Xenharmonic Wiki

Hemifamity temperaments: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,375 edits
+ abergravity
 
(39 intermediate revisions by 8 users not shown)
Line 1: Line 1:
The hemifamity temperaments temper out the hemifamity comma, {{monzo| 10 -6 1 -1 }} = [[5120/5103]], dividing an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]]~[[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same circle of fifths inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[hemifamity comma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the syntonic~septimal comma to the opposite sides. In addition we may identify [[10/7]] by the augmented fourth and [[50/49]] by the [[Pythagorean comma]].  


Belonging to it and considered below are undecental, leapday, mystery, quanic, septiquarter, countriton, ketchup, and artoneutral. Other hemifamity temperaments are:
Temperaments belonging to this category and generated by the fifth are dominant, garibaldi, kwai, undecental, and leapday. Dominant has 5/4 mapped to M3. Garibaldi has 5/4 mapped to d4. Kwai has 5/4 mapped to 4A7. Undecental has 5/4 mapped to 5d7. Leapday has 5/4 mapped to 3A1.
* ''[[Dominant]]'' → [[Meantone family #Dominant|Meantone family]] (+36/35, the 12 & 17c temperament, generated by the fifth with 5/4 mapped to the M3)
* [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+225/224, the 41 & 53 temperament, generated by the fifth with 5/4 mapped to the d4)
* ''[[Kwai]]'' → [[Mirkwai clan #Kwai|Mirkwai clan]] (+16875/16807, the 41 & 70 temperament, generated by the fifth with 5/4 mapped to the 4A7 aka m3 + 3 Pyth. commas)
* ''[[Diaschismic]]'' → [[Diaschismic family #Diaschismic|Diaschismic family]] (+126/125, the 46 & 58 temperament, generated by the fifth and using a semioctave period)
* [[Hemififths]] → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]] (+2401/2400, the 41 & 58 temperament, fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma)
* [[Rodan]] → [[Gamelismic clan #Rodan|Gamelismic clan]] (+245/243, the 41 & 46 temperament, fifth sliced into three as does slendric)
* ''[[Trimot]]'' → [[Tricot family #Trimot|Tricot family]] (+2430/2401, the 53 & 70 temperament, twelfth sliced into three as does tricot)
* ''[[Monkey]]'' → [[Tetracot family #Monkey|Tetracot family]] (+875/864, the 34 & 41 temperament, fifth sliced into four as does tetracot)
* [[Buzzard]] → [[Vulture family #Buzzard|Vulture family]] (+1728/1715, the 53 & 58 temperament, twelfth sliced into four as does vulture)
* [[Misty]] → [[Misty family #Misty|Misty family]] (+3136/3125, the 12 & 99 temperament, generated by the fifth and using a 1/3-octave period)
* ''[[Supers]]'' → [[Stearnsmic clan #Supers|Stearnsmic clan]] (+118098/117649, the 36c & 58 temperament, fifth sliced into three and using a semioctave period)
* ''[[Undim]]'' → [[Undim family #Septimal undim|Undim family]] (+390625/388962, the 12 & 140 temperament, generated by the fifth and using a 1/4-octave period)
* ''[[Quinticosiennic]]'' → [[Quintaleap family #Quinticosiennic|Quintaleap family]] (+395136/390625, the 12 & 145 temperament, fourth sliced into five)
* ''[[Quintakwai]]'' → [[Quindromeda family #Quintakwai|Quindromeda family]] (+9765625/9680832, the 12 & 181 temperament, fourth sliced into five)
* [[Amity]] → [[Amity family #Septimal amity|Amity family]] (+4375/4374, the 46 & 53 temperament, eleventh sliced into five)
* ''[[Countercata]]'' → [[Kleismic family #Countercata|Kleismic family]] (+15625/15552, the 34 & 53 temperament, twelfth sliced into six as does hanson)
* ''[[Warrior]]'' → [[Sensipent family #Warrior|Sensipent family]] (+78732/78125, the 46 & 65d temperament, 6th harmonic sliced into seven as does sensi)
* ''[[Alphaquarter]]'' → [[Escapade family #Alphaquarter|Escapade family]] (+29360128/29296875, the 65d & 87 temperament, fourth sliced into nine as does escapade)


== Undecental ==
Diaschismic is generated by the fifth with a semi-octave period. Hemififths has the fifth sliced into two and 5/4 mapped to the hemififth + Pyth. comma. Hemidromeda has the fourth sliced into two and 5/4 mapped to the hemifourth + 3d4. Rodan has the fifth sliced into three as does slendric. Alphatrimot has the twelfth sliced into three as does alphatricot. Monkey has the fifth sliced into four as does tetracot. Buzzard has the twelfth sliced into four as does vulture. Misty is generated by the fifth with a 1/3-octave period. Supers has the fifth sliced into three with a semi-octave period. Undim is generated by the fifth with a 1/4-octave period. Quinticosiennic and quintakwai have the fourth sliced into five. Amity has the eleventh sliced into five. Countercata has the twelfth sliced into six as does hanson. Warrior has the 6th harmonic sliced into seven as does sensi. Finally, alphaquarter has the fourth sliced into nine as does escapade.
Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple diminished seventh (5d7) or equivalently the perfect fourth (P4) - 3 Pyth. commas. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting choice is the argent fifth, 2<sup>(2 - sqrt (2))</sup>.  
 
