Quintosec family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''quintosec family''' tempers out the quintosec comma, 140737488355328/140126044921875 = {{monzo| 47 -15 -10 }}.
The '''quintosec family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quintosec comma]] ({{monzo|legend=1| 47 -15 -10 }}, [[ratio]]: 140 737 488 355 328 / 140 126 044 921 875).


== Quintosec ==
== Quintosec ==
Quintosec is naturally a 2.3.5.11-subgroup temperament. It was documented as ''qintosec'' on the temperament finder, which was probably due to a typo<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. The original, intended spelling is now restored, and the distinctive no-''u'' form is relegated to the lower-accuracy 7-limit extension considered below.  
Quintosec has a 1/5-octave period and a generator of around 831 [[cent]]s such that two of them plus a period make the [[3/1|3rd harmonic]]. However, an equivalent generator is the neutral third, in addition to the minimal generator, [[~]][[16/15]]. The [[ploidacot]] of this temperament is pentaploid dicot.
 
Quintosec is naturally a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament. It was documented as ''qintosec'' on the [https://x31eq.com/temper/ Temperament Finder], which was probably due to a typo<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. The original, intended spelling is restored here, and the distinctive no-''u'' form is relegated to the lower-accuracy 7-limit extension considered below.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 12:


{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}
{{Mapping|legend=1| 5 1 22 | 0 2 -3 }}
 
: mapping generators: ~524288/455625, ~16384/10125
: Mapping generators: ~524288/455625, ~16384/10125


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~524288/455625 = 1＼5, ~16384/10125 = 831.1061 (~16/15 = 111.1061)
* [[WE]]: ~524288/455625 = 239.9786{{c}}, ~16384/10125 = 831.0310{{c}} (~16/15 = 111.0952{{c}})
* [[POTE]]: ~524288/455625 = 1＼5, ~16384/10125 = 831.1051 (~16/15 = 111.1051)
: [[error map]]: {{val| -0.107 +0.086 +0.123 }}
* [[CWE]]: ~524288/455625 = 240.0000{{c}}, ~16384/10125 = 831.1054{{c}} (~16/15 = 111.1054{{c}})
: error map: {{val| 0.000 +0.256 +0.370 }}


{{Optimal ET sequence|legend=1| 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc }}
{{Optimal ET sequence|legend=1| 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc }}


[[Badness]]: 0.139191
[[Badness]] (Sintel): 3.27


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
Line 26: Line 29:
Comma list: 5632/5625, 26214400/26198073
Comma list: 5632/5625, 26214400/26198073


Mapping: {{mapping| 5 1 22 45 | 0 2 -3 -8 }}
{{Mapping|legend=0| 5 1 22 45 | 0 2 -3 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~1024/891 = 1＼5, ~160/99 = 831.0932 (~16/15 = 111.0932)
* WE: ~1024/891 = 239.9820{{c}}, ~160/99 = 831.0062{{c}} (~16/15 = 111.0603{{c}})
* CWE: ~1024/891 = 1＼5, ~160/99 = 831.0743 (~16/15 = 111.0743)
* CWE: ~1024/891 = 240.0000{{c}}, ~160/99 = 831.0743{{c}} (~16/15 = 111.0743{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}
{{Optimal ET sequence|legend=0| 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c }}


Badness: 0.0195
Badness (Sintel): 0.608


== Qintosec ==
== Qintosec ==
Qintosec can be described as the 10 &amp; 65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.  
Qintosec can be described as the 10 & 65d temperament, tempering out the [[marvel comma]], 225/224 in the 7-limit.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 45: Line 48:
{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}
{{Mapping|legend=1| 5 1 22 21 | 0 2 -3 -2 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~400/343 = 1＼5, ~80/49 = 831.7095 (~15/14 = 111.7095)
* [[WE]]: ~400/343 = 239.8397{{c}}, ~80/49 = 830.9971{{c}} (~16/15 = 111.7095{{c}})
* POTE: ~400/343 = 1＼5, ~80/49 = 831.5525 (~15/14 = 111.5525)
: [[error map]]: {{val| -0.802 -0.121 -2.832 +5.813 }}
* [[CWE]]: ~400/343 = 240.0000{{c}}, ~80/49 = 831.5851{{c}} (~16/15 = 111.5851{{c}})
: error map: {{val| 0.000 +1.215 -1.069 +8.004 }}


