Septiennealimmal clan: Difference between revisions

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== No-five septiennealimmal ==
== No-five septiennealimmal ==
This rank-2 temperament is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the 2.3.7 subgroup, such as [[36edo]], which adds the [[1029/1024|gamelisma]], or [[63edo]], which in the 7-limit can be used for [[magic]] and in higher limits for [[parapyth]] among other things.
This rank-2 temperament simply equates a stack of nine [[7/6]] subminor thirds with two octaves. It is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the [[2.3.7 subgroup]], such as [[36edo]], which adds the [[1029/1024|gamelisma]], or [[63edo]], which in the 7-limit can be used for [[magic]] and in higher limits for [[parapyth]] among other things.


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 18: Line 18:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2592/2401 = 133.3333, ~3/2 = 702.0044
* [[WE]]: ~2592/2401 = 133.3357{{c}}, ~3/2 = 701.9772{{c}}
: [[error map]]: {{val| 0.0000 +0.0494 -0.1549 }}
: [[error map]]: {{val| +0.021 +0.043 -0.135 }}
* [[POTE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9649
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}
: error map: {{val| 0.0000 +0.0099 -0.1944 }}
: error map: {{val| 0.000 +0.028 -0.176 }}


{{Optimal ET sequence|legend=1| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}
{{Optimal ET sequence|legend=1| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}


=== Septiennealic ===
[[Badness]] (Sintel): 0.191
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 tuning]] ([[CTE]]): ~13/12 = 133.333, ~3/2 = 702.638
 
{{Optimal ET sequence|legend=1| 27, 36, 99, 135f, 171f }}
 
[[Badness]]:
* Smith: 0.0134
* Dirichlet: 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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~121/112 = 133.3333, 343/198 = 951.0143 (~99/98 = 17.6810)
* WE: ~121/112 = 133.3392{{c}}, 343/198 = 951.0013{{c}} (~99/98 = 17.6266{{c}})
* POTE: ~121/112 = 133.3333, 343/198 = 950.9591 (~99/98 = 17.6258)
* CWE: ~121/112 = 133.3333{{c}}, 343/198 = 950.9799{{c}} (~99/98 = 17.6466{{c}})


{{Optimal ET sequence|legend=0| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
{{Optimal ET sequence|legend=0| 63, 72, 135, 342, 477, 1089, 1566 }}
 
Badness (Sintel): 0.161


== Ennealimmal ==
== Ennealimmal ==
Line 67: Line 50:
: ''For the 5-limit version, see [[Ennealimma #Ennealimmal]].''
: ''For the 5-limit version, see [[Ennealimma #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).
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]]}.  


In the 5-limit, it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2), and [[ploidacot]] enneaploid dicot. While [[27/25]] is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.  
In the 5-limit, it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2), and [[ploidacot]] enneaploid dicot. While [[27/25]] is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.  


Aside from 10/9 which has already been mentioned, possible generators include [[36/35]], [[21/20]], [[6/5]], [[7/5]] and the neutral thirds pair [[49/40]][[~]][[60/49]], all of which have their own interesting advantages. Possible tunings are [[441edo|441-]], [[612edo|612-]], or [[3600edo]], though its hardly likely anyone could tell the difference.
Aside from 10/9 which has already been mentioned, possible generators include [[36/35]], [[21/20]], [[6/5]], [[7/5]] and the neutral thirds pair [[49/40]][[~]][[60/49]], all of which have their own interesting advantages. Possible tunings are [[441edo|441-]], [[612edo|612-]], or [[3600edo]], though it is hardly likely anyone could tell the difference.


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28- or 43-note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17{{frac|2|3}} to the octave mos.
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28- or 43-note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17{{frac|2|3}} to the octave mos.


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].  
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].  
7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 84: Line 65:


{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
{{Multival|legend=1| 18 27 18 1 -22 -34 }}
: mapping generators: ~27/25, ~5/3
: mapping generators: ~27/25, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~27/25 = 133.3333, ~5/3 = 884.3317 (~36/35 = 49.0016)
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}} (~36/35 = 49.0214{{c}})
: [[error map]]: {{val| 0.0000 +0.0416 +0.0146 -0.1626 }}
: [[error map]]: {{val| +0.022 +0.038 +0.009 -0.139 }}
* [[POTE]]: ~27/25 = 133.3333, ~5/3 = 884.3129 (~36/35 = 49.0205)
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}} (~36/35 = 49.0118{{c}})
: error map: {{val| 0.0000 +0.0040 -0.0418 -0.2002 }}
: error map: {{val| 0.000 +0.021 -0.016 -0.183 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 102: Line 80:
{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}


