Hemimage temperaments: Difference between revisions

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== Chromat ==
== Chromat ==
The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[Amity family|amity extension]] with third-octave period.
The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[amity family|amity extension]] with third-octave period.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27
: mapping generators: ~63/50, ~28/27


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~28/27 = 60.528{{c}}


{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]]: 0.057499
[[Badness]] (Smith): 0.057499


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.430{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258, 417d }}


Badness: 0.050379
Badness (Smith): 0.050379


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.428{{c}}


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417d }}


Badness: 0.046006
Badness (Smith): 0.046006


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.438{{c}}


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}


Badness: 0.031678
Badness (Smith): 0.031678


==== Catachrome ====
==== Catachrome ====
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Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.378{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.043844
Badness (Smith): 0.043844


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~28/27 = 60.377{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.030218
Badness (Smith): 0.030218


==== Chromic ====
==== Chromic ====
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Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~27/26 = 60.456{{c}}


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f, 258ff }}


Badness: 0.049857
Badness (Smith): 0.049857


===== 17-limit =====
===== 17-limit =====
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Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459
Optimal tuning (POTE): ~63/50 = 400.000{{c}}, ~27/26 = 60.459{{c}}


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f, 258ff }}


Badness: 0.031043
Badness (Smith): 0.031043


=== Hemichromat ===
=== Hemichromat ===
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Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~55/54 = 30.2511{{c}}


{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.067173
Badness (Smith): 0.067173


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
Optimal tuning (CTE): ~63/50 = 400.0000{{c}}, ~55/54 = 30.2527{{c}}


{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.033420
Badness (Smith): 0.033420


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
: mapping generators: ~1225/864, ~192/175
: mapping generators: ~1225/864, ~192/175


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 600.000{{c}}, ~192/175 = 162.806{{c}}


{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.065492
[[Badness]] (Smith): 0.065492


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~11/10 = 162.773{{c}}


{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de }}


Badness: 0.032080
Badness (Smith): 0.032080


== Bicommatic ==
== Bicommatic ==
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{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80
: mapping generators: ~567/400, ~81/80


[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000{{c}}, ~81/80 = 20.377{{c}}


{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}


[[Badness]]: 0.084317
[[Badness]] (Smith): 0.084317


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~81/80 = 20.390{{c}}


{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}


Badness: 0.030461
Badness (Smith): 0.030461


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~66/65 = 20.427{{c}}


{{Optimal ET sequence|legend=1| 58, 118, 176f }}
{{Optimal ET sequence|legend=0| 58, 118, 176f }}


Badness: 0.026336
Badness (Smith): 0.026336


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~66/65 = 20.378{{c}}


{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}
{{Optimal ET sequence|legend=0| 58, 118, 294ffg, 412dffgg }}


Badness: 0.022396
Badness (Smith): 0.022396


== Degrees ==
== Degrees ==
{{About|the regular temperament|scale degrees|degree}}
{{About|the regular temperament|scale degrees|degree}}
{{ See also | 20th-octave temperaments }}
{{See also| 20th-octave temperaments }}
 
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.  
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.  


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{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~3
: mapping generators: ~28/27, ~3


[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
[[Optimal tuning]] ([[POTE]]): ~28/27 = 60.000{{c}}, ~3/2 = 703.015{{c}} (~126/125 = 16.985{{c}})


{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}


[[Badness]]: 0.106471
[[Badness]] (Sintel): 2.69
 
Badness (Sintel): 2.694


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)
Optimal tuning (POTE): ~28/27 = 60.000{{c}}, ~3/2 = 703.231{{c}} (~100/99 = 16.769{{c}})
 
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}


Badness: 0.046770
{{Optimal ET sequence|legend=0| 20cd, 60e, 80, 140, 360 }}


Badness (Sintel): 1.546
Badness (Sintel): 1.55


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)
Optimal tuning (POTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.080{{c}} (~100/99 = 16.920{{c}})
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


Badness: 0.032718
{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}


Badness (Sintel): 1.352
Badness (Sintel): 1.35


=== 17-limit ===
=== 17-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.107{{c}} (~100/99 = 16.893{{c}})


{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}


Badness (Sintel): 1.171
Badness (Sintel): 1.17


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.107{{c}} (~100/99 = 16.893{{c}})


{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cde, 60e, 80, 140 }}


Badness (Sintel): 1.273
Badness (Sintel): 1.27


=== 23-limit ===
=== 23-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
Optimal tuning (CTE): ~28/27 = 600.000{{c}}, ~3/2 = 703.169{{c}} (~100/99 = 16.831{{c}})


