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{{Technical data page}}
{{Technical data page}}
<div style="float: right;">
[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]
</div>
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].


Line 15: Line 11:
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Pelogic family #Mohavila|Pelogic family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Mavila family #Mohavila|Mavila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
Line 23: Line 19:
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Decistearn]]'' (+118098/117649) → [[Trisedodge family #Decistearn|Trisedodge family]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Septischismic clan #Quintagar|Septischismic clan]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]


= 2.5.7 subgroup =
= 2.5.7 subgroup =
== Didacus ==
== Didacus ==
{{main|Didacus}}
{{Main| Didacus }}
 
See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 39: Line 37:
[[Comma list]]: [[3136/3125]]
[[Comma list]]: [[3136/3125]]


{{Mapping|legend=2| 1 0 -3 | 0 2 5 }}
{{Mapping|legend=3| 1 0 -3 | 0 2 5 }}
 
: sval mapping generators: ~2, ~56/25
 
{{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}


: [[gencom]]: [2 56/25; 3136/3125]
{{Mapping|legend=5| 1 0 0 -3 | 0 0 2 5 }}
: mapping generators: ~2, ~56/25


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8194{{c}}, ~28/25 = 193.7425{{c}}
: [[error map]]: {{val| -0.181 +0.810 -0.474 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 193.7167{{c}}
: error map: {{val| 0.000 +1.120 -0.243 }}


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Badness]] (Sintel): 0.0908
 
[[Badness]] (Dirichlet): 0.091


= Strong extensions =
= Strong extensions =
Line 69: Line 66:
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
|}
<nowiki />* Defined by intersection with other documented extensions
<nowiki/>* Defined by intersection with other documented extensions


== Hemiwürschmidt ==
== Hemiwürschmidt ==
{{See also| Würschmidt family }}
{{See also| Würschmidt family }}


'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.
Hemiwürschmidt (sometimes spelled ''hemiwuerschmidt'') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings. 130edo in particular shows how hemiwürschmidt may be extended to a higher limit, mapping 11 to 40 generators which is the same as würschmidt.
 
Hemiwürschmidt was named by [[Gene Ward Smith]] in 2004 as ''hemi-'' + ''würschmidt''.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html Yahoo! Tuning Group (archive) | ''Names for important high-complexity temperaments'']</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 80: Line 79:
[[Comma list]]: 2401/2400, 3136/3125
[[Comma list]]: 2401/2400, 3136/3125


{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}
{{Mapping|legend=1| 1 -1 2 2 | 0 16 2 5 }}
: mapping generators: ~2, ~28/25


Mapping generators: ~2, ~25/14
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.7153{{c}}, ~28/25 = 193.8522{{c}}
{{Multival|legend=1| 16 2 5 -34 -37 6 }}
: [[error map]]: {{val| -0.285 -0.035 +0.821 -0.134 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 193.8883{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
: error map: {{val| 0.000 +0.258 +1.463 +0.616 }}


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}


[[Badness]]: 0.020307
[[Badness]] (Sintel): 0.514


=== 2.3.5.7.23 subgroup ===
=== 11-limit ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7.23
Comma list: 243/242, 441/440, 3136/3125


[[Comma list]]: 576/575, 736/735, 1127/1125
{{Mapping|legend=0| 1 -1 2 2 -3 | 0 16 2 5 40 }}


{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}
Optimal tunings:
* WE: ~2 = 1199.7863{{c}}, ~28/25 = 193.8058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8342{{c}}


Mapping generators: ~2, ~25/14
{{Optimal ET sequence|legend=0| 31, 68e, 99e, 130 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901
Badness (Sintel): 0.697


{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{todo|verify this & get badness|inline=1}}
Comma list: 243/242, 351/350, 441/440, 3584/3575


=== 11-limit ===
{{Mapping|legend=0| 1 -1 2 2 -3 10 | 0 16 2 5 40 -39 }}
Subgroup: 2.3.5.7.11


Comma list: 243/242, 441/440, 3136/3125
Optimal tunings:  
* WE: ~2 = 1199.8626{{c}}, ~28/25 = 193.8063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8277{{c}}


Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}
{{Optimal ET sequence|legend=0| 31, 99e, 130, 291, 421e, 551ce }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840
Badness (Sintel): 0.953


{{Optimal ET sequence|legend=1| 31, 99e, 130, 650ce, 811ce }}
===== 2.3.5.7.11.13.23 subgroup =====
As described at the page for [[würschmidt]], there is an extension for prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).


Badness: 0.021069
Subgroup: 2.3.5.7.11.13.23


==== 13-limit ====
Comma list: 243/242, 276/275, 351/350, 441/440, 3584/3575
Subgroup: 2.3.5.7.11.13


Comma list: 243/242, 351/350, 441/440, 3584/3575
{{Mapping|legend=0| 1 -1 2 2 -3 10 0 | 0 16 2 5 40 -39 28 }}


Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}
Optimal tunings:  
* WE: ~2 = 1199.9052{{c}}, ~28/25 = 193.8170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8316{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829
{{Optimal ET sequence|legend=0| 31, 99e, 130 }}


{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}
Badness (Sintel): 0.839
 
Badness: 0.023074


==== Hemithir ====
==== Hemithir ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 144/143, 196/195, 243/242, 625/624


Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}
{{Mapping|legend=0| 1 -1 2 2 -3 5 | 0 16 2 5 40 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918
Optimal tunings:
* WE: ~2 = 1199.1395{{c}}, ~28/25 = 193.7788{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.9023{{c}}


{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}
{{Optimal ET sequence|legend=0| 31, 68e, 99ef }}


