Ragismic microtemperaments: Difference between revisions

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{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  


Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:  
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:  
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
Line 28: Line 29:
* ''[[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 ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/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 is not enough notes for you, there is always the 171-note mos.
The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/2<sup>22</sup>, 46 give 5/2<sup>27</sup>, and 75 give 7/2<sup>45</sup>. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 37: Line 38:
[[Comma list]]: 4375/4374, 52734375/52706752
[[Comma list]]: 4375/4374, 52734375/52706752


{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
: mapping generators: ~2, ~14/9


{{Multival|legend=1| 37 46 75 -13 15 45 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}
: [[error map]]: {{val| +0.007 +0.019 -0.074 +0.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}
: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}


{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}
[[Badness]] (Sintel): 0.274
 
[[Badness]]: 0.010836


=== Semisupermajor ===
=== Semisupermajor ===
Line 52: Line 56:
Comma list: 3025/3024, 4375/4374, 35156250/35153041
Comma list: 3025/3024, 4375/4374, 35156250/35153041


Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
: mapping generators: ~99/70, ~11/10


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
Optimal tunings:
* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}


{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}


Badness: 0.012773
Badness (Sintel): 0.422


== 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)<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.
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''


''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
Enneadecal 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 [[6/5|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 }}
{{Multival|legend=1| 19 19 57 -14 37 79 }}
: mapping generators: ~28/27, ~3
: mapping generators: ~28/27, ~3


[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})
: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}
* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})
: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}


{{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 }}


[[Badness]]: 0.010954
[[Badness]] (Sintel): 0.277


=== 11-limit ===
=== 11-limit ===
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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 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
Optimal tunings:
* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})


{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}


Badness: 0.043734
Badness (Sintel): 1.45


==== 13-limit ====
==== 13-limit ====
Line 101: Line 113:
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 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
Optimal tunings:
* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})


{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}


Badness: 0.033545
Badness (Sintel): 1.39


=== Hemienneadecal ===
=== Hemienneadecal ===
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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
: mapping generators: ~55/54, ~3


Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
Optimal tunings:
* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})


{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}


Badness: 0.009985
Badness (Sintel): 0.330


==== Hemienneadecalis ====
==== Hemienneadecalis ====
Line 129: Line 144:
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 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
Optimal tunings:
* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})


{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}


Badness: 0.020782
Badness (Sintel): 0.859


==== Hemienneadec ====
==== Hemienneadec ====
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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 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
Optimal tunings:
* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})


{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}


Badness: 0.030391
Badness (Sintel): 1.26


==== Semihemienneadecal ====
==== Semihemienneadecal ====
Line 154: Line 173:


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


: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
Optimal tunings:  
 
* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})


{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}


Badness: 0.014694
Badness (Sintel): 0.607


=== Kalium ===
=== Kalium ===
Line 172: Line 192:
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 = 1\19, ~6545/5928 = 171.244
Optimal tunings:
* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}
 
{{Optimal ET sequence|legend=0| 855, 988, 1843 }}


{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
Badness (Sintel): 3.15


== Semidimi ==
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
: ''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 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4375/4374, 3955078125/3954653486
[[Comma list]]: 4375/4374, 3955078125/3954653486


{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
: mapping generators: ~2, ~35/27


{{Multival|legend=1| 55 73 93 -12 -7 11 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}
: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
{{Optimal ET sequence|legend=1| 8d, …, 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 }}
[[Badness]] (Sintel): 0.382


[[Badness]]: 0.015075
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament.  


== Brahmagupta ==
Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|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{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta 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{{c}}) and 1/21-apotome generator (5.4136{{c}}).
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  
{{todo | expand | inline=1 | comment=Explain the relationship to Sagittal's Athenian system.}}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 70368744177664/70338939985125
[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}


{{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
: mapping generators: ~1157625/1048576, ~27/20


{{Multival|legend=1| 21 56 -77 40 -181 -336 }}
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}


[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}


{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
[[Badness]] (Sintel): 0.737
 
[[Badness]]: 0.029122


=== 11-limit ===
=== 11-limit ===
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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 = 1\7, ~27/20 = 519.704
Optimal tunings:
* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}


{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}


Badness: 0.052190
Badness (Sintel): 1.73


=== 13-limit ===
=== 13-limit ===
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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 = 1\7, ~27/20 = 519.706
Optimal tunings:
* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}


{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}


Badness: 0.023132
Badness (Sintel): 0.956


== Abigail ==
== 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>
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''
 
Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings.  


''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "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." —Gene Ward Smith</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 250: Line 285:
[[Comma list]]: 4375/4374, 2147483648/2144153025
[[Comma list]]: 4375/4374, 2147483648/2144153025


{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
: mapping generators: ~46305/32768, ~1536/1225


: mapping generators: ~46305/32768, ~27/20
[[Optimal tuning]]s:  
 
* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
{{Multival|legend=1| 22 48 -38 25 -122 -223 }}
: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
 
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}


{{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, 6428bcdd, 8226bbcddd }}


[[Badness]]: 0.037000
[[Badness]] (Sintel): 0.936


=== 11-limit ===
=== 11-limit ===
Line 267: Line 303:
Comma list: 3025/3024, 4375/4374, 131072/130977
Comma list: 3025/3024, 4375/4374, 131072/130977


Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
Optimal tunings:
* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}


{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}


Badness: 0.012860
Badness (Sintel): 0.425


=== 13-limit ===
=== 13-limit ===
Line 280: Line 318:
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
Optimal tunings:
* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}


{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}


Badness: 0.008856
Badness (Sintel): 0.366


== Gamera ==
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 295: Line 335:
[[Comma list]]: 4375/4374, 589824/588245
[[Comma list]]: 4375/4374, 589824/588245


{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
: mapping generators: ~2, ~7/4


: mapping generators: ~2, ~8/7
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
{{Multival|legend=1| 23 40 1 10 -63 -110 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336


{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}


[[Badness]]: 0.037648
[[Badness]] (Sintel): 0.953


=== Hemigamera ===
=== Hemigamera ===
Line 312: Line 351:
Comma list: 3025/3024, 4375/4374, 589824/588245
Comma list: 3025/3024, 4375/4374, 589824/588245


Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
: mapping generators: ~99/70, ~99/80


: mapping generators: ~99/70, ~8/7
Optimal tunings:  
* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}


Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}


{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}
Badness (Sintel): 1.35
 
Badness: 0.040955


==== 13-limit ====
==== 13-limit ====
Line 327: Line 367:
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024


Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
Optimal tunings:
* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}


{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}


Badness: 0.020416
Badness (Sintel): 0.844


=== Semigamera ===
=== Semigamera ===
Line 340: Line 382:
Comma list: 4375/4374, 14641/14580, 15488/15435
Comma list: 4375/4374, 14641/14580, 15488/15435


Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
: mapping generators: ~2, ~144/77


: mapping generators: ~2, ~77/72
Optimal tunings:  
* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}


{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
Badness (Sintel): 2.59
 
Badness: 0.078


==== 13-limit ====
==== 13-limit ====
Line 355: Line 398:
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580


Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
Optimal tunings:
* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}


{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}


Badness: 0.044
Badness (Sintel): 1.82


== Crazy ==
== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an obvious tuning.  
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.
 
Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 373: Line 420:


{{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
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
* [[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| 0.000 +0.024 -0.051 -0.022 }}


{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}


[[Badness]] (Smith): 0.0394
[[Badness]] (Sintel): 0.998


=== 11-limit ===
=== 11-limit ===
Line 394: Line 440:


Optimal tunings:
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481
* CWE: ~99/70 = 600.0000{{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 }}


Badness (Smith): 0.0170
Badness (Sintel): 0.562


== Orga ==
== Orga ==
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 54975581388800/54936068900769
[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}


{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
: mapping generators: ~7411887/5242880, ~8/7


: mapping generators: ~7411887/5242880, ~1310720/1058841
[[Optimal tuning]]s:  
* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}


{{Multival|legend=1| 58 102 -2 27 -166 -291 }}
{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}


[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
[[Badness]] (Sintel): 1.02
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
 
[[Badness]]: 0.040236


=== 11-limit ===
=== 11-limit ===
Line 423: Line 472:
Comma list: 3025/3024, 4375/4374, 5767168/5764801
Comma list: 3025/3024, 4375/4374, 5767168/5764801


Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Optimal tunings:
* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}


{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}


Badness: 0.016188
Badness (Sintel): 0.535


=== 13-limit ===
=== 13-limit ===
Line 436: Line 487:
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360


Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
Optimal tunings:
* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}


{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}


Badness: 0.021762
Badness (Sintel): 0.899


== Seniority ==
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].


Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 453: Line 506:
[[Comma list]]: 4375/4374, 201768035/201326592
[[Comma list]]: 4375/4374, 201768035/201326592


{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087


{{Multival|legend=1| 35 62 -3 17 -103 -181 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}


{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
[[Badness]] (Sintel): 1.14
 
[[Badness]]: 0.044877


=== Senator ===
=== Senator ===
The senator temperament (26 &amp; 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.
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and 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 470: Line 526:
Comma list: 441/440, 4375/4374, 65536/65219
Comma list: 441/440, 4375/4374, 65536/65219


Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Optimal tunings:
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}


{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}


Badness: 0.092238
Badness (Sintel): 3.05


==== 13-limit ====
==== 13-limit ====
Line 483: Line 541:
Comma list: 364/363, 441/440, 2200/2197, 4375/4374
Comma list: 364/363, 441/440, 2200/2197, 4375/4374


Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
Optimal tunings:
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}


{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}


Badness: 0.044662
Badness (Sintel): 1.85


==== 17-limit ====
==== 17-limit ====
Line 496: Line 556:
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197


Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}


Optimal tuning (POTE): ~77/64 = 322.793
Optimal tunings:  
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}


{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}


Badness: 0.026562
Badness (Sintel): 1.35


== Monzismic ==
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].  
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  


The monzismic temperament (53 &amp; 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]].  
Monzismic 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]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. 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 513: Line 575:
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}


{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}


{{Multival|legend=1| 2 37 -134 54 -218 -415 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}


{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}


[[Badness]]: 0.046569
[[Badness]] (Sintel): 1.18


=== Monzism ===
=== Monzism ===
Line 528: Line 593:
Comma list: 4375/4374, 41503/41472, 184549376/184528125
Comma list: 4375/4374, 41503/41472, 184549376/184528125


Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}


Optimal tuning (POTE): ~231/200 = 249.0193
Optimal tunings:  
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}


{{Optimal ET sequence|legend=1| 53, 559, 612 }}
{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}


Badness: 0.057083
Badness (Sintel): 1.89


==== 13-limit ====
==== 13-limit ====
Line 541: Line 608:
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625


Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}


Optimal tuning (POTE): ~231/200 = 249.0199
Optimal tunings:  
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}


{{Optimal ET sequence|legend=1| 53, 559, 612 }}
{{Optimal ET sequence|legend=0| 53, 559, 612 }}


Badness: 0.053780
Badness (Sintel): 2.22


== Semidimfourth ==
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''


The semidimfourth 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 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 558: Line 627:
[[Comma list]]: 4375/4374, 235298/234375
[[Comma list]]: 4375/4374, 235298/234375


[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
: mapping generators: ~2, ~35/27


{{Multival|legend=1| 31 41 53 -7 -3 8 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}


{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
[[Badness]] (Sintel): 1.40
 
[[Badness]]: 0.055249


=== Neusec ===
=== Neusec ===
Line 573: Line 645:
Comma list: 3025/3024, 4375/4374, 235298/234375
Comma list: 3025/3024, 4375/4374, 235298/234375


Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
: mapping generators: ~99/70, ~35/27


Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
Optimal tunings:
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}


{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}


Badness: 0.059127
Badness (Sintel): 1.95


==== 13-limit ====
==== 13-limit ====
Line 586: Line 661:
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374


Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
Optimal tunings:
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}


{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}


Badness: 0.030941
Badness (Sintel): 1.28


== Acrokleismic ==
== Acrokleismic ==
Line 599: Line 676:
[[Comma list]]: 4375/4374, 2202927104/2197265625
[[Comma list]]: 4375/4374, 2202927104/2197265625


{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
: mapping generators: ~2, ~5/3


: mapping generators: ~2, ~6/5
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
{{Multival|legend=1| 32 33 92 -22 56 121 }}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}


{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}


[[Badness]]: 0.056184
[[Badness]] (Sintel): 1.42


=== 11-limit ===
=== 11-limit ===
Line 616: Line 694:
Comma list: 4375/4374, 41503/41472, 172032/171875
Comma list: 4375/4374, 41503/41472, 172032/171875


Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
Optimal tunings:
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}


{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}


Badness: 0.036878
Badness (Sintel): 1.22


==== 13-limit ====
==== 13-limit ====
Line 629: Line 709:
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976


Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
Optimal tunings:
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}


{{Optimal ET sequence|legend=1| 19, 251, 270 }}
{{Optimal ET sequence|legend=0| 19, 251, 270 }}


Badness: 0.026818
Badness (Sintel): 1.11


=== Counteracro ===
=== Counteracro ===
Line 642: Line 724:
Comma list: 4375/4374, 5632/5625, 117649/117612
Comma list: 4375/4374, 5632/5625, 117649/117612


Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
Optimal tunings:
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}


{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}


Badness: 0.042572
Badness (Sintel): 1.41


==== 13-limit ====
==== 13-limit ====
Line 655: Line 739:
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374


Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
Optimal tunings:
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}


{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}


Badness: 0.026028
Badness (Sintel): 1.08


== 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}}.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 55 -64 20 }}
 
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
 
: mapping generators: ~51200000/43046721, ~1594323/1280000
 
[[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 }}
Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, 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 it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.


[[Badness]]: 0.099519
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 686: Line 759:


{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
: mapping generators: ~65536/55125, ~5103/4096


{{Multival|legend=1| 20 64 -116 55 -240 -449 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}


{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}


[[Badness]]: 0.061813
[[Badness]] (Sintel): 1.56


=== 11-limit ===
=== 11-limit ===
Line 702: Line 778:
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 tunings:
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})


{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}


Badness: 0.021125
Badness (Sintel): 0.698


=== 13-limit ===
=== 13-limit ===
Line 715: Line 793:
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 tunings:
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})


{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}


Badness: 0.029501
Badness (Sintel): 1.22


== Deca ==
== Deca ==
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''
: ''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 has a period of 1/10 octave and tempers out the neon comma {{monzo| 21 60 -50 }} in the 5-limit, the [[linus comma]]{{monzo| 11 -10 -10 10 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 731: Line 811:


{{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, ~460992/390625


: mapping generators: ~15/14, ~6/5
[[Optimal tuning]]s:  
 
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
{{Multival|legend=1| 50 60 110 -21 34 87 }}
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
 
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}


{{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]]: 0.080637
[[Badness]] (Sintel): 2.04
 
Badness (Dirichlet): 2.041


=== 11-limit ===
=== 11-limit ===
Line 751: Line 830:
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 = 1\10, ~6/5 = 315.582
Optimal tunings:
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})


{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}


Badness: 0.024329
Badness (Sintel): 0.804
 
Badness (Dirichlet): 0.804


=== 13-limit ===
=== 13-limit ===
Line 766: Line 845:
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 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
Optimal tunings:
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})


{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}


Badness: 0.016810
Badness (Sintel): 0.695


Badness (Dirichlet): 0.695
=== 2.3.5.7.11.13.19 subgroup ===
 
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13.19


Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374


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 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
Optimal tunings:
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})


{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}


Badness (Dirichlet): 0.556
Badness (Sintel): 0.556


== Keenanose ==
== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 795: Line 876:


{{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 }}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}
: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}
: error map: {{val| 0.000 +0.025 -0.043 -0.050 }}


{{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 }}


[[Badness]]: 0.0858
[[Badness]] (Sintel): 2.17


=== 11-limit ===
=== 11-limit ===
Line 811: Line 895:
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 = 1\1, ~385/384 = 4.4465
Optimal tunings:
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}


{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}


Badness: 0.0308
Badness (Sintel): 1.02


=== 13-limit ===
=== 13-limit ===
Line 824: Line 910:
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 = 1\1, ~385/384 = 4.4466
Optimal tunings:
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}


{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}


Badness: 0.0213
Badness (Sintel): 0.879


== Aluminium ==
== 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.
: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''


[[Subgroup]]: 2.3.5
Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.


[[Comma list]]: {{monzo| 92 -39 -13 }}
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
: mapping generators: ~135/128, ~3
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
[[Badness]]: 0.123
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 853: Line 928:


[[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


[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024
[[Optimal tuning]]s:
* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}}
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}
: error map: {{val| 0.000 +0.048 -0.015 +0.001 }}


{{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 }}


[[Badness]]: 0.126
[[Badness]] (Sintel): 3.20


=== 11-limit ===
=== 11-limit ===
Line 867: Line 947:
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 = 1\13, ~3/2 = 702.0042
Optimal tunings:
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}


{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}


Badness: 0.0421
Badness (Sintel): 1.39


=== 13-limit ===
=== 13-limit ===
Line 880: Line 962:
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 = 1\13, ~3/2 = 702.0099
Optimal tunings:
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}
 
{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}


{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
Badness (Sintel): 1.18


Badness: 0.0286
== Ragitritonic ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


== Countritonic ==
Ragitritonic may be described as the {{nowrap| 53 & 369 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[422edo]] makes for a strong tuning.  
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''


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.  
Ragitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''ragismic'' and ''tritonic''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 895: Line 981:
[[Comma list]]: 4375/4374, 68719476736/68356598625
[[Comma list]]: 4375/4374, 68719476736/68356598625


{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
: mapping generators: ~2, ~65536/45927


: mapping generators: ~2, ~45927/32768
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}


{{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 }}


[[Badness]]: 0.133
[[Badness]] (Sintel): 3.37


=== 11-limit ===
=== 11-limit ===
Line 910: Line 999:
Comma list: 4375/4374, 5632/5625, 2621440/2614689
Comma list: 4375/4374, 5632/5625, 2621440/2614689


Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}


Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
Optimal tunings:
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}


{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}


Badness: 0.0707
Badness (Sintel): 2.34


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625


Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}


Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
Optimal tunings:
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}


{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}


Badness: 0.0366
Badness (Sintel): 1.51


== Quatracot ==
== Quatracot ==
Line 938: Line 1,031:
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}


{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
: mapping generators: ~2278125/1605632, ~7168/5625


: mapping generators: ~2278125/1605632, ~448/405
[[Optimal tuning]]s:  
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}


{{Multival|legend=1| 26 16 118 -35 114 229 }}
{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}


[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
[[Badness]] (Sintel): 4.45
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
 
[[Badness]]: 0.175982


=== 11-limit ===
=== 11-limit ===
Line 955: Line 1,049:
Comma list: 3025/3024, 4375/4374, 1265625/1261568
Comma list: 3025/3024, 4375/4374, 1265625/1261568


Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
Optimal tunings:
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}


{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}


Badness: 0.041043
Badness (Sintel): 1.36


=== 13-limit ===
=== 13-limit ===
Line 968: Line 1,064:
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
Comma list: 625/624, 729/728, 1575/1573, 2200/2197


Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
Optimal tunings:
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}


{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}


Badness: 0.022643
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 can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 983: Line 1,081:
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}


{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
: mapping generators: ~2, ~3796875/3211264


: mapping generators: ~2, ~6422528/3796875
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}


{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
[[Badness]] (Sintel): 5.93
 
[[Badness]]: 0.234


=== 11-limit ===
=== 11-limit ===
Line 998: Line 1,099:
Comma list: 4375/4374, 759375/758912, 100663296/100656875
Comma list: 4375/4374, 759375/758912, 100663296/100656875


Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}


Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}


{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness: 0.0678
Badness (Sintel): 2.24


=== 13-limit ===
=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078


Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}


Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}


{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}


Badness: 0.0271
Badness (Sintel): 1.12


== Palladium ==
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.
: ''For the 5-limit version, 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 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
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 1,031: Line 1,134:


{{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


{{Multival|legend=1| 46 92 -46 39 -202 -365 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}


{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}


[[Badness]]: 0.308505
[[Badness]] (Sintel): 7.81


=== 11-limit ===
=== 11-limit ===
Line 1,049: Line 1,153:
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 = 1\46, ~3/2 = 701.5951
Optimal tunings:
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}


{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}


Badness: 0.073783
Badness (Sintel): 2.44


=== 13-limit ===
=== 13-limit ===
Line 1,062: Line 1,168:
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 = 1\46, ~3/2 = 701.6419
Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}


{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}


Badness: 0.040751
Badness (Sintel): 1.68


=== 17-limit ===
=== 17-limit ===
Line 1,075: Line 1,183:
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 = 1\46, ~3/2 = 701.6425
Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}
 
{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}
 
Badness (Sintel): 1.14


{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
== Counterorson ==
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]].