{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}
{{Optimal ET sequence|legend=1| 10, …, 55d, 65d, 75, 140d }}


[[Badness]]: 0.156764
[[Badness]] (Sintel): 3.97


=== 11-limit ===
=== 11-limit ===
Line 58: Line 63:
Comma list: 225/224, 243/242, 3840/3773
Comma list: 225/224, 243/242, 3840/3773


Mapping: {{mapping| 5 1 22 21 0 | 0 2 -3 -2 5 }}
{{Mapping|legend=0| 5 1 22 21 0 | 0 2 -3 -2 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 1＼5, ~80/49 = 831.1939 (~15/14 = 111.1939)
* WE: ~55/48 = 239.8038{{c}}, ~80/49 = 830.6313{{c}} (~16/15 = 111.2200{{c}})
* POTE: ~55/48 = 1＼5, ~80/49 = 831.3110 (~15/14 = 111.3110)
* CWE: ~55/48 = 240.0000{{c}}, ~80/49 = 831.2911{{c}} (~16/15 = 111.2911{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55d, 65d, 140de }}
{{Optimal ET sequence|legend=0| 10, 55d, 65d, 140de }}


Badness: 0.077907
Badness (Sintel): 2.58


=== Qinto ===
=== Qinto ===
This alternative extension can be tuned to [[75edo]] and [[85edo]], both in [[patent val]]s.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 332750/321489
Comma list: 225/224, 385/384, 332750/321489


Mapping: {{mapping| 5 1 22 21 -7 | 0 2 -3 -2 7 }}
{{Mapping|legend=0| 5 1 22 21 -7 | 0 2 -3 -2 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~220/189 = 1＼5, ~44/27 = 832.4051 (~15/14 = 112.4051)
* WE: ~220/189 = 239.9453{{c}}, ~44/27 = 832.2583{{c}} (~16/15 = 112.4224{{c}})
* POTE: ~220/189 = 1＼5, ~44/27 = 832.4480 (~15/14 = 112.4480)
* CWE: ~220/189 = 240.0000{{c}}, ~44/27 = 832.4403{{c}} (~16/15 = 112.4403{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 75 }}
{{Optimal ET sequence|legend=0| 10, 75 }}


Badness: 0.127732
Badness (Sintel): 4.22


== Sengasec ==
== Sengasec ==
Sengasec can be described as the 10 &amp; 55 temperament, tempering out the [[cloudy comma]], 16807/16384 and the sengic comma, 686/675 in the 7-limit.
Sengasec can be described as the 10 & 65 temperament, tempering out the [[cloudy comma]], 16807/16384 and the sengic comma, 686/675 in the 7-limit. It was named by [[Xenllium]] in 2021.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 93: Line 100:


[[Optimal tuning]]:  
[[Optimal tuning]]:  
* [[CTE]]: ~8/7 = 1＼5, ~45/28 = 831.1061 (~16/15 = 111.1061)
* [[WE]]: ~8/7 = 240.1396{{c}}, ~45/28 = 831.5960{{c}} (~16/15 = 111.1772{{c}})
* [[POTE]]: ~8/7 = 1＼5, ~45/28 = 831.1126 (~16/15 = 111.1126)
: [[error map]]: {{val| +0.698 +1.377 +1.969 -6.872 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~45/28 = 831.1113{{c}} (~16/15 = 111.1113{{c}})
: error map: {{val| 0.000 +0.268 +0.352 -8.826 }}


{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}
{{Optimal ET sequence|legend=1| 10, …, 55, 65, 140dd }}


[[Badness]]: 0.156259
[[Badness]] (Sintel): 3.95


=== 11-limit ===
=== 11-limit ===
Line 105: Line 114:
Comma list: 243/242, 385/384, 686/675
Comma list: 243/242, 385/384, 686/675


Mapping: {{mapping| 5 1 22 14 0 | 0 2 -3 0 5 }}
{{Mapping|legend=0| 5 1 22 14 0 | 0 2 -3 0 5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1＼5, ~45/28 = 830.7772 (~16/15 = 110.7772)
* WE: ~8/7 = 240.0876{{c}}, ~45/28 = 830.9623{{c}} (~16/15 = 110.7772{{c}})
* POTE: ~8/7 = 1＼5, ~45/28 = 830.6592 (~16/15 = 110.6592)
* CWE: ~8/7 = 240.0000{{c}}, ~45/28 = 830.6798{{c}} (~16/15 = 110.6798{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
{{Optimal ET sequence|legend=0| 10, 55, 65 }}


