Ragismic microtemperaments: Difference between revisions
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Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Maja family|maja]], [[Schismatic family #Pontiac|pontiac]], [[Tricot family|trillium]], [[Würschmidt family #Whirrschmidt|whirrschmidt]], [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]]. | Temperaments not discussed here include [[Jubilismic clan #Crepuscular|crepuscular]], [[Meantone family #Flattone|flattone]], [[Porcupine family #Hystrix|hystrix]], [[Starling temperaments #Sensi|sensi]], [[Gamelismic clan #Unidec|unidec]], [[Orwellismic temperaments #Quartonic|quartonic]], [[Kleismic family #Catakleismic|catakleismic]], [[Tetracot family #Modus|modus]], [[Maja family|maja]], [[Schismatic family #Pontiac|pontiac]], [[Tricot family|trillium]], [[Würschmidt family #Whirrschmidt|whirrschmidt]], [[Gravity family #Zarvo|zarvo]], [[Vishnuzmic family #Vishnu|vishnu]], and [[Vulture family #Vulture|vulture]]. | ||
== Ennealimmal | == Ennealimmal == | ||
{{main|Ennealimmal}} | {{main|Ennealimmal}} | ||
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Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference. | Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 EDOs, though its hardly likely anyone could tell the difference. | ||
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example. | If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS. | ||
[[Comma list]]: 2401/2400, 4375/4374 | [[Comma list]]: 2401/2400, 4375/4374 | ||
[[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}] | [[Mapping]]: [{{val|9 1 1 12}}, {{val|0 2 3 2}}] | ||
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[[POTE generator]]s: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980 | [[POTE generator]]s: ~36/35 = 49.0205; ~10/9 = 182.354; ~6/5 = 315.687; ~49/40 = 350.980 | ||
[[Tuning ranges]]: | |||
* [[Diamond monotone]] range: [26.667, 66.667] (1\45 to 1\18) | |||
* [[Diamond tradeoff]] range: [48.920, 49.179] | |||
* Diamond monotone and tradeoff: [48.920, 49.179] | |||
{{Val list|legend=1| 27, 45, 72, 99, 171, 441, 612 }} | {{Val list|legend=1| 27, 45, 72, 99, 171, 441, 612 }} | ||
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[[Badness]]: 0.003610 | [[Badness]]: 0.003610 | ||
=== 11-limit | === 11-limit === | ||
The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma). | The ennealimmal temperament can be described as 99e&270 temperament, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma). | ||
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Badness: 0.027332 | Badness: 0.027332 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374 | Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374 | ||
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Badness: 0.029404 | Badness: 0.029404 | ||
=== Ennealimmia | === Ennealimmia === | ||
Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&270, named by [[User:Xenllium|Xenllium]]). | Ennealimmal temperament has various extensions to the 11-limit. Tempering out 131072/130977 (salururu comma) leads to the ''ennealimmia'' temperament (171&270, named by [[User:Xenllium|Xenllium]]). | ||
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Badness: 0.026463 | Badness: 0.026463 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374 | Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374 | ||
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Badness: 0.016607 | Badness: 0.016607 | ||
=== Ennealimnic | === Ennealimnic === | ||
Comma list: 243/242, 441/440, 4375/4356 | Comma list: 243/242, 441/440, 4375/4356 | ||
Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}] | Mapping: [{{val|9 1 1 12 -2}}, {{val|0 2 3 2 5}}] | ||
POTE generator: ~36/35 = 49.395 | POTE generator: ~36/35 = 49.395 | ||
Tuning ranges: | |||
* Diamond monotone range: [44.444, 53.333] (1\27 to 2\45) | |||
* Diamond tradeoff range: [48.920, 52.592] | |||
