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Septiennealimmal clan: Difference between revisions

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[[Badness]] (Sintel): 0.191
[[Badness]] (Sintel): 0.191
=== Septiennealic ===
Septiennealic finds a somewhat high-damage but very simple and intuitive mapping of prime 13 by fixing 13/12~14/13 at 1\9.
A notable tuning of septiennealic not appearing in the optimal ET sequence is [[63edo]]. If we include a somewhat more complex mapping for 11 via {{nowrap| 36e & 63 }}, it will become the optimal patent val and largest in the sequence.
Subgroup: 2.3.7.13
Comma list: 169/168, 31213/31104
Subgroup-val mapping: {{mapping| 9 0 11 19 | 0 1 1 1 }}
Optimal tunings:
* WE: ~13/12 = 133.3847{{c}}, ~3/2 = 701.9342{{c}}
* CWE: ~13/12 = 133.3333{{c}}, ~3/2 = 702.0763{{c}}
{{Optimal ET sequence|legend=0| 27, 36, 99, 135f, 171f }}
Badness (Sintel): 0.540


=== Ennea ===
=== Ennea ===
{{See also| No-fives subgroup temperaments }}
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11


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[[Badness]] (Sintel): 0.0914
[[Badness]] (Sintel): 0.0914


=== 11-limit ===
=== Enneabiotic ===
The undecimal ennealimmal temperament can be described as {{nowrap| 99e & 171e }}, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
Enneabiotic ({{nowrap| 99e & 171e }}) tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma). It is catalogued as ''undecimal ennealimmal'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 121: Line 100:


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
Optimal tunings:
* WE: ~27/25 = 133.3215{{c}}, ~5/3 = 884.4027{{c}} (~36/35 = 48.8479{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4745{{c}} (~36/35 = 48.8589{{c}})
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
Badness (Sintel): 0.912
==== Enneabio ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 128: Line 122:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3321{{c}}, ~5/3 = 884.4225{{c}} (~36/35 = 48.9025{{c}})
* WE: ~27/25 = 133.3321{{c}}, ~5/3 = 884.4225{{c}} (~36/35 = 48.9025{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4301{{c}} (~36/35 = 48.9033{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4301{{c}} (~36/35 = 48.9033{{c}})


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Badness (Sintel): 1.25
Badness (Sintel): 1.25


==== Ennealimmalis ====
=== Ennealympic ===
Subgroup: 2.3.5.7.11.13
Ennealympic ({{nowrap| 99 & 171 }}, formerly ''ennealimmia'') is an alternative extension which tempers out [[131072/130977]] (olympia).  
 
Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
 
Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
 
Optimal tunings:
* WE: ~27/25 = 133.3215{{c}}, ~5/3 = 884.4027{{c}} (~36/35 = 48.8479{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4745{{c}} (~36/35 = 48.8589{{c}})
 
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
 
Badness (Sintel): 0.912
 
=== Ennealimmia ===
The ennealimmia temperament is an alternative extension and can be described as {{nowrap| 99 & 171 }}, which tempers out [[131072/130977]] (olympia).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Ennealimnic ===
=== Ennealimnic ===
{{Distinguish| Ennealimmic }}
{{Distinguish| Ennealimmic }}
{{See also| Chords of ennealimnic }}


Ennealimnic ({{nowrap| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
Ennealimnic ({{nowrap| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
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Badness (Sintel): 0.673
Badness (Sintel): 0.673
See also: [[Chords of ennealimnic]]


==== 13-limit ====
==== 13-limit ====
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=== Hemiennealimmal ===
=== Hemiennealimmal ===
Hemiennealimmal ({{nowrap| 72 & 198 }}) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, [[3025/3024]], 4375/4374, and [[9801/9800]]. Tempering out 9801/9800 leads to an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.
Hemiennealimmal ({{nowrap| 72 & 198 }}) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, [[3025/3024]], 4375/4374, and [[9801/9800]]. Its [[S-expression]]-based comma list is {([[3025/3024|S22/S24 = S55 = S25/S27 × S99]]), [[4375/4374|S25/S27]], [[2401/2400|S49]], [[9801/9800|S33/S35 = S99]]}. Tempering out 9801/9800 leads to an octave split into two equal parts.  
 