Badness: 0.081607
Badness (Sintel): 2.70


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


Comma list: 105/104, 144/143, 243/242, 686/675
Comma list: 91/90, 169/168, 243/242, 385/384


Mapping: {{mapping| 5 1 22 14 0 22 | 0 2 -3 0 5 -1 }}
{{Mapping|legend=0| 5 1 22 14 0 15 | 0 2 -3 0 5 1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 1＼5, ~45/28 = 830.8942 (~16/15 = 110.8942)
* WE: ~8/7 = 240.1611{{c}}, ~13/8 = 831.2123{{c}} (~16/15 = 110.7289{{c}})
* POTE: ~8/7 = 1＼5, ~45/28 = 830.9099 (~16/15 = 110.9099)
* CWE: ~8/7 = 240.0000{{c}}, ~13/8 = 830.6933{{c}} (~16/15 = 110.6933{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10, 55, 65 }}
{{Optimal ET sequence|legend=0| 10, 55f, 65f }}


Badness: 0.056390
Badness (Sintel): 2.08


== Decoid ==
== Decoid ==
{{See also| Breedsmic temperaments #Decoid }}
Decoid tempers out 2401/2400, the [[breedsma]], and 67108864/66976875, the [[decovulture comma]], as well as {{monzo| 11 -10 -10 10 }}, the [[linus comma]]. It may be described as the 130 & 270 temperament, and its [[ploidacot]] is decaploid dicot. The hemitwelfth or neutral third can be used as a generator, but also ~[[8/7]], ~[[16/15]], or the [[marvel comma]]. 181\940 or 233\1210 makes for an excellent tuning choice.  
 
Decoid tempers out 2401/2400 and 67108864/66976875, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. Either 8/7 or 16/15 can be used as its generator. It may be described as the 130 &amp; 270 temperament, and as one might expect, 181\940 or 233\1210 makes for an excellent tuning choice.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 140: Line 147:


{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
{{Mapping|legend=1| 10 0 47 36 | 0 2 -3 -1 }}
 
: mapping generators: ~15/14, ~8192/4725
: Mapping generators: ~15/14, ~8192/4725


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 1＼10, ~8192/4725 = 951.1086 (~16/15 = 111.1086, or ~225/224 = 8.8914)
* [[WE]]: ~15/14 = 119.9913{{c}}, ~8192/4725 = 951.0298{{c}} (~225/224 = 8.9006{{c}})
* [[POTE]]: ~15/14 = 1＼10, ~8192/4725 = 951.0987 (~16/15 = 111.0987, or ~225/224 = 8.9013)
: [[error map]]: {{val| -0.087 +0.105 +0.188 -0.169 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~8192/4725 = 951.1007{{c}} (~225/224 = 8.8993{{c}})
: error map: {{val| 0.000 +0.246 +0.384 +0.073 }}


{{Optimal ET sequence|legend=1| 10, …, 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}
{{Optimal ET sequence|legend=1| 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc }}


[[Badness]]: 0.033902
[[Badness]] (Sintel): 0.858


=== 11-limit ===
=== 11-limit ===
Line 156: Line 164:
Comma list: 2401/2400, 5632/5625, 9801/9800
Comma list: 2401/2400, 5632/5625, 9801/9800


Mapping: {{mapping| 10 0 47 36 98 | 0 2 -3 -1 -8 }}
{{Mapping|legend=0| 10 0 47 36 98 | 0 2 -3 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1＼10, ~400/231 = 951.0943 (~16/15 = 111.0943, or ~225/224 = 8.9057)
* WE: ~15/14 = 120.9921{{c}}, ~400/231 = 951.0074{{c}} (~225/224 = 8.9294{{c}})
* POTE: ~15/14 = 1＼10, ~400/231 = 951.0700 (~16/15 = 111.0700, or ~225/224 = 8.9300)
* CWE: ~15/14 = 120.0000{{c}}, ~400/231 = 951.0747{{c}} (~225/224 = 8.9253{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 670, 940, 1210, 2150c }}
{{Optimal ET sequence|legend=0| 130, 270, 670, 940, 1210, 2150c }}


Badness: 0.018735
Badness (Sintel): 0.619


==== 13-limit ====
==== 13-limit ====
Line 171: Line 179:
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095
Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095