Badness: 0.031199
Badness (Sintel): 1.29


=== Hemiwur ===
=== Hemiwur ===
Hemiwur may as well be described as 31 & 37, mapping prime 11 to 9 generator steps.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 1375/1372
Comma list: 121/120, 176/175, 1375/1372


Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}
{{Mapping|legend=0| 1 -1 2 2 2 | 0 16 2 5 9 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884
Optimal tunings:
* WE: ~2 = 1200.1742{{c}}, ~28/25 = 193.9117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.8896{{c}}


{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}
{{Optimal ET sequence|legend=0| 31, 68, 99, 130e }}


Badness: 0.029270
Badness (Sintel): 0.968


==== 13-limit ====
==== 13-limit ====
Line 166: Line 176:
Comma list: 121/120, 176/175, 196/195, 275/273
Comma list: 121/120, 176/175, 196/195, 275/273


Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}
{{Mapping|legend=0| 1 -1 2 2 2 5 | 0 16 2 5 9 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004
Optimal tunings:
* WE: ~2 = 1199.4972{{c}}, ~28/25 = 193.9231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.9942{{c}}


{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}
{{Optimal ET sequence|legend=0| 31, 68 }}


Badness: 0.028432
Badness (Sintel): 1.17


==== Hemiwar ====
==== Hemiwar ====
Line 179: Line 191:
Comma list: 66/65, 105/104, 121/120, 1375/1372
Comma list: 66/65, 105/104, 121/120, 1375/1372


Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}
{{Mapping|legend=0| 1 -1 2 2 2 0 | 0 16 2 5 9 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698
Optimal tunings:
* WE: ~2 = 1199.9465{{c}}, ~28/25 = 193.6889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 193.6956{{c}}


{{Optimal ET sequence|legend=1| 6f, 31 }}
{{Optimal ET sequence|legend=0| 6f, 25bf, 31 }}


Badness: 0.044886
Badness (Sintel): 1.85


=== Quadrawürschmidt ===
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
This extension has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense because it splits the generator in four.
 
The generator of quadrawürschmidt is essentially a [[meantone family #Meantone|meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 whole tones up. The small comma between the generator and 3/2 represents [[441/440]][[~]][[540/539]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 194: Line 210:
Comma list: 2401/2400, 3025/3024, 3136/3125
Comma list: 2401/2400, 3025/3024, 3136/3125


Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}
{{Mapping|legend=0| 1 -17 0 -3 -25 | 0 32 4 10 49 }}
 
: mapping generators: ~2, ~220/147
: mapping generators: ~2, ~147/110


Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404
Optimal tunings:
* WE: ~2 = 1199.7360{{c}}, ~220/147 = 696.8062{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~220/147 = 696.9548{{c}}


{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
{{Optimal ET sequence|legend=0| 31, 105be, 136e, 167, 198, 427c }}


Badness: 0.034814
Badness (Sintel): 1.15


=== Semihemiwür ===
=== Semihemiwürschmidt ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 3136/3125, 9801/9800
Comma list: 2401/2400, 3136/3125, 9801/9800


Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}
{{Mapping|legend=0| 2 -2 4 4 15 | 0 16 2 5 -25 }}
 
: mapping generators: ~99/70, ~28/25
: mapping generators: ~99/70, ~495/392


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021
Optimal tunings:
* WE: ~99/70 = 599.8485{{c}}, ~28/25 = 193.8531{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/25 = 193.9002{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198, 328 }}


Badness: 0.044848
Badness (Sintel): 1.48


==== 13-limit ====
==== 13-limit ====
Line 224: Line 242:
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125


Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 | 0 16 2 5 -25 26 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035
Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~28/25 = 193.8555{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/25 = 193.9006{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198, 328 }}


Badness: 0.023388
Badness (Sintel): 0.966


===== Semihemiwürat =====
===== Semihemiwürat =====
Line 237: Line 257:
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625


Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 3 | 0 16 2 5 -25 26 16 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112
Optimal tunings:
* WE: ~17/12 = 599.9069{{c}}, ~28/25 = 193.8811{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9092{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198 }}


Badness: 0.028987
Badness (Sintel): 1.48


====== 19-limit ======
====== 19-limit ======
Line 250: Line 272:
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625


Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 3 3 | 0 16 2 5 -25 26 16 17 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145
Optimal tunings:
* WE: ~17/12 = 599.9292{{c}}, ~19/17 = 193.8916{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~19/17 = 193.9128{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198 }}


Badness: 0.021707
Badness (Sintel): 1.32


===== Semihemiwüram =====
===== Semihemiwüram =====
Line 263: Line 287:
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224


Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 14 | 0 16 2 5 -25 26 -18 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112
Optimal tunings:  
* WE: ~17/12 = 599.7759{{c}}, ~28/25 = 193.8534{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9238{{c}}


{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}
{{Optimal ET sequence|legend=0| 68, 130g, 198g }}


Badness: 0.029718
Badness (Sintel): 1.51


====== 19-limit ======
====== 19-limit ======
Line 276: Line 302:
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224


Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}
{{Mapping|legend=0| 2 -2 4 4 15 -1 14 14 | 0 16 2 5 -25 26 -18 -17 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428
Optimal tunings:
* WE: ~17/12 = 599.7147{{c}}, ~28/25 = 193.8506{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9417{{c}}


{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}
{{Optimal ET sequence|legend=0| 68, 130gh, 198gh }}


Badness: 0.029545
Badness (Sintel): 1.80


== Hemithirds ==
== Hemithirds ==
Line 291: Line 319:
[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Mapping|legend=1| 1 -11 4 7 | 0 15 -2 -5 }}
: mapping generators: ~2, ~25/14