{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cdei, 60e, 80, 140 }}


Badness (Sintel): 1.209
Badness (Sintel): 1.21


=== 29-limit ===
=== 29-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.171{{c}} (~100/99 = 16.829{{c}})


{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cdeij, 60e, 80, 140 }}


Badness (Sintel): 1.134
Badness (Sintel): 1.13


=== no-31's 37-limit ===
=== no-31's 37-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}


Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.222{{c}} (~100/99 = 16.778{{c}})


{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cdeijl, 60el, 80, 140 }}


Badness (Sintel): 1.127
Badness (Sintel): 1.13


=== no-31's 41-limit ===
=== no-31's 41-limit ===
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Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}


Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
Optimal tuning (CTE): ~29/28 = 600.000{{c}}, ~3/2 = 703.207{{c}}


{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 20cdeijlm, 60el, 80, 140 }}


Badness (Sintel): 1.100
Badness (Sintel): 1.10


== Squarschmidt ==
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as {{nowrap| 118 & 239 }} temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 380: Line 371:


{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
: mapping generators: ~2, ~98304/78125
: mapping generators: ~2, ~98304/78125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98304/78125 = 396.621{{c}}


{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


[[Badness]]: 0.218314
[[Badness]] (Smith): 0.218314


=== 7-limit ===
=== 7-limit ===
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{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1125/896 = 396.643{{c}}


{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


[[Badness]]: 0.132821
[[Badness]] (Smith): 0.132821


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~44/35 = 396.644{{c}}


{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}


Badness: 0.038186
Badness (Smith): 0.038186


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 10:53, 28 April 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 hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:

Chromat

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an amity extension with third-octave period.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

mapping generators: ~63/50, ~28/27

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~28/27 = 60.528 ¢

Optimal ET sequence: 39d, 60, 99, 258, 357, 456

Badness (Smith): 0.057499

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 10976/10935

Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~28/27 = 60.430 ¢

Optimal ET sequence: 60e, 99e, 159, 258, 417d

Badness (Smith): 0.050379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 10976/10935

Mapping: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~28/27 = 60.428 ¢

Optimal ET sequence: 99ef, 159, 258, 417d

Badness (Smith): 0.046006

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757

Mapping: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~28/27 = 60.438 ¢

Optimal ET sequence: 99ef, 159, 258, 417dg

Badness (Smith): 0.031678

Catachrome

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 1001/1000, 10976/10935

Mapping: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~28/27 = 60.378 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Smith): 0.043844

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913

Mapping: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~28/27 = 60.377 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Smith): 0.030218

Chromic

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1875/1859

Mapping: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]

Optimal tuning (POTE): ~44/35 = 400.000 ¢, ~27/26 = 60.456 ¢

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness (Smith): 0.049857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 352/351, 375/374, 595/594

Mapping: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]

Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~27/26 = 60.459 ¢

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness (Smith): 0.031043

Hemichromat

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 10976/10935, 102487/102400

Mapping: [⟨3 4 5 6 10], ⟨0 10 26 32 5]]

Optimal tuning (CTE): ~63/50 = 400.0000 ¢, ~55/54 = 30.2511 ¢

Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b

Badness (Smith): 0.067173

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935

Mapping: [⟨3 4 5 6 10 8], ⟨0 10 26 32 5 41]]

Optimal tuning (CTE): ~63/50 = 400.0000 ¢, ~55/54 = 30.2527 ¢

Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b

Badness (Smith): 0.033420

Bisupermajor

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

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [⟨2 1 6 1], ⟨0 8 -5 17]]

mapping generators: ~1225/864, ~192/175

Optimal tuning (POTE): ~1225/864 = 600.000 ¢, ~192/175 = 162.806 ¢

Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d

Badness (Smith): 0.065492

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [⟨2 1 6 1 8], ⟨0 8 -5 17 -4]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~11/10 = 162.773 ¢

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de

Badness (Smith): 0.032080

Bicommatic

Used to be known simply as the commatic temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

mapping generators: ~567/400, ~81/80

Optimal tuning (POTE): ~567/400 = 600.000 ¢, ~81/80 = 20.377 ¢

Optimal ET sequence: 58, 118, 294, 412d, 530d

Badness (Smith): 0.084317

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~81/80 = 20.390 ¢

Optimal ET sequence: 58, 118, 294, 412d

Badness (Smith): 0.030461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1001/1000