Its [[S-expression]]-based comma list is {([[3025/3024|S22/S24 = S55 = S25/S27 × S99]],) [[4375/4374|S25/S27]], [[2401/2400|S49]], [[9801/9800|S33/S35 = S99]]}.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 438: Line 414:


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667{{c}}, ~400/231 = 950.9553{{c}}
* WE: ~80/77 = 66.6698{{c}}, ~400/231 = 950.9982{{c}}
* CWE: ~80/77 = 66.6667{{c}}, ~400/231 = 950.9736{{c}}


Tuning ranges:  
Tuning ranges:  
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* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]


{{Optimal ET sequence|legend=0| 72, 198, 270, 342, 612, 954, 1566 }}
{{Optimal ET sequence|legend=0| 72, 198, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}


Badness (Smith): 0.006283
Badness (Sintel): 0.208


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}
* WE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0838{{c}}
* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}


Tuning ranges:  
Tuning ranges:  
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{{Optimal ET sequence|legend=0| 72, 198, 270 }}
{{Optimal ET sequence|legend=0| 72, 198, 270 }}


Badness (Smith): 0.012505
Badness (Sintel): 0.517


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}
* WE: ~27/26 = 66.6681{{c}}, ~26/15 = 951.0200{{c}}
* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0063{{c}}


{{Optimal ET sequence|legend=0| 72, 198g, 270 }}
{{Optimal ET sequence|legend=0| 72, 198g, 270 }}
Badness (Sintel): 0.664


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}
* WE: ~27/26 = 66.6653{{c}}, ~26/15 = 951.0226{{c}}
* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0386{{c}}


{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
{{Optimal ET sequence|legend=0| 72, 198g, 270 }}
 
Badness (Sintel): 0.812


==== Semihemiennealimmal ====
==== Semihemiennealimmal ====
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Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
: mapping generators: ~80/77, ~1053/800
: mapping generators: ~80/77, ~1053/800


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}
* WE: ~80/77 = 66.6702{{c}}, ~1053/800 = 475.4979{{c}}
* CWE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4782{{c}}


{{Optimal ET sequence|legend=0| 126, 144, 270, 684, 954 }}
{{Optimal ET sequence|legend=0| 126, 144, 270, 684, 954 }}


Badness (Smith): 0.013104
Badness (Sintel): 0.541


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}
* WE: ~80/77 = 66.6698{{c}}, ~1053/800 = 475.5039{{c}}
* CWE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4837{{c}}


{{Optimal ET sequence|legend=0| 270, 684, 954 }}
{{Optimal ET sequence|legend=0| 270, 684g, 954, 1224, 2178ef }}


Badness (Smith): 0.013104
Badness (Sintel): 0.994


===== 19-limit =====
===== 19-limit =====
Line 530: Line 515:


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}
* WE: ~80/77 = 66.6702{{c}}, ~1053/800 = 475.5078{{c}}
* CWE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4854{{c}}


{{Optimal ET sequence|legend=0| 270, 684h, 954h, 1224 }}
{{Optimal ET sequence|legend=0| 270, 684gh, 954h, 1224, 2178efh }}


Badness (Smith): 0.013104
Badness (Sintel): 0.927


=== Semiennealimmal ===
=== Ennealimmapine ===
Semiennealimmal tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.  
Ennealimmapine (formerly ''semiennealimmal'') tempers out [[4000/3993]], and uses a ~140/121 semifourth generator, six of which and 1/3 octave give the 3rd harmonic. Perhaps a better generator is the [[secor]], ~77/72, six of which give the perfect fifth, or the [[ptolemisma]], six of which and 1/3 octave give the perfect fourth.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3367{{c}}
* WE: ~27/25 = 133.3264{{c}}, ~140/121 = 250.3236{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3283{{c}}


{{Optimal ET sequence|legend=0| 72, 369, 441 }}
{{Optimal ET sequence|legend=0| 72, …, 297e, 369, 441 }}


Badness (Smith): 0.034196
Badness (Sintel): 1.13


==== 13-limit ====
==== 13-limit ====
Line 561: Line 548:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3375{{c}}
* WE: ~27/25 = 133.3262{{c}}, ~140/121 = 250.3241{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3317{{c}}


{{Optimal ET sequence|legend=0| 72, 297ef, 369f, 441 }}
{{Optimal ET sequence|legend=0| 72, …, 297ef, 369f, 441 }}


Badness (Smith): 0.026122
Badness (Sintel): 1.08


=== Quadraennealimmal ===
=== Quadraennealimmal ===
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Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333{{c}}, ~25/22 = 221.0717{{c}}
* WE: ~27/25 = 133.3372{{c}}, ~25/22 = 221.0781{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~25/22 = 221.0746{{c}}


{{Optimal ET sequence|legend=0| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
{{Optimal ET sequence|legend=0| 27e, …, 342, 1053, 1395, 1737 }}


Badness (Smith): 0.021320
Badness (Sintel): 0.705


=== Trinealimmal ===
=== Trinealimmal ===
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2744/2673 = 44.4444{{c}}, ~2352/1375 = 928.8000{{c}}
* WE: ~2744/2673 = 44.4437{{c}}, ~2352/1375 = 928.7852{{c}}
* CWE: ~2744/2673 = 44.4444{{c}}, ~2352/1375 = 928.7985{{c}}


