revision		Revision as of 16:55, 2 July 2026
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	'''Diaschismic''', 2.3.5(.17)[10 & 12], is the temperament generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.		'''Diaschismic''', 2.3.5(.17)[10 & 12], is the temperament generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. (17 is added by a [[Canonical extension\|structurally induced extension]].) 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.

	[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].		[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].

Inthar: /* Generator chain */ make interval table isomorphic

2026-07-01T06:33:58Z

Generator chain: make interval table isomorphic

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 |40/27
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 |24
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 |576
 |7/5
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 |128
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 |472
 |21/16
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 |264
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 |160
 |35/32
 |}

Inthar at 06:29, 1 July 2026

2026-07-01T06:29:43Z

← Older revision		Revision as of 06:29, 1 July 2026
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	'''Diaschismic''', 2.3.5.17[10 & 12], is the temperament generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.		'''Diaschismic''', 2.3.5(.17)[10 & 12], is the temperament generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.

	[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].		[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].

Inthar at 06:23, 1 July 2026

2026-07-01T06:23:20Z

← Older revision		Revision as of 06:23, 1 July 2026
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	'''Diaschismic''', [10 & 12], is the temperament ~~(usually 2.3.5.17)~~ generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.		'''Diaschismic''', 2.3.5.17[10 & 12], is the temperament generated by a half-octave representing 45/32~17/12~24/17~64/45 and a semitone representing 135/128~18/17~17/16~16/15, which is tuned to around 105 cents. 3/2 is a period plus a generator; 5/4 is a period minus two generators. (Note that 3/2 can be taken as the generator instead of 17/16.) In other words, Diaschismic equates a stack of two 5/4's and two 9/8's to the octave; equivalently, it equates a stack of one 5/4 and two 16/15's to the half-octave.

	[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].		[[12edo]] and [[22edo]] are notable Diaschismic tunings. Some tunings of Diaschismic that are more accurate in the 5-limit are [[34edo]] and [[46edo]].

Inthar at 15:04, 6 June 2026

2026-06-06T15:04:31Z

← Older revision		Revision as of 15:04, 6 June 2026
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	==== Aberschismic (46 & 58) ====		==== Aberschismic (46 & 58) ====
	A more accurate but complex option is to interpret Diaschismic as a [[~~Hemifamity~~]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).		A more accurate but complex option is to interpret Diaschismic as an [[Aberschismic]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).

	In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in Pele; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.		In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in Pele; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.

Vector at 05:38, 23 May 2026

2026-05-23T05:38:03Z

← Older revision		Revision as of 05:38, 23 May 2026
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	Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between [[Gentle tuning\|gentle]] and [[superpyth]] tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. Overall, however, this largely ends up just functioning as an excuse to tune the fifth to 707 cents, which should be done directly if no particular JI interpretation is desired.		Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between [[Gentle tuning\|gentle]] and [[superpyth]] tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. Overall, however, this largely ends up just functioning as an excuse to tune the fifth to 707 cents, which should be done directly if no particular JI interpretation is desired.

	Interestingly, the comma 82/81 is the next superparticular up after the [[meantone]] comma 81/80. Fittingly, shrub tunings are ~~roughly opposite~~ 3/2 ~~from~~ meantone tunings.		Interestingly, the comma 82/81 is the next superparticular up after the [[meantone]] comma 81/80. Fittingly, shrub tunings are detuned the same amount from 3/2 as meantone tunings, but in the opposite direction.

Vector at 05:36, 23 May 2026

2026-05-23T05:36:55Z

← Older revision		Revision as of 05:36, 23 May 2026
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	== Shrub temperament ==		== Shrub temperament ==
	Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between [[Gentle tuning\|gentle]] and [[superpyth]] tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. {{Navbox regtemp}}		Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between [[Gentle tuning\|gentle]] and [[superpyth]] tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. Overall, however, this largely ends up just functioning as an excuse to tune the fifth to 707 cents, which should be done directly if no particular JI interpretation is desired.

			Interestingly, the comma 82/81 is the next superparticular up after the [[meantone]] comma 81/80. Fittingly, shrub tunings are roughly opposite 3/2 from meantone tunings.



			{{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Vector at 05:32, 23 May 2026

2026-05-23T05:32:42Z

← Older revision		Revision as of 05:32, 23 May 2026
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	\|}		\|}

	{{Navbox regtemp}}		== Shrub temperament ==
			Shrub temperament is a restriction of diaschismic temperament to the 2.3.25 subgroup, removing the tritone. It bridges the gap between [[Gentle tuning\|gentle]] and [[superpyth]] tunings, with reasonable tunings around 707 cents, which leads to a 428 cents major third representing 32/25 and a 279 cents minor third representing 75/64. It may also set the major third to 23/18 and 41/32. {{Navbox regtemp}}
	{{Cat\|temperaments}}		{{Cat\|temperaments}}

Inthar: /* Aberschismic (46 & 58) */

2026-04-26T00:41:51Z

Aberschismic (46 & 58)

← Older revision		Revision as of 00:41, 26 April 2026
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	A more accurate but complex option is to interpret Diaschismic as a [[Hemifamity]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).		A more accurate but complex option is to interpret Diaschismic as a [[Hemifamity]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).

	In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in ~~Akea~~; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.		In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in Pele; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.

	Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.		Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.

Inthar: /* Aberschismic (46 & 58) */

2026-04-26T00:41:17Z

Aberschismic (46 & 58)

← Older revision		Revision as of 00:41, 26 April 2026
Line 15:		Line 15:
	A more accurate but complex option is to interpret Diaschismic as a [[Hemifamity]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).		A more accurate but complex option is to interpret Diaschismic as a [[Hemifamity]] temperament. Here, the comma 81/80 represented by six generators is also 64/63 and half of 50/49, so that 8/7 is a comma above 9/8 (being itself a comma above 10/9).

	In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in ~~Hemifamity~~; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.		In the 11-limit, an obvious mapping is to treat the neogothic major third 81/64 as 14/11, as in Akea; this sets 11/8 to be offset from the tritone by two commas. As a result, 11/9 is a comma sharp of 6/5. 16/13 is a further comma sharp.

	Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.		Alternatively, another mapping offsets 11 from the fourth by two commas instead, finding 13 at 8 generators up and setting the tritone to 91/64.