[[Badness]] (Smith): 0.003610
[[Badness]] (Sintel): 0.0914


=== 11-limit ===
=== Enneabiotic ===
The 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 114: Line 92:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4385 (~36/35 = 48.8948)
* WE: ~27/25 = 133.3229{{c}}, ~5/3 = 884.3988 (~36/35 = 48.8616{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4679 (~36/35 = 48.8654)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4596 (~36/35 = 48.8737{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}


Badness (Smith): 0.027332
Badness (Sintel): 0.904


==== 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 129: Line 122:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4285 (~36/35 = 48.9048)
* WE: ~27/25 = 133.3321{{c}}, ~5/3 = 884.4225{{c}} (~36/35 = 48.9025{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4301{{c}} (~36/35 = 48.9033{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}


Badness (Smith): 0.029404
Badness (Sintel): 1.22


===== 17-limit =====
===== 17-limit =====
Line 144: Line 137:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4110 (~36/35 = 48.9223)
* WE: ~27/25 = 133.3268{{c}}, ~5/3 = 884.3797{{c}} (~36/35 = 48.9076{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4234 (~36/35 = 48.9099)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4215{{c}} (~36/35 = 48.9119{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
Badness (Sintel): 1.44


===== 19-limit =====
===== 19-limit =====
Line 157: Line 152:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4139 (~36/35 = 48.9194)
* WE: ~27/25 = 133.3271{{c}}, ~5/3 = 884.3856{{c}} (~36/35 = 48.9040{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4270 (~36/35 = 48.9063)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4251{{c}} (~36/35 = 48.9083{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}


==== Ennealimmalis ====
Badness (Sintel): 1.25
Subgroup: 2.3.5.7.11.13


Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625
=== Ennealympic ===
 
Ennealympic ({{nowrap| 99 & 171 }}, formerly ''ennealimmia'') is an alternative extension which tempers out [[131072/130977]] (olympia).  
Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
 
Optimal tunings:
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4560 (~36/35 = 48.8773)
* CWE: ~27/25 = 133.3333, ~5/3 = 884.4745 (~36/35 = 48.8588)
 
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
 
Badness (Smith): 0.022068
 
=== 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 169:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4113 (~36/35 = 48.9220)
* WE: ~27/25 = 133.3264{{c}}, ~5/3 = 884.3631{{c}} (~36/35 = 48.9219{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4089 (~36/35 = 48.9244)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4093{{c}} (~36/35 = 48.9240{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}


Badness (Smith): 0.026463
Badness (Sintel): 0.875


==== 13-limit ====
==== 13-limit ====
Line 202: Line 184:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4055 (~36/35 = 48.9278)
* WE: ~27/25 = 133.3281{{c}}, ~5/3 = 884.3647{{c}} (~36/35 = 48.9317{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4006{{c}} (~36/35 = 48.9328{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}


Badness (Smith): 0.016607
Badness (Sintel): 0.686


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


Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
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: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.3867 (~36/35 = 48.9466)
* WE: ~27/25 = 133.3227{{c}}, ~5/3 = 884.3102{{c}} (~36/35 = 48.9486{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3808 (~36/35 = 48.9525)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3816{{c}} (~36/35 = 48.9518{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270 }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 441, 711g }}
 
Badness (Sintel): 1.04


===== 19-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
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: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.3960 (~36/35 = 48.9373)
* WE: ~27/25 = 133.3255{{c}}, ~5/3 = 884.3467{{c}} (~36/35 = 48.9320{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3985 (~36/35 = 48.9348)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3982{{c}} (~36/35 = 48.9351{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270 }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 441 }}
 
Badness (Sintel): 1.16


=== Ennealimnic ===
=== Ennealimnic ===
{{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 245: Line 234:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0697 (~36/35 = 49.2636)
* WE: ~27/25 = 133.3514{{c}}, ~5/3 = 884.0582{{c}} (~36/35 = 49.4015{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9386 (~36/35 = 49.3948)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9977{{c}} (~36/35 = 49.3357{{c}})


Tuning ranges:  
Tuning ranges:  
Line 252: Line 241:
* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]
* 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]