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.3104{{c}}, ~25/14 = 1007.0160{{c}} (~28/25 = 193.2944{{c}})
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
: [[error map]]: {{val| +0.310 -0.129 +0.896 -1.733 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 1006.7606{{c}} (~28/25 = 193.2394{{c}})
: error map: {{val| 0.000 -0.545 +0.165 -2.629 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118, 267d, 385d }}


[[Badness]]: 0.044284
[[Badness]] (Sintel): 1.12


=== 11-limit ===
=== 11-limit ===
Line 314: Line 345:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}
{{Mapping|legend=0| 1 -11 4 7 -15 | 0 15 -2 -5 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
Optimal tunings:
* WE: ~2 = 1200.3576{{c}}, ~25/14 = 1007.0732{{c}} (~28/25 = 193.2844{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 1006.7807{{c}} (~28/25 = 193.2193{{c}})


Minimax tuning:
Minimax tuning:
Line 323: Line 356:
: Eigenmonzos (unchanged-intervals): 2, 11/7
: Eigenmonzos (unchanged-intervals): 2, 11/7


{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}
{{Optimal ET sequence|legend=0| 25e, 31, 87, 118 }}


Badness: 0.019003
Badness (Sintel): 0.628


=== 13-limit ===
=== 13-limit ===
Line 332: Line 365:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
{{Mapping|legend=0| 1 4 2 2 7 0 | 0 15 -2 -5 22 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
Optimal tunings:
* WE: ~2 = 1200.0602{{c}}, ~25/14 = 1006.8844{{c}} (~28/25 = 193.1757{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 1006.8343{{c}} (~28/25 = 193.1457{{c}})


{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=0| 31, 56, 87, 118, 205d }}


Badness: 0.021738
Badness (Sintel): 0.898


== Spell ==
== Spell ==
{{See also| Magic family }}
{{See also| Magic family }}
Spell tempers out [[49/48]] and may be described as the 19 & 25 temperament, though the optimum is much closer to [[19edo]] than to [[25edo]]. In the 11-limit the generator is taken as [[~]][[11/10]], and in the 13-limit the hemifourth is identified with [[~]][[15/13]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 349: Line 386:
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}


{{Multival|legend=1| 10 2 5 -20 -20 6 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.8089{{c}}, ~28/25 = 190.3712{{c}}
: [[error map]]: {{val| +2.809 +1.757 +0.047 -11.352 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 190.1078{{c}}
: error map: {{val| 0.000 -0.877 -6.098 -18.287 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
{{Optimal ET sequence|legend=1| 6, 13b, 19, 82dd }}


{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
[[Badness]] (Sintel): 2.05
 
[[Badness]]: 0.080958


=== 11-limit ===
=== 11-limit ===
Line 362: Line 401:
Comma list: 49/48, 56/55, 125/121
Comma list: 49/48, 56/55, 125/121


Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}
{{Mapping|legend=0| 1 0 2 2 3 | 0 10 2 5 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
Optimal tunings:
* WE: ~2 = 1200.4163{{c}}, ~11/10 = 190.3511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.3091{{c}}


{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=0| 6, 13b, 19 }}


Badness: 0.059791
Badness (Sintel): 1.98


==== 13-limit ====
==== 13-limit ====
This extension was known as ''cantrip'' in earlier materials such as the [https://x31eq.com/temper/ Temperament Finder]. It is considered the canonical extension here because of the naturalness of mapping 15/13 to the hemifourth.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 78/77, 125/121
Comma list: 49/48, 56/55, 91/90, 125/121
 
{{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}


Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
Optimal tunings:  
* WE: ~2 = 1200.4118{{c}}, ~11/10 = 190.4253{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.3835{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
{{Optimal ET sequence|legend=0| 6f, 19 }}


{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
Badness (Sintel): 1.72


Badness: 0.045591
==== Spellcraft ====
This extension was considered the canonical 13-limit spell, decanonized in 2026 in favor of the above one.  


==== Cantrip ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 125/121
Comma list: 49/48, 56/55, 78/77, 125/121


Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
{{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
Optimal tunings:
* WE: ~2 = 1201.8239{{c}}, ~11/10 = 190.2163{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.0020{{c}}


{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=0| 6, 13b, 19 }}


Badness: 0.041603
Badness (Sintel): 1.88


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


The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.  
The generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning. The name was given by [[Petr Pařízek]] in 2011.<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group (archive) | ''Suggested names for the unclasified temperaments'']</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 407: Line 455:
[[Comma list]]: 1728/1715, 3136/3125
[[Comma list]]: 1728/1715, 3136/3125


{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
{{Mapping|legend=1| 1 -5 0 -3 | 0 17 6 15 }}
 
: mapping generators: ~2, ~98/75
: mapping generators: ~2, ~75/49
 
{{Multival|legend=1| 17 6 15 -30 -24 18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2968{{c}}, ~98/75 = 464.5722{{c}}
: [[error map]]: {{val| -0.703 -0.712 +1.119 +1.867 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 464.8120{{c}}
: error map: {{val| 0.000 -0.150 +2.559 +3.355 }}


{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111, 364cdd, 475ccddd }}


[[Badness]]: 0.050472
[[Badness]] (Sintel): 1.28


=== 11-limit ===
=== 11-limit ===
Line 424: Line 473:
Comma list: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 | 0 17 6 15 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3073{{c}}, ~72/55 = 464.6063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8423{{c}}


{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.022476
Badness (Sintel): 0.743


