Ragismic microtemperaments: Difference between revisions
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* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]] | * ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]] | ||
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]] | * ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]] | ||
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]] | * ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]] | ||
== Supermajor == | == Supermajor == | ||
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{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }} | {{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 435.082{{c}} | ||
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }} | {{Optimal ET sequence|legend=1| 11, 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }} | ||
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Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }} | Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 435.082{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }} | ||
Badness (Sintel): 0.422 | Badness (Sintel): 0.422 | ||
== Enneadecal == | == Enneadecal == | ||
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].'' | |||
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)<sup>1/3</sup>. 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-limit, 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)<sup>1/3</sup>. 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-limit, 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }} | {{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }} | ||
: Mapping generators: ~28/27, ~3 | : Mapping generators: ~28/27, ~3 | ||
[[Optimal tuning]] ([[CTE]]): ~28/27 = | [[Optimal tuning]] ([[CTE]]): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9275{{c}} (~225/224 = 7.1907{{c}}) | ||
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }} | {{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }} | ||
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Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }} | Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }} | ||
Optimal tuning (CTE): ~28/27 = | Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 702.1483{{c}} (~225/224 = 7.4115{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }} | ||
Badness (Sintel): 1.446 | Badness (Sintel): 1.446 | ||
| Line 98: | Line 97: | ||
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }} | Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }} | ||
Optimal tuning (CTE): ~28/27 = | Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9258{{c}} (~225/224 = 7.1890{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }} | ||
Badness (Sintel): 1.386 | Badness (Sintel): 1.386 | ||
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Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }} | Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }} | ||
: mapping generators: ~55/54, ~3 | |||
Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9351{{c}} (~225/224 = 7.1983{{c}}) | |||
Optimal tuning (CTE): ~55/54 = | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }} | ||
Badness (Sintel): 0.330 | Badness (Sintel): 0.330 | ||
| Line 126: | Line 124: | ||
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }} | Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }} | ||
Optimal tuning (CTE): ~55/54 = | Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9955{{c}} (~225/224 = 7.2587{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152f, 342f, 494 }} | ||
Badness (Sintel): 0.859 | Badness (Sintel): 0.859 | ||
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Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }} | Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }} | ||
Optimal tuning (CTE): ~55/54 = | Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9812{{c}} (~225/224 = 7.2444{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }} | ||
Badness (Sintel): 1.256 | Badness (Sintel): 1.256 | ||
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Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }} | Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }} | ||
: mapping generators: ~55/54, ~429/250 | |||
Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~429/250 = 935.1789{{c}} (~144/143 = 12.1895{{c}}) | |||
{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }} | |||
{{Optimal ET sequence|legend= | |||
Badness (Sintel): 0.607 | Badness (Sintel): 0.607 | ||
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Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }} | Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }} | ||
Optimal tuning (CTE): ~28/27 = | Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.244{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 855, 988, 1843 }} | ||
Badness (Sintel): 3.153 | Badness (Sintel): 3.153 | ||
== Semidimi == | == Semidimi == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #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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{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }} | {{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}} | ||
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }} | {{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }} | ||
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== Brahmagupta == | == Brahmagupta == | ||
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] | The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}). | ||
Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/ | Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286 ¢) and 1/21-apotome generator (5.4136 ¢). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }} | {{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }} | ||
: mapping generators: ~1157625/1048576, ~27/20 | |||
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 171.429{{c}}, ~27/20 = 519.716{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = | |||
{{Optimal ET sequence|legend=1| 217, 224, 441, 1106, 1547 }} | {{Optimal ET sequence|legend=1| 217, 224, 441, 1106, 1547 }} | ||
| Line 218: | Line 214: | ||
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }} | Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }} | ||
Optimal tuning (POTE): ~243/220 = | Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.704{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771ee }} | ||
