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 | ||
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[[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 | ||
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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 | ||
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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 | ||
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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 | ||
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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 | ||
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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. | ||
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[[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}}] | ||
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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 | ||
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[[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>". | ||
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[[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. | ||
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[[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. | ||
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[[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 | ||