Badness: 0.022441
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
 
{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}
 
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
 
[[Badness]] (Sintel): 7.92


== Oviminor ==
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}
: ''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 was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,090: Line 1,220:
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}


{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
: mapping generators: ~2, ~5/3


: mapping generators: ~2, ~6/5
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}


{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}


[[Badness]]: 0.582
[[Badness]] (Sintel): 14.7


== Octoid ==
== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
: {{Main| Octoid }}
: ''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 comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
 
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 minimizing the average 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, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,110: Line 1,246:


{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
{{Multival|legend=1| 24 32 40 -5 -4 3 }}
: mapping generators: ~49/45, ~7/5
: mapping generators: ~49/45, ~7/5


[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
[[Optimal tuning]]s:
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,123: Line 1,260:
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}


[[Badness]]: 0.042670
[[Badness]] (Sintel): 1.08
 
Scales: [[octoid72]], [[octoid80]]


=== 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 — not octoid — as 80edo doesn't 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 1,138: Line 1,271:
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 = 1\8, ~7/5 = 583.962
Optimal tunings:
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}


Tuning ranges:  
Tuning ranges:  
Line 1,144: Line 1,279:
* 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=1| 72, 152, 224 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
 
Badness: 0.014097


Scales: [[octoid72]], [[octoid80]]
Badness (Sintel): 0.466


==== 13-limit ====
==== 13-limit ====
Line 1,157: Line 1,290:
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 = 1\8, ~7/5 = 583.905
Optimal tunings:
 
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
 
Badness: 0.015274


Scales: [[octoid72]], [[octoid80]]
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}


; Music
Badness (Sintel): 0.631
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning


===== 17-limit =====
===== 17-limit =====
Line 1,175: Line 1,305:
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 = 1\8, ~7/5 = 583.842
Optimal tunings:
 
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}


Badness: 0.014304
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}


Scales: [[octoid72]], [[octoid80]]
Badness (Sintel): 0.729


===== 19-limit =====
===== 19-limit =====
Line 1,190: Line 1,320:
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 = 1\8, ~7/5 = 583.932
Optimal tunings:
 
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}


Badness: 0.016036
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}


Scales: [[octoid72]], [[octoid80]]
Badness (Sintel): 0.975


==== Octopus ====
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{cent}}.
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,207: Line 1,337:
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 = 1\8, ~7/5 = 583.892
Optimal tunings:
 
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
{{Optimal ET sequence|legend=1| 72, 152, 224f }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}


Badness: 0.021679
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}


Scales: [[octoid72]], [[octoid80]]
Badness (Sintel): 0.896


===== 17-limit =====
===== 17-limit =====
Line 1,222: Line 1,352:
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 = 1\8, ~7/5 = 583.811
Optimal tunings:
 
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


Badness: 0.015614
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Scales: [[Octoid72]], [[Octoid80]]
Badness (Sintel): 0.795


===== 19-limit =====
===== 19-limit =====
Line 1,237: Line 1,367:
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 = 1\8, ~7/5 = 584.064
Optimal tunings:
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}


Badness: 0.016321
Badness (Sintel): 0.993


Scales: [[Octoid72]], [[Octoid80]]
Scales: [[Octoid72]], [[Octoid80]]


==== Hexadecoid ====
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}
{{See also| 16th-octave temperaments }}


Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
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 1,255: Line 1,387:


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 = 1\16, ~13/8 = 841.015
Optimal tunings:
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


{{Optimal ET sequence|legend=1| 80, 144, 224 }}
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Badness: 0.030818
Badness (Sintel): 1.27


===== 17-limit =====
===== 17-limit =====
Line 1,271: Line 1,404:
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 = 1\16, ~13/8 = 840.932
Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}


Badness: 0.028611
Badness (Sintel): 1.46


===== 19-limit =====
===== 19-limit =====
Line 1,282: Line 1,417:
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


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 = 1\16, ~13/8 = 840.896
Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


Badness: 0.023731
Badness (Sintel): 1.44


== Parakleismic ==
== Parakleismic ==
{{Main| Parakleismic }}
{{Main| Parakleismic }}
: ''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, with the 7-limit wedgie being {{multival| 13 14 35 -8 19 42 }} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival| 13 14 35 -36 -8 19 -102 42 -132 -222 }} 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. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.  
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1224440064/1220703125
 
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
 
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}


[[Badness]]: 0.043279
Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).  


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 4375/4374
[[Comma list]]: 3136/3125, 4375/4374


{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
: mapping generators: ~2, ~5/3


{{Multival|legend=1| 13 14 35 -8 19 42 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}


{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
[[Badness]] (Sintel): 0.694
 
[[Badness]]: 0.027431


=== 11-limit ===
=== 11-limit ===
Line 1,329: Line 1,457:
Comma list: 385/384, 3136/3125, 4375/4374
Comma list: 385/384, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
Optimal tunings:
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}


{{Optimal ET sequence|legend=1| 19, 99, 118 }}
{{Optimal ET sequence|legend=0| 19, 99, 118 }}


Badness: 0.049711
Badness (Sintel): 1.64


=== Paralytic ===
=== Paralytic ===
The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
Paralytic (99e & 118) 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 1,344: Line 1,474:
Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 441/440, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
Optimal tunings:
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}


{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}
{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}


Badness: 0.036027
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 1,357: Line 1,489:
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
Optimal tunings:
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}


{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}
{{Optimal ET sequence|legend=0| 99e, 118, 217 }}


Badness: 0.044710
Badness (Sintel): 1.85


==== Paraklein ====
==== Paraklein ====
The ''paraklein'' temperament (19e &amp; 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]].
Paraklein (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 1,372: Line 1,506:
Comma list: 196/195, 352/351, 625/624, 729/728
Comma list: 196/195, 352/351, 625/624, 729/728


Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}


{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}


Badness: 0.037618
Badness (Sintel): 1.55


=== Parkleismic ===
=== Parkleismic ===
Line 1,385: Line 1,521:
Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 176/175, 1375/1372, 2200/2187


Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
Optimal tunings:
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}


{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}


Badness: 0.055884
Badness (Sintel): 1.85


==== 13-limit ====
==== 13-limit ====
Line 1,398: Line 1,536:
Comma list: 169/168, 176/175, 325/324, 1375/1372
Comma list: 169/168, 176/175, 325/324, 1375/1372


Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
Optimal tunings:
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}


{{Optimal ET sequence|legend=1| 19e, 80, 179 }}
{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}


Badness: 0.036559
Badness (Sintel): 1.51


=== Paradigmic ===
=== Paradigmic ===
Line 1,411: Line 1,551:
Comma list: 540/539, 896/891, 3136/3125
Comma list: 540/539, 896/891, 3136/3125


Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
Optimal tunings:
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}


{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}


Badness: 0.041720
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 1,424: Line 1,566:
Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 169/168, 325/324, 540/539, 832/825


Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
Optimal tunings:
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}


{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}


Badness: 0.035781
Badness (Sintel): 1.48


=== Semiparakleismic ===
=== Semiparakleismic ===
Line 1,437: Line 1,581:
Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 3025/3024, 3136/3125, 4375/4374


Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
: mapping generators: ~99/70, ~33/28


Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181
Optimal tunings:
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}


{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}
{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}


Badness: 0.034208
Badness (Sintel): 1.13


==== Semiparamint ====
==== Semiparamint ====
Line 1,452: Line 1,599:
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
Optimal tunings:
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}


{{Optimal ET sequence|legend=1| 80, 118, 198 }}
{{Optimal ET sequence|legend=0| 80, 118, 198 }}


Badness: 0.033775
Badness (Sintel): 1.40


==== Semiparawolf ====
==== Semiparawolf ====
Line 1,467: Line 1,616:
Comma list: 169/168, 325/324, 364/363, 3136/3125
Comma list: 169/168, 325/324, 364/363, 3136/3125


Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
Optimal tunings:
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}


{{Optimal ET sequence|legend=1| 80, 118f, 198f }}
{{Optimal ET sequence|legend=0| 80, 118f, 198f }}


Badness: 0.040467
Badness (Sintel): 1.67


== Counterkleismic ==
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}
: ''For the 5-limit version, 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 &amp; 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
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)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,484: Line 1,635:
[[Comma list]]: 4375/4374, 158203125/157351936
[[Comma list]]: 4375/4374, 158203125/157351936


{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
: mapping generators: ~2, ~6/5


: mapping generators: ~2, ~5/3
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}
{{Multival|legend=1| 25 24 79 -20 55 116 }}
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}


{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
[[Badness]] (Sintel): 2.29
 
[[Badness]]: 0.090553


=== 11-limit ===
=== 11-limit ===
Line 1,501: Line 1,653:
Comma list: 540/539, 4375/4374, 2097152/2096325
Comma list: 540/539, 4375/4374, 2097152/2096325


Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
Optimal tunings:
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}


{{Optimal ET sequence|legend=1| 19, 205, 224 }}
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


Badness: 0.070952
Badness (Sintel): 2.35


==== 13-limit ====
==== 13-limit ====
Line 1,514: Line 1,668:
Comma list: 540/539, 625/624, 729/728, 10985/10976
Comma list: 540/539, 625/624, 729/728, 10985/10976


Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
Optimal tunings:
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}


{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


Badness: 0.033874
Badness (Sintel): 1.40


=== Counterlytic ===
=== Counterlytic ===
Line 1,527: Line 1,683:
Comma list: 1375/1372, 4375/4374, 496125/495616
Comma list: 1375/1372, 4375/4374, 496125/495616


Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Optimal tunings:
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}


{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}


Badness: 0.065400
Badness (Sintel): 2.16


==== 13-limit ====
==== 13-limit ====
Line 1,540: Line 1,698:
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
Optimal tunings:
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}


{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}


Badness: 0.029782
Badness (Sintel): 1.23


== Quincy ==
== Quincy ==
Line 1,554: Line 1,714:


{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2, ~1728/1715


{{Multival|legend=1| 30 49 14 8 -62 -105 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}


{{Optimal ET sequence|legend=1| 72, 217, 289 }}
[[Badness]] (Sintel): 2.02
 
[[Badness]]: 0.079657


=== 11-limit ===
=== 11-limit ===
Line 1,570: Line 1,733:
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 = 1\1, ~100/99 = 16.613
Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}


{{Optimal ET sequence|legend=1| 72, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 217, 289 }}


Badness: 0.030875
Badness (Sintel): 1.02


=== 13-limit ===
=== 13-limit ===
Line 1,583: Line 1,748:
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 = 1\1, ~100/99 = 16.602
Optimal tunings:
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}


{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}


Badness: 0.023862
Badness (Sintel): 0.986


=== 17-limit ===
=== 17-limit ===
Line 1,596: Line 1,763:
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 = 1\1, ~100/99 = 16.602
Optimal tunings:
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}


{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}


Badness: 0.014741
Badness (Sintel): 0.751


=== 19-limit ===
=== 19-limit ===
Line 1,609: Line 1,778:
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 = 1\1, ~100/99 = 16.594
Optimal tunings:
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}


{{Optimal ET sequence|legend=1| 72, 145, 217 }}
{{Optimal ET sequence|legend=0| 72, 145, 217 }}


Badness: 0.015197
Badness (Sintel): 0.924


== Sfourth ==
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,623: Line 1,794:


{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
: mapping generators: ~2, ~49/48


{{Multival|legend=1| 19 31 9 5 -39 -66 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}}
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }}


{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


[[Badness]]: 0.123291
[[Badness]] (Sintel): 3.12


=== 11-limit ===
=== 11-limit ===
Line 1,639: Line 1,813:
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 = 1\1, ~49/48 = 26.286
Optimal tunings:
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}


{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}


Badness: 0.054098
Badness (Sintel): 1.78


==== 13-limit ====
==== 13-limit ====
Line 1,652: Line 1,828:
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 = 1\1, ~49/48 = 26.310
Optimal tunings:
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}


{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}


Badness: 0.033067
Badness (Sintel): 1.37


=== Sfour ===
=== Sfour ===
Line 1,665: Line 1,843:
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 = 1\1, ~49/48 = 26.246
Optimal tunings:
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}


{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness: 0.076567
Badness (Sintel): 2.53


==== 13-limit ====
==== 13-limit ====
Line 1,678: Line 1,858:
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 = 1\1, ~49/48 = 26.239
Optimal tunings:
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}


{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness: 0.051893
Badness (Sintel): 2.14


== Trideci ==
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''
: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''


The trideci temperament (26 &amp; 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]]").
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 "thirteen").


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,694: Line 1,876:


{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3


[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410
[[Optimal tuning]]s:
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}


{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
{{Optimal ET sequence|legend=1| 26, 65, 91 }}


[[Badness]]: 0.184585
[[Badness]] (Sintel): 4.67


=== 11-limit ===
=== 11-limit ===
Line 1,708: Line 1,895:
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 = 1\13, ~3/2 = 699.6179
Optimal tunings:
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}


{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}
{{Optimal ET sequence|legend=0| 26, 65, 91 }}


Badness: 0.084590
Badness (Sintel): 2.80


=== 13-limit ===
=== 13-limit ===
Line 1,721: Line 1,910:
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 = 1\13, ~3/2 = 699.2969
Optimal tunings:
 
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}


Badness: 0.052366
{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
 
== Counterorson ==
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
 
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}
 
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113
 
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


Badness: 0.312806
Badness (Sintel): 2.16


== Notes ==
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Ennealimmal]]
[[Category:Gamera]]
[[Category:Mitonic]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]

Latest revision as of 10:42, 20 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 ([-1 -7 4 1⟩). The ragisma is the smallest 7-limit superparticular ratio.

Since (10/9)4 = (4375/4374)⋅(32/21), the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = (4375/4374)⋅(27/25)2, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below, sorted by badness, are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, ragitritonic, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

Supermajor

The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its octave complement. 37 of these give 3/222, 46 give 5/227, and 75 give 7/245. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106edo or 1277edo can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 52734375/52706752

Mapping: [⟨1 -22 -27 -45], ⟨0 37 46 75]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1200.0067 ¢, ~14/9 = 764.9222 ¢
error map: ⟨+0.007 +0.019 -0.074 +0.037]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 764.9181 ¢
error map: ⟨0.000 +0.013 -0.083 +0.029]

Optimal ET sequence: 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214

Badness (Sintel): 0.274

Semisupermajor

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

Mapping: [⟨2 -7 -8 -15 -6], ⟨0 37 46 75 47]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

  • WE: ~99/70 = 600.0103 ¢, ~11/10 = 164.9205 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~11/10 = 164.9180 ¢

Optimal ET sequence: 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee

Badness (Sintel): 0.422

Enneadecal

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Enneadecal (5-limit).

Enneadecal tempers out the enneadeca, [-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-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

Comma list: 4375/4374, 703125/702464

Mapping: [⟨19 0 14 -37], ⟨0 1 1 3]]

mapping generators: ~28/27, ~3

Optimal tunings:

  • WE: ~28/27 = 63.1599 ¢, ~3/2 = 701.9027 ¢ (~225/224 = 7.1437 ¢)
error map: ⟨+0.038 -0.014 -0.134 +0.080]
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 701.9002 ¢ (~225/224 = 7.1634 ¢)
error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 19, …, 152, 171, 665, 836, 1007, 2185, 3192c

Badness (Sintel): 0.277

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: [⟨19 0 14 -37 126], ⟨0 1 1 3 -2]]

Optimal tunings:

  • WE: ~28/27 = 63.1431 ¢, ~3/2 = 702.1956 ¢ (~225/224 = 7.6216 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.3164 ¢ (~225/224 = 7.5795 ¢)

Optimal ET sequence: 19, 133d, 152, 323e, 475de, 627de

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: [⟨19 0 14 -37 126 -20], ⟨0 1 1 3 -2 3]]

Optimal tunings:

  • WE: ~28/27 = 63.1406 ¢, ~3/2 = 702.0192 ¢ (~225/224 = 7.4730 ¢)
  • CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.1539 ¢ (~225/224 = 7.4171 ¢)

Optimal ET sequence: 19, 133df, 152f, 323ef

Badness (Sintel): 1.39

Hemienneadecal

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

Mapping: [⟨38 0 28 -74 11], ⟨0 1 1 3 2]]

mapping generators: ~55/54, ~3

Optimal tunings:

  • WE: ~55/54 = 31.5800 ¢, ~3/2 = 701.9053 ¢ (~243/242 = 7.1448 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9034 ¢ (~243/242 = 7.1666 ¢)

Optimal ET sequence: 152, 342, 836, 1178, 2014, 3192ce, 5206ce

Badness (Sintel): 0.330

Hemienneadecalis

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: [⟨38 0 28 -74 11 -281], ⟨0 1 1 3 2 7]]

Optimal tunings:

  • WE: ~55/54 = 31.5785 ¢, ~3/2 = 701.9995 ¢ (~243/242 = 7.2727 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 702.0053 ¢ (~243/242 = 7.2685 ¢)

Optimal ET sequence: 152f, 342f, 494

Badness (Sintel): 0.859

Hemienneadec

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: [⟨38 0 28 -74 11 502], ⟨0 1 1 3 2 -6]]

Optimal tunings:

  • WE: ~55/54 = 31.5784 ¢, ~3/2 = 701.9736 ¢ (~243/242 = 7.2493 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9855 ¢ (~243/242 = 7.2487 ¢)

Optimal ET sequence: 152, 342, 494, 1330, 1824, 2318d

Badness (Sintel): 1.26

Semihemienneadecal

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: [⟨38 1 29 -71 13 111], ⟨0 2 2 6 4 1]]

mapping generators: ~55/54, ~429/250

Optimal tunings:

  • WE: ~55/54 = 31.5799 ¢, ~429/250 = 935.1824 ¢ (~144/143 = 12.2152 ¢)
  • CWE: ~55/54 = 31.5789 ¢, ~429/250 = 935.1617 ¢ (~144/143 = 12.2067 ¢)

Optimal ET sequence: 190, 304d, 494, 684, 1178, 2850, 4028ce

Badness (Sintel): 0.607

Kalium

Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. 19/16 can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: [⟨19 3 17 -28 82 92 159 78], ⟨0 10 10 30 -6 -8 -30 1]]

Optimal tunings:

  • WE: ~28/27 = 63.1582 ¢, ~6545/5928 = 171.2448 ¢
  • CWE: ~28/27 = 63.1579 ¢, ~6545/5928 = 171.2439 ¢

Optimal ET sequence: 855, 988, 1843

Badness (Sintel): 3.15

Semidimi

For the 5-limit version, see Miscellaneous 5-limit temperaments #Semidimi.