{{Optimal ET sequence|legend=0| 27, 243, 270, 783, 1053, 1323 }}
{{Optimal ET sequence|legend=0| 27, 243, 270, 783, 1053, 1323 }}


Badness (Smith): 0.029812
Badness (Sintel): 0.986


=== Rhodium ===
=== Rhodium ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6658{{c}} (~385/384 = 4.3325{{c}})
* WE: ~3072/3025 = 26.6668{{c}}, ~55/32 = 937.6664{{c}} (~385/384 = 4.3288{{c}})
* CWE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6630{{c}} (~385/384 = 4.3397{{c}})
* CWE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6630{{c}} (~385/384 = 4.3297{{c}})


{{Optimal ET sequence|legend=0| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
{{Optimal ET sequence|legend=0| 45, 225c, 270, 1125, 1395, 1665, 5265d }}


Badness (Smith): 0.0381
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 625: Line 615:


Optimal tunings:  
Optimal tunings:  
* CTE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6569{{c}} (~385/384 = 4.3236{{c}})
* WE: ~66/65 = 26.6670{{c}}, ~55/32 = 937.6633{{c}} (~385/384 = 4.3172{{c}})
* CWE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6515{{c}} (~385/384 = 4.3182{{c}})
* CWE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6515{{c}} (~385/384 = 4.3182{{c}})


{{Optimal ET sequence|legend=0| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
{{Optimal ET sequence|legend=0| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}


Badness (Smith): 0.0226
Badness (Sintel): 0.936


== Undecentic ==
== Undecentic ==
Undecentic ({{nowrap| 99 & 198 }}) has a period of 1/99 octave.
{{Distinguish| Undecental }}
 
Named by [[Xenllium]] in 2021, undecentic ({{nowrap| 99 & 198 }}) has a period of 1/99 octave.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 640: Line 632:


{{Mapping|legend=1| 99 157 230 278 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 99 157 230 278 0 | 0 0 0 0 1 }}
: mapping generators: ~126/125, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~126/125 = 12.121{{c}}, ~11/8 = 552.756{{c}}
* [[WE]]: ~126/125 = 12.1170{{c}}, ~11/8 = 552.5647{{c}}
* [[CWE]]: ~126/125 = 12.1212{{c}}, ~11/8 = 552.4684{{c}}


{{Optimal ET sequence|legend=1| 99e, 198, 297e, 495ce }}
{{Optimal ET sequence|legend=1| 99e, 198, 297e, 495ce }}


[[Badness]] (Smith): 0.058801
[[Badness]] (Sintel): 1.94


=== 13-limit ===
=== 13-limit ===
Line 656: Line 650:


Optimal tunings:  
Optimal tunings:  
* POTE: ~144/143 = 12.121{{c}}, ~11/8 = 552.024{{c}}
* WE: ~144/143 = 12.1170{{c}}, ~11/8 = 551.8308{{c}}
* CWE: ~144/143 = 12.1212{{c}}, ~11/8 = 551.7241{{c}}


{{Optimal ET sequence|legend=0| 99ef, 198 }}
{{Optimal ET sequence|legend=0| 99ef, 198, 693bcdefff }}


Badness (Smith): 0.042547
Badness (Sintel): 1.76


== Schisennealimmal ==
== Schisennealimmal ==
Schisennealimmal ({{nowrap| 171 & 342 }}) has a period of 1/171 octave. [[171edo]] and its multiples are members of both [[schismatic family|schismic]] and ennealimmal, and from this it derives its name.
Schisennealimmal ({{nowrap| 171 & 342 }}) has a period of 1/171 octave. It was named by [[Xenllium]] in 2021 for the fact that [[171edo]] and its multiples are members of both [[schismic]] and ennealimmal.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 670: Line 665:


{{Mapping|legend=1| 171 271 397 480 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 171 271 397 480 0 | 0 0 0 0 1 }}
: mapping generators: ~225/224, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~225/224 = 7.018{{c}}, ~11/8 = 550.954{{c}}
* [[WE]]: ~225/224 = 7.0182{{c}}, ~11/8 = 551.0022{{c}}
* [[CWE]]: ~225/224 = 7.0175{{c}}, ~11/8 = 551.0267{{c}}


{{Optimal ET sequence|legend=1| 171, 342 }}
{{Optimal ET sequence|legend=1| 171, 342 }}


[[Badness]] (Smith): 0.031739
[[Badness]] (Sintel): 1.05


=== 13-limit ===
=== 13-limit ===
Line 686: Line 683:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.322{{c}}
* WE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.3212{{c}}
* CWE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.3210{{c}}