Badness (Sintel): 1.725 | Badness (Sintel): 1.725 | ||
| Line 231: | Line 227: | ||
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }} | Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }} | ||
Optimal tuning (POTE): ~243/220 = | Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.706{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771eef }} | ||
Badness (Sintel): 0.956 | Badness (Sintel): 0.956 | ||
== Abigail == | == Abigail == | ||
: ''For the 5-limit versino, see [[Miscellaneous 5-limit temperaments #Abigail]].'' | |||
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref> | Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref> | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }} | {{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }} | ||
: Mapping generators: ~46305/32768, ~27/20 | : Mapping generators: ~46305/32768, ~27/20 | ||
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = | [[Optimal tuning]] ([[POTE]]): ~46305/32768 = 600.000{{c}}, ~6912/6125 = 208.899{{c}} | ||
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }} | {{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }} | ||
| Line 263: | Line 258: | ||
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }} | Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~1155/1024 = 208.901{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }} | ||
Badness (Sintel): 0.425 | Badness (Sintel): 0.425 | ||
| Line 276: | Line 271: | ||
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }} | Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~44/39 = 208.903{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }} | ||
Badness: 0.366 | Badness (Sintel): 0.366 | ||
== Gamera == | == Gamera == | ||
''For the 5-limit | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].'' | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }} | {{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }} | ||
: mapping generators: ~2, ~8/7 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 230.336{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = | |||
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }} | {{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }} | ||
| Line 305: | Line 299: | ||
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }} | Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }} | ||
: mapping generators: ~99/70, ~8/7 | |||
Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3370{{c}} | |||
{{Optimal ET sequence|legend=0| 26, 198, 224, 422, 646, 1068d }} | |||
{{Optimal ET sequence|legend= | |||
Badness (Sintel): 1.354 | Badness (Sintel): 1.354 | ||
| Line 321: | Line 314: | ||
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }} | Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3373{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 198, 224, 422, 646f, 1068df }} | ||
Badness: 0.844 | Badness (Sintel): 0.844 | ||
=== Semigamera === | === Semigamera === | ||
| Line 333: | Line 326: | ||
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }} | Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }} | ||
: mapping generators: ~2, ~77/72 | |||
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1642{{c}} | |||
Optimal tuning (POTE): ~2 = | |||
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }} | {{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }} | ||
| Line 349: | Line 341: | ||
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }} | Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1628{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }} | ||
Badness (Sintel): 1.821 | Badness (Sintel): 1.821 | ||
| Line 365: | Line 357: | ||
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }} | {{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }} | ||
: Mapping generators: ~332150625/234881024, ~1125/1024 | : Mapping generators: ~332150625/234881024, ~1125/1024 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475 | * [[CTE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7475{{c}} | ||
* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }} | * [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }} | ||
* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474 | * [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}} | ||
* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }} | * error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }} | ||
| Line 386: | Line 377: | ||
Optimal tunings: | Optimal tunings: | ||
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485 | * CTE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7485{{c}} | ||
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481 | * CWE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7481{{c}} | ||
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }} | {{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }} | ||
| Line 399: | Line 390: | ||
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }} | {{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }} | ||
: Mapping generators: ~7411887/5242880, ~1310720/1058841 | : Mapping generators: ~7411887/5242880, ~1310720/1058841 | ||
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = | [[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 600.000{{c}}, ~8/7 = 231.104{{c}} | ||
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }} | {{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }} | ||
| Line 415: | Line 405: | ||
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }} | Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }} | ||
Badness (Sintel): 0.535 | Badness (Sintel): 0.535 | ||
| Line 428: | Line 418: | ||
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }} | Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }} | ||
Badness (Sintel): 0.899 | Badness (Sintel): 0.899 | ||
== Seniority == | == Seniority == | ||
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]]. | |||
Aside from the ragisma, the seniority temperament (26 & | Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }} | {{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3087/2560 = 322.804{{c}} | ||
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }} | {{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }} | ||
| Line 452: | Line 442: | ||
=== Senator === | === Senator === | ||
The senator temperament (26 & | The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 460: | Line 450: | ||
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }} | Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e, 487ee }} | ||