{{Optimal ET sequence|legend=0| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 27e, 45e, 72, 171, 243 }}


Badness (Smith): 0.020347
Badness (Sintel): 0.673
 
See also: [[Chords of ennealimnic]]


==== 13-limit ====
==== 13-limit ====
Line 266: Line 253:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0603 (~36/35 = 49.2730)
* WE: ~27/25 = 133.3467{{c}}, ~5/3 = 884.0809{{c}} (~36/35 = 49.3463{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9920 (~36/35 = 49.3414)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0160{{c}} (~36/35 = 49.3173{{c}})


Tuning ranges:  
Tuning ranges:  
Line 275: Line 262:
{{Optimal ET sequence|legend=0| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}


Badness (Smith): 0.023250
Badness (Sintel): 0.961


===== 17-limit =====
===== 17-limit =====
Line 285: Line 272:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0742 (~36/35 = 49.2591)
* WE: ~27/25 = 133.3479{{c}}, ~5/3 = 884.0943{{c}} (~36/35 = 49.3406{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9981 (~36/35 = 49.3353)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0247{{c}} (~36/35 = 49.3087{{c}})


Tuning ranges:  
Tuning ranges:  
Line 294: Line 281:
{{Optimal ET sequence|legend=0| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}


Badness (Smith): 0.014602
Badness (Sintel): 0.744


===== 19-limit =====
===== 19-limit =====
Line 304: Line 291:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0366 (~36/35 = 49.2967)
* WE: ~27/25 = 133.3562{{c}}, ~5/3 = 884.0991{{c}} (~36/35 = 49.3941{{c}})
* CWE: ~27/25 = 133.3333, ~5/3 = 883.9630 (~36/35 = 49.3703)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9630{{c}} (~36/35 = 49.3703{{c}})


{{Optimal ET sequence|legend=0| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}
Badness (Sintel): 1.18


==== Ennealim ====
==== Ennealim ====
Line 317: Line 306:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.2055 (~36/35 = 49.1278)
* WE: ~13/12 = 133.4086{{c}}, ~5/3 = 884.1245{{c}} (~36/35 = 49.7357{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8556{{c}} (~36/35 = 49.4777{{c}})


{{Optimal ET sequence|legend=0| 27e, 45ef, 72 }}
{{Optimal ET sequence|legend=0| 27e, 45ef, 72 }}


Badness (Smith): 0.020697
Badness (Sintel): 0.855


===== 17-limit =====
===== 17-limit =====
Line 332: Line 321:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1935 (~36/35 = 49.1398)
* WE: ~13/12 = 133.4072{{c}}, ~5/3 = 884.1439{{c}} (~36/35 = 49.7066{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8641{{c}} (~36/35 = 49.4692{{c}})


{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
Badness (Sintel): 0.774


===== 19-limit =====
===== 19-limit =====
Line 345: Line 336:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1388 (~36/35 = 49.1945)
* WE: ~13/12 = 133.3584{{c}}, ~5/3 = 884.1121{{c}} (~36/35 = 49.3967{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.0107{{c}} (~36/35 = 49.3226{{c}})


{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
Badness (Sintel): 0.927


=== Ennealiminal ===
=== Ennealiminal ===
Line 358: Line 351:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0925 (~36/35 = 49.2408)
* WE: ~27/25 = 133.3883{{c}}, ~5/3 = 884.1944{{c}} (~36/35 = 49.5240{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.8298 (~36/35 = 49.5036)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.8853{{c}} (~36/35 = 49.4480{{c}})


{{Optimal ET sequence|legend=0| 27, 45, 72, 171e, 243e, 315e }}
{{Optimal ET sequence|legend=0| 27, 45, 72, 171e, 243e, 315e, 873bccdeeee }}


Badness (Smith): 0.031123
Badness (Sintel): 1.03


==== 13-limit ====
==== 13-limit ====
Line 373: Line 366:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.2648 (~36/35 = 49.0685)
* WE: ~13/12 = 133.4091{{c}}, ~5/3 = 884.3500{{c}} (~36/35 = 49.5139{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.9276{{c}} (~36/35 = 49.4057{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72, 171ef, 243eff }}
{{Optimal ET sequence|legend=0| 27, 45f, 72, 171ef, 243eff }}