Temperaments discussed elsewhere are:
* [[Dominant (temperament)|Dominant]] (+36/35) → [[Meantone family #Dominant|Meantone family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Hemififths]] (+2401/2400) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Rodan]] (+245/243) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Alphatrimot]]'' (+2430/2401) → [[Alphatricot family #Alphatrimot|Alphatricot family]]
* [[Misty]] (+3136/3125) → [[Misty family #Misty|Misty family]]
* [[Monkey]] (+875/864) → [[Tetracot family #Monkey|Tetracot family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Buzzard|Buzzardsmic clan]]
* ''[[Undim]]'' (+390625/388962) → [[Undim family #Septimal undim|Undim family]]
* ''[[Quinticosiennic]]'' (+395136/390625) → [[Quintaleap family #Quinticosiennic|Quintaleap family]]
* ''[[Quintakwai]]'' (+9765625/9680832) → [[Quindromeda family #Quintakwai|Quindromeda family]]
* [[Amity]] (+4375/4374) → [[Amity family #Septimal amity|Amity family]]
* ''[[Countercata]]'' (+15625/15552) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Abergravity]]'' (+177147/175000) → [[Gravity family #Abergravity|Gravity family]]
* ''[[Supers]]'' (+118098/117649) → [[Stearnsmic clan #Supers|Stearnsmic clan]]
* ''[[Warrior]]'' (+78732/78125) → [[Sensipent family #Warrior|Sensipent family]]
* ''[[Alphaquarter]]'' (+29360128/29296875) → [[Escapade family #Alphaquarter|Escapade family]]
 
Considered below are septiquarter, kwai, ketchup, undecental, leapday, mystery, hemidromeda, countriton, artoneutral, quanic and jorgensen, in the order of increasing [[TE logflat badness]].  
 
== Septiquarter ==
Septiquarter tempers out [[420175/419904]] and may be described as the {{nowrap| 94 & 99 }} temperament. Its [[ploidacot]] is epsilon-heptacot. [[99edo]] makes for an excellent tuning, and [[292edo]] an even better one. [[94edo]] and [[104edo]] in the 104c val are also among the possibilities.  


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


[[Comma list]]: 5120/5103, 235298/234375
[[Comma list]]: 5120/5103, 420175/419904
 
{{Mapping|legend=1| 1 -4 -28 6 | 0 7 38 -4 }}
: mapping generators: ~2, ~243/140
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7212{{c}}, ~243/140 = 957.3250{{c}}
: [[error map]]: {{val| -0.279 +0.435 -0.158 +0.201 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/140 = 957.5424{{c}}
: error map: {{val| 0.000 +0.842 +0.298 +1.004 }}
 
{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
 
[[Badness]] (Sintel): 1.36
 
=== Semiseptiquarter ===
Subgroup: 2.3.5.7.11
 
Comma list: 5120/5103, 9801/9800, 14641/14580
 
Mapping: {{mapping| 2 -8 -56 12 -25 | 0 7 38 -4 20 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8953{{c}}, ~210/121 = 957.3819{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5449{{c}}
 
{{Optimal ET sequence|legend=0| 94, 198, 292, 490 }}
 
Badness (Sintel): 2.12
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 0 61 71 }}, {{val| 0 1 -37 -43 }}]
Comma list: 352/351, 847/845, 1716/1715, 14641/14580


{{Multival|legend=1| 1 -37 -43 -61 -71 4 }}
Mapping: {{mapping| 2 -8 -56 12 -25 9 | 0 7 38 -4 20 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.039
Optimal tunings:
* WE: ~99/70 = 599.8565{{c}}, ~210/121 = 957.3261{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~210/121 = 957.5508{{c}}


{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc, 1118bbcc, 1217bbcc, 1316bbccd }}
{{Optimal ET sequence|legend=0| 94, 198, 490f }}


[[Badness]]: 0.094603
Badness (Sintel): 1.44


== Leapday ==
== Kwai ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kwai]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Leapday]].''


Leapday tempers out the leapday comma, {{monzo| 31 -21 1 }}, in the 5-limit, mapping 5/4 to the triple augmented unison (3A1) or equivalently the minor third (m3) + 2 Pyth. commas. This temperament can be described as the 29 &amp; 46 temperament, which tempers out the hemifamity and [[686/675]] (senga). The alternative extension [[Porwell temperaments #Polypyth|polypyth]] (46 &amp; 121) tempers out the same 5-limit comma as the leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity tempered out.
Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a [[3/2|perfect fifth]], and can be described as {{nowrap| 41 & 70 }}.  


