Breedsmic temperaments: Difference between revisions
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'''Breedsmic temperaments''' are rank two temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = [[2401/2400]]. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ( | '''Breedsmic temperaments''' are rank two temperaments tempering out the [[breedsma]], {{monzo|-5 -1 -2 4}} = [[2401/2400]]. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma. | ||
It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system. | It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system. | ||
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Temperaments not discussed here include [[Dicot family #Decimal|decimal]], [[Archytas clan #Beatles|beatles]], [[Meantone family #Squares|squares]], [[Starling temperaments #Myna|myna]], [[Keemic temperaments #Quasitemp|quasitemp]], [[Gamelismic clan #Miracle|miracle]], [[Magic family #Quadrimage|quadrimage]], [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], [[Tetracot family #Octacot|octacot]], [[Kleismic family #Quadritikleismic|quadritikleismic]], [[Schismatic family #Sesquiquartififths|sesquiquartififths]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], and [[Gammic family #Neptune|neptune]]. | Temperaments not discussed here include [[Dicot family #Decimal|decimal]], [[Archytas clan #Beatles|beatles]], [[Meantone family #Squares|squares]], [[Starling temperaments #Myna|myna]], [[Keemic temperaments #Quasitemp|quasitemp]], [[Gamelismic clan #Miracle|miracle]], [[Magic family #Quadrimage|quadrimage]], [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], [[Tetracot family #Octacot|octacot]], [[Kleismic family #Quadritikleismic|quadritikleismic]], [[Schismatic family #Sesquiquartififths|sesquiquartififths]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], and [[Gammic family #Neptune|neptune]]. | ||
= Hemififths = | == Hemififths == | ||
{{main|Hemififths}} | {{main|Hemififths}} | ||
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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 36: | Line 23: | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* 7 and 9-limit minimax | * 7 and 9-limit minimax: ~49/40 = {{monzo|1/5 0 1/25}} | ||
: [{{monzo|1 0 0 0}}, {{monzo|7/5 | : [{{monzo|1 0 0 0}}, {{monzo|7/5 0 2/25 0}}, {{monzo|0 0 1 0}}, {{monzo|8/5 0 13/25 0}}] | ||
: | : Eigenmonzos: 2, 5 | ||
[[Algebraic generator]]: (2 + sqrt(2))/2 | [[Algebraic generator]]: (2 + sqrt(2))/2 | ||
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[[Badness]]: 0.022243 | [[Badness]]: 0.022243 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 59: | Line 46: | ||
Badness: 0.023498 | Badness: 0.023498 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 72: | Line 59: | ||
Badness: 0.019090 | Badness: 0.019090 | ||
== Semihemi == | === Semihemi === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 85: | Line 72: | ||
Badness: 0.042487 | Badness: 0.042487 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 98: | Line 85: | ||
Badness: 0.021188 | Badness: 0.021188 | ||
= Tertiaseptal = | == Tertiaseptal == | ||
{{main|Tertiaseptal}} | {{main|Tertiaseptal}} | ||
| Line 117: | Line 104: | ||
[[Badness]]: 0.012995 | [[Badness]]: 0.012995 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 130: | Line 117: | ||
Badness: 0.035576 | Badness: 0.035576 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 143: | Line 130: | ||
Badness: 0.036876 | Badness: 0.036876 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 156: | Line 143: | ||
Badness: 0.027398 | Badness: 0.027398 | ||
== Tertia == | === Tertia === | ||
Subgroup:2.3.5.7.11 | Subgroup:2.3.5.7.11 | ||
| Line 169: | Line 156: | ||
Badness: 0.030171 | Badness: 0.030171 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 182: | Line 169: | ||
Badness: 0.028384 | Badness: 0.028384 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 195: | Line 182: | ||
Badness: 0.022416 | Badness: 0.022416 | ||
== Hemitert == | === Hemitert === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 208: | Line 195: | ||
Badness: 0.015633 | Badness: 0.015633 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 221: | Line 208: | ||
Badness: 0.033573 | Badness: 0.033573 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 234: | Line 221: | ||