* Diamond monotone and tradeoff: [48.920, 52.592] | |||
Vals: {{Val list| 72, 171, 243 }} | Vals: {{Val list| 72, 171, 243 }} | ||
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Badness: 0.020347 | Badness: 0.020347 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 243/242, 364/363, 441/440, 625/624 | Comma list: 243/242, 364/363, 441/440, 625/624 | ||
Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}] | Mapping: [{{val|9 1 1 12 -2 -33}}, {{val|0 2 3 2 5 10}}] | ||
POTE generator: ~36/35 = 49.341 | POTE generator: ~36/35 = 49.341 | ||
Tuning ranges: | |||
* Diamond monotone range: [48.485, 50.000] (4\99 to 3\72) | |||
* Diamond tradeoff range: [48.825, 52.592] | |||
* Diamond monotone and tradeoff: [48.825, 50.000] | |||
Vals: {{Val list| 72, 171, 243 }} | Vals: {{Val list| 72, 171, 243 }} | ||
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Badness: 0.023250 | Badness: 0.023250 | ||
===== 17-limit | ===== 17-limit ===== | ||
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594 | Comma list: 243/242, 364/363, 375/374, 441/440, 595/594 | ||
Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}] | Mapping: [{{val|9 1 1 12 -2 -33 -3}}, {{val|0 2 3 2 5 10 6}}] | ||
POTE generator: ~36/35 = 49.335 | POTE generator: ~36/35 = 49.335 | ||
Tuning ranges: | |||
* Diamond monotone range: [48.485, 50.000] (4\99 to 3\72) | |||
* Diamond tradeoff range: [46.363, 52.592] | |||
* Diamond monotone and tradeoff: [48.485, 50.000] | |||
Vals: {{Val list| 72, 171, 243 }} | Vals: {{Val list| 72, 171, 243 }} | ||
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Badness: 0.014602 | Badness: 0.014602 | ||
==== Ennealim | ==== Ennealim ==== | ||
Comma list: 169/168, 243/242, 325/324, 441/440 | Comma list: 169/168, 243/242, 325/324, 441/440 | ||
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Badness: 0.020697 | Badness: 0.020697 | ||
=== Ennealiminal | === Ennealiminal === | ||
Comma list: 385/384, 1375/1372, 4375/4374 | Comma list: 385/384, 1375/1372, 4375/4374 | ||
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Badness: 0.031123 | Badness: 0.031123 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 169/168, 325/324, 385/384, 1375/1372 | Comma list: 169/168, 325/324, 385/384, 1375/1372 | ||
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Badness: 0.030325 | Badness: 0.030325 | ||
=== Hemiennealimmal | === Hemiennealimmal === | ||
Comma list: 2401/2400, 3025/3024, 4375/4374 | Comma list: 2401/2400, 3025/3024, 4375/4374 | ||
Mapping: [{{val|18 0 -1 22 48}}, {{val|0 2 3 2 1}}] | Mapping: [{{val|18 0 -1 22 48}}, {{val|0 2 3 2 1}}] | ||
POTE generator: ~99/98 = 17.6219 | POTE generator: ~99/98 = 17.6219 | ||
Tuning ranges: | |||
* Diamond monotone range: [13.333, 22.222] (1\90 to 1\54) | |||
* Diamond tradeoff range: [17.304, 17.985] | |||
* Diamond monotone and tradeoff: [17.304, 17.985] | |||
Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }} | Vals: {{Val list| 72, 198, 270, 342, 612, 954, 1566 }} | ||
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Badness: 0.006283 | Badness: 0.006283 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024 | Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024 | ||
Mapping: [{{val|18 0 -1 22 48 -19}}, {{val|0 2 3 2 1 6}}] | Mapping: [{{val|18 0 -1 22 48 -19}}, {{val|0 2 3 2 1 6}}] | ||
POTE generator ~99/98 = 17.7504 | POTE generator ~99/98 = 17.7504 | ||
Tuning ranges: | |||
* Diamond monotone range: [16.667, 22.222] (1\72 to 1\54) | |||
* Diamond tradeoff range: [17.304, 18.309] | |||
* Diamond monotone and tradeoff: [17.304, 18.309] | |||
Vals: {{Val list| 72, 198, 270 }} | Vals: {{Val list| 72, 198, 270 }} | ||
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Badness: 0.012505 | Badness: 0.012505 | ||
==== Semihemiennealimmal | ==== Semihemiennealimmal ==== | ||
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374 | Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374 | ||