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


[[Comma list]]: 686/675, 5120/5103
[[Comma list]]: 5120/5103, 16875/16807
 
[[Mapping]]: [{{val| 1 0 -31 -21 }}, {{val| 0 1 21 15 }}]


{{Multival|legend=1| 1 21 15 31 21 -24 }}
{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.263
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7337{{c}}, ~3/2 = 702.4600{{c}}
: [[error map]]: {{val| -0.266 +0.239 -0.607 +1.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6085{{c}}
: error map: {{val| 0.000 +0.653 -0.234 +1.603 }}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 167d, 213d, 259cdd, 305bcdd }}
{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}


[[Badness]]: 0.096123
[[Badness]] (Sintel): 1.38


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 441/440, 686/675
Comma list: 540/539, 1375/1372, 5120/5103


Mapping: [{{val| 1 0 -31 -21 -14 }}, {{val| 0 1 21 15 11 }}]
Mapping: {{mapping| 1 0 -50 -40 32 | 0 1 33 27 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.250
Optimal tunings:
* WE: ~2 = 1199.6672{{c}}, ~3/2 = 702.4282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6189{{c}}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 167de, 213de, 259cdde }}
{{Optimal ET sequence|legend=0| 41, 111, 152, 497de, 649dde }}


Badness: 0.038624
Badness (Sintel): 0.867


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


Comma list: 91/90, 121/120, 169/168, 352/351
Comma list: 352/351, 540/539, 729/728, 1375/1372


Mapping: [{{val| 1 0 -31 -21 -14 -9 }}, {{val| 0 1 21 15 11 8 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 | 0 1 33 27 -18 -21 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.214
Optimal tunings:
* WE: ~2 = 1199.4772{{c}}, ~3/2 = 702.3379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6409{{c}}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 121def, 167def, 213deff }}
{{Optimal ET sequence|legend=0| 41, 111, 152f, 415dff }}


Badness: 0.024732
Badness (Sintel): 1.01


=== 17-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 }}, {{val| 0 1 21 15 11 8 24 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 58 | 0 1 33 27 -18 -21 -34 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.229
Optimal tunings:
* WE: ~2 = 1199.3537{{c}}, ~3/2 = 702.2850{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6589{{c}}


{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 121defg, 167defg, 213deffg }}
{{Optimal ET sequence|legend=0| 41, 70, 111, 152fg, 263dfg }}


Badness: 0.017863
Badness (Sintel): 1.12


==== 19-limit ====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 169/168
Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 }}, {{val| 0 1 21 15 11 8 24 -3 }}]
Mapping: {{mapping| 1 0 -50 -40 32 27 58 -56 | 0 1 33 27 -18 -21 -34 38 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.135
Optimal tunings:
* WE: ~2 = 1199.3401{{c}}, ~3/2 = 702.2705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6548{{c}}


{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75dfgh, 121defgh }}
{{Optimal ET sequence|legend=0| 41, 70h, 111, 152fg, 263dfgh }}


Badness: 0.017356
Badness (Sintel): 1.03


===== 23-limit =====
==== Hemikwai ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 133/132, 136/135, 154/153, 161/160, 169/168
Comma list: 540/539, 676/675, 1375/1372, 5120/5103


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 9 -5 }}, {{val| 0 1 21 15 11 8 24 -3 6 }}]
Mapping: {{mapping| 1 0 -50 -40 32 -51 | 0 2 66 54 -36 69 }}
: mapping generators: ~2, ~26/15


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.141
Optimal tunings:
* WE: ~2 = 1199.6968{{c}}, ~26/15 = 951.0740{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3123{{c}}


{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75dfgh, 121defgh }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness: 0.014065
Badness (Sintel): 1.82


==== Leapling ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 77/76, 91/90, 121/120, 136/135, 153/152, 169/168
Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 }}, {{val| 0 1 21 15 11 8 24 26 }}]
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 | 0 2 66 54 -36 69 43 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.123
Optimal tunings:
* WE: ~2 = 1199.6861{{c}}, ~26/15 = 951.0654{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3120{{c}}


{{Optimal ET sequence|legend=1| 17cgh, 29g, 46h, 75dfg, 121defghh }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304d }}


Badness: 0.019065
Badness (Sintel): 1.31


===== 23-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 91/90, 115/114, 121/120, 136/135, 153/152, 161/160
Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444


Mapping: [{{val| 1 0 -31 -21 -14 -9 -34 -37 -5 }}, {{val| 0 1 21 15 11 8 24 26 6 }}]
Mapping: {{mapping| 1 0 -50 -40 32 -51 -30 -56 | 0 2 66 54 -36 69 43 76 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.114
Optimal tunings:
* WE: ~2 = 1199.6718{{c}}, ~26/15 = 951.0526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.3103{{c}}


{{Optimal ET sequence|legend=1| 17cgh, 29g, 46h, 75dfg, 121defghh }}
{{Optimal ET sequence|legend=0| 82, 111, 193, 304dh }}


Badness: 0.016067
Badness (Sintel): 1.16


== Mystery ==
== Ketchup ==
{{Main| Mystery }}
Ketchup may be described as the {{nowrap| 46 & 94 }} temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its [[ploidacot]] is diploid gamma-tetracot. [[140edo]] is an obvious tuning for this temperament.  
: ''For the 5-limit version of this temperament, see [[29th-octave temperaments #Mystery]].''
 