==== 13-limit ====
==== 13-limit ====
Line 437: Line 488:
Comma list: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 351/350, 540/539, 1375/1372


Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 | 0 17 6 15 27 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
Optimal tunings:
* WE: ~2 = 1199.3145{{c}}, ~72/55 = 464.6080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8636{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.025204
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 450: Line 503:
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 | 0 17 6 15 27 -24 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3067{{c}}, ~17/13 = 464.6061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8699{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.019919
Badness (Sintel): 1.01


===== 19-limit =====
===== 19-limit =====
Line 463: Line 518:
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 | 0 17 6 15 27 -24 -23 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116
Optimal tunings:
* WE: ~2 = 1199.2346{{c}}, ~17/13 = 464.5878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8816{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.016301
Badness (Sintel): 0.992


===== 23-limit =====
===== 23-limit =====
Line 476: Line 533:
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 -4 | 0 17 6 15 27 -24 -23 -20 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106
Optimal tunings:
* WE: ~2 = 1199.3227{{c}}, ~17/13 = 464.6320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8912{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=0| 31, 80, 111, 191cdh }}


Badness: 0.014957
Badness (Sintel): 1.07


==== Semishly ====
==== Semishly ====
Line 489: Line 548:
Comma list: 144/143, 176/175, 196/195, 275/273
Comma list: 144/143, 176/175, 196/195, 275/273


Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 1 | 0 17 6 15 27 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
Optimal tunings:
* WE: ~2 = 1198.3103{{c}}, ~13/10 = 464.3248{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 464.9110{{c}}


{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
{{Optimal ET sequence|legend=0| 31, 49f, 80f }}


Badness: 0.028408
Badness (Sintel): 1.17


== Emka ==
== Emka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''
 
Emka may be described as the 37 & 50 temperament, which tempers out the hemimean comma and 84035/82944 (quinzo-ayo). It was named by [[Petr Pařízek]] in 2011.<ref name="petr's long post"/>


Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 &amp; 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 &amp; 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
A related temperament of emka is [[horwell temperaments #Emkay|emkay]] (87 & 224), which is more accurate but more complex, and only differs from emka in the mapping of 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 506: Line 569:
[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}
{{Mapping|legend=1| 1 -13 -2 -8 | 0 27 8 20 }}
: mapping generators: ~2, ~35/24


: mapping generators: ~2, ~48/35
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4701{{c}}, ~35/24 = 648.4717{{c}}
{{Multival|legend=1| 27 8 20 -50 -44 24 }}
: [[error map]]: {{val| +0.470 +0.669 +0.520 -3.153 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 648.2307{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
: error map: {{val| 0.000 +0.274 -0.468 -4.212 }}


{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}


[[Badness]]: 0.144338
[[Badness]] (Sintel): 3.65


=== 11-limit ===
=== 11-limit ===
Line 523: Line 587:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 | 0 27 8 20 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
Optimal tunings:
* WE: ~2 = 1200.2552{{c}}, ~16/11 = 648.3727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2399{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=0| 37, 50, 87 }}


Badness: 0.054744
Badness (Sintel): 1.81


=== 13-limit ===
=== 13-limit ===
Line 536: Line 602:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 1 | 0 27 8 20 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
Optimal tunings:
* WE: ~2 = 1200.1506{{c}}, ~16/11 = 648.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2442{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=0| 37, 50, 87, 224d, 311d, 398d }}


Badness: 0.029741
Badness (Sintel): 1.23


== Decipentic ==
== Decipentic ==
The generator for the decipentic temperament (43 &amp; 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.
Decipentic may be described as the 43 & 56 temperament. The minimal generator for this temperament is the 10th root of the [[5/1|5th harmonic]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.
 
Decipentic was named by [[Xenllium]] in 2021 for the fact that it splits the 5th harmonic into ten equal parts.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 551: Line 621:
[[Comma list]]: 3136/3125, 2097152/2083725
[[Comma list]]: 3136/3125, 2097152/2083725


{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 -13 10 22 | 0 19 -10 -25 }}
: mapping generators: ~2, ~128/75


{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5804{{c}}, ~128/75 = 920.8777{{c}} (~75/64 = 278.7027{{c}})
: [[error map]]: {{val| -0.420 +0.175 +0.714 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/75 = 921.2002{{c}} (~75/64 = 278.7998{{c}})
: error map: {{val| 0.000 +0.849 +1.684 +1.169 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
{{Optimal ET sequence|legend=1| 43, 56, 99, 538cd, 637cd, …, 1033bccdd }}


{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
[[Badness]] (Sintel): 2.21
 
[[Badness]]: 0.087325


=== 11-limit ===
=== 11-limit ===
Line 566: Line 639:
Comma list: 441/440, 1344/1331, 3136/3125
Comma list: 441/440, 1344/1331, 3136/3125


Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
Optimal tunings:
* WE: ~2 = 1199.3002{{c}}, ~128/75 = 920.6639{{c}} (~75/64 = 278.6364{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~128/75 = 921.2019{{c}} (~75/64 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 43, 56, 99e }}


Badness: 0.061413
Badness (Sintel): 2.03


==== 13-limit ====
==== 13-limit ====
Line 579: Line 654:
Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 | 0 19 -10 -25 -2 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
Optimal tunings:
* WE: ~2 = 1199.5986{{c}}, ~22/13 = 920.8900{{c}} (~13/11 = 278.7085{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 921.1988{{c}} (~13/11 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.047611
Badness (Sintel): 1.97


==== 17-limit ====
==== 17-limit ====
Line 592: Line 669:
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374


Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 | 0 19 -10 -25 -2 -3 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
Optimal tunings:
* WE: ~2 = 1199.5089{{c}}, ~17/10 = 920.8252{{c}} (~13/11 = 278.6837{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2034{{c}} (~13/11 = 278.7966{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.031191
Badness (Sintel): 1.59


==== 19-limit ====
==== 19-limit ====
Line 605: Line 684:
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285


Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 15 | 0 19 -10 -25 -2 -3 -9 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790
Optimal tunings:
* WE: ~2 = 1199.4107{{c}}, ~17/10 = 920.7575{{c}} (~13/11 = 278.6532{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2020{{c}} (~13/11 = 278.7980{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.023899
Badness (Sintel): 1.45


=== Quasijerome ===
=== Quasijerome ===
Line 618: Line 699:
Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}
{{Mapping|legend=0| 1 -32 20 47 45 | 0 38 -20 -50 -47 }}
: mapping generators: ~2, ~825/448


Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6276{{c}}, ~825/448 = 1060.2861{{c}} (~896/825 = 139.3595{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~825/448 = 1060.5982{{c}} (~896/825 = 139.4016{{c}})


{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}
{{Optimal ET sequence|legend=0| 43, 112, 155, 198, 439cd, 637cd }}


Badness: 0.092996
Badness (Sintel): 3.07


==== 13-limit ====
==== 13-limit ====
Line 631: Line 715:
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
{{Mapping|legend=0| 1 -32 20 47 45 -29 | 0 38 -20 -50 -47 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~24/13 = 1060.2595{{c}} (~13/12 = 139.3589{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~24/13 = 1060.5971{{c}} (~13/12 = 139.4029{{c}})


{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}
{{Optimal ET sequence|legend=0| 43, 155, 198, 439cdf, 637cdf }}


Badness: 0.044328
Badness (Sintel): 1.83


== Sengagen ==
== Sengagen ==
Sengagen tempers out 420175/419904, the [[wizma]], and may be described as the 49 & 50 temperament. Its generator is a comma-sized interval, most notably representing the [[senga]], hence the name of the temperament. 29 generator steps make a [[3/2|perfect fifth]], so the [[ploidacot]] of the temperament is 29-cot.
A simple way to extend sengagen to the 11-limit is to map [[11/8]] to 23 steps, a mapping common to [[49edo]] and [[50edo]], and to extend it to the 13-limit, [[13/8]] can be similarly mapped to 35 steps, which requires the 49f val. 49f and 50 form a twin as 49f tunes [[39/32]] a step wider than [[16/13]] while 50 tunes [[11/9]] a step wider than [[27/22]], and then 99ef is where they all converge to the true neutral third.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 645: Line 735:


{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
: mapping generators: ~2, ~686/675


{{Multival|legend=1| 29 16 40 -42 -18 48 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.8393{{c}}, ~686/675 = 24.2133{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
: [[error map]]: {{val| -0.161 +0.070 +0.778 -0.615 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~686/675 = 24.2119{{c}}
: error map: {{val| 0.000 +0.191 +1.077 -0.349 }}


{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}
{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}


[[Badness]]: 0.057978
[[Badness]] (Sintel): 1.47


=== 11-limit ===
=== 11-limit ===
Line 659: Line 752:
Comma list: 540/539, 1344/1331, 3136/3125
Comma list: 540/539, 1344/1331, 3136/3125


Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}
{{Mapping|legend=0| 1 1 2 2 3 | 0 29 16 40 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235
Optimal tunings:
* WE: ~2 = 1199.3714{{c}}, ~99/98 = 24.2224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.2176{{c}}


{{Optimal ET sequence|legend=1| 49, 50, 99e }}
{{Optimal ET sequence|legend=0| 49, 50, 99e }}


Badness: 0.053828
Badness (Sintel): 1.78


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


Comma list: 351/350, 540/539, 975/968, 1344/1331
Comma list: 144/143, 196/195, 364/363, 625/624
 
{{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}


Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}
Optimal tunings:  
* WE: ~2 = 1199.1402{{c}}, ~99/98 = 24.2164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.2092{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181
{{Optimal ET sequence|legend=0| 49f, 50, 99ef }}


{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}
Badness (Sintel): 1.55


Badness: 0.053531
==== Sengageny ====
This extension used to be considered the canonical 13-limit sengagen, decanonized in 2026 in favor of the above one.  


==== Sengage ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 196/195, 364/363, 625/624
Comma list: 351/350, 540/539, 975/968, 1344/1331


Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}
{{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
Optimal tunings:
* WE: ~2 = 1200.0364{{c}}, ~99/98 = 24.1817{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.1815{{c}}


{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}
{{Optimal ET sequence|legend=0| 49, 50, 99e, 149e }}


Badness: 0.037416
Badness (Sintel): 2.21


== Mowglic ==
== Mowglic ==
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–enneadecal equivalence continuum|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Mowgli]].''
 