Badness: 0.025298 | Badness: 0.025298 | ||
= Harry = | == Harry == | ||
{{main|Harry}} | {{main|Harry}} | ||
{{see also|Gravity family #Harry}} | {{see also|Gravity family #Harry}} | ||
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[[Badness]]: 0.034077 | [[Badness]]: 0.034077 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 271: | Line 258: | ||
Badness: 0.015867 | Badness: 0.015867 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 284: | Line 271: | ||
Badness: 0.013046 | Badness: 0.013046 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 297: | Line 284: | ||
Badness: 0.012657 | Badness: 0.012657 | ||
= Quasiorwell = | == Quasiorwell == | ||
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = {{monzo|22 -1 -10 1}}. It has a wedgie {{multival|38 -3 8 -93 -94 27}}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths. | In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = {{monzo|22 -1 -10 1}}. It has a wedgie {{multival|38 -3 8 -93 -94 27}}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths. | ||
| Line 314: | Line 301: | ||
[[Badness]]: 0.035832 | [[Badness]]: 0.035832 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 327: | Line 314: | ||
Badness: 0.017540 | Badness: 0.017540 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095 | |||
Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}] | Mapping: [{{val|1 31 0 9 53 -59}}, {{val|0 -38 3 -8 -64 81}}] | ||
| Line 340: | Line 327: | ||
Badness: 0.017921 | Badness: 0.017921 | ||
= Decoid = | == Decoid == | ||
{{see also|Qintosec family #Decoid}} | {{see also|Qintosec family #Decoid}} | ||
In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of [[10edo| | In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of [[10edo|10EDO]]. 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 359: | Line 346: | ||
[[Badness]]: 0.033902 | [[Badness]]: 0.033902 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 372: | Line 359: | ||
Badness: 0.018735 | Badness: 0.018735 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 385: | Line 372: | ||
Badness: 0.013475 | Badness: 0.013475 | ||
= Neominor = | == Neominor == | ||
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as "<b>Neo-gothic minor third</b>". | The generator for neominor temperament is tridecimal minor third [[13/11]], also known as "<b>Neo-gothic minor third</b>". | ||
| Line 402: | Line 389: | ||
[[Badness]]: 0.088221 | [[Badness]]: 0.088221 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 415: | Line 402: | ||
Badness: 0.027959 | Badness: 0.027959 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 428: | Line 415: | ||
Badness: 0.026942 | Badness: 0.026942 | ||
= 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. | ||
| Line 445: | Line 432: | ||
[[Badness]]: 0.016736 | [[Badness]]: 0.016736 | ||
= Quinmite = | == Quinmite == | ||
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. | The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. | ||
| Line 462: | Line 449: | ||
[[Badness]]: 0.037322 | [[Badness]]: 0.037322 | ||
= Unthirds = | == Unthirds == | ||
The generator for unthirds temperament is undecimal major third, 14/11. | The generator for unthirds temperament is undecimal major third, 14/11. | ||
| Line 479: | Line 466: | ||
[[Badness]]: 0.075253 | [[Badness]]: 0.075253 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 492: | Line 479: | ||
Badness: 0.022926 | Badness: 0.022926 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 505: | Line 492: | ||
Badness: 0.020888 | Badness: 0.020888 | ||
= 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]]. | ||
| Line 522: | Line 509: | ||
[[Badness]]: 0.041878 | [[Badness]]: 0.041878 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 535: | Line 522: | ||
Badness: 0.019461 | Badness: 0.019461 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 548: | Line 535: | ||
Badness: 0.013830 | Badness: 0.013830 | ||
= Amicable = | == Amicable == | ||
{{see also| Amity family }} | {{see also| Amity family }} | ||
| Line 565: | Line 552: | ||
[[Badness]]: 0.045473 | [[Badness]]: 0.045473 | ||
= Septidiasemi = | == Septidiasemi == | ||
{{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. | ||
| Line 583: | Line 570: | ||
[[Badness]]: 0.044115 | [[Badness]]: 0.044115 | ||
== Sedia == | === Sedia === | ||
