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Septiennealimmal clan: Difference between revisions

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[[Comma list]]: 40353607/40310784
[[Comma list]]: 40353607/40310784


{{Mapping|legend=2| 9 0 11 | 0 1 1 }}
{{Mapping|legend=3| 9 0 11 | 0 1 1 }}
: mapping generators: ~2592/2401, ~3
: mapping generators: ~2592/2401, ~3


Line 29: Line 29:


[[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


Comma list: 41503/41472, 43923/43904
Comma list: 41503/41472, 43923/43904


Subgroup-val mapping: {{mapping| 9 0 11 24 | 0 2 2 1 }}
{{Mapping|legend=2| 9 0 11 24 | 0 2 2 1 }}
: mapping generators: ~121/112, ~343/198
: mapping generators: ~121/112, ~343/198


Line 69: Line 48:
== Ennealimmal ==
== Ennealimmal ==
{{Main| Ennealimmal }}
{{Main| Ennealimmal }}
: ''For the 5-limit version, see [[Ennealimma #Ennealimmal]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Ennealimmal]].''


Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, [[2401/2400]] and [[4375/4374]], leading to a temperament of unusual [[efficiency]]. It also tempers out the [[landscape comma]], which is (2401/2400)/(4375/4374), and the [[wizma]], which is (2401/2400)⋅(4375/4374). 7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}.  
Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, [[2401/2400]] and [[4375/4374]], leading to a temperament of unusual [[efficiency]]. It also tempers out the [[landscape comma]], which is (2401/2400)/(4375/4374), and the [[wizma]], which is (2401/2400)⋅(4375/4374). 7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}.  
Line 103: Line 82:
[[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 110: Line 89:
Comma list: 2401/2400, 4375/4374, 5632/5625
Comma list: 2401/2400, 4375/4374, 5632/5625


Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
{{Mapping|legend=0| 9 1 1 12 -75 | 0 2 3 2 16 }}


Optimal tunings:  
Optimal tunings:  
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|legend=0| 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
===== 2.3.5.7.11.13.19 subgroup =====
Subgroup: 2.3.5.7.11.13.19
Comma list: 1216/1215, 1540/1539, 2080/2079, 2376/2375, 2401/2400
{{Mapping|legend=2| 9 1 1 12 -75 -106 -48 | 0 2 3 2 16 21 13 }}
Optimal tunings:
* WE: ~27/25 = 133.3209{{c}}, ~5/3 = 884.3968{{c}} (~36/35 = 48.8494{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4722{{c}} (~36/35 = 48.8612{{c}})
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639h, 909h, 1179h }}
Badness (Sintel): 0.684
==== Enneabio ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374


Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
{{Mapping|legend=0| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 140: Line 149:
Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374
Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374


Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
{{Mapping|legend=0| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}


Optimal tunings:  
Optimal tunings:  
Line 155: Line 164:
Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374
Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374


Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
{{Mapping|legend=0| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}


Optimal tunings:  
Optimal tunings:  
Line 165: Line 174:
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
Line 187: Line 181:
Comma list: 2401/2400, 4375/4374, 131072/130977
Comma list: 2401/2400, 4375/4374, 131072/130977


Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
{{Mapping|legend=0| 9 1 1 12 124 | 0 2 3 2 -14 }}


Optimal tunings:  
Optimal tunings:  
Line 202: Line 196:
Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374
Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374


Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
{{Mapping|legend=0| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}


Optimal tunings:  
Optimal tunings:  
Line 217: Line 211:
Comma list: 936/935, 1225/1224, 1701/1700, 2401/2400, 4096/4095
Comma list: 936/935, 1225/1224, 1701/1700, 2401/2400, 4096/4095


Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
{{Mapping|legend=0| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}


Optimal tunings:  
Optimal tunings:  
Line 232: Line 226:
Comma list: 936/935, 1216/1215, 1225/1224, 1701/1700, 1729/1728, 2401/2400
Comma list: 936/935, 1216/1215, 1225/1224, 1701/1700, 1729/1728, 2401/2400


Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
{{Mapping|legend=0| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}