Mystery has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step. [[145edo]] or [[232edo]] are good candidates for tunings.  


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


[[Comma list]]: 5120/5103, 50421/50000
[[Comma list]]: 5120/5103, 1071875/1062882


[[Mapping]]: [{{val| 29 46 0 14 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 2 3 4 6 | 0 4 15 -9 }}
: mapping generators: ~1225/864, ~64/63


{{Multival|legend=1| 0 29 29 46 46 -14 }}
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 599.9685{{c}}, ~64/63 = 25.7181{{c}}
: [[error map]]: {{val| -0.063 +0.823 -0.668 -0.478 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~64/63 = 25.7181{{c}}
: error map: {{val| 0.000 +0.917 -0.543 -0.288 }}


[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\29, ~5/4 = 388.646
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
[[Badness]] (Sintel): 2.14
 
[[Badness]]: 0.103734


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 896/891, 3388/3375
Comma list: 385/384, 1331/1323, 2200/2187


Mapping: [{{val| 29 46 0 14 33 }}, {{val| 0 0 1 1 1 }}]
Mapping: {{mapping| 2 3 4 6 7 | 0 4 15 -9 -2 }}


Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.460
Optimal tunings:
* WE: ~99/70 = 600.0678{{c}}, ~64/63 = 25.6963{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~64/63 = 25.6956{{c}}


{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
{{Optimal ET sequence|legend=0| 46, 94, 140 }}


Badness: 0.034291
Badness (Sintel): 1.31


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


Comma list: 196/195, 352/351, 364/363, 676/675
Comma list: 325/324, 352/351, 385/384, 1331/1323
 
Mapping: {{mapping| 2 3 4 6 7 8 | 0 4 15 -9 -2 -14 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0612{{c}}, ~66/65 = 25.7000{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 25.6978{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 1.03
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 385/384, 442/441
 
Mapping: {{mapping| 2 3 4 6 7 8 8 | 0 4 15 -9 -2 -14 4 }}
 
Optimal tunings:
* WE: ~17/12 = 600.0896{{c}}, ~66/65 = 25.7048{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7017{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.845
 
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390
 
Mapping: {{mapping| 2 3 4 6 7 8 8 9 | 0 4 15 -9 -2 -14 4 1 }}
 
Optimal tunings:
* WE: ~17/12 = 600.1139{{c}}, ~66/65 = 25.7053{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 25.7013{{c}}
 
{{Optimal ET sequence|legend=0| 46, 94, 140 }}
 
Badness (Sintel): 0.772
 
== Undecental ==
Undecental adds the triwellisma to the comma list and may be described as the {{nowrap| 29 & 70 }} temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three [[diesis (scale theory)|dieses]]. [[99edo|58\99]] is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, {{nowrap| 2<sup>(2 - sqrt (2))</sup> }}.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 235298/234375
 
{{Mapping|legend=1| 1 0 61 71 | 0 1 -37 -43 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6543{{c}}, ~3/2 = 702.8370{{c}}
: [[error map]]: {{val| -0.346 +0.536 +0.423 -0.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.0465{{c}}
: error map: {{val| 0.000 +1.092 +0.966 +0.175 }}
 
{{Optimal ET sequence|legend=1| 29, 70, 99, 722bc, 821bc, 920bc, 1019bc }}
 
[[Badness]] (Sintel): 2.39


Mapping: [{{val| 29 46 0 14 33 40 }}, {{val| 0 0 1 1 1 1 }}]
== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''


Optimal tuning (POTE): ~45/44 = 1\29, ~5/4 = 388.354
Leapday tempers out [[686/675]], the senga, in addition to the hemifamity comma, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.  


{{Optimal ET sequence|legend=1| 29, 58, 87, 145, 232, 377cef }}
Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.


Badness: 0.018591
Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell ([[6144/6125]]) rather than the hemifamity comma tempered out.