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Badness: 0.013104 | Badness: 0.013104 | ||
=== Semiennealimmal | === Semiennealimmal === | ||
Comma list: 2401/2400, 4000/3993, 4375/4374 | Comma list: 2401/2400, 4000/3993, 4375/4374 | ||
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Badness: 0.034196 | Badness: 0.034196 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374 | Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374 | ||
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Badness: 0.026122 | Badness: 0.026122 | ||
=== Quadraennealimmal | === Quadraennealimmal === | ||
Comma list: 2401/2400, 4375/4374, 234375/234256 | Comma list: 2401/2400, 4375/4374, 234375/234256 | ||
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Badness: 0.021320 | Badness: 0.021320 | ||
=== Trinealimmal | === Trinealimmal === | ||
Comma list: 2401/2400, 4375/4374, 2097152/2096325 | Comma list: 2401/2400, 4375/4374, 2097152/2096325 | ||
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Badness: 0.029812 | Badness: 0.029812 | ||
== Gamera | == Gamera == | ||
[[Comma list]]: 4375/4374, 589824/588245 | [[Comma list]]: 4375/4374, 589824/588245 | ||
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[[Badness]]: 0.037648 | [[Badness]]: 0.037648 | ||
=== Hemigamera | === Hemigamera === | ||
Comma list: 3025/3024, 4375/4374, 589824/588245 | Comma list: 3025/3024, 4375/4374, 589824/588245 | ||
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Badness: 0.040955 | Badness: 0.040955 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024 | Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024 | ||
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Badness: 0.020416 | Badness: 0.020416 | ||
== Supermajor | == Supermajor == | ||
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS. | The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of {{multival|37 46 75 -13 15 45}}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS. | ||
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[[Badness]]: 0.010836 | [[Badness]]: 0.010836 | ||
=== Semisupermajor | === Semisupermajor === | ||
Comma list: 3025/3024, 4375/4374, 35156250/35153041 | Comma list: 3025/3024, 4375/4374, 35156250/35153041 | ||
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Badness: 0.012773 | Badness: 0.012773 | ||
== Enneadecal | == Enneadecal == | ||
Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning. | Enneadecal temperament tempers out the enneadeca, {{monzo|-14 -19 19}}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo|19EDO]] up to just ones. [[171edo|171EDO]] is a good tuning for either the 5 or 7 limits, and [[494edo|494EDO]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo|665EDO]] for a tuning. | ||
[[Comma list]]: 4375/4374, 703125/702464 | [[Comma list]]: 4375/4374, 703125/702464 | ||
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[[Badness]]: 0.010954 | [[Badness]]: 0.010954 | ||
=== Hemienneadecal | === Hemienneadecal === | ||
Comma list: 3025/3024, 4375/4374, 234375/234256 | Comma list: 3025/3024, 4375/4374, 234375/234256 | ||
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Badness: 0.009985 | Badness: 0.009985 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213 | Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213 | ||
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Badness: 0.030391 | Badness: 0.030391 | ||
== Deca | == Deca == | ||
Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu). | Deca temperament has a period of 1/10 octave and tempers out the [[15/14ths equal temperament #Linus temperaments|linus comma]], {{monzo|11 -10 -10 10}} and {{monzo|12 -3 -14 9}} = 165288374272/164794921875 (satritrizo-asepbigu). | ||
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[[Badness]]: 0.080637 | [[Badness]]: 0.080637 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 4375/4374, 422576/421875 | Comma list: 3025/3024, 4375/4374, 422576/421875 | ||
