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Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Catafourth]]'' (+78732/78125) → [[Sensipent family #Catafourth|Sensipent family]]
* ''[[Neptune (temperament)|Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* ''[[Maviloid]]'' (+1224440064/1220703125) → [[Parakleismic family #Maviloid|Parakleismic family]]
* ''[[Subneutral]]'' (+274877906944/274658203125) → [[Luna family #Subneutral|Luna family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
* ''[[Hemicountercomp]]'' (+52613349376/52301766015) → [[Countercomp family #Hemicountercomp|Countercomp family]]


Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, septidiasemi, subneutral, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, lockerbie, unthirds, neominor, cotritone, fibo, quasimoha, mintone, gorgik, retrotonic, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  


== Tertiaseptal ==
== Tertiaseptal ==
{{Main| Tertiaseptal }}
{{See also| Tertiaseptal extensions }}


Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  


[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
Tertiaseptal does not extend well to the the [[11-limit]], with three possible mappings for the [[11/1|11th harmonic]]: tertia (109 & 140), tertiaseptia (140 & 171), and septia (171 & 202). Their useful tuning ranges are indicated by the equal-temperament joins. However, since the entire range can use the same mapping for [[13/1|13]] and [[17/1|17]], we can consider tertiaseptal a no-11 13- and 17-limit temperament, in which [[625/624]], [[833/832]], [[1225/1224]], and [[4096/4095]] are tempered out.
 
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and for the add-13 add-17 extension [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-3's harmony, and the 31-note mos gives plenty of room for those as well.


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


Line 52: Line 60:
[[Badness]] (Sintel): 0.329
[[Badness]] (Sintel): 0.329


=== 11-limit ===
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.13
 
Comma list: 625/624, 2401/2400, 4096/4095


Comma list: 243/242, 441/440, 65625/65536
{{Mapping|legend=2| 1 -19 7 0 43 | 0 22 -5 3 -42 }}


Mapping: {{mapping| 1 -19 7 0 -48 | 0 22 -5 3 55 }}
{{Mapping|legend=4| 1 -19 7 0 0 43 | 0 22 -5 3 0 -42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* WE: ~2 = 1199.9931{{c}}, ~224/117 = 1122.8203{{c}} (~117/112 = 77.1729{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8267{{c}} (~117/112 = 77.1733{{c}})
 
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}
 
Badness (Sintel): 1.18
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}


Optimal tunings:
{{Optimal ET sequence|legend=0| 31, 109, 140, 171, 311, 793f }}
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}
Badness (Sintel): 0.682


Badness (Sintel): 1.52
==== 2.3.5.7.13.17 subgroup ====
Subgroup: 2.3.5.7.13.17


==== 17-limit ====
Comma list: 625/624, 833/832, 1225/1224, 4096/4095
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
{{Mapping|legend=2| 1 -19 7 0 43 49 | 0 22 -5 3 -42 -48 }}


Mapping: {{mapping| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}
{{Mapping|legend=4| 1 -19 7 0 0 43 49 | 0 22 -5 3 0 -42 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* WE: ~2 = 1200.0139{{c}}, ~65/34 = 1122.8361{{c}} (~68/65 = 77.1779{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8230{{c}} (~68/65 = 77.1770{{c}})


{{Optimal ET sequence|legend=0| 31, 140e, 171 }}
{{Optimal ET sequence|legend=0| 31, 109g, 140, 171, 311, 482 }}


Badness (Sintel): 1.40
Badness (Sintel): 0.574


=== Tertia ===
=== Tertia ===
Subgroup:2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 1331/1323, 1375/1372
Comma list: 385/384, 1331/1323, 1375/1372


Mapping: {{mapping| 1 -19 7 0 -19 | 0 22 -5 3 24 }}
{{Mapping|legend=0| 1 -19 7 0 -19 | 0 22 -5 3 24 }}


Optimal tunings:  
Optimal tunings:  
Line 117: Line 114:
Comma list: 352/351, 385/384, 625/624, 1331/1323
Comma list: 352/351, 385/384, 625/624, 1331/1323


Mapping: {{mapping| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}
{{Mapping|legend=0| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}


Optimal tunings:  
Optimal tunings:  
Line 132: Line 129:
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714


Mapping: {{mapping| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}
{{Mapping|legend=0| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}