Mowglic may be described as the 19 & 161 temperament, named by [[Xenllium]] in 2021 as a variant of ''mowgli''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 701: Line 804:


{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
: mapping generators: ~2, ~27/25


{{Multival|legend=1| 15 22 55 0 45 66 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0862{{c}}, ~27/25 = 126.7147{{c}}
: [[error map]]: {{val| +0.086 -1.234 +1.411 +0.226 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 126.7081{{c}}
: error map: {{val| 0.000 -1.333 +1.265 +0.122 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
{{Optimal ET sequence|legend=1| 19, …, 123d, 142, 161 }}


{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}
[[Badness]] (Sintel): 3.29
 
[[Badness]]: 0.129915


=== 11-limit ===
=== 11-limit ===
Line 715: Line 821:
Comma list: 540/539, 3136/3125, 72171/71680
Comma list: 540/539, 3136/3125, 72171/71680


Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}
{{Mapping|legend=0| 1 0 0 -3 8 | 0 15 22 55 -43 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711
Optimal tunings:
* WE: ~2 = 1200.0198{{c}}, ~27/25 = 126.7134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/25 = 126.7114{{c}}


{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}
{{Optimal ET sequence|legend=0| 19, 142, 161 }}


Badness: 0.094032
Badness (Sintel): 3.11


=== 13-limit ===
=== 13-limit ===
Line 728: Line 836:
Comma list: 351/350, 540/539, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 1701/1690, 3136/3125


Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tunings:
* WE: ~2 = 1199.9743{{c}}, ~14/13 = 126.7022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7047{{c}}


{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.051571
Badness (Sintel): 2.13


=== 17-limit ===
=== 17-limit ===
Line 741: Line 851:
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125
Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125


Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1200.0016{{c}}, ~14/13 = 126.7029{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7027{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 123defg, 142f, 161 }}


Badness: 0.041918
Badness (Sintel): 2.14


=== 19-limit ===
=== 19-limit ===
Line 754: Line 866:
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690
Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690


Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tunings:
* WE: ~2 = 1199.9713{{c}}, ~14/13 = 126.7018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7048{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 123defg, 142f, 161 }}


Badness: 0.032168
Badness (Sintel): 1.96


=== 23-limit ===
=== 23-limit ===
Line 767: Line 881:
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1200.0296{{c}}, ~14/13 = 126.7061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7030{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.026117
Badness (Sintel): 1.87


=== 29-limit ===
=== 29-limit ===
Line 780: Line 896:
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704
Optimal tunings:
* WE: ~2 = 1200.0491{{c}}, ~14/13 = 126.7094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7043{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.021398
Badness (Sintel): 1.79


=== 31-limit ===
=== 31-limit ===
Line 793: Line 911:
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539
Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539


Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1199.9986{{c}}, ~14/13 = 126.7027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7028{{c}}


{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}
{{Optimal ET sequence|legend=0| 19, 142fk, 161 }}


Badness: 0.019331
Badness (Sintel): 1.85


== Tremka ==
== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 &amp; 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.
Tremka may be described as the 50 & 111 temperament. The name ''tremka'' was devised by [[Petr Pařízek]] in 2011, initially used for the [[no-sevens subgroup temperaments #Tremka|no-7's version]], especially in the 2.3.5.11.13 subgroup<ref name="petr's long post"/>, but extending to the full 13-limit or higher does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.


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


Line 810: Line 929:


{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
: mapping generators: ~2, ~4375/3888


{{Multival|legend=1| 31 24 60 -34 8 72 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.7671{{c}}, ~4375/3888 = 216.1302{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
: [[error map]]: {{val| -0.233 -0.987 +1.277 +0.849 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4375/3888 = 216.1665{{c}}
: error map: {{val| 0.000 -0.794 +1.682 +1.162 }}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}


[[Badness]]: 0.179925
[[Badness]] (Sintel): 4.55


=== 11-limit ===
=== 11-limit ===
Line 824: Line 946:
Comma list: 540/539, 3136/3125, 35937/35840
Comma list: 540/539, 3136/3125, 35937/35840


Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
Optimal tunings:
* WE: ~2 = 1199.8503{{c}}, ~112/99 = 216.1406{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/99 = 216.1652{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272, 433c }}


Badness: 0.068825
Badness (Sintel): 2.28


=== 13-limit ===
=== 13-limit ===
Line 837: Line 961:
Comma list: 351/350, 540/539, 847/845, 3136/3125
Comma list: 351/350, 540/539, 847/845, 3136/3125


Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
Optimal tunings:
* WE: ~2 = 1199.7479{{c}}, ~112/99 = 216.1263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/99 = 216.1681{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272 }}


Badness: 0.036070
Badness (Sintel): 1.49


=== 17-limit ===
=== 17-limit ===
Line 850: Line 976:
Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088
Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088


Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
Optimal tunings:
* WE: ~2 = 1199.7537{{c}}, ~17/15 = 216.1281{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/15 = 216.1688{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272 }}


Badness: 0.022528
Badness (Sintel): 1.15


=== 19-limit ===
=== 19-limit ===
Line 863: Line 991:
Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539
Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539


Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170
Optimal tunings:
* WE: ~2 = 1199.7496{{c}}, ~17/15 = 216.1244{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/15 = 216.1658{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272h, 433cfh }}


Badness: 0.016900
Badness (Sintel): 1.03


== Undetrita ==
== Undetrita ==
The undetrita temperament (111 &amp; 118) tempers out the hemimean comma (3136/3125) and [[skeetsma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 &amp; 118.
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''
 
Undetrita tempers out 14348907/14336000, the [[scheme comma]], in addition to the hemimean comma, and may be described as the 111 & 118 temperament. It splits 3rd harmonic into eleven parts; its [[ploidacot]] is thus alpha-hendecacot. It tempers out [[3025/3024]], [[3388/3375]], and [[8019/8000]] in the [[11-limit]], which relates itself to [[no-sevens subgroup temperaments #Twentcufo|twentcufo]], the no-7's version of 111 & 118.
 