{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342 }}


Badness (Smith): 0.054029
Badness (Sintel): 2.23


==== 17-limit ====
==== 17-limit ====
Line 700: Line 698:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.365{{c}}
* WE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.3583{{c}}
* CWE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.3578{{c}}


{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}


Badness (Smith): 0.031323
Badness (Sintel): 1.60


=== Schisennealimmic ===
=== Schisennealimmic ===
Line 714: Line 713:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.625{{c}}
* WE: ~225/224 = 7.0182{{c}}, ~11/8 = 551.6748{{c}}
* CWE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.7024{{c}}


{{Optimal ET sequence|legend=1| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=1| 171, 342f, 513 }}


Badness (Smith): 0.046843
Badness (Sintel): 1.94


==== 17-limit ====
==== 17-limit ====
Line 728: Line 728:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.756{{c}}
* WE: ~225/224 = 7.0180{{c}}, ~11/8 = 551.7893{{c}}
* CWE: ~225/224 = 7.0175{{c}}, ~11/8 = 551.7990{{c}}


{{Optimal ET sequence|legend=0| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=0| 171, 342f, 513 }}


Badness (Smith): 0.030622
Badness (Sintel): 1.56


== Lunennealimmal ==
== Lunennealimmal ==
Lunennealimmal ({{nowrap| 441 & 882 }}) has has a period of 1/441 octave. [[441edo]] and its multiples are members of both [[luna family|luna]] and ennealimmal, and from this it derives its name.
Lunennealimmal ({{nowrap| 441 & 882 }}) has has a period of 1/441 octave. It was named by [[Xenllium]] in 2021 for the fact that [[441edo]] and its multiples are members of both [[luna]] and ennealimmal.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 742: Line 743:


{{Mapping|legend=1| 441 699 1024 1238 1526 | 0 0 0 0 -1 }}
{{Mapping|legend=1| 441 699 1024 1238 1526 | 0 0 0 0 -1 }}
: mapping generators: ~32805/32768, ~11


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~32805/32768 = 2.7211{{c}}, ~11/8 = 551.3584{{c}}
* [[WE]]: ~32805/32768 = 2.7211{{c}}, ~11/8 = 551.3530{{c}}
* [[CWE]]: ~32805/32768 = 2.7211{{c}}, ~11/8 = 551.3503{{c}}


{{Optimal ET sequence|legend=1| 441, 882, 1323, 2205, 3528 }}
{{Optimal ET sequence|legend=1| 441, 882, 1323, 2205, 3528 }}


[[Badness]] (Smith): 0.091939
[[Badness]] (Sintel): 3.04


=== 13-limit ===
=== 13-limit ===
Line 758: Line 761:


Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.4043{{c}}
* WE: ~729/728 = 2.7210{{c}}, ~11/8 = 551.3928{{c}}
* CWE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.3899{{c}}


{{Optimal ET sequence|legend=0| 441, 882, 1323, 3528f, 4851ff, 6174dff }}
{{Optimal ET sequence|legend=0| 441, 882, 1323 }}


Badness (Smith): 0.042975
Badness (Sintel): 1.78


=== 17-limit ===
=== 17-limit ===
Line 772: Line 776:


Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.3688{{c}}
* WE: ~729/728 = 2.7210{{c}}, ~11/8 = 551.3572{{c}}
* CWE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.3532{{c}}
 
{{Optimal ET sequence|legend=0| 441, 882, 1323, 2205f }}
 
Badness (Sintel): 1.49
 
== Other subgroup extensions ==
=== Septiennealic (2.3.7.13) ===
Septiennealic finds a somewhat high-damage but very simple and intuitive mapping of prime 13 by fixing 13/12~14/13 at 1\9.


{{Optimal ET sequence|legend=0| 441, 882, 1323, 2205f, 3528f }}
A notable tuning of septiennealic not appearing in the optimal ET sequence is [[63edo]]. If we include a somewhat more complex mapping for 11 via {{nowrap| 36e & 63 }}, it will become the optimal patent val and largest in the sequence.


Badness (Smith): 0.029334
Subgroup: 2.3.7.13
 
Comma list: 169/168, 31213/31104
 
Subgroup-val mapping: {{mapping| 9 0 11 19 | 0 1 1 1 }}
 
Optimal tunings:
* WE: ~13/12 = 133.3847{{c}}, ~3/2 = 701.9342{{c}}
* CWE: ~13/12 = 133.3333{{c}}, ~3/2 = 702.0763{{c}}
 
{{Optimal ET sequence|legend=0| 27, 36, 99, 135f, 171f }}
 
Badness (Sintel): 0.540


[[Category:Septiennealimmal clan| ]] <!-- main article -->
[[Category:Septiennealimmal clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
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