Optimal tunings:  
Optimal tunings:  
Line 142: Line 139:
Badness (Sintel): 1.14
Badness (Sintel): 1.14


=== Tertiaseptia ===
=== Septia ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 6250/6237, 65625/65536
Comma list: 243/242, 441/440, 65625/65536


Mapping: {{mapping| 1 -19 7 0 112 | 0 22 -5 3 -116 }}
{{Mapping|legend=0| 1 -19 7 0 -48 | 0 22 -5 3 55 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})


{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}


Badness (Sintel): 1.88
Badness (Sintel): 1.18


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


Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
Comma list: 243/242, 441/440, 625/624, 3584/3575


Mapping: {{mapping| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}
{{Mapping|legend=0| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})


{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}


Badness (Sintel): 1.14
Badness (Sintel): 1.52


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


Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575


Mapping: {{mapping| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}
{{Mapping|legend=0| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})


{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}


Badness (Sintel): 0.956
Badness (Sintel): 1.40


==== 19-limit ====
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11.13.17.19
This extension is enormously complex, but preserves much of the accuracy. It was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.  


Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 | 0 22 -5 3 -116 -42 -48 105 }}
Comma list: 2401/2400, 6250/6237, 65625/65536
 
Optimal tunings:
* WE: ~2 = 1200.0187{{c}}, ~65/34 = 1122.8489{{c}} (~68/65 = 77.1698{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8313{{c}} (~68/65 = 77.1687{{c}})
 
{{Optimal ET sequence|legend=0| 140, 171, 311 }}
 
Badness (Sintel): 1.07
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 | 0 22 -5 3 -116 -42 -48 105 -117 }}
{{Mapping|legend=0| 1 -19 7 0 112 | 0 22 -5 3 -116 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0101{{c}}, ~44/23 = 1122.8418{{c}} (~23/22 = 77.1683{{c}})
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8323{{c}} (~23/22 = 77.1677{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})


{{Optimal ET sequence|legend=0| 140, 311, 762g }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness (Sintel): 1.08
Badness (Sintel): 1.88


==== 29-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29
Subgroup: 2.3.5.7.11.13


Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 | 0 22 -5 3 -116 -42 -48 105 -117 -60 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0007{{c}}, ~44/23 = 1122.8332{{c}} (~23/22 = 77.1675{{c}})
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8326{{c}} (~23/22 = 77.1674{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})


{{Optimal ET sequence|legend=0| 140, 311, 762g }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}


Badness (Sintel): 1.02
Badness (Sintel): 1.14


==== 31-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Subgroup: 2.3.5.7.11.13.17


Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 -83 | 0 22 -5 3 -116 -42 -48 105 -117 -60 94 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9721{{c}}, ~44/23 = 1122.8047{{c}} (~23/22 = 77.1673{{c}})
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8309{{c}} (~23/22 = 77.1691{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})


{{Optimal ET sequence|legend=0| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}


Badness (Sintel): 1.18
Badness (Sintel): 0.956


==== 37-limit ====
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Subgroup: 2.3.5.7.11.13.17.23


Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 -83 81 | 0 22 -5 3 -116 -42 -48 105 -117 -60 94 -81 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9824{{c}}, ~44/23 = 1122.8139{{c}} (~23/22 = 77.1685{{c}})
* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8304{{c}} (~23/22 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})


{{Optimal ET sequence|legend=0| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}


Badness (Sintel): 1.19
Badness (Sintel): 0.944


==== 41-limit ====
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Subgroup: 2.3.5.7.11.13.17.23.29


Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155


Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 -83 81 -4 | 0 22 -5 3 -116 -42 -48 105 -117 -60 94 -81 10 }}
{{Mapping|legend=0| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9957{{c}}, ~44/23 = 1122.8266{{c}} (~23/22 = 77.1691{{c}})
* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8306{{c}} (~23/22 = 77.1694{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})


{{Optimal ET sequence|legend=0| 140, 171, 311 }}
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}


Badness (Sintel): 1.20
Badness (Sintel): 0.858


=== Hemitert ===
=== Hemitert ===
Line 282: Line 266:
Comma list: 2401/2400, 3025/3024, 65625/65536
Comma list: 2401/2400, 3025/3024, 65625/65536


Mapping: {{mapping| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
: mapping generators: ~2, ~88/45
: mapping generators: ~2, ~88/45