The ''sedia'' temperament (10&161, named by [[User:Xenllium|Xenllium]]) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440. | The ''sedia'' temperament (10&161, named by [[User:Xenllium|Xenllium]]) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440. | ||
| Line 598: | Line 585: | ||
Badness: 0.090687 | Badness: 0.090687 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 611: | Line 598: | ||
Badness: 0.045773 | Badness: 0.045773 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 624: | Line 611: | ||
Badness: 0.027322 | Badness: 0.027322 | ||
= Maviloid = | == Maviloid == | ||
{{see also| Ragismic microtemperaments #Parakleismic }} | {{see also| Ragismic microtemperaments #Parakleismic }} | ||
| Line 641: | Line 628: | ||
[[Badness]]: 0.057632 | [[Badness]]: 0.057632 | ||
= Subneutral = | == Subneutral == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 656: | Line 643: | ||
[[Badness]]: 0.045792 | [[Badness]]: 0.045792 | ||
= Osiris = | == Osiris == | ||
{{see also|Metric microtemperaments #Geb}} | {{see also|Metric microtemperaments #Geb}} | ||
| Line 673: | Line 660: | ||
[[Badness]]: 0.028307 | [[Badness]]: 0.028307 | ||
= Gorgik = | == Gorgik == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 688: | Line 675: | ||
[[Badness]]: 0.158384 | [[Badness]]: 0.158384 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 701: | Line 688: | ||
Badness: 0.059260 | Badness: 0.059260 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 714: | Line 701: | ||
Badness: 0.032205 | Badness: 0.032205 | ||
= Fibo = | == Fibo == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 729: | Line 716: | ||
Badness: 0.100511 | Badness: 0.100511 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 742: | Line 729: | ||
Badness: 0.056514 | Badness: 0.056514 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 755: | Line 742: | ||
Badness: 0.027429 | Badness: 0.027429 | ||
= 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. | ||
| Line 772: | Line 759: | ||
[[Badness]]: 0.125672 | [[Badness]]: 0.125672 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 785: | Line 772: | ||
Badness: 0.039962 | Badness: 0.039962 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 798: | Line 785: | ||
Badness: 0.021849 | Badness: 0.021849 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845 | Comma list: 243/242, 351/350, 441/440, 561/560, 847/845 | ||
| Line 811: | Line 798: | ||
Badness: 0.020295 | Badness: 0.020295 | ||
= Catafourth = | == Catafourth == | ||
{{see also| Sensipent family }} | {{see also| Sensipent family }} | ||
| Line 828: | Line 815: | ||
Badness: 0.079579 | Badness: 0.079579 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 841: | Line 828: | ||
Badness: 0.036785 | Badness: 0.036785 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 854: | Line 841: | ||
Badness: 0.021694 | Badness: 0.021694 | ||
= Cotritone = | == Cotritone == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 869: | Line 856: | ||
[[Badness]]: 0.098322 | [[Badness]]: 0.098322 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 882: | Line 869: | ||
Badness: 0.032225 | Badness: 0.032225 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 12:18, 3 June 2021
Breedsmic temperaments are rank two temperaments tempering out the breedsma, [-5 -1 -2 4⟩ = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12EDO, for example) which does not possess a neutral third cannot be tempering out the breedsma.
It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
Temperaments not discussed here include decimal, beatles, squares, myna, quasitemp, miracle, quadrimage, ennealimmal, octacot, quadritikleismic, sesquiquartififths, hemiwürschmidt, and neptune.
Hemififths
Hemififths tempers out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator, with 99EDO and 140EDO providing good tunings, and 239EDO an even better one; and other possible tunings are (160)^(1/25), giving just 5s, the 7 and 9 limit minimax tuning, or 14^(1/13), giving just 7s. It may be called the 41&58 temperament and has wedgie ⟨⟨ 2 25 13 35 15 -40 ]], which tells us that it requires 25 generator steps to get to the class for major thirds, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17 and 24 note MOS are suited. The full force of this highly accurate temperament can be found using the 41 note MOS or even the 34 note 2MOS.