Badness (Sintel): 3.049 | Badness (Sintel): 3.049 | ||
| Line 473: | Line 463: | ||
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }} | Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }} | ||
Badness (Sintel): 1.845 | Badness (Sintel): 1.845 | ||
| Line 486: | Line 476: | ||
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }} | Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }} | ||
Optimal tuning (POTE): ~77/64 = 322.793 | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }} | ||
Badness (Sintel): 1.353 | Badness (Sintel): 1.353 | ||
== Monzismic == | == Monzismic == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]]. | ||
The monzismic temperament (53 & | The monzismic temperament (53 & 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 503: | Line 493: | ||
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }} | {{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| -27 11 3 1 }} = 249.0207{{c}} | ||
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }} | {{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }} | ||
| Line 516: | Line 506: | ||
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }} | Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }} | ||
Optimal tuning (POTE): ~231/200 = 249.0193 | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0193{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 559, 612 }} | ||
Badness (Sintel): 1.887 | Badness (Sintel): 1.887 | ||
| Line 529: | Line 519: | ||
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }} | Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }} | ||
Optimal tuning (POTE): ~231/200 = 249.0199 | Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0199{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 559, 612 }} | ||
Badness (Sintel): 2.222 | Badness (Sintel): 2.222 | ||
== Semidimfourth == | == Semidimfourth == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].'' | ||
The semidimfourth temperament is featured by a | The semidimfourth temperament is featured by a semidiminished 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 546: | Line 536: | ||
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }} | [[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/27 = 448.456{{c}} | ||
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }} | {{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }} | ||
| Line 559: | Line 549: | ||
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }} | Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~12/11 = 151.547{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8d, 190, 388 }} | ||
Badness (Sintel): 1.955 | Badness (Sintel): 1.955 | ||
| Line 572: | Line 562: | ||
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }} | Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 1200.000{{c}}, ~12/11 = 151.545{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 8d, 190, 198, 388 }} | ||
Badness: 1.279 | Badness (Sintel): 1.279 | ||
== Acrokleismic == | == Acrokleismic == | ||
| Line 584: | Line 574: | ||
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }} | {{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }} | ||
: mapping generators: ~2, ~6/5 | : mapping generators: ~2, ~6/5 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}} | ||
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }} | {{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }} | ||
| Line 600: | Line 589: | ||
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }} | Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.558{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }} | ||
Badness (Sintel): 1.219 | Badness (Sintel): 1.219 | ||
| Line 613: | Line 602: | ||
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }} | Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 251, 270 }} | ||
Badness (Sintel): 1.108 | Badness (Sintel): 1.108 | ||
| Line 626: | Line 615: | ||
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }} | Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.553{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }} | ||
Badness (Sintel): 1.407 | Badness (Sintel): 1.407 | ||
| Line 639: | Line 628: | ||
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }} | Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.554{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }} | ||
Badness (Sintel): 1.076 | Badness (Sintel): 1.076 | ||
| Line 653: | Line 642: | ||
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }} | {{Mapping|legend=1| 4 0 -11 | 0 5 16 }} | ||
: mapping generators: ~51200000/43046721, ~1594323/1280000 | |||
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721 = 300.000{{c}}, ~1594323/1280000 = 380.395{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395 | |||
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }} | {{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }} | ||
| Line 669: | Line 657: | ||
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }} | {{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }} | ||
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = | [[Optimal tuning]] ([[POTE]]): ~65536/55125 = 300.000{{c}}, ~5103/4096 = 380.388 {{c}} | ||
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }} | {{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }} | ||
| Line 682: | Line 670: | ||
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }} | Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }} | ||
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387) | Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~5103/4096 = 380.387{{c}} (or ~22/21 = 80.387{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448 }} | ||
Badness (Sintel): 0.698 | Badness (Sintel): 0.698 | ||
| Line 695: | Line 683: | ||
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }} | Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }} | ||
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385) | Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~81/65 = 380.385{{c}} (or ~22/21 = 80.385{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }} | ||
Badness (Sintel): 1.219 | Badness (Sintel): 1.219 | ||
== Deca == | == Deca == | ||
: ''For 5-limit version | : ''For 5-limit version, see [[10th-octave temperaments #Neon]].'' | ||
Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu). | Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu). | ||
| Line 711: | Line 699: | ||