== Quanic ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 5832000/5764801
[[Comma list]]: 686/675, 5120/5103


[[Mapping]]: [{{val| 1 1 -4 0 }}, {{val| 0 5 54 24 }}]
{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~160/147 = 140.493
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=1| 17c, 29, 46 }}


[[Badness]]: 0.179475
[[Badness]] (Sintel): 2.43


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 1331/1323, 5120/5103
Comma list: 121/120, 441/440, 686/675


Mapping: [{{val| 1 1 -4 0 1 }}, {{val| 0 5 54 24 21 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.489
Optimal tunings:
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17c, 29, 46 }}


Badness: 0.058678
Badness (Sintel): 1.28


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


Comma list: 352/351, 540/539, 729/728, 1331/1323
Comma list: 91/90, 121/120, 169/168, 352/351


Mapping: [{{val| 1 1 -4 0 1 3 }}, {{val| 0 5 54 24 21 6 }}]
Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496
Optimal tunings:
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}


Badness: 0.032481
Badness (Sintel): 1.02


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 442/441, 540/539, 715/714, 847/845
Comma list: 91/90, 121/120, 136/135, 154/153, 169/168
 
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}


Mapping: [{{val| 1 1 -4 0 1 3 -2 }}, {{val| 0 5 54 24 21 6 52 }}]
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.497
Badness (Sintel): 0.910


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23


Badness: 0.021112
Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168


=== 19-limit ===
Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
Optimal tunings:  
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}


Mapping: [{{val| 1 1 -4 0 1 3 -2 -5 }}, {{val| 0 5 54 24 21 6 52 79 }}]
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.496
Badness (Sintel): 0.872


{{Optimal ET sequence|legend=1| 94, 111, 205 }}
== Mystery ==
{{Main| Mystery }}
: ''For the 5-limit version, see [[29th-octave temperaments #Mystery]].''


Badness: 0.017273
Mystery tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 58 }} temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. [[145edo]] or [[232edo]] are good candidates for tunings.  


== Septiquarter ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 420175/419904
[[Comma list]]: 5120/5103, 50421/50000
 
{{Mapping|legend=1| 29 46 0 14 | 0 0 1 1 }}
: mapping generators: ~50/49, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~50/49 = 41.3652{{c}}, ~5/4 = 388.5128{{c}}
: [[error map]]: {{val| -0.410 +0.842 +1.378 -2.022 }}
* [[CWE]]: ~50/49 = 41.3793{{c}}, ~5/4 = 388.3030{{c}}
: error map: {{val| 0.000 +1.493 +1.989 -1.213 }}
 
{{Optimal ET sequence|legend=1| 29, 58, 87, 145 }}
 
[[Badness]] (Sintel): 2.63
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 896/891, 3388/3375
 
Mapping: {{mapping| 29 46 0 14 33 | 0 0 1 1 1 }}
 
Optimal tunings:
* WE: ~45/44 = 41.3637{{c}}, ~5/4 = 388.3136{{c}}
* CWE: ~45/44 = 41.3793{{c}}, ~5/4 = 388.0598{{c}}
 
{{Optimal ET sequence|legend=0| 29, 58, 87, 145 }}
 
Badness (Sintel): 1.13
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 364/363, 676/675
 
Mapping: {{mapping| 29 46 0 14 33 40 | 0 0 1 1 1 1 }}
 
Optimal tunings:
* WE: ~45/44 = 41.3623{{c}}, ~5/4 = 388.1942{{c}}
* CWE: ~40/39 = 41.3793{{c}}, ~5/4 = 387.9017{{c}}
 
{{Optimal ET sequence|legend=0| 29, 58, 87, 145, 232 }}
 
Badness (Sintel): 0.768
 
== Hemidromeda ==
Hemidromeda may be described as the {{nowrap| 29 & 111 }} temperament. Named by [[Xenllium]] in 2023, ''hemidromeda'' comes from ''hemi-'' (Ancient Greek for "one half") and ''[[andromeda]]'', because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 3 10 2 }}, {{val| 0 -7 -38 4 }}]
[[Comma list]]: 5120/5103, 52734375/52706752


{{Multival|legend=1| 7 38 -4 44 -26 -116 }}
{{Mapping|legend=1| 1 0 38 48 | 0 2 -45 -57 }}
: mapping generator: ~2, ~12500/7203


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~147/128 = 242.453
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7236{{c}}, ~12500/7203 = 951.1864{{c}}
: [[error map]]: {{val| -0.276 +0.418 -0.205 +0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12500/7203 = 951.4098{{c}}
: error map: {{val| 0.000 +0.865 +0.243 +0.813 }}


{{Optimal ET sequence|legend=1| 94, 99, 292, 391, 881bd, 1272bcd }}
{{Optimal ET sequence|legend=1| 29, 82cd, 111, 140, 251, 391, 1424bbcdd }}


[[Badness]]: 0.053760
[[Badness]] (Sintel): 2.93


=== Semiseptiquarter ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 5120/5103, 9801/9800, 14641/14580
Comma list: 1331/1323, 1375/1372, 5120/5103


Mapping: [{{val| 2 6 20 4 15 }}, {{val| 0 -7 -38 4 -20 }}]
Mapping: {{mapping| 1 0 38 48 32 | 0 2 -45 -57 -36 }}


Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4511
Optimal tunings:
* WE: ~2 = 1199.8767{{c}}, ~400/231 = 951.3065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 951.4063{{c}}


{{Optimal ET sequence|legend=1| 94, 198, 292, 490 }}
{{Optimal ET sequence|legend=0| 29, 82cd, 111, 140, 251, 391e }}


Badness: 0.064160
Badness (Sintel): 2.01


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


Comma list: 352/351, 847/845, 1716/1715, 14641/14580
Comma list: 352/351, 676/675, 847/845, 1331/1323
 
Mapping: {{mapping| 1 0 38 48 32 37 | 0 2 -45 -57 -36 -42 }}
 
Optimal tunings:
* WE: ~2 = 1199.8753{{c}}, ~26/15 = 951.3054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4064{{c}}
 
{{Optimal ET sequence|legend=0| 29, 82cdf, 111, 140, 251, 391e }}
 
Badness (Sintel): 1.18
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 442/441, 561/560, 676/675, 715/714
 
Mapping: {{mapping| 1 0 38 48 32 37 58 | 0 2 -45 -57 -36 -42 -68 }}
 
Optimal tunings:
* WE: ~2 = 1199.8770{{c}}, ~26/15 = 951.3039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4035{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 82cdfg, 111, 140, 251, 391e }}
 
Badness (Sintel): 0.971
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560
 
Mapping: {{mapping| 1 0 38 48 32 37 58 32 | 0 2 -45 -57 -36 -42 -68 -35 }}
 
Optimal tunings:
* WE: ~2 = 1199.7534{{c}}, ~26/15 = 951.2024{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4020{{c}}
 
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}
 
Badness (Sintel): 1.01
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459


Mapping: [{{val| 2 6 20 4 15 7 }}, {{val| 0 -7 -38 4 -20 1 }}]
Mapping: {{mapping| 1 0 38 48 32 37 58 32 18 | 0 2 -45 -57 -36 -42 -68 -35 -17 }}


Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 242.4448
Optimal tunings:
* WE: ~2 = 1199.9128{{c}}, ~26/15 = 951.3371{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.4076{{c}}


{{Optimal ET sequence|legend=1| 94, 198, 490f }}
{{Optimal ET sequence|legend=0| 29g, 82cdfgh, 111, 140 }}


Badness: 0.034834
Badness (Sintel): 1.10


== Countriton ==
== Countriton ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Countriton]].''
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
Countriton may be described as the {{nowrap| 51c & 53 }} temperament. It splits the [[24/1|24th harmonic]] into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are [[157edo]] and [[210edo]], as well as [[104edo]] in the 104c val.
 
Countriton was named by [[Xenllium]] in 2022 as a counterpart of [[untriton]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 308: Line 552:
[[Comma list]]: 5120/5103, 7558272/7503125
[[Comma list]]: 5120/5103, 7558272/7503125


[[Mapping]]: [{{val| 1 6 19 -7 }}, {{val| 0 -9 -34 20 }}]
{{Mapping|legend=1| 1 -3 -15 13 | 0 9 34 -20 }}
 
: mapping generators: ~2, ~1225/864
{{Multival|legend=1| 9 34 -20 33 -57 -142 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1225 = 588.582
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4179{{c}}, ~1225/864 = 611.1213{{c}}
: [[error map]]: {{val| -0.582 -0.117 +0.541 +1.181 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1225/864 = 611.4120{{c}}
: error map: {{val| 0.000 +0.753 +1.695 +2.934 }}


{{Optimal ET sequence|legend=1| 53, 157, 210 }}
{{Optimal ET sequence|legend=1| 51c, 53, 157, 210, 473cdd }}


[[Badness]]: 0.131191
[[Badness]] (Sintel): 3.32


=== 11-limit ===
=== 11-limit ===
Line 323: Line 570:
Comma list: 176/175, 5120/5103, 41503/41472
Comma list: 176/175, 5120/5103, 41503/41472


Mapping: [{{val| 1 6 19 -7 27 }}, {{val| 0 -9 -34 20 -48 }}]
Mapping: {{mapping| 1 -3 -15 13 -21 | 0 9 34 -20 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.545
Optimal tunings:
* WE: ~2 = 1199.5178{{c}}, ~77/54 = 611.2097{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4495{{c}}


{{Optimal ET sequence|legend=1| 53, 104c, 157 }}
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Badness: 0.084782
Badness (Sintel): 2.80


=== 13-limit ===
=== 13-limit ===
Line 336: Line 585:
Comma list: 176/175, 351/350, 847/845, 2197/2187
Comma list: 176/175, 351/350, 847/845, 2197/2187


Mapping: [{{val| 1 6 19 -7 27 14 }}, {{val| 0 -9 -34 20 -48 -21 }}]
Mapping: {{mapping| 1 -3 -15 13 -21 -7 | 0 9 34 -20 48 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.5944{{c}}, ~77/54 = 611.2491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/54 = 611.4506{{c}}
 
{{Optimal ET sequence|legend=0| 51ce, 53, 104c, 157 }}


Optimal tuning (POTE): ~2 = 1\1, ~108/77 = 588.544
Badness (Sintel): 1.75


{{Optimal ET sequence|legend=1| 53, 104c, 157 }}
== Artoneutral ==
Artoneutral can be described as the {{nowrap| 87 & 94 }} temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the [[12/1|12th harmonic]]; its ploidacot is thus beta-enneacot. [[181edo]] may be recommended as a tuning.