Line 298: Line 282:
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095


Mapping: {{mapping| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}


Optimal tunings:  
Optimal tunings:  
Line 313: Line 297:
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095


Mapping: {{mapping| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}
{{Mapping|legend=0| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}


Optimal tunings:  
Optimal tunings:  
Line 328: Line 312:
Comma list: 2401/2400, 9801/9800, 65625/65536
Comma list: 2401/2400, 9801/9800, 65625/65536


Mapping: {{mapping| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
{{Mapping|legend=0| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
: mapping generators: ~99/70, ~693/512
: mapping generators: ~99/70, ~693/512


Line 366: Line 350:
Comma list: 243/242, 441/440, 1792000/1771561
Comma list: 243/242, 441/440, 1792000/1771561


Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}
{{Mapping|legend=0| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}


Optimal tunings:  
Optimal tunings:  
Line 381: Line 365:
Comma list: 243/242, 364/363, 441/440, 2200/2197
Comma list: 243/242, 364/363, 441/440, 2200/2197


Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}


Optimal tunings:  
Optimal tunings:  
Line 396: Line 380:
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197


Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
{{Mapping|legend=0| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 442: Line 426:
Comma list: 243/242, 441/440, 896/891
Comma list: 243/242, 441/440, 896/891


Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
{{Mapping|legend=0| 1 1 -5 -1 2 | 0 2 25 13 5 }}


Optimal tunings:  
Optimal tunings:  
Line 457: Line 441:
Comma list: 144/143, 196/195, 243/242, 364/363
Comma list: 144/143, 196/195, 243/242, 364/363


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


Optimal tunings:  
Optimal tunings:  
Line 472: Line 456:
Comma list: 2401/2400, 3388/3375, 5120/5103
Comma list: 2401/2400, 3388/3375, 5120/5103


Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
{{Mapping|legend=0| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231
: mapping generators: ~99/70, ~400/231


Line 488: Line 472:
Comma list: 352/351, 676/675, 847/845, 1716/1715
Comma list: 352/351, 676/675, 847/845, 1716/1715


Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
{{Mapping|legend=0| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}


Optimal tunings:  
Optimal tunings:  
Line 505: Line 489:
Comma list: 2401/2400, 3025/3024, 5120/5103
Comma list: 2401/2400, 3025/3024, 5120/5103


Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
{{Mapping|legend=0| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220
: mapping generators: ~2, ~243/220


Line 521: Line 505:
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
Comma list: 352/351, 847/845, 2401/2400, 3025/3024


Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
{{Mapping|legend=0| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}


Optimal tunings:  
Optimal tunings:  
Line 530: Line 514:


Badness (Sintel): 1.29
Badness (Sintel): 1.29
=== Cutefourths ===
This extension splits the neutral third plus an octave in three, with a ploidacot signature of beta-hexacot. The generator is an acute fourth in size (but not representing [[27/20]]), hence the name.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4000/3993, 5120/5103
{{Mapping|legend=0| 1 -1 -30 -14 -28 | 0 6 75 39 73 }}
: mapping generators: ~2, ~66/49
Optimal tunings:
* WE: ~2 = 1199.7345{{c}}, ~66/49 = 517.0436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1543{{c}}
{{Optimal ET sequence|legend=0| 58, 181, 239, 1014bcee }}
Badness (Sintel): 1.71
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 1575/1573, 2401/2400
{{Mapping|legend=0| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}
Optimal tunings:
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}
{{Optimal ET sequence|legend=0| 58, 181, 239f }}
Badness (Sintel): 1.45


== Osiris ==
== Osiris ==
{{See also| Metric microtemperaments #Geb }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 550: Line 565:


[[Badness]] (Sintel): 0.716
[[Badness]] (Sintel): 0.716
=== Semiosiris ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 1771561/1771470
{{Mapping|legend=0| 2 -6 -20 -9 -14 | 0 32 86 51 73 }}
: mapping generators: ~99/70, ~243/220
Optimal tunings:
* WE: ~99/70 = 600.0152{{c}}, ~243/220 = 171.9383{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~243/220 = 171.9344{{c}}
{{Optimal ET sequence|legend=0| 328, 342, 1012, 1354, 1696 }}
Badness (Sintel): 0.703