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }} | {{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }} | ||
: mapping generators: ~15/14, ~6/5 | |||
[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~6/5 = 315.577{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~15/14 = | |||
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }} | {{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }} | ||
[[Badness]] | [[Badness]] (Sintel): 2.041 | ||
=== 11-limit === | === 11-limit === | ||
| Line 729: | Line 714: | ||
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }} | Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }} | ||
Optimal tuning (POTE): ~15/14 = | Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.582{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }} | ||
Badness (Sintel): 0.804 | Badness (Sintel): 0.804 | ||
| Line 744: | Line 727: | ||
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }} | Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }} | ||
Optimal tuning (POTE): ~15/14 = | Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.602{{c}} (~40/39 = 44.398{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }} | ||
Badness (Sintel): 0.695 | Badness (Sintel): 0.695 | ||
=== 2.3.5.7.11.13.19 === | === 2.3.5.7.11.13.19 subgroup === | ||
Subgroup: 2.3.5.7.11.13.19 | Subgroup: 2.3.5.7.11.13.19 | ||
| Line 759: | Line 740: | ||
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }} | Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }} | ||
Optimal tuning (CTE): ~15/14 = | Optimal tuning (CTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.581{{c}} (~39/38 = 44.419{{c}}) | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }} | ||
Badness (Sintel): 0.556 | Badness (Sintel): 0.556 | ||
| Line 773: | Line 754: | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }} | {{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }} | ||
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4465{{c}} | |||
[[Optimal tuning]] ([[CTE]]): ~2 = | |||
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }} | {{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }} | ||
| Line 789: | Line 769: | ||
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }} | Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }} | ||
Badness (Sintel): 1.020 | Badness (Sintel): 1.020 | ||
| Line 802: | Line 782: | ||
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }} | Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4466{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }} | ||
Badness (Sintel): 0.879 | Badness (Sintel): 0.879 | ||
== Aluminium == | == Aluminium == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].'' | ||
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave. | Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave. | ||
| Line 818: | Line 798: | ||
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }} | [[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }} | ||
: Mapping generators: ~135/128, ~3 | : Mapping generators: ~135/128, ~3 | ||
[[Optimal tuning]] ([[CTE]]): ~135/128 = | [[Optimal tuning]] ([[CTE]]): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0024{{c}} | ||
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }} | {{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }} | ||
| Line 834: | Line 813: | ||
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }} | Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }} | ||
Optimal tuning (CTE): ~135/128 = | Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0042{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }} | ||
Badness (Sintel): 1.393 | Badness (Sintel): 1.393 | ||
| Line 847: | Line 826: | ||
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }} | Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }} | ||
Optimal tuning (CTE): ~135/128 = | Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0099{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }} | ||
Badness (Sintel): 1.180 | Badness (Sintel): 1.180 | ||
== Countritonic == | == Countritonic == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].'' | ||
Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. | Countritonic (''co-un-tritonic'') can be described as the {{nowrap| 53 & 422 }} temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 863: | Line 842: | ||
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }} | {{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }} | ||
: mapping generators: ~2, ~45927/32768 | |||
[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~45927/32768 = 588.6216{{c}} | |||
[[Optimal tuning]] (CTE): ~2 = | |||
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }} | {{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }} | ||
| Line 879: | Line 857: | ||
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }} | Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~539/384 = 588.6258{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }} | ||
Badness (Sintel): 2.336 | Badness (Sintel): 2.336 | ||
| Line 892: | Line 870: | ||
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }} | Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~128/91 = 588.6277{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }} | ||
Badness (Sintel): 1.514 | Badness (Sintel): 1.514 | ||
| Line 906: | Line 884: | ||
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }} | {{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }} | ||
: mapping generators: ~2278125/1605632, ~448/405 | |||
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 600.000{{c}}, ~448/405 = 176.805{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = | |||
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }} | {{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }} | ||
| Line 922: | Line 899: | ||
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }} | Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~448/405 = 176.806{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1052c }} | ||
Badness (Sintel): 1.357 | Badness (Sintel): 1.357 | ||
| Line 935: | Line 912: | ||
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }} | Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~195/176 = 176.804{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 190, 224, 414, 638, 1690bcc, 2328bccde }} | ||
Badness (Sintel): 0.936 | Badness (Sintel): 0.936 | ||
== Moulin == | == Moulin == | ||