Optimal tunings:  
Optimal tunings:  
Line 244: Line 238:
=== 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.
Line 251: Line 246:
Comma list: 243/242, 441/440, 4375/4356
Comma list: 243/242, 441/440, 4375/4356


Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
{{Mapping|legend=0| 9 1 1 12 -2 | 0 2 3 2 5 }}


Optimal tunings:  
Optimal tunings:  
Line 264: Line 259:


Badness (Sintel): 0.673
Badness (Sintel): 0.673
See also: [[Chords of ennealimnic]]


==== 13-limit ====
==== 13-limit ====
Line 272: Line 265:
Comma list: 243/242, 364/363, 441/440, 625/624
Comma list: 243/242, 364/363, 441/440, 625/624


Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
{{Mapping|legend=0| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}


Optimal tunings:  
Optimal tunings:  
Line 291: Line 284:
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594
Comma list: 243/242, 364/363, 375/374, 441/440, 595/594


Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
{{Mapping|legend=0| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}


Optimal tunings:  
Optimal tunings:  
Line 310: Line 303:
Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594
Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594


Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}
{{Mapping|legend=0| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 325: Line 318:
Comma list: 169/168, 243/242, 325/324, 441/440
Comma list: 169/168, 243/242, 325/324, 441/440


Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
{{Mapping|legend=0| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}


Optimal tunings:  
Optimal tunings:  
Line 340: Line 333:
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440


Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
{{Mapping|legend=0| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}


Optimal tunings:  
Optimal tunings:  
Line 355: Line 348:
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440
Comma list: 169/168, 221/220, 243/242, 325/324, 441/440


Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
{{Mapping|legend=0| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}


Optimal tunings:  
Optimal tunings:  
Line 370: Line 363:
Comma list: 385/384, 1375/1372, 4375/4374
Comma list: 385/384, 1375/1372, 4375/4374


Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
{{Mapping|legend=0| 9 1 1 12 51 | 0 2 3 2 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 385: Line 378:
Comma list: 169/168, 325/324, 385/384, 1375/1372
Comma list: 169/168, 325/324, 385/384, 1375/1372


Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
{{Mapping|legend=0| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}


Optimal tunings:  
Optimal tunings:  
Line 400: Line 393:
Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372
Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372


Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
{{Mapping|legend=0| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 415: Line 408:
Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372
Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372


Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
{{Mapping|legend=0| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}


Optimal tunings:  
Optimal tunings:  
Line 426: Line 419:


=== 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 434: Line 425:
Comma list: 2401/2400, 3025/3024, 4375/4374
Comma list: 2401/2400, 3025/3024, 4375/4374


Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
{{Mapping|legend=0| 18 0 -1 22 48 | 0 2 3 2 1 }}
: mapping generators: ~80/77, ~400/231
: mapping generators: ~80/77, ~400/231


Line 454: Line 445:
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024
Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024


Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
{{Mapping|legend=0| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}


Optimal tunings:  
Optimal tunings:  
Line 475: Line 466:
Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024
Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024


Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
{{Mapping|legend=0| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}


Optimal tunings:  
Optimal tunings:  
Line 490: Line 481:
Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024
Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024


Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}
{{Mapping|legend=0| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 496: Line 487:
* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0386{{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
Badness (Sintel): 0.812
Line 505: Line 496:
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374


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


Line 521: Line 512:
Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374
Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374


Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
{{Mapping|legend=0| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}


Optimal tunings:  
Optimal tunings:  
Line 536: Line 527:
Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374
Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374


Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
{{Mapping|legend=0| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}


Optimal tunings:  
Optimal tunings:  
Line 546: Line 537:
Badness (Sintel): 0.927
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
Line 553: Line 544:
Comma list: 2401/2400, 4000/3993, 4375/4374
Comma list: 2401/2400, 4000/3993, 4375/4374


Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
{{Mapping|legend=0| 9 3 4 14 18 | 0 6 9 6 7 }}
: mapping generators: ~27/25, ~140/121
: mapping generators: ~27/25, ~140/121


Line 569: Line 560:
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374
Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374


Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
{{Mapping|legend=0| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}


Optimal tunings:  
Optimal tunings:  
Line 584: Line 575:
Comma list: 2401/2400, 4375/4374, 234375/234256
Comma list: 2401/2400, 4375/4374, 234375/234256


Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
{{Mapping|legend=0| 9 1 1 12 -7 | 0 8 12 8 23 }}
: mapping generators: ~27/25, ~25/22
: mapping generators: ~27/25, ~25/22


Line 600: Line 591:
Comma list: 2401/2400, 4375/4374, 2097152/2096325
Comma list: 2401/2400, 4375/4374, 2097152/2096325


Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
{{Mapping|legend=0| 27 1 0 34 177 | 0 2 3 2 -4 }}
: mapping generators: ~2744/2673, ~2352/1375
: mapping generators: ~2744/2673, ~2352/1375


Line 620: Line 611:
Comma list: 2401/2400, 4375/4374, 117440512/117406179
Comma list: 2401/2400, 4375/4374, 117440512/117406179


Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
{{Mapping|legend=0| 45 1 -1 56 226 | 0 2 3 2 -2 }}
: mapping generators: ~3072/3025, ~55/32
: mapping generators: ~3072/3025, ~55/32


Line 636: Line 627:
Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655
Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655


Mapping: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}
{{Mapping|legend=0| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 671: Line 662:
Comma list: 352/351, 847/845, 2401/2400, 3136/3125
Comma list: 352/351, 847/845, 2401/2400, 3136/3125


Mapping: {{mapping| 99 157 230 278 0 24 | 0 0 0 0 1 1 }}
{{Mapping|legend=0| 99 157 230 278 0 24 | 0 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 704: Line 695:
Comma list: 625/624, 729/728, 2205/2197, 2401/2400
Comma list: 625/624, 729/728, 2205/2197, 2401/2400


Mapping: {{mapping| 171 271 397 480 0 633 | 0 0 0 0 1 0 }}
{{Mapping|legend=0| 171 271 397 480 0 633 | 0 0 0 0 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 719: Line 710:
Comma list: 625/624, 729/728, 833/832, 1225/1224, 2205/2197
Comma list: 625/624, 729/728, 833/832, 1225/1224, 2205/2197


Mapping: {{mapping| 171 271 397 480 0 633 699 | 0 0 0 0 1 0 0 }}
{{Mapping|legend=0| 171 271 397 480 0 633 699 | 0 0 0 0 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
Line 734: Line 725:
Comma list: 2080/2079, 2401/2400, 4375/4374, 32805/32768
Comma list: 2080/2079, 2401/2400, 4375/4374, 32805/32768


Mapping: {{mapping| 171 271 397 480 0 41 | 0 0 0 0 1 1 }}
{{Mapping|legend=0| 171 271 397 480 0 41 | 0 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 749: Line 740:
Comma list: 936/935, 1225/1224, 1701/1700, 2025/2023, 11271/11264
Comma list: 936/935, 1225/1224, 1701/1700, 2025/2023, 11271/11264


Mapping: {{mapping| 171 271 397 480 0 41 699 | 0 0 0 0 1 1 0 }}
{{Mapping|legend=0| 171 271 397 480 0 41 699 | 0 0 0 0 1 1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 782: Line 773:
Comma list: 2401/2400, 4096/4095, 4375/4374, 85750/85683
Comma list: 2401/2400, 4096/4095, 4375/4374, 85750/85683


Mapping: {{mapping| 441 699 1024 1238 1526 1632 | 0 0 0 0 -1 0 }}
{{Mapping|legend=0| 441 699 1024 1238 1526 1632 | 0 0 0 0 -1 0 }}


Optimal tunings:  
Optimal tunings:  
Line 797: Line 788:
Comma list: 2401/2400, 4096/4095, 4375/4374, 8624/8619, 14161/14157
Comma list: 2401/2400, 4096/4095, 4375/4374, 8624/8619, 14161/14157


Mapping: {{mapping| 441 699 1024 1238 1526 1632 1803 | 0 0 0 0 -1 0 -1 }}
{{Mapping|legend=0| 441 699 1024 1238 1526 1632 1803 | 0 0 0 0 -1 0 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 806: Line 797:


Badness (Sintel): 1.49
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.
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
{{Mapping|legend=2| 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]]
[[Category:Catalogs of rank-2 temperaments]]
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