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
Comma list: 2401/2400, 5120/5103
Mapping: [⟨1 1 -5 -1], ⟨0 2 25 13]]
Wedgie: ⟨⟨ 2 25 13 35 15 -40 ]]
POTE generator: ~49/40 = 351.477
- 7 and 9-limit minimax: ~49/40 = [1/5 0 1/25⟩
- [[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
- Eigenmonzos: 2, 5
Algebraic generator: (2 + sqrt(2))/2
Badness: 0.022243
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 896/891
Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]
POTE generator: ~11/9 = 351.521
Vals: Template:Val list
Badness: 0.023498
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 243/242, 364/363
Mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]
POTE generator: ~11/9 = 351.573
Vals: Template:Val list
Badness: 0.019090
Semihemi
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3388/3375, 9801/9800
Mapping: [⟨2 0 -35 -15 -47], ⟨0 2 25 13 34]]
POTE generator: ~49/40 = 351.505
Vals: Template:Val list
Badness: 0.042487
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 1716/1715
Mapping: [⟨2 0 -35 -15 -47 -37], ⟨0 2 25 13 34 28]]
POTE generator: ~49/40 = 351.502
Vals: Template:Val list
Badness: 0.021188
Tertiaseptal
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 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
Comma list: 2401/2400, 65625/65536
Mapping: [⟨1 3 2 3], ⟨0 -22 5 -3]]
Wedgie: ⟨⟨ 22 -5 3 -59 -57 21 ]]
POTE generator: ~256/245 = 77.191
Badness: 0.012995
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 65625/65536
Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]
POTE generator: ~256/245 = 77.227
Vals: Template:Val list
Badness: 0.035576
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 625/624, 3584/3575
Mapping: [⟨1 3 2 3 7 1], ⟨0 -22 5 -3 -55 42]]
POTE generator: ~117/112 = 77.203
Vals: Template:Val list
Badness: 0.036876
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
Mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]
POTE generator: ~68/65 = 77.201
Vals: Template:Val list
Badness: 0.027398
Tertia
Subgroup:2.3.5.7.11
Comma list: 385/384, 1331/1323, 1375/1372
Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]
POTE generator: ~22/21 = 77.173
Vals: Template:Val list
Badness: 0.030171
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 1331/1323
Mapping: [⟨1 3 2 3 5 1], ⟨0 -22 5 -3 -24 42]]
POTE generator: ~22/21 = 77.158
Vals: Template:Val list
Badness: 0.028384
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]
POTE generator: ~22/21 = 77.162
Vals: Template:Val list
Badness: 0.022416
Hemitert
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 65625/65536
Mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]
POTE generator: ~45/44 = 38.596
Vals: Template:Val list
Badness: 0.015633
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Mapping: [⟨1 3 2 3 6 1], ⟨0 -44 10 -6 -79 84]]
POTE generator: ~45/44 = 38.588
Vals: Template:Val list
Badness: 0.033573
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Mapping: [⟨1 3 2 3 6 1 1], ⟨0 -44 10 -6 -79 84 96]]
POTE generator: ~45/44 = 38.589
Vals: Template:Val list
Badness: 0.025298
Harry
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&72 temperament, with wedgie ⟨⟨ 12 34 20 26 -2 -49 ]]. The period is half an octave, and the generator 21/20, with generator tunings of 9\130 or 14\202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.
Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is ⟨⟨ 12 34 20 30 ... ]].
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with ⟨⟨ 12 34 20 30 52 ... ]] as the octave wedgie. 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
Comma list: 2401/2400, 19683/19600
Mapping: [⟨2 4 7 7], ⟨0 -6 -17 -10]]
Wedgie: ⟨⟨ 12 34 20 26 -2 -49 ]]
POTE generator: ~21/20 = 83.156
Badness: 0.034077
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4000/3993
Mapping: [⟨2 4 7 7 9], ⟨0 -6 -17 -10 -15]]
POTE generator: ~21/20 = 83.167
Vals: Template:Val list
Badness: 0.015867
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 676/675
Mapping: [⟨2 4 7 7 9 11], ⟨0 -6 -17 -10 -15 -26]]
POTE generator: ~21/20 = 83.116
Vals: Template:Val list
Badness: 0.013046
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 289/288, 351/350, 441/440
Mapping: [⟨2 4 7 7 9 11 9], ⟨0 -6 -17 -10 -15 -26 -6]]
POTE generator: ~21/20 = 83.168
Vals: Template:Val list
Badness: 0.012657
Quasiorwell
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = [22 -1 -10 1⟩. It has a wedgie ⟨⟨ 38 -3 8 -93 -94 27 ]]. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.
Adding 3025/3024 extends to the 11-limit and gives ⟨⟨ 38 -3 8 64 ... ]] for the initial wedgie, and as expected, 270 remains an excellent tuning.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 29360128/29296875
Mapping: [⟨1 31 0 9], ⟨0 -38 3 -8]]
POTE generator: ~1024/875 = 271.107
Badness: 0.035832
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 5632/5625
Mapping: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]
POTE generator: ~90/77 = 271.111
Vals: Template:Val list
Badness: 0.017540
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Mapping: [⟨1 31 0 9 53 -59], ⟨0 -38 3 -8 -64 81]]
POTE generator: ~90/77 = 271.107
Vals: Template:Val list
Badness: 0.017921
Decoid
In addition to 2401/2400, decoid tempers out 67108864/66976875 to temper 15/14 into one degree of 10EDO. 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 temperament.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 67108864/66976875
Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]
Wedgie: ⟨⟨ 20 -30 -10 -94 -72 61 ]]
POTE generator: ~8/7 = 231.099
Badness: 0.033902
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 5832/5825, 9801/9800
Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]
POTE generator: ~8/7 = 231.070
Vals: Template:Val list
Badness: 0.018735
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 4225/4224
Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]
POTE generator: ~8/7 = 231.083
Vals: Template:Val list
Badness: 0.013475
Neominor
The generator for neominor temperament is tridecimal minor third 13/11, also known as "Neo-gothic minor third".
