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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
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{{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]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"