Badness: 0.042321
Artoneutral was named by [[Flora Canou]] in 2023 for its generator's quality.  


== Ketchup ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 1071875/1062882
[[Comma list]]: 5120/5103, 3828125/3779136


[[Mapping]]: [{{val| 2 3 4 6 }}, {{val| 0 4 15 -9 }}]
{{Mapping|legend=1| 1 -1 -4 12 | 0 9 22 -32 }}
: mapping generators: ~2, ~128/105


{{Multival|legend=1| 8 30 -18 29 -51 -126 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1400{{c}}, ~128/105 = 344.7929{{c}}
: [[error map]]: {{val| +0.140 +1.041 -1.430 -0.518 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 344.7531{{c}}
: error map: {{val| 0.000 +0.823 -1.746 -0.925 }}


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~64/63 = 25.719
{{Optimal ET sequence|legend=1| 87, 94, 181 }}
 
{{Optimal ET sequence|legend=1| 46, 94, 140 }}


[[Badness]]: 0.084538
[[Badness]] (Sintel): 3.98


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1331/1323, 2200/2187
Comma list: 385/384, 2200/2187, 4000/3993


Mapping: [{{val| 2 3 4 6 7 }}, {{val| 0 4 15 -9 -2 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 | 0 9 22 -32 19 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~64/63 = 25.693
Optimal tunings:  
* WE: ~2 = 1200.1668{{c}}, ~11/9 = 344.8027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7557{{c}}


{{Optimal ET sequence|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.039555
Badness (Sintel): 1.52


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


Comma list: 325/324, 352/351, 385/384, 1331/1323
Comma list: 325/324, 352/351, 385/384, 1575/1573


Mapping: [{{val| 2 3 4 6 7 8 }}, {{val| 0 4 15 -9 -2 -14 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 | 0 9 22 -32 19 -8 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 25.697
Optimal tunings:  
* WE: ~2 = 1200.0662{{c}}, ~11/9 = 344.7804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7617{{c}}


{{Optimal ET sequence|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 181 }}


Badness: 0.024824
Badness (Sintel): 1.08


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 289/288, 325/324, 352/351, 385/384, 561/560
Comma list: 325/324, 352/351, 375/374, 385/384, 595/594


Mapping: [{{val| 2 3 4 6 7 8 8 }}, {{val| 0 4 15 -9 -2 -14 4 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 | 0 9 22 -32 19 -8 56 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.701
Optimal tunings:  
* WE: ~2 = 1200.0346{{c}}, ~11/9 = 344.7589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7492{{c}}


{{Optimal ET sequence|legend=1| 46, 94, 140 }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.016591
Badness (Sintel): 1.16


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 190/189, 209/208, 289/288, 352/351, 385/384, 561/560
Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594


Mapping: [{{val| 2 3 4 6 7 8 8 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 | 0 9 22 -32 19 -8 56 67 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.660
Optimal tunings:  
* WE: ~2 = 1200.0282{{c}}, ~11/9 = 344.7532{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7453{{c}}


{{Optimal ET sequence|legend=1| 46, 94, 140h, 234eh }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.018170
Badness (Sintel): 1.19


=== 23-limit ===
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 190/189, 209/208, 253/252, 289/288, 323/322, 352/351, 385/384
Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483


Mapping: [{{val| 2 3 4 6 7 8 8 9 9 }}, {{val| 0 4 15 -9 -2 -14 4 -12 1 }}]
Mapping: {{mapping| 1 -1 -4 12 -2 6 -12 -15 -13 | 0 9 22 -32 19 -8 56 67 61 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 25.661
Optimal tunings:  
* WE: ~2 = 1200.0163{{c}}, ~11/9 = 344.7461{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 344.7416{{c}}


{{Optimal ET sequence|legend=1| 46, 94, 140h, 234ehi }}
{{Optimal ET sequence|legend=0| 87, 94, 181 }}


Badness: 0.014033
Badness (Sintel): 1.17


== Artoneutral ==
== Quanic ==
Artoneutral is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11) and can be described as the 87 & 94 temperament. [[181edo]] is a recommendable tuning.  
Quanic may be described as the {{nowrap| 94 & 111 }} temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. [[205edo]] may be recommended as a tuning.  