Subgroup: 2.3.5.7
Comma list: 2401/2400, 177147/175616
Mapping: [⟨1 3 12 8], ⟨0 -6 -41 -22]]
Wedgie: ⟨⟨ 6 41 22 51 18 -64 ]]
POTE generator: ~189/160 = 283.280
Badness: 0.088221
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 35937/35840
Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]
POTE generator: ~33/28 = 283.276
Vals: Template:Val list
Badness: 0.027959
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 243/242, 364/363, 441/440
Mapping: [⟨1 3 12 8 7 7], ⟨0 -6 -41 -22 -15 -14]]
POTE generator: ~13/11 = 283.294
Vals: Template:Val list
Badness: 0.026942
Emmthird
The generator for emmthird temperament is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 14348907/14336000
Mapping: [⟨1 -3 -17 -8], ⟨0 14 59 33]]
Wedgie: ⟨⟨ 14 59 33 61 13 -89 ]]
POTE generator: ~2744/2187 = 392.988
Badness: 0.016736
Quinmite
The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1959552/1953125
Mapping: [⟨1 -7 -5 -3], ⟨0 34 29 23]]
Wedgie: ⟨⟨ 34 29 23 -33 -59 -28 ]]
POTE generator: ~25/21 = 302.997
Badness: 0.037322
Unthirds
The generator for unthirds temperament is undecimal major third, 14/11.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 68359375/68024448
Mapping: [⟨1 -13 -14 -9], ⟨0 42 47 34]]
Wedgie: ⟨⟨ 42 47 34 -23 -64 -53 ]]
POTE generator: ~3969/3125 = 416.717
Badness: 0.075253
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4000/3993
Map: [⟨1 -13 -14 -9 -8], ⟨0 42 47 34 33]]
POTE generator: ~14/11 = 416.718
Vals: Template:Val list
Badness: 0.022926
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Mapping: [⟨1 -13 -14 -9 -9 -47], ⟨0 42 47 34 33 146]]
POTE generator: ~14/11 = 416.716
Vals: Template:Val list
Badness: 0.020888
Newt
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
Comma list: 2401/2400, 33554432/33480783
Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]
Wedgie: ⟨⟨ 2 -57 -28 -95 -50 95 ]]
POTE generator: ~49/40 = 351.113
Badness: 0.041878
11-limit
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 19712/19683
Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]
POTE generator: ~49/40 = 351.115
Vals: Template:Val list
Badness: 0.019461
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]
POTE generator: ~49/40 = 351.117
Vals: Template:Val list
Badness: 0.013830
Amicable
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1600000/1594323
Mapping: [⟨1 3 6 5], ⟨0 -20 -52 -31]]
Wedgie: ⟨⟨ 20 52 31 36 -7 -74 ]]
POTE generator: ~21/20 = 84.880
Badness: 0.045473
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.
Subgroup: 2.3.5.7
Comma list: 2401/2400, 2152828125/2147483648
Mapping: [⟨1 -1 6 4], ⟨0 26 -37 -12]]
Wedgie: ⟨⟨ 26 -37 -12 -119 -92 76 ]]
POTE generator: ~15/14 = 119.297
Badness: 0.044115
Sedia
The sedia temperament (10&161, named by Xenllium) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 939524096/935859375
Mapping: [⟨1 -1 6 4 -3], ⟨0 26 -37 -12 65]]
POTE generator: ~15/14 = 119.279
Vals: Template:Val list
Badness: 0.090687
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Mapping: [⟨1 -1 6 4 -3 4], ⟨0 26 -37 -12 65 -3]]
POTE generator: ~15/14 = 119.281
Vals: Template:Val list
Badness: 0.045773
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Mapping: [⟨1 -1 6 4 -3 4 2], ⟨0 26 -37 -12 65 -3 21]]
POTE generator: ~15/14 = 119.281
Vals: Template:Val list
Badness: 0.027322
Maviloid
Subgroup: 2.3.5.7
Comma list: 2401/2400, 1224440064/1220703125
Mapping: [⟨1 31 34 26], ⟨0 -52 -56 -41]]
Wedgie: ⟨⟨ 52 56 41 -32 -81 -62 ]]
POTE generator: ~1296/875 = 678.810
Badness: 0.057632
Subneutral
Subgroup: 2.3.5.7
Comma list: 2401/2400, 274877906944/274658203125