It was named by [[Xenllium]] in 2021 as a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 879: Line 1,013:


{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
: mapping generators: ~2, ~448/405


{{Multival|legend=1| 11 30 75 22 88 90 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.8545{{c}}, ~448/405 = 172.8961{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
: error map: {{val| -0.146 -0.098 +0.859 -0.458 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 172.9143{{c}}
: error map: {{val| 0.000 +0.102 +1.115 -0.255 }}


{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}
{{Optimal ET sequence|legend=0| 111, 118, 229, 347, 576c }}


[[Badness]]: 0.114188
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 11-limit ===
Line 893: Line 1,030:
Comma list: 3025/3024, 3136/3125, 8019/8000
Comma list: 3025/3024, 3136/3125, 8019/8000


Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}
{{Mapping|legend=0| 1 0 -2 -8 0 | 0 11 30 75 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912
Optimal tunings:
* WE: ~2 = 1200.0112{{c}}, ~400/363 = 172.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/363 = 172.9123{{c}}


{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}
{{Optimal ET sequence|legend=0| 111, 118, 229, 347 }}


Badness: 0.043883
Badness (Sintel): 1.45


==== 13-limit ====
==== 13-limit ====
Line 906: Line 1,045:
Comma list: 352/351, 729/728, 1001/1000, 3025/3024
Comma list: 352/351, 729/728, 1001/1000, 3025/3024


Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}
{{Mapping|legend=0| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930
Optimal tunings:
* WE: ~2 = 1199.7099{{c}}, ~72/65 = 172.8882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 172.9265{{c}}


{{Optimal ET sequence|legend=1| 111, 229f }}
{{Optimal ET sequence|legend=0| 111, 229f }}


Badness: 0.038771
Badness (Sintel): 1.60


==== Undetritoid ====
==== Undetritoid ====
Line 919: Line 1,060:
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125
Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125


Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}
{{Mapping|legend=0| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
Optimal tunings:
* WE: ~2 = 1199.8901{{c}}, ~400/363 = 172.9167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/363 = 172.9310{{c}}


{{Optimal ET sequence|legend=1| 111, 229 }}
{{Optimal ET sequence|legend=0| 111, 229 }}


Badness: 0.042744
Badness (Sintel): 1.77


= Subgroup extensions =
= Subgroup extensions =
== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension for prime 11 by equating the sharply tuned [[25/16]] with [[11/7]], tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>, tempering out [[1375/1372]].


== Undecimal didacus ==
In the no-3's 19-limit extension mediantone, this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.


Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: [[176/175]], [[1375/1372]]
Comma list: 176/175, 1375/1372


Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
{{Mapping|legend=2| 1 0 -3 -7 | 0 2 5 9 }}
: mapping generators: ~2, ~56/25


: sval mapping generators: ~2, ~56/25
Optimal tunings:  
* WE: ~2 = 1199.3680{{c}}, ~28/25 = 194.4533{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 194.4278{{c}}


Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
{{Optimal ET sequence|legend=0| 6, 19e, 25, 31, 37 }}


Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}
Badness (Sintel): 0.195
 
RMS error: 0.5567 cents
 
Badness (Dirichlet): 0.195


=== Tridecimal didacus ===
=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension of didacus if prime 3 is not included.


Subgroup: 2.5.7.11.13
Subgroup: 2.5.7.11.13
Line 955: Line 1,098:
Comma list: 176/175, 640/637, 1375/1372
Comma list: 176/175, 640/637, 1375/1372


Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
{{Mapping|legend=2| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
 
: sval mapping generators: ~2, ~56/25
 
Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}


: gencom: [2 28/25; 176/175 1375/1372 640/637]
{{Mapping|legend=4| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}


Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
Optimal tunings:  
* WE: ~2 = 1199.3069{{c}}, ~28/25 = 194.4820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 194.5685{{c}}


Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
{{Optimal ET sequence|legend=0| 6, 25, 31, 37 }}


Badness (Dirichlet): 0.324
Badness (Sintel): 0.324


==== Mediantone ====
==== Mediantone ====
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on ''meantone'', as the context one is most likely to first discover this logic is when the tone also represents 9/8~10/9.
 
In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.
 
Subgroup: 2.5.7.11.13.17
 
Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]


Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[meantone family #Undecimal meantone (huygens)|undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of 19/17~28/25 as ([[3/2]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.
 
: sval mapping generators: ~2, ~56/25
 
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
 
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}
 
Badness (Dirichlet): 0.612


===== 2.5.7.11.13.17.19 subgroup =====
===== 2.5.7.11.13.17.19 subgroup =====
Subgroup: 2.5.7.11.13.17.19
Subgroup: 2.5.7.11.13.17.19


Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]
Comma list: 176/175, 221/220, 476/475, 640/637, 1375/1372


Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}
{{Mapping|legend=2| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}


: sval mapping generators: ~2, ~56/25
Optimal tunings:  
* WE: ~2 = 1199.3213{{c}}, ~19/17 = 194.8469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/17 = 194.9266{{c}}


Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
{{Optimal ET sequence|legend=0| 6h, 31gh, 37, 80 }}


Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}
Badness (Sintel): 0.618
 
Badness (Dirichlet): 0.618


==== Roulette ====
==== Roulette ====
{{See also | Chromatic pairs #Roulette }}
{{See also | Chromatic pairs #Roulette }}


Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
Roulette is an alternative no-3's 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
 
Subgroup: 2.5.7.11.13.17
 
Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
 
Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
 
: sval mapping generators: ~2, ~28/25
 
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
 
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
 
Badness (Dirichlet): 0.685


===== 2.5.7.11.13.17.19 subgroup =====
===== 2.5.7.11.13.17.19 subgroup =====
Subgroup: 2.5.7.11.13.17.19
Subgroup: 2.5.7.11.13.17.19


Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
Comma list: 170/169, 176/175, 476/475, 640/637, 1375/1372
 
Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}


: sval mapping generators: ~2, ~28/25
{{Mapping|legend=2| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}


Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
Optimal tunings:
* WE: ~2 = 1199.7001{{c}}, ~19/17 = 194.2060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/17 = 194.2588{{c}}


Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
{{Optimal ET sequence|legend=0| 6g, 31, 37, 68, 105 }}


Badness (Dirichlet): 0.676
Badness (Sintel): 0.676


== Rectified hebrew ==
== Rectified hebrew (2.5.7.13) ==
{{Main| Rectified hebrew }}
{{Main| Rectified hebrew }}


Rectified hebrew (37 &amp; 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three.
Rectified hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11{{clarify}}. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-3 no-11 restriction of [[Sycamore family #Septimal sycamore|sycamore]].