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


[[Comma list]]: 5120/5103, 3828125/3779136
[[Comma list]]: 5120/5103, 5832000/5764801


[[Mapping]]: [{{val| 1 8 18 -20 }}, {{val| 0 -9 -22 32 }}]
{{Mapping|legend=1| 1 1 -4 0 | 0 5 54 24 }}
: mapping generators: ~2, ~160/147


: mapping generators: ~2, ~105/64
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.6159{{c}}, ~160/147 = 140.4483{{c}}
: [[error map]]: {{val| -0.384 -0.098 -0.570 +1.933 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 140.4862{{c}}
: error map: {{val| 0.000 +0.476 -0.061 +2.842 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~105/64 = 855.2452
{{Optimal ET sequence|legend=1| 94, 111, 205 }}


{{Optimal ET sequence|legend=1| 87, 94, 181 }}
[[Badness]] (Sintel): 4.54
 
[[Badness]]: 0.157


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 2200/2187, 4000/3993
Comma list: 540/539, 1331/1323, 5120/5103


Mapping: [{{val| 1 8 18 -20 17 }}, {{val| 0 -9 -22 32 -19 }}]
Mapping: {{mapping| 1 1 -4 0 1 | 0 5 54 24 21 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2397
Optimal tunings:
* WE: ~2 = 1199.7834{{c}}, ~88/81 = 140.4635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.4850{{c}}


{{Optimal ET sequence|legend=1| 87, 181 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.0459
Badness (Sintel): 1.94


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


Comma list: 325/324, 352/351, 385/384, 1575/1573
Comma list: 352/351, 540/539, 729/728, 1331/1323


Mapping: [{{val| 1 8 18 -20 17 -2 }}, {{val| 0 -9 -22 32 -19 8 }}]
Mapping: {{mapping| 1 1 -4 0 1 3 | 0 5 54 24 21 6 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2369
Optimal tunings:
* WE: ~2 = 1199.6639{{c}}, ~13/12 = 140.4562{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4904{{c}}


{{Optimal ET sequence|legend=1| 87, 181 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.0263
Badness (Sintel): 1.34


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 325/324, 352/351, 375/374, 385/384, 595/594
Comma list: 352/351, 442/441, 540/539, 715/714, 847/845


Mapping: [{{val| 1 8 18 -20 17 -2 44 }}, {{val| 0 -9 -22 32 -19 8 -56 }}]
Mapping: {{mapping| 1 1 -4 0 1 3 -2 | 0 5 54 24 21 6 52 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2495
Optimal tunings:
* WE: ~2 = 1199.6699{{c}}, ~13/12 = 140.4586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4920{{c}}


{{Optimal ET sequence|legend=1| 87, 94, 181 }}
{{Optimal ET sequence|legend=0| 94, 111, 205 }}


Badness: 0.0227
Badness (Sintel): 1.08


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594
Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714
 
Mapping: {{mapping| 1 1 -4 0 1 3 -2 -5 | 0 5 54 24 21 6 52 79 }}
 
Optimal tunings:
* WE: ~2 = 1199.6745{{c}}, ~13/12 = 140.4574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.4908{{c}}
 
{{Optimal ET sequence|legend=0| 94, 111, 205 }}
 
Badness (Sintel): 1.05


Mapping: [{{val| 1 8 18 -20 17 -2 44 52 }}, {{val| 0 -9 -22 32 -19 8 -56 -67 }}]
== Jorgensen ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Jorgensen]].''


Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2534
Jorgensen tempers out the [[linus comma]] in addition to the hemifamity comma, and may be described as the {{nowrap| 70 & 140 }} temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.  


{{Optimal ET sequence|legend=1| 87, 94, 181 }}
It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by [[Mike Battaglia]] in 2012 for historical interests<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103982.html Yahoo! Tuning Group | ''Jorgensen Temperament'']</ref>.


Badness: 0.0196
[[Subgroup]]: 2.3.5.7


=== 23-limit ===
[[Comma list]]: 5120/5103, 578509309952/576650390625
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483
{{Mapping|legend=1| 70 111 0 34 | 0 0 1 1 }}
: mapping generators: ~50421/50000, ~5


Mapping: [{{val| 1 8 18 -20 17 -2 44 52 48 }}, {{val| 0 -9 -22 32 -19 8 -56 -67 -61 }}]
[[Optimal tuning]]s:
* [[WE]]: ~50421/50000 = 17.1387{{c}}, ~5/4 = 386.8071{{c}}
: [[error map]]: {{val| -0.288 +0.445 -0.084 +0.121 }}
* [[CWE]]: ~50421/50000 = 17.1429{{c}}, ~5/4 = 386.6593{{c}}
: error map: {{val| 0.000 +0.902 +0.346 +0.690 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/11 = 855.2576
{{Optimal ET sequence|legend=1| 70, 140, 350, 490 }}


{{Optimal ET sequence|legend=1| 87, 94, 181 }}
[[Badness]] (Sintel): 5.40


Badness: 0.0163
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Hemifamity temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemifamity_temperaments"