Diaschismic straddle temperaments - Revision history

Ground at 14:59, 30 August 2026

2026-08-30T14:59:38Z

Ground at 19:02, 28 August 2026

2026-08-28T19:02:23Z

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	== Diaschismic oneirotonic ==		== Diaschismic oneirotonic ==

	Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46.		Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46, which may be called '''Diaschismic Sensibeta''' after its 2.9.5.7 commas.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1

Ground at 18:59, 28 August 2026

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	== Diaschismic oneirotonic ==		== Diaschismic oneirotonic ==

	Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to ~~tine~~ 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic.		Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1

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	It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.		It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

			== Diaschismic oneirotonic ==

			Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tine 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic.

			17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1

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	'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.		'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

	Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17 \|\| 0 9 -8 -1 4 9 -8		Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3

	It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.		It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

Ground at 14:27, 28 August 2026

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	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7

			== Diaschismic Hemiseven ==

			[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.

			Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1

	== Diaschismic whole-tone ==		== Diaschismic whole-tone ==

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	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7

			== Diaschismic whole-tone ==

			'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

			Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17 \|\| 0 9 -8 -1 4 9 -8

			It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

Ground at 12:34, 28 August 2026

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	These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a. This can be converted to standard RTT to avoid straddling, resulting in 2.9.5.7.11.289 22&30.		These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a. This can be converted to standard RTT to avoid straddling, resulting in 2.9.5.7.11.289 22&30.

	Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration a is redundant because ~~there is~~ no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.		Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

	== Diaschismic Porcupine ==		== Diaschismic Porcupine ==
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	Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.		Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

	One possible ~~GTO string~~ resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.<17.>>17 \|\| 0 8 -3 -5 6 -4 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3

			One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

	== Diaschismic Semioceanfront ==		== Diaschismic Semioceanfront ==

	'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.		'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

			17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7

Ground at 06:28, 28 August 2026

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	* Septimal Diaschismic tempers out 126/125, a 2.9.5.7 comma requiring no straddling because it has 9 rather than 3.		* Septimal Diaschismic tempers out 126/125, a 2.9.5.7 comma requiring no straddling because it has 9 rather than 3.
	* Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1.		* Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1.
	These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a.		These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a. This can be converted to standard RTT to avoid straddling, resulting in 2.9.5.7.11.289 22&30.

	Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration a is redundant because there is no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.		Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration a is redundant because there is no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

Ground at 06:25, 28 August 2026

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	{{proposed}}		{{proposed}}
	~~{{stub}}~~

	[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. ~~Normally the~~ subgroup is interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.		[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

	Pajara, Hedgehog, and Echidna are related, but they ~~only use the sharp 3 to~~ temper out 50/49. Straddling 3 ~~provides more accuracy while keeping the Archy portion~~.		Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.

	== Straddle Diaschismic ==		== Straddle Diaschismic ==

	'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp 3. Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, ~~another~~ 2.9.5.7 comma requiring no straddling~~, and~~ Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. These both result in the same extension. As in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3.		'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:
			* Septimal Diaschismic tempers out 126/125, a 2.9.5.7 comma requiring no straddling because it has 9 rather than 3.
			* Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1.
			These both result in the same extension. And as in Diaschismic, the half-octave can be interpreted as 17/12, implying that 17 is straddled like 3. The final resulting SRTT subgroup is 2.3a.5.7.17a.a^2, where 3a/(a^2) = 3/a.

	Much of what is useful here can be found in a ~~non~~-~~straddle~~ restriction ~~with sharp 3 and 17~~, 2.3.7.17 10&22 ~~which~~ tempers out 64/62 and 289/288~~. It~~ adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.		Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration a is redundant because there is no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

	== Diaschismic Porcupine ==		== Diaschismic Porcupine ==
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	'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.		'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.

	The simplest interpretation of the generator is 9/7. Straddle Diaschismic is equivalent to Stearn in the 2.9.7 subgroup, tempering out 118098/117649 = ((9/7)/(7/6)*(9/7))^2/2.		The simplest interpretation of the generator is 9/7, which also generates Hedgehog and Stearn. Straddle Diaschismic is equivalent to Stearn in the 2.9.7 subgroup, tempering out 118098/117649 = ((9/7)/(7/6)*(9/7))^2/2. This is an extremely efficient temperament.

	Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping of 81/64 a^4 = 13/10.		Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

	One possible GTO string resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.<17.>>17 \|\| 0 8 -3 -5 6 -4 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.		One possible GTO string resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.<17.>>17 \|\| 0 8 -3 -5 6 -4 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

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	[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.		[[Diaschismic]] tempers out 2048/2025 = 2/((9/8)(5/4))^2. The subgroup is usually interpreted as 2.3.5, but 2.9.5 is also possible. This provides a great opportunity for [[Straddle primes\|straddling]] 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9.

	Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.		Pajara, Hedgehog, and Echidna are related temperaments, but they temper out 50/49, which damages 5 and 7. Straddling 3 allows both 2048/2025 and 64/63 a^-1 to be tempered out without doing this.

	~~== Straddle Diaschismic ==~~

	'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:		'''Straddle Diaschismic''' is the simplest temperament in this category. It is rank-3 and generated by the flat and sharp versions of 3. Of the Diaschismic extensions to the 7-limit:
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	Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.		Much of what is useful here can be found in a rank-2 restriction, 2.3a.7.17a (the alteration ''a'' is redundant because sharp 3 and 17 are used with no straddling). 2.3.7.17 10&22 tempers out 64/63 and 289/288 and adds 17/14 to Archy, creating a useful symmetry between subminor/supermajor and supraminor/submajor chords.

	== Diaschismic Porcupine ==		== Rank-2 Temperaments ==

			[todo: table]

			=== Diaschismic Porcupine ===

	'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.		'''Diaschismic Porcupine''' tempers out 2048/2025 and 250/243 a^-5, the Porcupine comma with 3 replaced with 3a, as 3 is usually tuned quite sharp. Without straddling 3, only 22edo tempers out 2048/2025 and 250/243 in 2.3.5.
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	Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.		Mappings of 5.7.11 are the same as Porcupine, and the Diaschismic mapping adds prime 17. 13 may be added using the Oceanfront mapping equating 81/64 a^4 to 13/10.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3; 162.287¢

	One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.		One possible erac group resembling the 74edo tuning is 600 \| 2.<3.>>3.>5.>7.11.>13.<17.>>17 \|\| 0 8 -3 -5 6 -4 -17 8 -3. This focuses on approximating 9/5, 9/7, 17/5, 17/7, and 11/9 about equally well, with >17/14 equated to <11/9.

	== Diaschismic Semioceanfront ==		=== Diaschismic Semioceanfront ===

	'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.		'''Diaschismic Semioceanfront''' has a generator of half of an Oceanfront fourth rather than a third. The generator is also equivalent to 13/8 due to the half-octave period.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -2 -7 9 4 -15 1 -2 -7; 243.180¢

	== Diaschismic Hemiseven ==		=== Diaschismic Hemiseven ===

	[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.		[[Hemiseven]] may already be interpreted as straddle-3, but being a 5edo cluster temperament, all versions have a complex 5. This is fixed using Diaschismic with the 9 made by stacking the sharp 3 generator and the flatter 3.

	Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1		Subgroup GTO string: 600 \| 2.<3.>3.5.7.<13.>13.<17.>17 \|\| 0 6 1 -7 -2 2 -3 6 1; 115.976 ¢

	== Diaschismic whole-tone ==		=== Diaschismic whole-tone ===

	'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.		'''Diaschismic whole-tone''' is a significant whole-tone scale containing the otonal triads 12:15:17 and 8:10:13. It can be written without straddling as 2.9.5.13.17/3 6&28, but the 3 can be separated in the same way as the other temperaments in this article.

	Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3		Subgroup GTO string: 600 \| 2.<3.>3.5.13.<17.>17.23 \|\| 0 9 -8 -1 4 9 -8 3; 210.145¢

	It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.		It is related to Antikythera, but that temperament is 2.9.5.7 and once again tempers out 50/49 as it is specifically a subset of Pajara.

	== Diaschismic oneirotonic ==		=== Diaschismic oneirotonic ===

	Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46, which may be called '''Diaschismic Sensibeta''' after its 2.9.5.7 commas.		Some tunings of oneirotonic generate a slightly sharp 9/8 that is good for Diaschismic, providing an opportunity to tune 5 well besides A-team. In this case, neither of the straddled versions of 3 are diatonic. This allows the generator to correspond to a simple ratio, 13/7. The non-straddle version of this temperament is 2.9.5.7.11.13 28&46, which may be called '''Diaschismic Sensibeta''' after its 2.9.5.7 commas.

	17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1		17-limit GTO string: 600 \| 2.<3.>3.5.7.11.13.<17.>17 \|\| 0 -4 1 3 -11 -5 -12 -4 1; 129.835 ¢