Mapping: [⟨2 3 4 5 6 7], ⟨0 5 19 18 27 12]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~66/65 = 20.427 ¢

Optimal ET sequence: 58, 118, 176f

Badness (Smith): 0.026336

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

Mapping: [⟨2 3 4 5 6 7 8], ⟨0 5 19 18 27 12 5]]

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~66/65 = 20.378 ¢

Optimal ET sequence: 58, 118, 294ffg, 412dffgg

Badness (Smith): 0.022396

Degrees

This page is about the regular temperament. For scale degrees, see degree.
See also: 20th-octave temperaments

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with 23/20, 6\20 = 3\10 with 69/56, 7\20 with 23/18, etc. By observing that 1\20 works as 30/29~29/28~28/27, with 29/28 being especially accurate, and by equating 29/22 with 2\5 = 240 ¢, we get a uniquely elegant extension to the 29-limit which tempers out (33/25)/(29/22) = 726/725, S28 = 784/783 and S29 = 841/840. An edo as large as 220 supports it by patent val, though it does not appear in the optimal ET sequence, and 80edo and 140edo are both much more recommendable tunings.

By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.

By looking at the mapping, we observe an 80-note mos scale is ideal, so that 80edo is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of 20edo by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in 140edo.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

mapping generators: ~28/27, ~3

Optimal tuning (POTE): ~28/27 = 60.000 ¢, ~3/2 = 703.015 ¢ (~126/125 = 16.985 ¢)

Optimal ET sequence: 20cd, 60, 80, 140, 640b, 780b

Badness (Sintel): 2.69

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]

Optimal tuning (POTE): ~28/27 = 60.000 ¢, ~3/2 = 703.231 ¢ (~100/99 = 16.769 ¢)

Optimal ET sequence: 20cd, 60e, 80, 140, 360

Badness (Sintel): 1.55

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]

Optimal tuning (POTE): ~28/27 = 600.000 ¢, ~3/2 = 703.080 ¢ (~100/99 = 16.920 ¢)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.35

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

Mapping: [⟨20 0 -17 -39 -26 74 50], ⟨0 1 2 3 3 0 1]]

Optimal tuning (CTE): ~28/27 = 600.000 ¢, ~3/2 = 703.107 ¢ (~100/99 = 16.893 ¢)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.17

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

Mapping: [⟨20 0 -17 -39 -26 74 50 85], ⟨0 1 2 3 3 0 1 0]]

Optimal tuning (CTE): ~28/27 = 600.000 ¢, ~3/2 = 703.107 ¢ (~100/99 = 16.893 ¢)

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.27

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27], ⟨0 1 2 3 3 0 1 0 2]]

Optimal tuning (CTE): ~28/27 = 600.000 ¢, ~3/2 = 703.169 ¢ (~100/99 = 16.831 ¢)

Optimal ET sequence: 20cdei, 60e, 80, 140

Badness (Sintel): 1.21

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2], ⟨0 1 2 3 3 0 1 0 2 3]]

Optimal tuning (CTE): ~29/28 = 600.000 ¢, ~3/2 = 703.171 ¢ (~100/99 = 16.829 ¢)

Optimal ET sequence: 20cdeij, 60e, 80, 140

Badness (Sintel): 1.13

no-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9], ⟨0 1 2 3 3 0 1 0 2 3 3]]

Optimal tuning (CTE): ~29/28 = 600.000 ¢, ~3/2 = 703.222 ¢ (~100/99 = 16.778 ¢)

Optimal ET sequence: 20cdeijl, 60el, 80, 140

Badness (Sintel): 1.13

no-31's 41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9 12], ⟨0 1 2 3 3 0 1 0 2 3 3 3]]

Optimal tuning (CTE): ~29/28 = 600.000 ¢, ~3/2 = 703.207 ¢

Optimal ET sequence: 20cdeijlm, 60el, 80, 140

Badness (Sintel): 1.10

Squarschmidt

A generator for the squarschimidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents. The squarschimidt temperament can be described as 118 & 239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

Subgroup: 2.3.5

Comma list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98304/78125 = 396.621 ¢

Optimal ET sequence: 118, 593, 711, 829, 947

Badness (Smith): 0.218314

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~1125/896 = 396.643 ¢

Optimal ET sequence: 118, 239, 357, 596, 1549bd

Badness (Smith): 0.132821

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: [⟨1 -8 1 -20 -21], ⟨0 29 4 69 74]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~44/35 = 396.644 ¢

Optimal ET sequence: 118, 239, 357, 596

Badness (Smith): 0.038186

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