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Badness: 0.024329 | Badness: 0.024329 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374 | ||
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Badness: 0.016810 | Badness: 0.016810 | ||
== Mitonic | == Mitonic == | ||
{{see also|Minortonic family #Mitonic}} | {{see also|Minortonic family #Mitonic}} | ||
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[[Badness]]: 0.025184 | [[Badness]]: 0.025184 | ||
== Sfourth | == Sfourth == | ||
[[Comma list]]: 4375/4374, 64827/64000 | [[Comma list]]: 4375/4374, 64827/64000 | ||
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[[Badness]]: 0.123291 | [[Badness]]: 0.123291 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 121/120, 441/440, 4375/4374 | Comma list: 121/120, 441/440, 4375/4374 | ||
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Badness: 0.054098 | Badness: 0.054098 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 121/120, 169/168, 325/324, 441/440 | Comma list: 121/120, 169/168, 325/324, 441/440 | ||
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Badness: 0.033067 | Badness: 0.033067 | ||
=== Sfour | === Sfour === | ||
Comma list: 385/384, 2401/2376, 4375/4374 | Comma list: 385/384, 2401/2376, 4375/4374 | ||
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Badness: 0.076567 | Badness: 0.076567 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 196/195, 364/363, 385/384, 4375/4374 | Comma list: 196/195, 364/363, 385/384, 4375/4374 | ||
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Badness: 0.051893 | Badness: 0.051893 | ||
== Abigail | == Abigail == | ||
[[Comma list]]: 4375/4374, 2147483648/2144153025 | [[Comma list]]: 4375/4374, 2147483648/2144153025 | ||
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[[Badness]]: 0.037000 | [[Badness]]: 0.037000 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 4375/4374, 20614528/20588575 | Comma list: 3025/3024, 4375/4374, 20614528/20588575 | ||
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Badness: 0.012860 | Badness: 0.012860 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095 | Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095 | ||
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Badness: 0.008856 | Badness: 0.008856 | ||
== Semidimi | == Semidimi == | ||
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374. | The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo|-12 -73 55}} and 7-limit 3955078125/3954653486, as well as 4375/4374. | ||
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[[Badness]]: 0.754866 | [[Badness]]: 0.754866 | ||
=== 7-limit | === 7-limit === | ||
[[Comma list]]: 4375/4374, 3955078125/3954653486 | [[Comma list]]: 4375/4374, 3955078125/3954653486 | ||
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[[Badness]]: 0.015075 | [[Badness]]: 0.015075 | ||
== Brahmagupta | == Brahmagupta == | ||
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625. | The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo|47 -7 -7 -7}} = 140737488355328 / 140710042265625. | ||
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[[Badness]]: 0.029122 | [[Badness]]: 0.029122 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 4000/3993, 4375/4374, 131072/130977 | Comma list: 4000/3993, 4375/4374, 131072/130977 | ||
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Badness: 0.052190 | Badness: 0.052190 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374 | Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374 | ||
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Badness: 0.023132 | Badness: 0.023132 | ||
== Quasithird | == Quasithird == | ||
The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}. | The '''quasithird''' temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}. | ||
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[[Badness]]: 0.099519 | [[Badness]]: 0.099519 | ||
=== 7-limit | === 7-limit === | ||