Badness (Smith): 0.030325
Badness (Sintel): 1.25


===== 17-limit =====
===== 17-limit =====
Line 388: Line 381:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1032 (~36/35 = 49.2301)
* WE: ~13/12 = 133.4276{{c}}, ~5/3 = 884.3160{{c}} (~36/35 = 49.6770{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.7517{{c}} (~36/35 = 49.5816{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
{{Optimal ET sequence|legend=0| 27, 45f, 72, 243effgg }}
 
Badness (Sintel): 1.26


===== 19-limit =====
===== 19-limit =====
Line 401: Line 396:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.0186 (~36/35 = 49.3147)
* WE: ~13/12 = 133.4067{{c}}, ~5/3 = 884.1374{{c}} (~36/35 = 49.7094{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.7008{{c}} (~36/35 = 49.6326{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
Badness (Sintel): 1.56


=== 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 416: Line 411:


Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
Mapping: {{mapping| 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, ~400/231 = 950.9553
* 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 426: Line 421:
* 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 438: Line 433:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* 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 448: Line 444:
{{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 458: Line 454:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* 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 470: Line 469:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* 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 ====
Line 480: Line 482:


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, ~1053/800 = 475.4727
* 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 496: Line 498:


Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
: mapping generators: ~80/77, ~1053/800


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~1053/800 = 475.4727
* 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 512: Line 513:


Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
: mapping generators: ~80/77, ~1053/800


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~1053/800 = 475.4727
* 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 530: Line 530:


Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
Mapping: {{mapping| 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, ~140/121 = 250.3367
* 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 548: Line 548:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333, ~140/121 = 250.3375
* 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 560: Line 561:


Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
Mapping: {{mapping| 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, ~25/22 = 221.0717
* 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 576: Line 577:


Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
Mapping: {{mapping| 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, ~2352/1375 = 928.8000
* 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 596: Line 597:


Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
Mapping: {{mapping| 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, ~55/32 = 937.6658 (~385/384 = 4.3325)
* WE: ~3072/3025 = 26.6668{{c}}, ~55/32 = 937.6664{{c}} (~385/384 = 4.3288{{c}})
* CWE: ~3072/3025 = 26.6667, ~55/32 = 937.6630 (~385/384 = 4.3397)
* 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 615: Line 615:


Optimal tunings:  
Optimal tunings:  
* CTE: ~66/65 = 26.6667, ~55/32 = 937.6569 (~385/384 = 4.3236)
* WE: ~66/65 = 26.6670{{c}}, ~55/32 = 937.6633{{c}} (~385/384 = 4.3172{{c}})
* CWE: ~66/65 = 26.6667, ~55/32 = 937.6515 (~385/384 = 4.3182)
* 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 630: 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, ~11/8 = 552.756
* [[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 646: Line 650:


Optimal tunings:  
Optimal tunings:  
* POTE: ~144/143 = 12.121, ~11/8 = 552.024
* 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 660: 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, ~11/8 = 550.954
* [[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 676: Line 683:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.322
* 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 690: Line 698:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.365
* 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 704: Line 713:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.625
* 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 718: Line 728:


Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.756
* 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 732: 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, ~11/8 = 551.3584
* [[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 748: Line 761:


Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211, ~11/8 = 551.4043
* 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 762: Line 776:


Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211, ~11/8 = 551.3688
* 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.
 
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| 441, 882, 1323, 2205f, 3528f }}
{{Optimal ET sequence|legend=0| 27, 36, 99, 135f, 171f }}


Badness (Smith): 0.029334
Badness (Sintel): 0.540


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

Latest revision as of 08:42, 19 May 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 septiennealimmal clan of temperaments tempers out the septimal ennealimma (monzo: [-11 -9 0 9⟩, ratio: 40353607/40310784). Primarily, this clan includes the 7-limit ennealimmal temperament and extensions of it.

Temperaments discussed elsewhere are:

No-five septiennealimmal

This rank-2 temperament simply equates a stack of nine 7/6 subminor thirds with two octaves. It is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the 2.3.7 subgroup, such as 36edo, which adds the gamelisma, or 63edo, which in the 7-limit can be used for magic and in higher limits for parapyth among other things.