== Quasiorwell ==
== Quasiorwell ==
Line 578: Line 609:
Comma list: 2401/2400, 3025/3024, 5632/5625
Comma list: 2401/2400, 3025/3024, 5632/5625


Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}
{{Mapping|legend=0| 1 -7 3 1 -11 | 0 38 -3 8 64 }}


Optimal tunings:  
Optimal tunings:  
Line 593: Line 624:
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095


Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}
{{Mapping|legend=0| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}


Optimal tunings:  
Optimal tunings:  
Line 622: Line 653:


[[Badness]] (Sintel): 0.945
[[Badness]] (Sintel): 0.945
== Newt ==
Newt tempers out the [[garischisma]] and can be described as the {{nowrap| 41 & 270 }} temperament. It has a generator of a neutral third (0.2 cents flat of [[49/40]]) with a [[ploidacot]] signature of dicot. It extends into a strong [[13-limit]] temperament and eventually a no-17 [[19-limit]] temperament, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work even better.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 33554432/33480783
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
: mapping generators: ~2, ~49/40
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9315{{c}}, ~49/40 = 351.0932{{c}}
: [[error map]]: {{val| -0.068 +0.163 +0.075 -0.188 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1141{{c}}
: error map: {{val| 0.000 +0.273 +0.180 -0.022 }}
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
[[Badness]] (Sintel): 1.06
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 19712/19683
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
Optimal tunings:
* WE: ~2 = 1199.9603{{c}}, ~49/40 = 351.1038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1155{{c}}
{{Optimal ET sequence|legend=0| 41, 188, 229, 270, 581, 851, 1121, 1972 }}
Badness (Sintel): 0.643
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
Optimal tunings:
* WE: ~2 = 1199.9747{{c}}, ~49/40 = 351.1094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1168{{c}}
{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851, 2283b }}
Badness (Sintel): 0.571
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
Optimal tunings:
* WE: ~2 = 1199.9782{{c}}, ~49/40 = 351.1102{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1166{{c}}
{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851 }}
Badness (Sintel): 0.438


== Septidiasemi ==
== Septidiasemi ==
Line 717: Line 683:
Comma list: 243/242, 441/440, 939524096/935859375
Comma list: 243/242, 441/440, 939524096/935859375


Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}
{{Mapping|legend=0| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}


Optimal tunings:  
Optimal tunings:  
Line 732: Line 698:
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 2200/2197, 3584/3575


Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}
{{Mapping|legend=0| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 747: Line 713:
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575


Mapping: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}
{{Mapping|legend=0| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}


Optimal tunings:  
Optimal tunings:  
Line 756: Line 722:


Badness (Sintel): 1.39
Badness (Sintel): 1.39
== Subneutral ==
{{See also| Luna family }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 274877906944/274658203125
{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
: mapping generators: ~2, ~46875/28672
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9998{{c}}, ~46875/28672 = 851.6994 (~57344/46875 = 348.3005{{c}})
: [[error map]]: {{val| -0.000 +0.013 +0.090 -0.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46875/28672 = 851.6995{{c}} (~57344/46875 = 348.3005{{c}})
: error map: {{val| 0.000 +0.014 +0.090 -0.132 }}
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
[[Badness]] (Sintel): 1.16
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 1224440064/1220703125
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
[[Badness]] (Sintel): 1.46


== Lockerbie ==
== Lockerbie ==
Line 802: Line 728:
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  


The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament has a [[temperament merging|join]] 103 & 270, hence the name. The name is proposed in 2022 by [[Eliora]], who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament has a [[temperament merging|join]] 103 & 270, hence the name. The name was proposed in 2022 by [[Eliora]], who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  


Lockerbie also has a unique extension that adds the [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
Lockerbie also has a unique extension that adds the [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
Line 828: Line 754:
Comma list: 2401/2400, 3025/3024, 766656/765625
Comma list: 2401/2400, 3025/3024, 766656/765625


Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
{{Mapping|legend=0| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}


Optimal tunings:  
Optimal tunings:  
Line 843: Line 769:
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}


Optimal tunings:  
Optimal tunings:  
Line 858: Line 784:
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224


Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
{{Mapping|legend=0| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}


Optimal tunings:  
Optimal tunings:  
Line 867: Line 793:


Badness (Sintel): 1.07
Badness (Sintel): 1.07
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
Optimal tunings:
* WE: ~2 = 1199.8693{{c}}, ~41/32 = 431.0650{{c}}
* CWE: ~2 = 1200.000{{c}}, ~41/32 = 431.1109{{c}}
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
Badness (Sintel): 1.25


== Unthirds ==
== Unthirds ==
Line 910: Line 821:
Comma list: 2401/2400, 3025/3024, 4000/3993
Comma list: 2401/2400, 3025/3024, 4000/3993


Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
{{Mapping|legend=0| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}


Optimal tunings:  
Optimal tunings:  
Line 925: Line 836:
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400


Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
{{Mapping|legend=0| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}


Optimal tunings:  
Optimal tunings:  
Line 960: Line 871:
Comma list: 243/242, 441/440, 35937/35840
Comma list: 243/242, 441/440, 35937/35840


Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}
{{Mapping|legend=0| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}


Optimal tunings:  
Optimal tunings:  
Line 975: Line 886:
Comma list: 169/168, 243/242, 364/363, 441/440
Comma list: 169/168, 243/242, 364/363, 441/440


Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}
{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}


Optimal tunings:  
Optimal tunings:  
Line 990: Line 901:
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363


Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}
{{Mapping|legend=0| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}


Optimal tunings:  
Optimal tunings:  
Line 999: Line 910:


Badness (Sintel): 0.918
Badness (Sintel): 0.918
== Catafourth ==
{{See also| Sensipent family }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, 78732/78125
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~189/125 = 710.7220{{c}}
: [[error map]]: {{val| -0.072 -0.656 +1.050 +0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/125 = 710.7626{{c}}
: error map: {{val| 0.000 -0.603 +1.139 +0.238 }}
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
[[Badness]] (Sintel): 2.01
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 78408/78125
Mapping: {{mapping| 1 -15 -19 -12 -38 | 0 28 36 25 70 }}
Optimal tunings:
* WE: ~2 = 1200.0219{{c}}, ~189/125 = 710.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/125 = 710.7487{{c}}
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}
Badness (Sintel): 1.22
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 10985/10976
Mapping: {{mapping| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}
Optimal tunings:
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}
Badness (Sintel): 0.896


== Cotritone ==
== Cotritone ==
Line 1,073: Line 934:
Comma list: 385/384, 1375/1372, 4000/3993
Comma list: 385/384, 1375/1372, 4000/3993


Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}
{{Mapping|legend=0| 1 -13 -4 -4 2 | 0 30 13 14 3 }}


Optimal tunings:  
Optimal tunings:  
Line 1,088: Line 949:
Comma list: 169/168, 364/363, 385/384, 625/624
Comma list: 169/168, 364/363, 385/384, 625/624


Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}
{{Mapping|legend=0| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}


Optimal tunings:  
Optimal tunings:  
Line 1,121: Line 982:
Comma list: 385/384, 1375/1372, 43923/43750
Comma list: 385/384, 1375/1372, 43923/43750


Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}
{{Mapping|legend=0| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}


Optimal tunings:  
Optimal tunings:  
Line 1,136: Line 997:
Comma list: 385/384, 625/624, 847/845, 1375/1372
Comma list: 385/384, 625/624, 847/845, 1375/1372


Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}
{{Mapping|legend=0| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,171: Line 1,032:
Comma list: 243/242, 441/440, 1815/1792
Comma list: 243/242, 441/440, 1815/1792


Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
{{Mapping|legend=0| 1 1 9 6 2 | 0 2 -23 -11 5 }}


Optimal tunings:  
Optimal tunings:  
Line 1,206: Line 1,067:
Comma list: 243/242, 441/440, 43923/43750
Comma list: 243/242, 441/440, 43923/43750


Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}
{{Mapping|legend=0| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}


Optimal tunings:  
Optimal tunings:  
Line 1,221: Line 1,082:
Comma list: 243/242, 351/350, 441/440, 847/845
Comma list: 243/242, 351/350, 441/440, 847/845


Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}
{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}


Optimal tunings:  
Optimal tunings:  
Line 1,236: Line 1,097:
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845


Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}
{{Mapping|legend=0| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 1,247: Line 1,108:


== Gorgik ==
== Gorgik ==
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,269: Line 1,132:
Comma list: 176/175, 2401/2400, 2560/2541
Comma list: 176/175, 2401/2400, 2560/2541