[[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976 | [[Comma list]]: 4375/4374, 1153470752371588581/1152921504606846976 | ||
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[[Badness]]: 0.061813 | [[Badness]]: 0.061813 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296 | Comma list: 3025/3024, 4375/4374, 4296700485/4294967296 | ||
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Badness: 0.021125 | Badness: 0.021125 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195 | Comma list: 2200/2197, 3025/3024, 4375/4374, 468512/468195 | ||
| Line 602: | Line 602: | ||
Badness: 0.029501 | Badness: 0.029501 | ||
== Semidimfourth | == Semidimfourth == | ||
The '''semidimifourth''' temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375. | The '''semidimifourth''' temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375. | ||
| Line 615: | Line 615: | ||
[[Badness]]: 0.233376 | [[Badness]]: 0.233376 | ||
=== 7-limit | === 7-limit === | ||
[[Comma list]]: 4375/4374, 235298/234375 | [[Comma list]]: 4375/4374, 235298/234375 | ||
| Line 628: | Line 628: | ||
[[Badness]]: 0.055249 | [[Badness]]: 0.055249 | ||
=== Neusec | === Neusec === | ||
Comma list: 3025/3024, 4375/4374, 235298/234375 | Comma list: 3025/3024, 4375/4374, 235298/234375 | ||
| Line 639: | Line 639: | ||
Badness: 0.059127 | Badness: 0.059127 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374 | Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374 | ||
| Line 650: | Line 650: | ||
Badness: 0.030941 | Badness: 0.030941 | ||
== Acrokleismic | == Acrokleismic == | ||
[[Comma list]]: 4375/4374, 2202927104/2197265625 | [[Comma list]]: 4375/4374, 2202927104/2197265625 | ||
| Line 663: | Line 663: | ||
[[Badness]]: 0.056184 | [[Badness]]: 0.056184 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 4375/4374, 41503/41472, 172032/171875 | Comma list: 4375/4374, 41503/41472, 172032/171875 | ||
| Line 674: | Line 674: | ||
Badness: 0.036878 | Badness: 0.036878 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976 | Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976 | ||
| Line 685: | Line 685: | ||
Badness: 0.026818 | Badness: 0.026818 | ||
=== Counteracro | === Counteracro === | ||
Comma list: 4375/4374, 5632/5625, 117649/117612 | Comma list: 4375/4374, 5632/5625, 117649/117612 | ||
| Line 696: | Line 696: | ||
Badness: 0.042572 | Badness: 0.042572 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374 | Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374 | ||
| Line 707: | Line 707: | ||
Badness: 0.026028 | Badness: 0.026028 | ||
== Seniority | == Seniority == | ||
{{see also|Very high accuracy temperaments#Senior}} | {{see also|Very high accuracy temperaments#Senior}} | ||
| Line 722: | Line 722: | ||
[[Badness]]: 0.044877 | [[Badness]]: 0.044877 | ||
== Orga | == Orga == | ||
[[Comma list]]: 4375/4374, 54975581388800/54936068900769 | [[Comma list]]: 4375/4374, 54975581388800/54936068900769 | ||
| Line 735: | Line 735: | ||
[[Badness]]: 0.040236 | [[Badness]]: 0.040236 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 4375/4374, 5767168/5764801 | Comma list: 3025/3024, 4375/4374, 5767168/5764801 | ||
| Line 746: | Line 746: | ||
Badness: 0.016188 | Badness: 0.016188 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360 | Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360 | ||
| Line 757: | Line 757: | ||
Badness: 0.021762 | Badness: 0.021762 | ||
== Quatracot | == Quatracot == | ||
[[Comma list]]: 4375/4374, 1483154296875/1473173782528 | [[Comma list]]: 4375/4374, 1483154296875/1473173782528 | ||
| Line 770: | Line 770: | ||
[[Badness]]: 0.175982 | [[Badness]]: 0.175982 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 4375/4374, 1265625/1261568 | Comma list: 3025/3024, 4375/4374, 1265625/1261568 | ||
| Line 781: | Line 781: | ||
Badness: 0.041043 | Badness: 0.041043 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 625/624, 729/728, 1575/1573, 2200/2197 | Comma list: 625/624, 729/728, 1575/1573, 2200/2197 | ||