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament. | Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the {{nowrap| 494 & 1619 }} temperament. Since 11/8 is within 23 generators, the 25-tone mos (4L 21s) of this temperament contains the 8:11:13 triad. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 949: | Line 926: | ||
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }} | {{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }} | ||
: Mapping generators: ~2, ~6422528/3796875 | : Mapping generators: ~2, ~6422528/3796875 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6422528/3796875 = 910.9323{{c}} | ||
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }} | {{Optimal ET sequence|legend=1| 494, 1125, 1619 }} | ||
| Line 965: | Line 941: | ||
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }} | Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~1024/605 = 910.9323{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }} | ||
Badness (Sintel): 2.240 | Badness (Sintel): 2.240 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 980: | Line 954: | ||
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }} | Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }} | ||
Optimal tuning (CTE): ~2 = | Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~22/13 = 910.9323{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }} | ||
Badness (Sintel): 1.118 | Badness (Sintel): 1.118 | ||
== Palladium == | == Palladium == | ||
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''. | : ''For the 5-limit version of this temperament, see [[46th-octave temperaments #Palladium]]''. | ||
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 | The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 996: | Line 970: | ||
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }} | {{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }} | ||
: Mapping generators: ~83349/81920, ~3 | : Mapping generators: ~83349/81920, ~3 | ||
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = | [[Optimal tuning]] ([[POTE]]): ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6074{{c}} | ||
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }} | {{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }} | ||
| Line 1,012: | Line 985: | ||
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }} | Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }} | ||
Optimal tuning (POTE): ~8192/8085 = | Optimal tuning (POTE): ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.5951{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de }} | ||
Badness (Sintel): 2.439 | Badness (Sintel): 2.439 | ||
| Line 1,025: | Line 998: | ||
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }} | Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }} | ||
Optimal tuning (POTE): ~65/64 = | Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6419{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334de }} | ||
Badness (Sintel): 1.684 | Badness (Sintel): 1.684 | ||
| Line 1,038: | Line 1,011: | ||
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }} | Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }} | ||
Optimal tuning (POTE): ~65/64 = | Optimal tuning (POTE): ~65/64 = 26.0870{{c}}, ~3/2 = 701.6425{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334deg }} | ||
Badness (Sintel): 1.143 | Badness (Sintel): 1.143 | ||
== Oviminor == | == Oviminor == | ||
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].'' | |||
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate. | Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate. | ||
| Line 1,054: | Line 1,027: | ||
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }} | {{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }} | ||
: Mapping generators: ~2, ~6/5 | : Mapping generators: ~2, ~6/5 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~6/5 = 315.7501{{c}} | ||
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }} | {{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }} | ||
| Line 1,064: | Line 1,036: | ||
== Octoid == | == Octoid == | ||
: ''For the 5-limit | : ''For the 5-limit version, see [[8th-octave temperaments #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. | ||
| Line 1,073: | Line 1,045: | ||
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }} | {{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }} | ||
: Mapping generators: ~49/45, ~7/5 | : Mapping generators: ~49/45, ~7/5 | ||
[[Optimal tuning]] ([[POTE]]): ~49/45 = | [[Optimal tuning]] ([[POTE]]): ~49/45 = 150.000{{c}}, ~7/5 = 583.940{{c}} | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 1,091: | Line 1,062: | ||
=== 11-limit === | === 11-limit === | ||
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus | The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus – not octoid – as 80edo does not temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,099: | Line 1,070: | ||
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }} | Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.962{{c}} | ||
Tuning ranges: | Tuning ranges: | ||
| Line 1,105: | Line 1,076: | ||
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084] | * 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152, 224 }} | ||
Badness (Sintel): 0.466 | Badness (Sintel): 0.466 | ||
| Line 1,118: | Line 1,089: | ||
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }} | Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.905{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152f, 224 }} | ||
Badness (Sintel): 0.631 | Badness (Sintel): 0.631 | ||
| Line 1,136: | Line 1,107: | ||
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }} | Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.842{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }} | ||
Badness (Sintel): 0.729 | Badness (Sintel): 0.729 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 1,151: | Line 1,120: | ||
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }} | Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.932{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152fg, 224 }} | ||
Badness (Sintel): 0.975 | Badness (Sintel): 0.975 | ||
==== Octopus ==== | ==== Octopus ==== | ||
| Line 1,168: | Line 1,135: | ||
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }} | Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.892{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152, 224f }} | ||
Badness (Sintel): 0.0896 | Badness (Sintel): 0.0896 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 1,183: | Line 1,148: | ||