Mapping: {{mapping| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}
{{Mapping|legend=0| 10 0 47 36 98 37 | 0 2 -3 -1 -8 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~15/14 = 1＼10, ~26/15 = 951.0943 (~16/15 = 111.0943, or ~196/195 = 8.9057)
* WE: ~15/14 = 119.9964{{c}}, ~26/15 = 951.0546{{c}} (~196/195 = 8.9165{{c}})
* POTE: ~15/14 = 1＼10, ~26/15 = 951.0832 (~16/15 = 111.0832, or ~196/195 = 8.9168)
* CWE: ~15/14 = 120.0000{{c}}, ~26/15 = 951.0850{{c}} (~196/195 = 8.9150{{c}})


Optimal ET sequence: {{Optimal ET sequence| 10e, …, 130, 270, 940, 1210f, 1480cf }}
{{Optimal ET sequence|legend=0| 130, 270, 940, 1210f, 1480cf }}


Badness: 0.013475
Badness (Sintel): 0.557


=== Triacontoid ===
=== Triacontoid ===
Line 186: Line 194:
Comma list: 2401/2400, 42592/42525, 1771561/1769472
Comma list: 2401/2400, 42592/42525, 1771561/1769472


Mapping: {{mapping| 30 0 141 108 80 | 0 2 -3 -1 1 }}
{{Mapping|legend=0| 30 0 141 108 80 | 0 2 -3 -1 1 }}
 
: mapping generators: ~45/44, ~8192/4725
: Mapping generators: ~45/44, ~8192/4725


Optimal tunings:  
Optimal tunings:  
* CTE: ~45/44 = 1＼30, ~8192/4725 = 951.1137 (~225/224 = 8.8863)
* WE: ~45/44 = 39.9981{{c}}, ~8192/4725 = 951.0672{{c}} (~225/224 = 8.8875{{c}})
* POTE: ~45/44 = 1＼30, ~8192/4725 = 951.1121 (~225/224 = 8.8879)
* CWE: ~45/44 = 40.0000{{c}}, ~8192/4725 = 951.1124{{c}} (~225/224 = 8.8876{{c}})


Optimal ET sequence: {{Optimal ET sequence| 120, 150, 270 }}
{{Optimal ET sequence|legend=0| 120, 150, 270 }}


Badness: 0.075991
Badness (Sintel): 2.51


==== 13-limit ====
==== 13-limit ====
Line 203: Line 210:
Comma list: 676/675, 2401/2400, 4096/4095, 6656/6655
Comma list: 676/675, 2401/2400, 4096/4095, 6656/6655


Mapping: {{mapping| 30 0 141 108 80 111 | 0 2 -3 -1 1 0 }}
{{Mapping|legend=0| 30 0 141 108 80 111 | 0 2 -3 -1 1 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~45/44 = 1＼30, ~26/15 = 951.1137 (~196/195 = 8.8863)
* WE: ~45/44 = 39.9992{{c}}, ~26/15 = 951.0944{{c}} (~196/195 = 8.8868{{c}})
* POTE: ~45/44 = 1＼30, ~26/15 = 951.1130 (~196/195 = 8.8870)
* CWE: ~45/44 = 40.0000{{c}}, ~26/15 = 951.1131{{c}} (~196/195 = 8.8869{{c}})


Optimal ET sequence: {{Optimal ET sequence| 120, 150, 270 }}
{{Optimal ET sequence|legend=0| 120, 150, 270 }}


Badness: 0.040873
Badness (Sintel): 1.69


== Notes ==
== References ==


[[Category:Temperament families]]
[[Category:Quintosec family| ]] <!-- main article -->
[[Category:Quintosec family| ]] <!-- main article -->
[[Category:Quintosec| ]] <!-- key article -->
[[Category:Quintosec| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:32, 24 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quintosec family of temperaments tempers out the quintosec comma (monzo: [47 -15 -10⟩, ratio: 140 737 488 355 328 / 140 126 044 921 875).

Quintosec

Quintosec has a 1/5-octave period and a generator of around 831 cents such that two of them plus a period make the 3rd harmonic. However, an equivalent generator is the neutral third, in addition to the minimal generator, ~16/15. The ploidacot of this temperament is pentaploid dicot.

Quintosec is naturally a 2.3.5.11-subgroup temperament. It was documented as qintosec on the Temperament Finder, which was probably due to a typo[1]. The original, intended spelling is restored here, and the distinctive no-u form is relegated to the lower-accuracy 7-limit extension considered below.

