Diaschismic straddle temperaments: Difference between revisions
Created page with "{{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 straddling 3, where flat 3 (3/a in SRTT) and sharp 3 (3a) cancel out to make 9. Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, another 2.9.5.7 comma, and Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. Th..." |
No edit summary |
||
| Line 4: | Line 4: | ||
[[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. 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. | ||
Of the Diaschismic extensions to the 7-limit, Septimal Diaschismic tempers out 126/125, another 2.9.5.7 comma, and Pajara uses the sharp 3, tempering out 64/(3a*7) = 64/63 a^-1. These | 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. | ||
== 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. | |||
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. | |||
== Diaschismic Porcupine == | == Diaschismic Porcupine == | ||
| Line 10: | Line 16: | ||
'''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. | |||
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. | |||
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. | ||
Revision as of 06:09, 28 August 2026
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 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.
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.
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.
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. Straddle Diaschismic is equivalent to Stearn in the 2.9.7 subgroup, tempering out 118098/117649 = ((9/7)/(7/6)*(9/7))^2/2.
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.
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.
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.