The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit [-12 -73 55⟩ and 7-limit 3955078125/3954653486, as well as 4375/4374.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 3955078125/3954653486

Mapping: [⟨1 -19 -25 -32], ⟨0 55 73 93]]

mapping generators: ~2, ~35/27

Optimal tunings:

  • WE: ~2 = 1200.0018 ¢, ~35/27 = 449.1277 ¢
error map: ⟨+0.002 +0.031 -0.040 -0.012]
  • CWE: ~2 = 1200.0000 ¢, ~35/27 = 449.1270 ¢
error map: ⟨0.000 +0.030 -0.043 -0.015]

Optimal ET sequence: 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

Badness (Sintel): 0.382

Brahmagupta

The brahmagupta temperament has a period of 1/7 octave, tempering out the akjaysma ([47 -7 -7 -7⟩), and may be described as the 217 & 224 temperament.

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 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

Comma list: 4375/4374, [46 -14 -3 -6⟩

Mapping: [⟨7 2 -8 53], ⟨0 3 8 -11]]

mapping generators: ~1157625/1048576, ~27/20

Optimal tunings:

  • WE: ~1157625/1048576 = 171.4275 ¢, ~27/20 = 519.7125 ¢
error map: ⟨-0.007 +0.037 -0.034 -0.004]
  • CWE: ~1157625/1048576 = 171.4286 ¢, ~27/20 = 519.7156 ¢
error map: ⟨0.000 +0.049 -0.018 +0.017]

Optimal ET sequence: 7, …, 217, 224, 441, 1106, 1547

Badness (Sintel): 0.737

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]

Optimal tunings:

  • WE: ~243/220 = 171.4208 ¢, ~27/20 = 519.6807 ¢
  • CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7034 ¢

Optimal ET sequence: 7, 217, 224, 441, 665

Badness (Sintel): 1.73

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: [⟨7 2 -8 53 3 35], ⟨0 3 8 -11 7 -3]]

Optimal tunings:

  • WE: ~243/220 = 171.4197 ¢, ~27/20 = 519.6789 ¢
  • CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7052 ¢

Optimal ET sequence: 7, 217, 224, 441, 665, 1106e

Badness (Sintel): 0.956

Abigail

For the 5-limit version, see Miscellaneous 5-limit temperaments #Abigail.

Abigail tempers out the pessoalisma in addition to the ragisma in the 7-limit, and may be described as the 46 & 224 temperament, with a ploidacot signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. 494edo, 764edo and 1258edo are among the possible tunings.

Abigail was named by Gene Ward Smith in 2010 after the birthday of First Lady Abigail Fillmore.[1]

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

Mapping: [⟨2 -4 -11 18], ⟨0 11 24 -19]]

mapping generators: ~46305/32768, ~1536/1225

Optimal tunings:

  • WE: ~46305/32768 = 599.9699 ¢, ~1536/1225 = 391.0818 ¢
error map: ⟨-0.060 +0.065 -0.021 +0.079]
  • CWE: ~46305/32768 = 600.0000 ¢, ~1536/1225 = 391.1007 ¢
error map: ⟨0.000 +0.152 +0.102 +0.262]

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: [⟨2 -4 -11 18 18], ⟨0 11 24 -19 -17]]

Optimal tunings:

  • WE: ~99/70 = 599.9782 ¢, ~1536/1225 = 391.0852 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~1536/1225 = 391.0992 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764

Badness (Sintel): 0.425

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: [⟨2 -4 -11 18 18 25], ⟨0 11 24 -19 -17 -27]]

Optimal tunings:

  • WE: ~99/70 = 599.9862 ¢, ~351/280 = 391.0879 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~351/280 = 391.0969 ¢

Optimal ET sequence: 46, 178, 224, 270, 494, 764, 1258

Badness (Sintel): 0.366

Gamera

For the 5-limit version, see Miscellaneous 5-limit temperaments #Gamera.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 589824/588245

Mapping: [⟨1 -17 -30 2], ⟨0 23 40 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1199.8483 ¢, ~7/4 = 969.5415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 969.6608 ¢

Optimal ET sequence: 26, 73, 99, 224, 323, 422, 745d

Badness (Sintel): 0.953

Hemigamera

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: [⟨2 -11 -20 5 10], ⟨0 23 40 1 -5]]

mapping generators: ~99/70, ~99/80

Optimal tunings:

  • WE: ~99/70 = 599.9323 ¢, ~99/80 = 369.6212 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~99/80 = 369.6610 ¢

Optimal ET sequence: 26, 172c, 198, 224, 422, 646, 1068d

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: [⟨2 -11 -20 5 10 -8], ⟨0 23 40 1 -5 25]]

Optimal tunings:

  • WE: ~99/70 = 599.9207 ¢, ~26/21 = 369.6139 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~26/21 = 369.6603 ¢

Optimal ET sequence: 26, 172cf, 198, 224, 422, 646f, 1068df

Badness (Sintel): 0.844

Semigamera

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: [⟨1 -40 -70 1 -77], ⟨0 46 80 2 89]]

mapping generators: ~2, ~144/77

Optimal tunings:

  • WE: ~2 = 1199.8845 ¢, ~144/77 = 1084.7314 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8345 ¢

Optimal ET sequence: 73, 125, 198, 323, 521

Badness (Sintel): 2.59

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: [⟨1 -40 -70 1 -77 -131], ⟨0 46 80 2 89 149]]

Optimal tunings:

  • WE: ~2 = 1199.8726 ¢, ~144/77 = 1084.7220 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8359 ¢

Optimal ET sequence: 73f, 125f, 198, 323, 521

Badness (Sintel): 1.82

Crazy

For the 5-limit version, see Very high accuracy temperaments #Kwazy.

Crazy tempers out the kwazy comma in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the 118 & 494 temperament, with a ploidacot of diploid alpha-octacot. 1106edo gives a strong tuning.

Crazy was named by Flora Canou in 2025 by removing the mutation from kwazy, the name for the 5-limit microtemperament.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-53 10 16⟩

Mapping: [⟨2 1 6 -15], ⟨0 8 -5 76]]

mapping generators: ~332150625/234881024, ~1125/1024

Optimal tunings:

  • WE: ~332150625/234881024 = 600.0019 ¢, ~1125/1024 = 162.7479 ¢
error map: ⟨+0.004 +0.030 -0.042 -0.014]
  • CWE: ~332150625/234881024 = 600.0000 ¢, ~1125/1024 = 162.7474 ¢
error map: ⟨0.000 +0.024 -0.051 -0.022]

Optimal ET sequence: 118, 376, 494, 612, 1106, 1718

Badness (Sintel): 0.998

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: [⟨2 1 6 -15 -8], ⟨0 8 -5 76 55]]

Optimal tunings:

  • WE: ~99/70 = 600.0047 ¢, ~1125/1024 = 162.7493 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~1125/1024 = 162.7481 ¢

Optimal ET sequence: 118, 376, 494, 612, 1106, 2824, 3930e

Badness (Sintel): 0.562

Orga

Orga may be described as the 26 & 270 temperament, and 1106edo gives a strong tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [41 -4 2 -14⟩

Mapping: [⟨2 -8 -15 6], ⟨0 29 51 -1]]

mapping generators: ~7411887/5242880, ~8/7

Optimal tunings:

  • WE: ~7411887/5242880 = 599.9927 ¢, ~8/7 = 231.1012 ¢
error map: ⟨-0.015 +0.037 -0.045 +0.029]
  • CWE: ~7411887/5242880 = 600.0000 ¢, ~8/7 = 231.1037 ¢
error map: ⟨0.000 +0.053 -0.023 +0.070]

Optimal ET sequence: 26, …, 244, 270, 836, 1106, 1376, 2482

Badness (Sintel): 1.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: [⟨2 -8 -15 6 10], ⟨0 29 51 -1 -8]]

Optimal tunings:

  • WE: ~99/70 = 600.0025 ¢, ~8/7 = 231.1039 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1030 ¢

Optimal ET sequence: 26, 244, 270, 566, 836, 1106

Badness (Sintel): 0.535

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

Mapping: [⟨2 -8 -15 6 10 -3], ⟨0 29 51 -1 -8 27]]

Optimal tunings:

  • WE: ~99/70 = 600.0192 ¢, ~8/7 = 231.1102 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1033 ¢

Optimal ET sequence: 26, 244, 270, 566, 836f, 1106f

Badness (Sintel): 0.899

Seniority

For the 5-limit version, see Very high accuracy temperaments #Senior.

Aside from the ragisma, the seniority temperament tempers out the wadisma, 201768035/201326592, and may be described as 26 & 145. It is so named because the senior comma ([-17 62 -35⟩) is tempered out.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 -24 -43 5], ⟨0 35 62 -3]]

mapping generators: ~2, ~5120/3087

Optimal tunings:

  • WE: ~2 = 1200.0745 ¢, ~5120/3087 = 877.2500 ¢
error map: ⟨+0.075 +0.008 -0.016 -0.203]
  • CWE: ~2 = 1200.0000 ¢, ~5120/3087 = 877.1965 ¢
error map: ⟨0.000 -0.077 -0.130 -0.415]

Optimal ET sequence: 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d

Badness (Sintel): 1.14

Senator

Senator (26 & 145) extends seniority by tempering out 441/440 and 65536/65219, and 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

Comma list: 441/440, 4375/4374, 65536/65219

Mapping: [⟨1 -24 -43 5 2], ⟨0 35 62 -3 2]]

Optimal tunings:

  • WE: ~2 = 1199.7665 ¢, ~128/77 = 877.0367 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/77 = 877.2051 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316e

Badness (Sintel): 3.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

Mapping: [⟨1 -24 -43 5 2 -27], ⟨0 35 62 -3 2 42]]

Optimal tunings:

  • WE: ~2 = 1199.7136 ¢, ~108/65 = 877.9974 ¢
  • CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2038 ¢

Optimal ET sequence: 26, 119cf, 145, 171, 316ef

Badness (Sintel): 1.85

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

Mapping: [⟨1 -24 -43 5 2 -27 -31], ⟨0 35 62 -3 2 42 48]]

Optimal tunings:

  • WE: ~2 = 1199.7195 ¢, ~108/65 = 877.0018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2039 ¢

Optimal ET sequence: 26, 119cfg, 145, 171, 316ef

Badness (Sintel): 1.35

Monzismic

For the 5-limit version, see Very high accuracy temperaments #Monzismic.