Subgroup: 2.3.5

Comma list: [47 -15 -10⟩

Mapping: [⟨5 1 22], ⟨0 2 -3]]

mapping generators: ~524288/455625, ~16384/10125

Optimal tunings:

  • WE: ~524288/455625 = 239.9786 ¢, ~16384/10125 = 831.0310 ¢ (~16/15 = 111.0952 ¢)
error map: ⟨-0.107 +0.086 +0.123]
  • CWE: ~524288/455625 = 240.0000 ¢, ~16384/10125 = 831.1054 ¢ (~16/15 = 111.1054 ¢)
error map: ⟨0.000 +0.256 +0.370]

Optimal ET sequence: 10, 45c, 55, 65, 140, 205, 270, 1955c, 2225c, 2495bc, 2765bc, 3035bc, 3305bcc, 3575bcc

Badness (Sintel): 3.27

2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 5632/5625, 26214400/26198073

Mapping: [⟨5 1 22 45], ⟨0 2 -3 -8]]

Optimal tunings:

  • WE: ~1024/891 = 239.9820 ¢, ~160/99 = 831.0062 ¢ (~16/15 = 111.0603 ¢)
  • CWE: ~1024/891 = 240.0000 ¢, ~160/99 = 831.0743 ¢ (~16/15 = 111.0743 ¢)

Optimal ET sequence: 10e, 55e, 65, 205, 270, 335, 605, 940, 1545c

Badness (Sintel): 0.608

Qintosec

Qintosec can be described as the 10 & 65d temperament, tempering out the marvel comma, 225/224 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 225/224, 2560000/2470629

Mapping: [⟨5 1 22 21], ⟨0 2 -3 -2]]

Optimal tunings:

  • WE: ~400/343 = 239.8397 ¢, ~80/49 = 830.9971 ¢ (~16/15 = 111.7095 ¢)
error map: ⟨-0.802 -0.121 -2.832 +5.813]
  • CWE: ~400/343 = 240.0000 ¢, ~80/49 = 831.5851 ¢ (~16/15 = 111.5851 ¢)
error map: ⟨0.000 +1.215 -1.069 +8.004]

Optimal ET sequence: 10, …, 55d, 65d, 75, 140d

Badness (Sintel): 3.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3840/3773

Mapping: [⟨5 1 22 21 0], ⟨0 2 -3 -2 5]]

Optimal tunings:

  • WE: ~55/48 = 239.8038 ¢, ~80/49 = 830.6313 ¢ (~16/15 = 111.2200 ¢)
  • CWE: ~55/48 = 240.0000 ¢, ~80/49 = 831.2911 ¢ (~16/15 = 111.2911 ¢)

Optimal ET sequence: 10, 55d, 65d, 140de

Badness (Sintel): 2.58

Qinto

This alternative extension can be tuned to 75edo and 85edo, both in patent vals.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 332750/321489

Mapping: [⟨5 1 22 21 -7], ⟨0 2 -3 -2 7]]

Optimal tunings:

  • WE: ~220/189 = 239.9453 ¢, ~44/27 = 832.2583 ¢ (~16/15 = 112.4224 ¢)
  • CWE: ~220/189 = 240.0000 ¢, ~44/27 = 832.4403 ¢ (~16/15 = 112.4403 ¢)

Optimal ET sequence: 10, 75

Badness (Sintel): 4.22

Sengasec

Sengasec can be described as the 10 & 65 temperament, tempering out the cloudy comma, 16807/16384 and the sengic comma, 686/675 in the 7-limit. It was named by Xenllium in 2021.

Subgroup: 2.3.5.7

Comma list: 686/675, 16807/16384

Mapping: [⟨5 1 22 14], ⟨0 2 -3 0]]

Optimal tuning:

  • WE: ~8/7 = 240.1396 ¢, ~45/28 = 831.5960 ¢ (~16/15 = 111.1772 ¢)
error map: ⟨+0.698 +1.377 +1.969 -6.872]
  • CWE: ~8/7 = 240.0000 ¢, ~45/28 = 831.1113 ¢ (~16/15 = 111.1113 ¢)
error map: ⟨0.000 +0.268 +0.352 -8.826]

Optimal ET sequence: 10, …, 55, 65, 140dd

Badness (Sintel): 3.95

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 686/675

Mapping: [⟨5 1 22 14 0], ⟨0 2 -3 0 5]]