| Line 792: | Line 792: | ||
Badness: 0.022643 | Badness: 0.022643 | ||
== Octoid | == Octoid == | ||
The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99. | The '''octoid''' temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99. | ||
[[Comma list]]: 4375/4374, 16875/16807 | [[Comma list]]: 4375/4374, 16875/16807 | ||
[[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}] | [[Mapping]]: [{{val|8 1 3 3}}, {{val|0 3 4 5}}] | ||
| Line 809: | Line 804: | ||
[[POTE generator]]: ~7/5 = 583.940 | [[POTE generator]]: ~7/5 = 583.940 | ||
[[Tuning ranges]]: | |||
* [[Diamond monotone]] range: [578.571, 600.000] (27\56 to 4\8) | |||
* [[Diamond tradeoff]] range: [582.512, 584.359] | |||
* Diamond monotone and tradeoff: [582.512, 584.359] | |||
[[Vals]]: {{Val list| 8d, 72, 152, 224 }} | [[Vals]]: {{Val list| 8d, 72, 152, 224 }} | ||
| Line 816: | Line 816: | ||
Scales: [[Octoid72]], [[Octoid80]] | Scales: [[Octoid72]], [[Octoid80]] | ||
=== 11-limit | === 11-limit === | ||
Comma list: 540/539, 1375/1372, 4000/3993 | Comma list: 540/539, 1375/1372, 4000/3993 | ||
Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}] | Mapping: [{{val|8 1 3 3 16}}, {{val|0 3 4 5 3}}] | ||
POTE generator: ~7/5 = 583.962 | POTE generator: ~7/5 = 583.962 | ||
Tuning ranges: | |||
* Diamond monotone range: [581.250, 586.364] (31\64, 43\88) | |||
* Diamond tradeoff range: [582.512, 585.084] | |||
* diamond monotone and tradeoff: [582.512, 585.084] | |||
Vals: {{Val list| 72, 152, 224 }} | Vals: {{Val list| 72, 152, 224 }} | ||
| Line 834: | Line 834: | ||
Scales: [[Octoid72]], [[Octoid80]] | Scales: [[Octoid72]], [[Octoid80]] | ||
=== 13-limit | === 13-limit === | ||
Comma list: 540/539, 1375/1372, 4000/3993, 625/624 | Comma list: 540/539, 1375/1372, 4000/3993, 625/624 | ||
| Line 847: | Line 847: | ||
Scales: [[Octoid72]], [[Octoid80]] | Scales: [[Octoid72]], [[Octoid80]] | ||
==== Music | ==== Music ==== | ||
* [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] | * [http://www.archive.org/details/Dreyfus http://www.archive.org/details/Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] | ||
=== Octopus | === Octopus === | ||
Comma list: 169/168, 325/324, 364/363, 540/539 | Comma list: 169/168, 325/324, 364/363, 540/539 | ||
| Line 863: | Line 863: | ||
Scales: [[Octoid72]], [[Octoid80]] | Scales: [[Octoid72]], [[Octoid80]] | ||
== Amity | == Amity == | ||
{{main| Amity }} | {{main| Amity }} | ||
{{see also| Amity family #Amity }} | {{see also| Amity family #Amity }} | ||
| Line 883: | Line 883: | ||
[[Badness]]: 0.023649 | [[Badness]]: 0.023649 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 540/539, 4375/4374, 5120/5103 | Comma list: 540/539, 4375/4374, 5120/5103 | ||
| Line 894: | Line 894: | ||
Badness: 0.031506 | Badness: 0.031506 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 352/351, 540/539, 625/624, 847/845 | Comma list: 352/351, 540/539, 625/624, 847/845 | ||
| Line 905: | Line 905: | ||
Badness: 0.028008 | Badness: 0.028008 | ||
=== Hitchcock | === Hitchcock === | ||
{{see also|Amity family #Hitchcock}} | {{see also|Amity family #Hitchcock}} | ||
| Line 918: | Line 918: | ||
Badness: 0.035187 | Badness: 0.035187 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 121/120, 169/168, 176/175, 325/324 | Comma list: 121/120, 169/168, 176/175, 325/324 | ||
| Line 929: | Line 929: | ||
Badness: 0.022448 | Badness: 0.022448 | ||
=== Hemiamity | === Hemiamity === | ||
Comma list: 3025/3024, 4375/4374, 5120/5103 | Comma list: 3025/3024, 4375/4374, 5120/5103 | ||
| Line 940: | Line 940: | ||
Badness: 0.031307 | Badness: 0.031307 | ||
== Parakleismic | == Parakleismic == | ||
{{main| Parakleismic }} | {{main| Parakleismic }} | ||
| Line 955: | Line 955: | ||
[[Badness]]: 0.043279 | [[Badness]]: 0.043279 | ||
=== 7-limit | === 7-limit === | ||