Monzismic tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374. It may be described as the 53 & 612 temperament, with a ploidacot signature of alpha-dicot. 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

Comma list: 4375/4374, [-55 30 2 1⟩

Mapping: [⟨1 0 -27 109], ⟨0 2 37 -134]]

mapping generators: ~2, ~[28 -11 -3 -1⟩

Optimal tunings:

  • WE: ~2 = 1200.0128 ¢, ~[28 -11 -3 -1⟩ = 950.9895 ¢
error map: ⟨+0.013 +0.024 -0.049 -0.019]
  • CWE: ~2 = 1200.0000 ¢, ~[28 -11 -3 -1⟩ = 950.9793 ¢
error map: ⟨0.000 +0.004 -0.080 -0.050]

Optimal ET sequence: 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd

Badness (Sintel): 1.18

Monzism

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: [⟨1 0 -27 109 -159], ⟨0 2 37 -134 205]]

Optimal tunings:

  • WE: ~2 = 1200.0347 ¢, ~400/231 = 951.0082 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9807 ¢

Optimal ET sequence: 53, 559, 612, 3619de, 4231de, …, 6067ddee

Badness (Sintel): 1.89

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: [⟨1 0 -27 109 -159 -70], ⟨0 2 37 -134 205 93]]

Optimal tunings:

  • WE: ~2 = 1200.0036 ¢, ~400/231 = 950.9829 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9801 ¢

Optimal ET sequence: 53, 559, 612

Badness (Sintel): 2.22

Semidimfourth

For the 5-limit version, see Miscellaneous 5-limit temperaments #Semidimfourth.

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

Comma list: 4375/4374, 235298/234375

Mapping: [⟨1 -10 -13 -17], ⟨0 31 41 53]]

mapping generators: ~2, ~35/27

Optimal tunings:

  • WE: ~2 = 1199.9936 ¢, ~35/27 = 448.4533 ¢
error map: ⟨-0.007 +0.160 +0.353 -0.694]
  • CWE: ~2 = 1200.0000 ¢, ~35/27 = 448.4555 ¢
error map: ⟨0.000 +0.165 +0.361 -0.685]

Optimal ET sequence: 8d, …, 91, 99, 289, 388, 875

Badness (Sintel): 1.40

Neusec

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: [⟨2 -20 -26 -34 -17], ⟨0 31 41 53 32]]

mapping generators: ~99/70, ~35/27

Optimal tunings:

  • WE: ~99/70 = 600.0381 ¢, ~35/27 = 448.4812 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4546 ¢

Optimal ET sequence: 8d, …, 190, 388

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -20 -26 -34 -17 -21], ⟨0 31 41 53 32 38]]

Optimal tunings:

  • WE: ~99/70 = 600.0034 ¢, ~35/27 = 448.4573 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4549 ¢

Optimal ET sequence: 8d, …, 190, 198, 388

Badness (Sintel): 1.28

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

Mapping: [⟨1 -22 -22 -65], ⟨0 32 33 92]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.9305 ¢, ~5/3 = 884.3923 ¢
error map: ⟨-0.070 +0.126 +0.160 -0.221]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4423 ¢
error map: ⟨0.000 +0.198 +0.282 -0.136]

Optimal ET sequence: 19, …, 251, 270, 2449c, 2719c, 2989bc

Badness (Sintel): 1.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: [⟨1 -22 -22 -65 58], ⟨0 32 33 92 -74]]

Optimal tunings:

  • WE: ~2 = 1199.9698 ¢, ~5/3 = 884.4193 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4414 ¢

Optimal ET sequence: 19, 251, 270, 829, 1099, 1369, 1639

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: [⟨1 -22 -22 -65 58 -56], ⟨0 32 33 92 -74 81]]

Optimal tunings:

  • WE: ~2 = 1199.9939 ¢, ~5/3 = 884.4384 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4429 ¢

Optimal ET sequence: 19, 251, 270

Badness (Sintel): 1.11

Counteracro

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: [⟨1 -22 -22 -65 -141], ⟨0 32 33 92 196]]

Optimal tunings:

  • WE: ~2 = 1199.8877 ¢, ~5/3 = 884.3639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4457 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

Mapping: [⟨1 -22 -22 -65 -141 -56], ⟨0 32 33 92 196 81]]

Optimal tunings:

  • WE: ~2 = 1199.9285 ¢, ~5/3 = 884.3937 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4458 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1331c

Badness (Sintel): 1.08

Quasithird

For the 5-limit version, see Miscellaneous 5-limit temperaments #Quasithird.

Quasithird may be described as the 224 & 388 temperament, featured by a major third interval which is 1600000/1594323 (amity comma) or 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 it to temper out the ragisma and [-60 29 0 5⟩. Its ploidacot is tetraploid delta-pentacot.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

mapping generators: ~65536/55125, ~5103/4096

Optimal tunings:

  • WE: ~65536/55125 = 300.0052 ¢, ~5103/4096 = 380.3949 ¢
error map: ⟨+0.021 +0.020 -0.052 -0.031]
  • CWE: ~65536/55125 = 300.0000 ¢, ~5103/4096 = 380.3884 ¢
error map: ⟨0.000 -0.013 -0.100 -0.089]

Optimal ET sequence: 60d, 164, 224, 388, 612, 1448, 2060

Badness (Sintel): 1.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: [⟨4 0 -11 48 43], ⟨0 5 16 -29 -23]]

Optimal tunings:

  • WE: ~65536/51125 = 300.0073 ¢, ~5103/4096 = 380.3963 ¢ (or ~22/21 = 80.3890 ¢)
  • CWE: ~65536/51125 = 300.0000 ¢, ~5103/4096 = 380.3868 ¢ (or ~22/21 = 80.3868 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee

Badness (Sintel): 0.698

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨4 0 -11 48 43 11], ⟨0 5 16 -29 -23 3]]

Optimal tunings:

  • WE: ~65536/51125 = 299.9985 ¢, ~81/65 = 380.3833 ¢ (or ~22/21 = 80.3848 ¢)
  • CWE: ~65536/51125 = 300.0000 ¢, ~81/65 = 380.3852 ¢ (or ~22/21 = 80.3852 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836

Badness (Sintel): 1.22

Deca

For 5-limit version, see 10th-octave temperaments #Neon.

Deca has a period of 1/10 octave and tempers out the neon comma [21 60 -50⟩ in the 5-limit, the linus comma[11 -10 -10 10⟩ and [12 -3 -14 9⟩ (165288374272/164794921875) in the 7-limit. It may be described as the 80 & 190 temperament, and has a ploidacot of decaploid wau-pentacot.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

mapping generators: ~15/14, ~460992/390625

Optimal tunings:

  • WE: ~15/14 = 119.9966 ¢, ~460992/390625 = 284.4150 ¢ (5625/5488 = 44.4219 ¢)
error map: ⟨-0.034 +0.106 +0.145 -0.268]
  • CWE: ~15/14 = 120.0000 ¢, ~460992/390625 = 284.4182 ¢ (5625/5488 = 44.4182 ¢)
error map: ⟨0.000 +0.136 +0.195 -0.226]

Optimal ET sequence: 80, 190, 270, 1270, 1540, 1810, 2080

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: [⟨10 4 9 2 18], ⟨0 5 6 11 7]]

Optimal tunings:

  • WE: ~15/14 = 120.0004 ¢, ~33/28 = 284.4193 ¢ (77/75 = 44.4185 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4189 ¢ (77/75 = 44.4189 ¢)

Optimal ET sequence: 80, 190, 270, 1000, 1270, 1540e, 1810e

Badness (Sintel): 0.804

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨10 4 9 2 18 37], ⟨0 5 6 11 7 0]]

Optimal tunings:

  • WE: ~15/14 = 120.0067 ¢, ~33/28 = 284.4139 ¢ (~40/39 = 44.4006 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4048 ¢ (~40/39 = 44.4048 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.695

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨10 4 9 2 18 37 33], ⟨0 5 6 11 7 0 4]]

Optimal tunings:

  • WE: ~15/14 = 120.0045 ¢, ~33/28 = 284.4140 ¢ (~39/38 = 44.4050 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4075 ¢ (~39/38 = 44.4075 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.556

Keenanose

Keenanose, the 270 & 1889 temperament, was named by Eliora in 2022 for the fact that it uses 385/384, the keenanisma, as the generator.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-56 1 -8 26⟩

Mapping: [⟨1 2 3 3], ⟨0 -112 -183 -52]]

mapping generators: ~2, ~[21 3 1 -10⟩

Optimal tunings:

  • WE: ~2 = 1200.0068 ¢, ~[21 3 1 -10⟩ = 4.4467 ¢
error map: ⟨+0.007 +0.031 -0.035 -0.032]
  • CWE: ~2 = 1200.0000 ¢, ~[21 3 1 -10⟩ = 4.4466 ¢
error map: ⟨0.000 +0.025 -0.043 -0.050]

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd

Badness (Sintel): 2.17

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864

Mapping: [⟨1 2 3 3 3], ⟨0 -112 -183 -52 124]]

Optimal tunings:

  • WE: ~2 = 1199.9970 ¢, ~385/384 = 4.4465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4465 ¢

Optimal ET sequence: 270, 1349, 1619, 1889, 2159, 11065, 13224

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: [⟨1 2 3 3 3 3], ⟨0 -112 -183 -52 124 189]]

Optimal tunings:

  • WE: ~2 = 1200.0065 ¢, ~385/384 = 4.4467 ¢
  • CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4467 ¢

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 4048

Badness (Sintel): 0.879

Aluminium

For the 5-limit version, see 13th-octave temperaments #Aluminium.

