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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:  
* 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}})


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


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:  
Line 444: Line 436:
* 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 ====
Line 453: 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:  
* 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:  
Line 466: Line 459:
{{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 =====
Line 473: 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:  
* 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 =====
Line 485: 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:  
* 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=0| 72, 198g, 270 }}


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


==== Semihemiennealimmal ====
==== Semihemiennealimmal ====
Line 497: 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


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 =====
Line 513: 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:  
* 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 527: 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:  
* 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
Line 543: 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


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 558: 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:  
* 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 ===
Line 572: 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


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 ===
Line 587: 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


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 ===
Line 606: 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


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 622: 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:  
* 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 647:


{{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 653: 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:  
* 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 680:


{{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 683: 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:  
* 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 697: 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:  
* 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 711: 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:  
* 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 725: 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:  
* 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 758:


{{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 755: 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:  
* 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 769: 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:  
* 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
 
{{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]]
Retrieved from "https://en.xen.wiki/w/Septiennealimmal_clan"