[[Comma list]]: 3136/3125, 4375/4374 | [[Comma list]]: 3136/3125, 4375/4374 | ||
| Line 968: | Line 968: | ||
[[Badness]]: 0.027431 | [[Badness]]: 0.027431 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 385/384, 3136/3125, 4375/4374 | Comma list: 385/384, 3136/3125, 4375/4374 | ||
| Line 979: | Line 979: | ||
Badness: 0.049711 | Badness: 0.049711 | ||
=== Paralytic | === Paralytic === | ||
The ''paralytic'' temperament (118&217, named by [[User:FloraC|Flora Canou]]) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&217 tempers out 1001/1000, 1575/1573, and 3584/3575. | The ''paralytic'' temperament (118&217, named by [[User:FloraC|Flora Canou]]) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118&217 tempers out 1001/1000, 1575/1573, and 3584/3575. | ||
| Line 992: | Line 992: | ||
Badness: 0.036027 | Badness: 0.036027 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374 | Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374 | ||
| Line 1,003: | Line 1,003: | ||
Badness: 0.044710 | Badness: 0.044710 | ||
==== Paraklein | ==== Paraklein ==== | ||
The ''paraklein'' temperament (19e&118, named by [[User:Xenllium|Xenllium]]) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]]. | The ''paraklein'' temperament (19e&118, named by [[User:Xenllium|Xenllium]]) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]]. | ||
| Line 1,016: | Line 1,016: | ||
Badness: 0.037618 | Badness: 0.037618 | ||
=== Parkleismic | === Parkleismic === | ||
Comma list: 176/175, 1375/1372, 2200/2187 | Comma list: 176/175, 1375/1372, 2200/2187 | ||
| Line 1,027: | Line 1,027: | ||
Badness: 0.055884 | Badness: 0.055884 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 169/168, 176/175, 325/324, 1375/1372 | Comma list: 169/168, 176/175, 325/324, 1375/1372 | ||
| Line 1,038: | Line 1,038: | ||
Badness: 0.036559 | Badness: 0.036559 | ||
=== Paradigmic | === Paradigmic === | ||
Comma list: 540/539, 896/891, 3136/3125 | Comma list: 540/539, 896/891, 3136/3125 | ||
| Line 1,049: | Line 1,049: | ||
Badness: 0.041720 | Badness: 0.041720 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 169/168, 325/324, 540/539, 832/825 | Comma list: 169/168, 325/324, 540/539, 832/825 | ||
| Line 1,060: | Line 1,060: | ||
Badness: 0.035781 | Badness: 0.035781 | ||
=== Semiparakleismic | === Semiparakleismic === | ||
Comma list: 3025/3024, 3136/3125, 4375/4374 | Comma list: 3025/3024, 3136/3125, 4375/4374 | ||
| Line 1,071: | Line 1,071: | ||
Badness: 0.034208 | Badness: 0.034208 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374 | Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374 | ||
| Line 1,082: | Line 1,082: | ||
Badness: 0.033775 | Badness: 0.033775 | ||
==== Gentsemiparakleismic | ==== Gentsemiparakleismic ==== | ||
Comma list: 169/168, 325/324, 364/363, 3136/3125 | Comma list: 169/168, 325/324, 364/363, 3136/3125 | ||
| Line 1,093: | Line 1,093: | ||
Badness: 0.040467 | Badness: 0.040467 | ||
== Counterkleismic | == Counterkleismic == | ||
{{see also|Syntonic-enneadecal equivalence continuum #Counterhanson}} | {{see also|Syntonic-enneadecal equivalence continuum #Counterhanson}} | ||
| Line 1,110: | Line 1,110: | ||
[[Badness]]: 0.090553 | [[Badness]]: 0.090553 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 540/539, 4375/4374, 2097152/2096325 | Comma list: 540/539, 4375/4374, 2097152/2096325 | ||
| Line 1,121: | Line 1,121: | ||
Badness: 0.070952 | Badness: 0.070952 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 540/539, 625/624, 729/728, 10985/10976 | Comma list: 540/539, 625/624, 729/728, 10985/10976 | ||
| Line 1,132: | Line 1,132: | ||
Badness: 0.033874 | Badness: 0.033874 | ||
=== Counterlytic | === Counterlytic === | ||
Comma list: 1375/1372, 4375/4374, 496125/495616 | Comma list: 1375/1372, 4375/4374, 496125/495616 | ||
| Line 1,143: | Line 1,143: | ||
Badness: 0.065400 | Badness: 0.065400 | ||
==== 13-limit | ==== 13-limit ==== | ||
Comma list: 625/624, 729/728, 1375/1372, 10985/10976 | Comma list: 625/624, 729/728, 1375/1372, 10985/10976 | ||