Subgroup: 2.5.7.13
Subgroup: 2.5.7.13
Line 1,046: Line 1,159:
Comma list: 3136/3125, 4394/4375
Comma list: 3136/3125, 4394/4375


Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}
{{Mapping|legend=2| 1 2 2 3 | 0 6 15 13 }}
: mapping generators: ~2, ~26/25


: sval mapping generators: ~2, ~26/25
Optimal tunings:  
* WE: ~2 = 1199.8702{{c}}, ~26/25 = 64.6016{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/25 = 64.5947{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086
{{Optimal ET sequence|legend=0| 18, 19, 37, 93, 130 }}


{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}
Badness (Sintel): 0.329


== Isra ==
== Isra (2.9.5.7) ==
Isra results from taking every other generator of [[septimal meantone]], or from [[didacus]] if the generator is interpreted as 9/8. It is named after the Isrāʾ (''iss-RAH'') night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).
Isra (''iss-RAH'') results from taking every other generator of [[meantone family #Septimal meantone|septimal meantone]], or from didacus if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna family #Luna|luna]], a restriction of [[#Hemithirds|hemithirds]], which is in turn an extension of didacus.


[[Subgroup]]: 2.9.5.7
Subgroup: 2.9.5.7


[[Comma list]]: 81/80, 126/125
Comma list: 81/80, 126/125


{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}
{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}
: mapping generators: ~2, ~9


: sval mapping generators: ~2, ~9
Optimal tunings:  
* WE: ~2 = 1201.2358{{c}}, ~9/8 = 193.1885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 193.3124{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.9898
{{Optimal ET sequence|legend=0| 6, 19, 25, 31, 56b′, 87b′ }}


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 56b, 87b }}
[[Badness]] (Sintel): 0.173


=== Tutone ===
=== Tutone ===
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]], or undecimal [[didacus]] with the generator interpreted as 9/8.
Tutone is every other step of undecimal meantone, or undecimal didacus with the generator interpreted as 9/8.


[[Subgroup]]: 2.9.5.7.11
Subgroup: 2.9.5.7.11


[[Comma list]]: 81/80, 99/98, 126/125
Comma list: 81/80, 99/98, 126/125


{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}
{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}


{{Mapping|legend=3| 1 3/2 2 2 2 | 0 1/2 2 5 9 }}
{{Mapping|legend=4| 1 0 -4 -13 -25 | 0 1/2 2 5 9 }}


: [[gencom]]: [2 9/8; 81/80 99/98 126/125]
Optimal tunings:  
* WE: ~2 = 1200.7636{{c}}, ~9/8 = 194.0608{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 194.0629{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 193.937
{{Optimal ET sequence|legend=0| 6, 19e, 25, 31, 68b′, 99b′ }}


{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 68b, 99b }}
Badness (Sintel): 0.281
 
[[Badness]]: 0.00536


=== Leantone ===
=== Leantone ===
Line 1,093: Line 1,212:
Leantone is every other step of [[vincenzo]].  
Leantone is every other step of [[vincenzo]].  


[[Subgroup]]: 2.9.5.7.11
Subgroup: 2.9.5.7.11


[[Comma list]]: 45/44, 56/55, 81/80
Comma list: 45/44, 56/55, 81/80


{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}
{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}


{{Mapping|legend=3| 1 3/2 2 2 3 | 0 1/2 2 5 3 }}
{{Mapping|legend=4| 1 0 -4 -13 -6 | 0 1/2 2 5 3 }}


: [[gencom]]: [2 9/8; 45/44 56/55 81/80]
Optimal tunings:  
* WE: ~2 = 1199.6946{{c}}, ~9/8 = 192.4513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 192.4166{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500
{{Optimal ET sequence|legend=0| 6, 19, 25e }}


{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}
Badness (Sintel): 0.354
 
[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents


=== Deutone ===
=== Deutone ===
Line 1,114: Line 1,233:
Deutone is (also) every other step of [[vincenzo]].  
Deutone is (also) every other step of [[vincenzo]].  


[[Subgroup]]: 2.9.5.7.13
Subgroup: 2.9.5.7.13


[[Comma list]]: 65/64, 81/80, 91/90
Comma list: 65/64, 81/80, 91/90


{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}
{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}


{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}
{{Mapping|legend=4| 1 0 -4 -13 0 10 | 0 1/2 2 5 0 -2 }}


: [[gencom]]: [2 9/8; 65/64 81/80 91/90]
Optimal tunings:  
* WE: ~2 = 1203.3456{{c}}, ~9/8 = 191.5914{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 191.4621{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059
{{Optimal ET sequence|legend=0| 6, 7, 13, 19, 25f, 44df }}


{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}
Badness (Sintel): 0.325


[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
== References ==


[[Category:Temperament clans]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemimean_clan"