Aluminium was named by Eliora in 2023 after the 13th element. It tempers out the [92 -39 -13⟩ comma, which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

Mapping: [⟨13 0 92 -355], ⟨0 1 -3 19]]

Mapping generators: ~135/128, ~3

Optimal tunings:

  • WE: ~135/128 = 92.3072 ¢, ~3/2 = 701.9995 ¢
error map: ⟨-0.006 +0.038 -0.030 -0.013]
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0030 ¢
error map: ⟨0.000 +0.048 -0.015 +0.001]

Optimal ET sequence: 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b

Badness (Sintel): 3.20

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: [⟨13 0 92 -355 148], ⟨0 1 -3 19 -5]]

Optimal tunings:

  • WE: ~135/128 = 92.3062 ¢, ~3/2 = 701.9946 ¢
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0056 ¢

Optimal ET sequence: 494, 1053, 1547, 3588e, 5135e

Badness (Sintel): 1.39

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨13 0 92 -355 148 419], ⟨0 1 -3 19 -5 -18]]

Optimal tunings:

  • WE: ~135/128 = 92.3055 ¢, ~3/2 = 701.9928 ¢
  • CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0098 ¢

Optimal ET sequence: 494, 1547, 2041, 4576def

Badness (Sintel): 1.18

Ragitritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Ragitritonic may be described as the 53 & 369 temperament, splitting the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. 422edo makes for a strong tuning.

Ragitritonic was named by Flora Canou in 2026 as a contraction of ragismic and tritonic.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 -3 -15 40], ⟨0 9 34 -73]]

mapping generators: ~2, ~65536/45927

Optimal tunings:

  • WE: ~2 = 1199.8189 ¢, ~65536/45927 = 611.2850 ¢
error map: ⟨-0.181 +0.153 +0.094 +0.123]
  • CWE: ~2 = 1200.0000 ¢, ~65536/45927 = 611.3775 ¢
error map: ⟨0.000 +0.443 +0.522 +0.615]

Optimal ET sequence: 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd

Badness (Sintel): 3.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: [⟨1 -3 -15 40 -75], ⟨0 9 34 -73 154]]

Optimal tunings:

  • WE: ~2 = 1199.8147 ¢, ~768/539 = 611.2822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~768/539 = 611.3762 ¢

Optimal ET sequence: 53, 316e, 369, 422, 791e, 1213cde

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: [⟨1 -3 -15 40 -75 -34], ⟨0 9 34 -73 154 74]]

Optimal tunings:

  • WE: ~2 = 1199.7916 ¢, ~91/64 = 611.2698 ¢
  • CWE: ~2 = 1200.0000 ¢, ~91/64 = 611.3754 ¢

Optimal ET sequence: 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff

Badness (Sintel): 1.51

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 -6 -1 -36], ⟨0 13 8 59]]

mapping generators: ~2278125/1605632, ~7168/5625

Optimal tunings:

  • WE: ~2278125/1605632 = 600.0888 ¢, ~7168/5625 = 423.2574 ¢
error map: ⟨+0.178 -0.141 -0.343 +0.165]
  • CWE: ~2278125/1605632 = 600.0000 ¢, ~7168/5625 = 423.1986 ¢
error map: ⟨0.000 -0.374 -0.725 -0.111]

Optimal ET sequence: 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc

Badness (Sintel): 4.45

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: [⟨2 -6 -1 -36 -22], ⟨0 13 8 59 41]]

Optimal tunings:

  • WE: ~99/70 = 600.0847 ¢, ~225/176 = 423.2536 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~225/176 = 423.1977 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638, 1052c

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: [⟨2 -6 -1 -36 -22 -6], ⟨0 13 8 59 41 19]]

Optimal tunings:

  • WE: ~99/70 = 600.0571 ¢, ~143/112 = 423.2366 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~143/112 = 423.1987 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638

Badness (Sintel): 0.936

Moulin

Moulin can be described as the 494 & 1619 temperament. It has a generator of ~22/13, and it was named by Eliora in 2022 after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~13/11, and 73 of them octave reduced reach the perfect fifth. Since 11/8 is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 -16 -9 -75], ⟨0 73 47 323]]

mapping generators: ~2, ~3796875/3211264

Optimal tunings:

  • WE: ~2 = 1200.0272 ¢, ~3796875/3211264 = 289.0675 ¢
error map: ⟨+0.027 +0.007 -0.084 +0.013]
  • CWE: ~2 = 1200.0000 ¢, ~3796875/3211264 = 289.0675 ¢
error map: ⟨0.000 -0.029 -0.142 -0.029]

Optimal ET sequence: 494, 1125, 1619, 8589cc, 10208cc

Badness (Sintel): 5.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: [⟨1 -16 -9 -75 9], ⟨0 73 47 323 -23]]

Optimal tunings:

  • WE: ~2 = 1200.0043 ¢, ~605/512 = 289.0687 ¢
  • CWE: ~2 = 1200.0000 ¢, ~605/512 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 2.24

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨1 -16 -9 -75 9 9], ⟨0 73 47 323 -23 -22]]

Optimal tunings:

  • WE: ~2 = 1200.0043 ¢, ~13/11 = 289.0687 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/11 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 1.12

Palladium

For the 5-limit version, 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, [-39 92 -46⟩, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out [-51 8 2 12⟩ as well as the ragisma.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-51 8 2 12⟩

Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]

mapping generators: ~83349/81920, ~3

Optimal tunings:

  • WE: ~83349/81920 = 26.0910 ¢, ~3/2 = 701.7155 ¢
error map: ⟨+0.185 -0.055 -0.061 +0.349]
  • CWE: ~83349/81920 = 26.0870 ¢, ~3/2 = 701.6491 ¢
error map: ⟨0.000 -0.306 -0.407 -0.910]

Optimal ET sequence: 46, …, 368, 414, 460, 874d

Badness (Sintel): 7.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: [⟨46 0 -39 202 232], ⟨0 1 2 -1 -1]]

Optimal tunings:

  • WE: ~8192/8085 = 26.0912 ¢, ~3/2 = 701.7082 ¢
  • CWE: ~8192/8085 = 26.0870 ¢, ~3/2 = 701.6173 ¢

Optimal ET sequence: 46, …, 368, 414, 460, 874de

Badness (Sintel): 2.44

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: [⟨46 0 -39 202 232 316], ⟨0 1 2 -1 -1 -2]]

Optimal tunings:

  • WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7411 ¢
  • CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6465 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334dde

Badness (Sintel): 1.68

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: [⟨46 0 -39 202 232 316 188], ⟨0 1 2 -1 -1 -2 0]]

Optimal tunings:

  • WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7399 ¢
  • CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6464 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334ddeg

Badness (Sintel): 1.14

Counterorson

Counterorson tempers out the [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

Comma list: 4375/4374, [154 -54 -21 -7⟩

Mapping: [⟨1 0 -21 85], ⟨0 7 103 -363]]

mapping generators: ~2, ~[66 -23 -9 -3⟩

Optimal tunings:

  • WE: ~2 = 1200.0040 ¢, ~[66 -23 -9 -3⟩ = 271.7122 ¢
error map: ⟨+0.004 -0.303 -0.041 -0.015]
  • CWE: ~2 = 1200.0000 ¢, ~[66 -23 -9 -3⟩ = 271.7113 ¢
error map: ⟨0.000 +0.024 -0.051 -0.025]

Optimal ET sequence: 53, …, 1612, 1665, 1718

Badness (Sintel): 7.92

Oviminor

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Oviminor (5-limit).

Oviminor was named by Eliora in 2022 after the facts that it takes 184 minor thirds of 6/5 to reach the interval class of 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-100 53 48 -34⟩

Mapping: [⟨1 -134 -134 -401], ⟨0 184 185 548]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.0193 ¢, ~5/3 = 884.2638 ¢
error map: ⟨+0.019 +0.010 -0.085 +0.032]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.2497 ¢
error map: ⟨0.000 -0.011 -0.120 +0.008]

Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c

Badness (Sintel): 14.7

Octoid

Main article: Octoid
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 (ragisma) and 16875/16807 (mirkwai comma). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives ~12/11, two give ~25/21, three give ~35/27, and four give 99/70~140/99.

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 minimizing the average 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, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

mapping generators: ~49/45, ~7/5

Optimal tunings:

  • WE: ~49/45 = 150.0003 ¢, ~7/5 = 583.9416 ¢
error map: ⟨+0.002 -0.130 -0.547 +0.883]
  • CWE: ~49/45 = 150.0000 ¢, ~7/5 = 583.9411 ¢
error map: ⟨0.000 -0.132 -0.549 +0.880]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
  • 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
  • 7-odd-limit diamond tradeoff: ~7/5 = [582.512, 584.359]
  • 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tunings:

  • WE: ~12/11 = 149.9932 ¢, ~7/5 = 583.9356 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9477 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
  • 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224, 824d

Badness (Sintel): 0.466

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tunings:

  • WE: ~12/11 = 150.0005 ¢, ~7/5 = 583.9066 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9052 ¢

Optimal ET sequence: 72, 152f, 224

Badness (Sintel): 0.631

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tunings:

  • WE: ~12/11 = 150.0064 ¢, ~7/5 = 583.8666 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.8489 ¢

Optimal ET sequence: 72, 152fg, 224, 296, 520g

Badness (Sintel): 0.729

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tunings:

  • WE: ~12/11 = 149.9785 ¢, ~7/5 = 583.8482 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9138 ¢

Optimal ET sequence: 72, 152fg, 224

Badness (Sintel): 0.975

Octopus

A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is 80edo, which has a strong sharp tendency that can be thought of as matching the sharpness of mapping 19/16 to 1\4 = 300 ¢.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tunings:

  • WE: ~12/11 = 150.0313 ¢, ~7/5 = 584.0134 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9583 ¢

Optimal ET sequence: 8d, …, 72, 152, 224f

Badness (Sintel): 0.896

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tunings:

  • WE: ~12/11 = 150.0528 ¢, ~7/5 = 584.0161 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9166 ¢

Optimal ET sequence: 8d, …, 72, 152, 224fg, 296ffg

Badness (Sintel): 0.795

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tunings:

  • WE: ~12/11 = 150.0049 ¢, ~7/5 = 584.0833 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 584.0712 ¢

Optimal ET sequence: 8d, 72, 152

Badness (Sintel): 0.993

Scales: Octoid72, Octoid80

Hexadecoid

See also: 16th-octave temperaments

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tunings:

