Diaschismic straddle temperaments: Difference between revisions
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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''' 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 === | |||
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; 129.835 ¢ | |||
Latest revision as of 14:59, 30 August 2026
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 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.
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. 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 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.
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.
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 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; 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.
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.
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
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; 115.976 ¢
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.>17.23 || 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.
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.
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 ¢