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }} | Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.811{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152, 224fg, 296ffg }} | ||
Badness (Sintel): 0.795 | Badness (Sintel): 0.795 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 1,198: | Line 1,161: | ||
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }} | Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 584.064{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 152, 224fg, 376ffgh }} | ||
Badness (Sintel): 0.993 | Badness (Sintel): 0.993 | ||
| Line 1,207: | Line 1,170: | ||
==== Hexadecoid ==== | ==== Hexadecoid ==== | ||
{{ See also | 16th-octave temperaments }} | {{See also| 16th-octave temperaments }} | ||
Hexadecoid (80 & | Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,216: | Line 1,179: | ||
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }} | Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }} | ||
: mapping generators: ~448/429, ~7/5 | : mapping generators: ~448/429, ~7/5 | ||
Optimal tuning (POTE): ~448/429 = | Optimal tuning (POTE): ~448/429 = 75.000{{c}}, ~13/8 = 841.015{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 144, 224 }} | ||
Badness (Sintel): 1.273 | Badness (Sintel): 1.273 | ||
| Line 1,232: | Line 1,194: | ||
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }} | Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }} | ||
Optimal tuning (POTE): ~117/112 = | Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.932{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }} | ||
Badness (Sintel): 1.458 | Badness (Sintel): 1.458 | ||
| Line 1,245: | Line 1,207: | ||
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }} | Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }} | ||
Optimal tuning (POTE): ~117/112 = | Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.896{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }} | ||
Badness (Sintel): 1.443 | Badness (Sintel): 1.443 | ||
| Line 1,253: | Line 1,215: | ||
== Parakleismic == | == Parakleismic == | ||
{{Main| Parakleismic }} | {{Main| Parakleismic }} | ||
: ''For the 5-limit | : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].'' | ||
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118. | In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118. | ||
| Line 1,262: | Line 1,224: | ||
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }} | {{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }} | ||
: mapping generators: ~2, ~6/5 | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.181{{c}} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = | |||
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }} | {{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }} | ||
| Line 1,277: | Line 1,239: | ||
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }} | Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.251{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 99, 118 }} | ||
Badness (Sintel): 1.643 | Badness (Sintel): 1.643 | ||
=== Paralytic === | === Paralytic === | ||
The | The paralytic temperament (118 & 217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,292: | Line 1,254: | ||
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }} | Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.220{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 99e, 118, 217, 335, 552d, 887dd }} | ||
Badness (Sintel): 1.191 | Badness (Sintel): 1.191 | ||
| Line 1,305: | Line 1,267: | ||
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }} | Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.214{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 99e, 118, 217, 552d, 769de }} | ||
Badness (Sintel): 1.847 | Badness (Sintel): 1.847 | ||
==== Paraklein ==== | ==== Paraklein ==== | ||
The | The paraklein temperament (19e & 118) 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]]. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,320: | Line 1,282: | ||
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }} | Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.225{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 99ef, 118, 217ff, 335ff }} | ||
Badness (Sintel): 1.554 | Badness (Sintel): 1.554 | ||
| Line 1,333: | Line 1,295: | ||
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }} | Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.060{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 80, 179, 259cd }} | ||
Badness (Sintel): 1.848 | Badness (Sintel): 1.848 | ||
| Line 1,346: | Line 1,308: | ||
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }} | Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.075{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 80, 179 }} | ||
Badness (Sintel): 1.511 | Badness (Sintel): 1.511 | ||
| Line 1,359: | Line 1,321: | ||
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }} | Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.096{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }} | ||
Badness (Sintel): 1.379 | Badness (Sintel): 1.379 | ||
| Line 1,372: | Line 1,334: | ||
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }} | Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.080{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }} | ||
Badness (Sintel): 1.479 | Badness (Sintel): 1.479 | ||
| Line 1,385: | Line 1,347: | ||
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }} | Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.181{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c, 830c }} | ||
Badness (Sintel): 1.131 | Badness (Sintel): 1.131 | ||
| Line 1,400: | Line 1,362: | ||
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }} | Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }} | ||
Optimal tuning (POTE): ~99/70 = | Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.156{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 118, 198 }} | ||
Badness (Sintel): 1.396 | Badness (Sintel): 1.396 | ||
| Line 1,415: | Line 1,377: | ||
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }} | Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }} | ||
Optimal tuning (POTE): ~55/39 = | Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 315.184{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 80, 118f, 198f }} | ||
Badness (Sintel): 1.672 | Badness (Sintel): 1.672 | ||
| Line 1,424: | Line 1,386: | ||