  • WE: ~448/429 = 74.9943 ¢, ~7/5 = 583.9408 ¢
  • CWE: ~448/429 = 75.0000 ¢, ~7/5 = 583.9709 ¢

Optimal ET sequence: 80, 144, 224

Badness (Sintel): 1.27

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9626 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0463 ¢

Optimal ET sequence: 80, 144, 224, 528dg

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: [⟨16 2 6 6 32 67 81 68], ⟨0 3 4 5 3 -1 -2 0]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9642 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0803 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.44

Parakleismic

Main article: Parakleismic
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, [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. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, for which 99edo, 118edo, and especially 217edo are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23], ⟨0 13 14 35]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.7820 ¢, ~5/3 = 884.6581 ¢
error map: ⟨-0.218 +0.344 +0.644 -0.779]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8088 ¢
error map: ⟨0.000 +0.560 +1.010 -0.516]

Optimal ET sequence: 19, 61d, 80, 99, 217, 316, 415

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 30], ⟨0 13 14 35 -36]]

Optimal tunings:

  • WE: ~2 = 1200.3296 ¢, ~5/3 = 884.9921 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7519 ¢

Optimal ET sequence: 19, 99, 118

Badness (Sintel): 1.64

Paralytic

Paralytic (99e & 118) 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

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57], ⟨0 13 14 35 82]]

Optimal tunings:

  • WE: ~2 = 1199.9940 ¢, ~5/3 = 884.7757 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7800 ¢

Optimal ET sequence: 19e, …, 99e, 118, 217, 335

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57 59], ⟨0 13 14 35 82 -75]]

Optimal tunings:

  • WE: ~2 = 1199.9218 ¢, ~5/3 = 884.7285 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7858 ¢

Optimal ET sequence: 99e, 118, 217

Badness (Sintel): 1.85

Paraklein

Paraklein (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

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: [⟨1 -8 -8 -23 -57 -28], ⟨0 13 14 35 82 43]]

Optimal tunings:

  • WE: ~2 = 1199.8239 ¢, ~5/3 = 884.6449 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7709 ¢

Optimal ET sequence: 19e, …, 99ef, 118

Badness (Sintel): 1.55

Parkleismic

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [⟨1 -8 -8 -23 -43], ⟨0 13 14 35 63]]

Optimal tunings:

  • WE: ~2 = 1199.1848 ¢, ~5/3 = 884.3386 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9158 ¢

Optimal ET sequence: 19e, 61de, 80, 179, 259cd

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [⟨1 -8 -8 -23 -43 -14], ⟨0 13 14 35 63 24]]

Optimal tunings:

  • WE: ~2 = 1199.5318 ¢, ~5/3 = 884.5800 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9118 ¢

Optimal ET sequence: 19e, 61de, 80, 179

Badness (Sintel): 1.51

Paradigmic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: [⟨1 -8 -8 -23 16], ⟨0 13 14 35 -17]]

Optimal tunings:

  • WE: ~2 = 1199.0616 ¢, ~5/3 = 884.2124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8877 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e, 457bcddeeee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [⟨1 -8 -8 -23 16 -14], ⟨0 13 14 35 -17 24]]

Optimal tunings:

  • WE: ~2 = 1199.2683 ¢, ~5/3 = 884.3805 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9061 ¢

Optimal ET sequence: 19, 61d, 80, 99e

Badness (Sintel): 1.48

Semiparakleismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4], ⟨0 13 14 35 23]]

mapping generators: ~99/70, ~33/28

Optimal tunings:

  • WE: ~99/70 = 599.9270 ¢, ~33/28 = 284.7841 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8119 ¢

Optimal ET sequence: 80, 118, 198, 316, 514c

Badness (Sintel): 1.13

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4 15], ⟨0 13 14 35 23 -16]]

Optimal tunings:

  • WE: ~99/70 = 599.8253 ¢, ~33/28 = 284.7608 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8366 ¢

Optimal ET sequence: 80, 118, 198

Badness (Sintel): 1.40

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [⟨2 -3 -2 -11 -4 -4], ⟨0 13 14 35 23 24]]

Optimal tunings:

  • WE: ~99/70 = 600.0569 ¢, ~13/11 = 284.8431 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~13/11 = 284.8216 ¢

Optimal ET sequence: 80, 118f, 198f

Badness (Sintel): 1.67

Counterkleismic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Counterhanson.

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, [-20 -24 25⟩, the amount by which six major dieses ((648/625)6) fall short of the classic major third (5/4). It can be described as 19 & 224 temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to catakleismic and parakleismic)

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 -5 -4 -18], ⟨0 25 24 79]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.1778 ¢, ~6/5 = 316.1065 ¢
error map: ⟨+0.178 -0.181 -0.469 +0.388]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0631 ¢
error map: ⟨0.000 -0.377 -0.799 +0.161]

Optimal ET sequence: 19, …, 205, 224, 243, 467

Badness (Sintel): 2.29

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: [⟨1 -5 -4 -18 19], ⟨0 25 24 79 -59]]

Optimal tunings:

  • WE: ~2 = 1199.9944 ¢, ~6/5 = 316.0690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0705 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 2.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: [⟨1 -5 -4 -18 19 -15], ⟨0 25 24 79 -59 71]]

Optimal tunings:

  • WE: ~2 = 1199.9827 ¢, ~6/5 = 316.0650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0695 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 1.40

Counterlytic

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: [⟨1 -5 -4 -18 -40], ⟨0 25 24 79 165]]

Optimal tunings:

  • WE: ~2 = 1200.1247 ¢, ~6/5 = 316.0976 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0660 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: [⟨1 -5 -4 -18 -40 -15], ⟨0 25 24 79 165 71]]

Optimal tunings:

  • WE: ~2 = 1200.0987 ¢, ~6/5 = 316.0908 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0658 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 1.23

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

mapping generators: ~2, ~1728/1715

Optimal tunings:

  • WE: ~2 = 1200.2169 ¢, ~1728/1715 = 16.6160 ¢
error map: ⟨+0.217 +0.000 +0.155 -0.799]
  • CWE: ~2 = 1200.0000 ¢, ~1728/1715 = 16.6083 ¢
error map: ⟨0.000 -0.205 -0.122 -1.343]

Optimal ET sequence: 72, 217, 289, 650d, 939dd

Badness (Sintel): 2.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -30 -49 -14 -39]]

Optimal tunings:

  • WE: ~2 = 1200.1286 ¢, ~100/99 = 16.6147 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6101 ¢

Optimal ET sequence: 72, 217, 289

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: [⟨1 2 3 3 4 5], ⟨0 -30 -49 -14 -39 -94]]

Optimal tunings:

  • WE: ~2 = 1200.0554 ¢, ~100/99 = 16.6028 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6011 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.986

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: [⟨1 2 3 3 4 5 5], ⟨0 -30 -49 -14 -39 -94 -66]]

Optimal tunings:

  • WE: ~2 = 1200.0647 ¢, ~100/99 = 16.6025 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6004 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.751

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

Mapping: [⟨1 2 3 3 4 5 5 4], ⟨0 -30 -49 -14 -39 -94 -66 18]]

Optimal tunings:

  • WE: ~2 = 1199.9287 ¢, ~100/99 = 16.5930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.5948 ¢

Optimal ET sequence: 72, 145, 217

Badness (Sintel): 0.924

Sfourth

For the 5-limit version, see Miscellaneous 5-limit temperaments #Sfourth.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.8332 ¢, ~49/48 = 26.3053 ¢
error map: ⟨+0.833 -0.090 +0.721 -3.074]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2590 ¢
error map: ⟨0.000 -0.876 -0.343 -5.157]

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 3.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -19 -31 -9 -25]]

Optimal tunings:

  • WE: ~2 = 1201.1486 ¢, ~49/48 = 26.3112 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2461 ¢

Optimal ET sequence: 45e, 46, 91e, 137de

Badness (Sintel): 1.78

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -19 -31 -9 -25 -14]]

Optimal tunings:

  • WE: ~2 = 1201.4956 ¢, ~49/48 = 26.3423 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2614 ¢

Optimal ET sequence: 45ef, 46, 91ef, 137def, 228ddeeefff

Badness (Sintel): 1.37

Sfour

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: [⟨1 2 3 3 3], ⟨0 -19 -31 -9 21]]

Optimal tunings:

  • WE: ~2 = 1200.4402 ¢, ~49/48 = 26.2557 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2403 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: [⟨1 2 3 3 3 3], ⟨0 -19 -31 -9 21 32]]

Optimal tunings:

  • WE: ~2 = 1200.3796 ¢, ~49/48 = 26.2473 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2372 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.14

Trideci

For the 5-limit version, see 13th-octave temperaments #Tridecatonic.

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 tridecatonic temperament, but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name trideci comes from tridecim (Latin for "thirteen").

Subgroup: 2.3.5.7

Comma list: 4375/4374, 83349/81920

Mapping: [⟨13 0 -11 57], ⟨0 1 2 -1]]

mapping generators: ~256/245, ~3

Optimal tunings:

  • WE: ~256/245 = 92.4141 ¢, ~3/2 = 699.9466 ¢
error map: ⟨+1.383 -0.626 -0.210 -2.554]
  • CWE: ~256/245 = 92.3077 ¢, ~3/2 = 699.4521 ¢
error map: ⟨0.000 -2.503 -2.794 -6.740]

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 4.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: [⟨13 0 -11 57 45], ⟨0 1 2 -1 0]]

Optimal tunings:

  • WE: ~22/21 = 92.3729 ¢, ~3/2 = 700.1118 ¢
  • CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.7703 ¢

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: [⟨13 0 -11 57 45 48], ⟨0 1 2 -1 0 0]]

Optimal tunings:

  • WE: ~22/21 = 92.4003 ¢, ~3/2 = 699.9983 ¢
  • CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.4772 ¢

Optimal ET sequence: 26, 65f, 91f

Badness (Sintel): 2.16

References

  1. ↑ Yahoo! Tuning Group | 11-limit rank 2 using only wedgies "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." —Gene Ward Smith
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