Mapping: {{mapping| 1 -13 8 2 14 | 0 18 -7 1 -13 }}
{{Mapping|legend=0| 1 -13 8 2 14 | 0 18 -7 1 -13 }}


Optimal tunings:  
Optimal tunings:  
Line 1,284: Line 1,147:
Comma list: 176/175, 196/195, 364/363, 512/507
Comma list: 176/175, 196/195, 364/363, 512/507


Mapping: {{mapping| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}
{{Mapping|legend=0| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 1,293: Line 1,156:


Badness (Sintel): 1.33
Badness (Sintel): 1.33
== Retrotonic ==
Retrotonic may be described as the {{nowrap| 58 & 311 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot, the same as [[#Gorgik|gorgik]], but is strongly a 58-form. It is the generated by the retroptolemaic whole tone, [[~]][[729/640]], whose [[octave complement]] is the retroptolemaic minor seventh, ~[[1280/729]], given in the data below.
Retrotonic becomes much more interesting in the [[11-limit]], where it tempers out important commas such as the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), and the [[symbiotic comma]] (19712/19683). This leads to a 13-limit extension where [[1575/1573]], [[2080/2079]], and [[2200/2197]] are tempered out, enabling their corresponding [[essentially tempered chord]]s. Recommended retrotonic tunings include [[311edo]], [[369edo]] (in the 369f val) and [[680edo]] (in the 680ef val).
Retrotonic was named by [[Flora Canou]] in 2026 after one of its possible generators, the retroptolemaic whole tone.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2401/2400, {{monzo| 29 -28 3 3 }}
{{Mapping|legend=1| 1 -13 -86 -45 | 0 18 109 59 }}
: mapping generators: ~2, ~1280/729
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9156{{c}}, ~1280/729 = 972.2848{{c}}
: [[error map]]: {{val| -0.084 +0.269 -0.012 -0.225 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 972.3523{{c}}
: error map: {{val| 0.000 +0.386 +0.088 -0.040 }}
{{Optimal ET sequence|legend=1| 58, 195c, 253, 311, 369, 680 }}
[[Badness]] (Sintel): 4.25
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4000/3993, 19712/19683
{{Mapping|legend=0| 1 -13 -86 -45 -80 | 0 18 109 59 103 }}
Optimal tunings:
* WE: ~2 = 1199.9156{{c}}, ~135/77 = 972.1718{{c}}
: error map: {{val| -0.099 +0.221 -0.196 -0.346 +0.577 }}
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3507{{c}}
: error map: {{val| 0.000 +0.358 -0.083 -0.132 +0.808 }}
{{Optimal ET sequence|legend=0| 58, 195ce, 253, 311, 680e, 991e }}
Badness (Sintel): 1.69
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 2200/2197, 2401/2400
{{Mapping|legend=0| 1 -13 -86 -45 -80 -83 | 0 18 109 59 103 107 }}
Optimal tunings:
* WE: ~2 = 1199.8883{{c}}, ~135/77 = 972.2604{{c}}
: error map: {{val| -0.112 +0.184 -0.325 -0.436 +0.438 +0.605 }}
* CWE: ~2 = 1200.0000{{c}}, ~135/77 = 972.3497{{c}}
: error map: {{val| 0.000 +0.339 -0.201 -0.196 +0.697 +0.886 }}
{{Optimal ET sequence|legend=0| 58, 195cef, 253, 311, 680ef, 991ef }}
Badness (Sintel): 1.10


== Hemigoldis ==
== Hemigoldis ==
Line 1,339: Line 1,260:
Comma list: 2401/2400, 820125/819896, 2097152/2096325
Comma list: 2401/2400, 820125/819896, 2097152/2096325


Mapping: {{mapping| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}
{{Mapping|legend=0| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}


Optimal tunings:  
Optimal tunings:  
Line 1,354: Line 1,275:
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167


Mapping: {{mapping| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}


Optimal tunings:  
Optimal tunings:  
Line 1,369: Line 1,290:
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619


Mapping: {{mapping| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}


Optimal tunings:  
Optimal tunings:  
Line 1,384: Line 1,305:
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984


Mapping: {{mapping| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}
{{Mapping|legend=0| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}


Optimal tunings:  
Optimal tunings:  
Line 1,396: Line 1,317:
== References ==
== References ==


[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"