: ''For the 5-limit temperament, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].'' | : ''For the 5-limit temperament, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].'' | ||
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 & | In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,431: | Line 1,393: | ||
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }} | {{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }} | ||
: Mapping generators: ~2, ~5/3 | : Mapping generators: ~2, ~5/3 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.060{{c}} | ||
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }} | {{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }} | ||
| Line 1,447: | Line 1,408: | ||
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }} | Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.071{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 205, 224 }} | ||
Badness (Sintel): 2.346 | Badness (Sintel): 2.346 | ||
| Line 1,460: | Line 1,421: | ||
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }} | Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.070{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }} | ||
Badness (Sintel): 1.400 | Badness (Sintel): 1.400 | ||
| Line 1,473: | Line 1,434: | ||
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }} | Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}} | ||
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | {{Optimal ET sequence|legend=1| 19e, 205e, 224 }} | ||
| Line 1,486: | Line 1,447: | ||
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }} | Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 19e, 205e, 224 }} | ||
Badness (Sintel): 1.231 | Badness (Sintel): 1.231 | ||
| Line 1,499: | Line 1,460: | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }} | {{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.613{{c}} | ||
{{Optimal ET sequence|legend=1| 72, 217, 289 }} | {{Optimal ET sequence|legend=1| 72, 217, 289 }} | ||
| Line 1,512: | Line 1,473: | ||
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }} | Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.613{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 217, 289 }} | ||
Badness (Sintel): 1.021 | Badness (Sintel): 1.021 | ||
| Line 1,525: | Line 1,486: | ||
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }} | Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }} | ||
Badness (Sintel): 0.986 | Badness (Sintel): 0.986 | ||
| Line 1,538: | Line 1,499: | ||
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }} | Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 145, 217, 289 }} | ||
Badness (Sintel): 0.751 | Badness (Sintel): 0.751 | ||
| Line 1,551: | Line 1,512: | ||
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }} | Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.594{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 72, 145, 217 }} | ||
Badness (Sintel): 0.924 | Badness (Sintel): 0.924 | ||
| Line 1,566: | Line 1,527: | ||
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }} | {{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 26.287{{c}} | ||
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | {{Optimal ET sequence|legend=1| 45, 46, 91, 137d }} | ||
| Line 1,579: | Line 1,540: | ||
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }} | Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.286{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }} | ||
Badness (Sintel): 1.788 | Badness (Sintel): 1.788 | ||
| Line 1,592: | Line 1,553: | ||
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }} | Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.310{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def }} | ||
Badness (Sintel): 1.366 | Badness (Sintel): 1.366 | ||
| Line 1,605: | Line 1,566: | ||
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }} | Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.246{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 45, 46, 91, 137d }} | ||
Badness (Sintel): 2.531 | Badness (Sintel): 2.531 | ||
| Line 1,618: | Line 1,579: | ||
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }} | Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.239{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 45, 46, 91, 137d }} | ||
Badness (Sintel): 2.144 | Badness (Sintel): 2.144 | ||
| Line 1,627: | Line 1,588: | ||
: ''For the 5-limit version of this temperament, see [[13th-octave temperaments #Tridecatonic]].'' | : ''For the 5-limit version of this temperament, see [[13th-octave temperaments #Tridecatonic]].'' | ||
The trideci temperament (26 & | The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]"). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,635: | Line 1,596: | ||
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }} | {{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~256/245 = | [[Optimal tuning]] ([[POTE]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.1410{{c}} | ||
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }} | {{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }} | ||
| Line 1,648: | Line 1,609: | ||
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }} | Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }} | ||
Optimal tuning (POTE): ~22/21 = | Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.6179{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 65, 91, 156d, 247cdde }} | ||
Badness (Sintel): 2.796 | Badness (Sintel): 2.796 | ||
| Line 1,661: | Line 1,622: | ||
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }} | Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }} | ||
Optimal tuning (POTE): ~22/21 = | Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.2969{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 26, 65f, 91f, 156dff }} | ||
Badness (Sintel): 2.164 | Badness (Sintel): 2.164 | ||
| Line 1,670: | Line 1,631: | ||
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]]. | Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]]. | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }} | [[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }} | ||
{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }} | |||
Optimal tuning (CTE): ~2 = | [[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}} | ||
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }} | {{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }} | ||
Badness (Sintel): 7.916 | [[Badness]] (Sintel): 7.916 | ||
== | == References == | ||
[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||