Optimal tunings:

  • WE: ~8/7 = 240.0876 ¢, ~45/28 = 830.9623 ¢ (~16/15 = 110.7772 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~45/28 = 830.6798 ¢ (~16/15 = 110.6798 ¢)

Optimal ET sequence: 10, 55, 65

Badness (Sintel): 2.70

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 243/242, 385/384

Mapping: [⟨5 1 22 14 0 15], ⟨0 2 -3 0 5 1]]

Optimal tunings:

  • WE: ~8/7 = 240.1611 ¢, ~13/8 = 831.2123 ¢ (~16/15 = 110.7289 ¢)
  • CWE: ~8/7 = 240.0000 ¢, ~13/8 = 830.6933 ¢ (~16/15 = 110.6933 ¢)

Optimal ET sequence: 10, 55f, 65f

Badness (Sintel): 2.08

Decoid

Decoid tempers out 2401/2400, the breedsma, and 67108864/66976875, the decovulture comma, as well as [11 -10 -10 10⟩, the linus comma. It may be described as the 130 & 270 temperament, and its ploidacot is decaploid dicot. The hemitwelfth or neutral third can be used as a generator, but also ~8/7, ~16/15, or the marvel comma. 181\940 or 233\1210 makes for an excellent tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 67108864/66976875

Mapping: [⟨10 0 47 36], ⟨0 2 -3 -1]]

mapping generators: ~15/14, ~8192/4725

Optimal tunings:

  • WE: ~15/14 = 119.9913 ¢, ~8192/4725 = 951.0298 ¢ (~225/224 = 8.9006 ¢)
error map: ⟨-0.087 +0.105 +0.188 -0.169]
  • CWE: ~15/14 = 120.0000 ¢, ~8192/4725 = 951.1007 ¢ (~225/224 = 8.8993 ¢)
error map: ⟨0.000 +0.246 +0.384 +0.073]

Optimal ET sequence: 130, 270, 2020c, 2290c, 2560c, 2830bc, 3100bcc, 3370bcc, 3640bcc

Badness (Sintel): 0.858

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 5632/5625, 9801/9800

Mapping: [⟨10 0 47 36 98], ⟨0 2 -3 -1 -8]]

Optimal tunings:

  • WE: ~15/14 = 120.9921 ¢, ~400/231 = 951.0074 ¢ (~225/224 = 8.9294 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~400/231 = 951.0747 ¢ (~225/224 = 8.9253 ¢)

Optimal ET sequence: 130, 270, 670, 940, 1210, 2150c

Badness (Sintel): 0.619

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 4096/4095

Mapping: [⟨10 0 47 36 98 37], ⟨0 2 -3 -1 -8 0]]

Optimal tunings:

  • WE: ~15/14 = 119.9964 ¢, ~26/15 = 951.0546 ¢ (~196/195 = 8.9165 ¢)
  • CWE: ~15/14 = 120.0000 ¢, ~26/15 = 951.0850 ¢ (~196/195 = 8.9150 ¢)

Optimal ET sequence: 130, 270, 940, 1210f, 1480cf

Badness (Sintel): 0.557

Triacontoid

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 42592/42525, 1771561/1769472

Mapping: [⟨30 0 141 108 80], ⟨0 2 -3 -1 1]]

mapping generators: ~45/44, ~8192/4725

Optimal tunings:

  • WE: ~45/44 = 39.9981 ¢, ~8192/4725 = 951.0672 ¢ (~225/224 = 8.8875 ¢)
  • CWE: ~45/44 = 40.0000 ¢, ~8192/4725 = 951.1124 ¢ (~225/224 = 8.8876 ¢)

Optimal ET sequence: 120, 150, 270

Badness (Sintel): 2.51

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 2401/2400, 4096/4095, 6656/6655

Mapping: [⟨30 0 141 108 80 111], ⟨0 2 -3 -1 1 0]]

Optimal tunings:

  • WE: ~45/44 = 39.9992 ¢, ~26/15 = 951.0944 ¢ (~196/195 = 8.8868 ¢)
  • CWE: ~45/44 = 40.0000 ¢, ~26/15 = 951.1131 ¢ (~196/195 = 8.8869 ¢)

Optimal ET sequence: 120, 150, 270

Badness (Sintel): 1.69

References

  1. ↑ Yahoo Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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