Diaschismic straddle temperaments: Difference between revisions
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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. | ||
Revision as of 06:28, 28 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
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 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 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.
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.