Mapping: [⟨1 19 0 6], ⟨0 -60 8 -11]]
Wedgie: ⟨⟨ 60 -8 11 -152 -151 48 ]]
POTE generator: ~57344/46875 = 348.301
Badness: 0.045792
Osiris
Subgroup: 2.3.5.7
Comma list: 2401/2400, 31381059609/31360000000
Mapping: [⟨1 13 33 21], ⟨0 -32 -86 -51]]
Wedgie: ⟨⟨ 32 86 51 62 -9 -123 ]]
POTE generator: ~2800/2187 = 428.066
Badness: 0.028307
Gorgik
Subgroup: 2.3.5.7
Comma list: 2401/2400, 28672/28125
Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]
Wedgie: ⟨⟨ 18 -7 1 -53 -49 22 ]]
POTE generator: ~8/7 = 227.512
Badness: 0.158384
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 2401/2400, 2560/2541
Mapping: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]
POTE generator: ~8/7 = 227.500
Vals: Template:Val list
Badness: 0.059260
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 196/195, 364/363, 512/507
Mapping: [⟨1 5 1 3 1 2], ⟨0 -18 7 -1 13 9]]
POTE generator: ~8/7 = 227.493
Vals: Template:Val list
Badness: 0.032205
Fibo
Subgroup: 2.3.5.7
Comma list: 2401/2400, 341796875/339738624
Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]
Wedgie: ⟨⟨ 46 15 19 -83 -99 2 ]]
POTE generator: ~125/96 = 454.310
Badness: 0.100511
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 43923/43750
Mapping: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]
POTE generator: ~100/77 = 454.318
Vals: Template:Val list
Badness: 0.056514
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 385/384, 625/624, 847/845, 1375/1372
Mapping: [⟨1 19 8 10 8 9], ⟨0 -46 -15 -19 -12 -14]]
POTE generator: ~13/10 = 454.316
Vals: Template:Val list
Badness: 0.027429
Mintone
In addition to 2401/2400, mintone tempers out 177147/175000 = [-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
Comma list: 2401/2400, 177147/175000
Mapping: [⟨1 5 9 7], ⟨0 -22 -43 -27]]
Wedgie: ⟨⟨ 22 43 27 17 -19 -58 ]]
POTE generator: ~10/9 = 186.343
Badness: 0.125672
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 43923/43750
Mapping: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]
POTE generator: ~10/9 = 186.345
Vals: Template:Val list
Badness: 0.039962
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 847/845
Mapping: [⟨1 5 9 7 12 11], ⟨0 -22 -43 -27 -55 -47]]
POTE generator: ~10/9 = 186.347
Vals: Template:Val list
Badness: 0.021849
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Mapping: [⟨1 5 9 7 12 11 3], ⟨0 -22 -43 -27 -55 -47 7]]
POTE generator: ~10/9 = 186.348
Vals: Template:Val list
Badness: 0.020295
Catafourth
Subgroup: 2.3.5.7
Comma list: 2401/2400, 78732/78125
Mapping: [⟨1 13 17 13], ⟨0 -28 -36 -25]]
Wedgie: ⟨⟨ 28 36 25 -8 -39 -43 ]]
POTE generator: ~250/189 = 489.235
Badness: 0.079579
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 78408/78125
Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]
POTE generator: ~250/189 = 489.252
Vals: Template:Val list
Badness: 0.036785
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 10985/10976
Mapping: [⟨1 13 17 13 32 9], ⟨0 -28 -36 -25 -70 -13]]
POTE generator: ~65/49 = 489.256
Vals: Template:Val list
Badness: 0.021694
Cotritone
Subgroup: 2.3.5.7
Comma list: 2401/2400, 390625/387072
Mapping: [⟨1 -13 -4 -4], ⟨0 30 13 14]]
Wedgie: ⟨⟨ 30 13 14 -49 -62 -4 ]]
POTE generator: ~7/5 = 583.385
Badness: 0.098322
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4000/3993
Mapping: [⟨1 -13 -4 -4 2], ⟨0 30 13 14 3]]
POTE generator: ~7/5 = 583.387
Vals: Template:Val list
Badness: 0.032225
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 364/363, 385/384, 625/624
Mapping: [⟨1 -13 -4 -4 2 -7], ⟨0 30 13 14 3 22]]
POTE generator: ~7/5 = 583.387
Vals: Template:Val list
Badness: 0.028683