| Line 1,154: | Line 1,154: | ||
Badness: 0.029782 | Badness: 0.029782 | ||
== Quincy | == Quincy == | ||
[[Comma list]]: 4375/4374, 823543/819200 | [[Comma list]]: 4375/4374, 823543/819200 | ||
| Line 1,167: | Line 1,167: | ||
[[Badness]]: 0.079657 | [[Badness]]: 0.079657 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 441/440, 4000/3993, 4375/4374 | Comma list: 441/440, 4000/3993, 4375/4374 | ||
| Line 1,178: | Line 1,178: | ||
Badness: 0.030875 | Badness: 0.030875 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 364/363, 441/440, 676/675, 4375/4374 | Comma list: 364/363, 441/440, 676/675, 4375/4374 | ||
| Line 1,189: | Line 1,189: | ||
Badness: 0.023862 | Badness: 0.023862 | ||
=== 17-limit | === 17-limit === | ||
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155 | Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155 | ||
| Line 1,200: | Line 1,200: | ||
Badness: 0.014741 | Badness: 0.014741 | ||
=== 19-limit | === 19-limit === | ||
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675 | Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675 | ||
| Line 1,211: | Line 1,211: | ||
Badness: 0.015197 | Badness: 0.015197 | ||
== Chlorine | == Chlorine == | ||
The name of chlorine temperament comes from Chlorine, the 17th element. | The name of chlorine temperament comes from Chlorine, the 17th element. | ||
| Line 1,226: | Line 1,226: | ||
[[Badness]]: 0.077072 | [[Badness]]: 0.077072 | ||
=== 7-limit | === 7-limit === | ||
[[Comma list]]: 4375/4374, 193119049072265625/193091834023510016 | [[Comma list]]: 4375/4374, 193119049072265625/193091834023510016 | ||
| Line 1,239: | Line 1,239: | ||
[[Badness]]: 0.041658 | [[Badness]]: 0.041658 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192 | Comma list: 4375/4374, 41503/41472, 1879453125/1879048192 | ||
| Line 1,250: | Line 1,250: | ||
Badness: 0.063706 | Badness: 0.063706 | ||
== Palladium | == Palladium == | ||
The name of ''palladium temperament'' (named by [[User:Xenllium|Xenllium]]) comes from Palladium, the 46th element. | The name of ''palladium temperament'' (named by [[User:Xenllium|Xenllium]]) comes from Palladium, the 46th element. | ||
| Line 1,267: | Line 1,267: | ||
[[Badness]]: 0.308505 | [[Badness]]: 0.308505 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 3025/3024, 9801/9800, 134775333/134217728 | Comma list: 3025/3024, 9801/9800, 134775333/134217728 | ||
| Line 1,278: | Line 1,278: | ||
Badness: 0.073783 | Badness: 0.073783 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364 | Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364 | ||
| Line 1,289: | Line 1,289: | ||
Badness: 0.040751 | Badness: 0.040751 | ||
=== 17-limit | === 17-limit === | ||
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | ||
| Line 1,300: | Line 1,300: | ||
Badness: 0.022441 | Badness: 0.022441 | ||
== Monzism | == Monzism == | ||
The ''monzism'' temperament (53&612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]]. | The ''monzism'' temperament (53&612, named by [[User:Xenllium|Xenllium]]) is a rank-two temperament which tempers out the [[monzisma]], {{monzo|54 -37 2}} and the [[nanisma]], {{monzo|109 -67 0 -1}}, as well as the ragisma, [[4375/4374]]. | ||
| Line 1,315: | Line 1,315: | ||
[[Badness]]: 0.046569 | [[Badness]]: 0.046569 | ||
=== 11-limit | === 11-limit === | ||
Comma list: 4375/4374, 41503/41472, 184549376/184528125 | Comma list: 4375/4374, 41503/41472, 184549376/184528125 | ||
| Line 1,326: | Line 1,326: | ||
Badness: 0.057083 | Badness: 0.057083 | ||
=== 13-limit | === 13-limit === | ||
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625 | Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625 | ||
| Line 1,351: | Line 1,351: | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Ragismic microtemperaments| ]] <!-- main article --> | [[Category:Ragismic microtemperaments| ]] <!-- main article --> | ||