Breedsmic temperaments: Difference between revisions
→Hemififths: +quadrafifths |
No edit summary |
||
| Line 26: | Line 26: | ||
By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice. | By adding [[243/242]] (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99EDO is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 113: | Line 114: | ||
Vals: {{Val list| 41, 157, 198, 239, 676b, 915be }} | Vals: {{Val list| 41, 157, 198, 239, 676b, 915be }} | ||
Badness: 0. | Badness: 0.040170 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 126: | Line 127: | ||
Vals: {{Val list| 41, 157, 198, 437f, 635bcff }} | Vals: {{Val list| 41, 157, 198, 437f, 635bcff }} | ||
Badness: 0. | Badness: 0.031144 | ||
== Tertiaseptal == | == Tertiaseptal == | ||
| Line 132: | Line 133: | ||
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well. | Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31&171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo|171EDO]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 273: | Line 275: | ||
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival| 12 34 20 30 52 …}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies. | Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with {{multival| 12 34 20 30 52 …}} as the octave wedgie. [[130edo|130EDO]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 331: | Line 334: | ||
Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning. | Adding 3025/3024 extends to the 11-limit and gives {{multival| 38 -3 8 64 …}} for the initial wedgie, and as expected, 270 remains an excellent tuning. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 376: | Line 380: | ||
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14ths equal temperament|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]]. | Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[15/14ths equal temperament|linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used its generator. It may be described as the 130&270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice. It is also described as an extension of the [[Qintosec family|qintosec temperament]]. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 421: | Line 426: | ||
== Neominor == | == Neominor == | ||
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''. | The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 464: | Line 470: | ||
== Emmthird == | == Emmthird == | ||
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935. | The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 481: | Line 488: | ||
== Quinmite == | == Quinmite == | ||
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth". | The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth". | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 498: | Line 506: | ||
== Unthirds == | == Unthirds == | ||
The generator for unthirds temperament is undecimal major third, 14/11. | The generator for unthirds temperament is undecimal major third, 14/11. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 541: | Line 550: | ||
== Newt == | == Newt == | ||
This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. | This temperament has a generator of neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 588: | Line 598: | ||
While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two. | While it extends well into 2.3.5.7.13/11, there are multiple reasonable places for the prime 11 and 13 in the interval chain. Amical (311 & 410) does this with no compromise of accuracy, but is enormously complex. Amorous (212 & 311) has the new primes placed on the same side of the interval chain so blends smarter with the other harmonics. Pseudoamical (99 & 113) and pseudoamorous (14cf & 99ef) are the corresponding low-complexity interpretations. Floral (198 & 212) shares the semioctave period and the ~21/20 generator with harry, but in a complementary style, including a characteristic flat 11. Finally, humorous (198 & 311) is one of the best extensions out there and it splits the generator in two. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 762: | Line 773: | ||
{{Main| Septidiasemi }} | {{Main| Septidiasemi }} | ||
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit. | Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 953: | Line 965: | ||
== Mintone == | == Mintone == | ||
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice. | In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo|-3 11 -5 -1}} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58&103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 1,090: | Line 1,103: | ||
Badness: 0.028683 | Badness: 0.028683 | ||
== Quasimoha == | |||
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].'' | |||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 2401/2400, 3645/3584 | |||
[[Mapping]]: [{{Val|1 1 9 6}}, {{Val|0 2 -23 -11}}] | |||
[[POTE generator]]: ~49/40 = 348.603 | |||
{{Val list|legend=1| 31, 117c, 148bc, 179bc }} | |||
[[Badness]]: 0.110820 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 243/242, 441/440, 1815/1792 | |||
Mapping: [{{Val|1 1 9 6 2}}, {{Val|0 2 -23 -11 5}}] | |||
POTE generator: ~11/9 = 348.639 | |||
Vals: {{Val list| 31, 86ce, 117ce, 148bce }} | |||
Badness: 0.046181 | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||