Subgroup: 2.3.7

Comma list: 40353607/40310784

Subgroup-val mapping: [⟨9 0 11], ⟨0 1 1]]

mapping generators: ~2592/2401, ~3

Optimal tunings:

  • WE: ~2592/2401 = 133.3357 ¢, ~3/2 = 701.9772 ¢
error map: ⟨+0.021 +0.043 -0.135]
  • CWE: ~2592/2401 = 133.3333 ¢, ~3/2 = 701.9833 ¢
error map: ⟨0.000 +0.028 -0.176]

Optimal ET sequence: 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd

Badness (Sintel): 0.191

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Subgroup-val mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

mapping generators: ~121/112, ~343/198

Optimal tunings:

  • WE: ~121/112 = 133.3392 ¢, 343/198 = 951.0013 ¢ (~99/98 = 17.6266 ¢)
  • CWE: ~121/112 = 133.3333 ¢, 343/198 = 950.9799 ¢ (~99/98 = 17.6466 ¢)

Optimal ET sequence: 63, 72, 135, 342, 477, 1089, 1566

Badness (Sintel): 0.161

Ennealimmal

Main article: Ennealimmal
For the 5-limit version, see Ennealimma #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 {S25/S27, S49}.

In the 5-limit, it tempers out the ennealimma, [1 -27 18⟩, which leads to the identification of (27/25)9 with the octave, and gives ennealimmal a period of 1/9 octave. Its pergen is (P8/9, P5/2), and ploidacot enneaploid dicot. While 27/25 is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.

Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though it is hardly likely anyone could tell the difference.

If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28- or 43-note mos with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 172⁄3 to the octave mos.

Ennealimmal extensions discussed elsewhere include omicronbeta.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12], ⟨0 2 3 2]]

mapping generators: ~27/25, ~5/3

Optimal tunings:

  • WE: ~27/25 = 133.3357 ¢, ~5/3 = 884.3288 ¢ (~36/35 = 49.0214 ¢)
error map: ⟨+0.022 +0.038 +0.009 -0.139]
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3215 ¢ (~36/35 = 49.0118 ¢)
error map: ⟨0.000 +0.021 -0.016 -0.183]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
  • 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
  • 7- and 9-odd-limit diamond tradeoff: ~36/35 = [48.920, 49.179]

Optimal ET sequence: 27, 45, 72, 99, 171, 441, 612

Badness (Sintel): 0.0914

Enneabiotic

Enneabiotic (99e & 171e) tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma). It is catalogued as undecimal ennealimmal in Graham Breed's Temperament Finder.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 5632/5625

Mapping: [⟨9 1 1 12 -75], ⟨0 2 3 2 16]]

Optimal tunings:

  • WE: ~27/25 = 133.3229 ¢, ~5/3 = 884.3988 (~36/35 = 48.8616 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4596 (~36/35 = 48.8737 ¢)

Optimal ET sequence: 99e, 171e, 270, 909, 1179, 1449c, 1719c

Badness (Sintel): 0.904

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625

Mapping: [⟨9 1 1 12 -75 -106], ⟨0 2 3 2 16 21]]

Optimal tunings:

  • WE: ~27/25 = 133.3215 ¢, ~5/3 = 884.4027 ¢ (~36/35 = 48.8479 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4745 ¢ (~36/35 = 48.8589 ¢)

Optimal ET sequence: 99ef, 171ef, 270, 639, 909, 1179, 2088bce

Badness (Sintel): 0.912

Enneabio

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨9 1 1 12 -75 93], ⟨0 2 3 2 16 -9]]

Optimal tunings:

  • WE: ~27/25 = 133.3321 ¢, ~5/3 = 884.4225 ¢ (~36/35 = 48.9025 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4301 ¢ (~36/35 = 48.9033 ¢)

Optimal ET sequence: 99e, 171e, 270

Badness (Sintel): 1.22

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 -75 93 -3], ⟨0 2 3 2 16 -9 6]]

Optimal tunings:

  • WE: ~27/25 = 133.3268 ¢, ~5/3 = 884.3797 ¢ (~36/35 = 48.9076 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4215 ¢ (~36/35 = 48.9119 ¢)

Optimal ET sequence: 99e, 171e, 270

Badness (Sintel): 1.44

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 -75 93 -3 -48], ⟨0 2 3 2 16 -9 6 13]]

Optimal tunings:

  • WE: ~27/25 = 133.3271 ¢, ~5/3 = 884.3856 ¢ (~36/35 = 48.9040 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4251 ¢ (~36/35 = 48.9083 ¢)

Optimal ET sequence: 99e, 171e, 270

Badness (Sintel): 1.25

Ennealympic

Ennealympic (99 & 171, formerly ennealimmia) is an alternative extension which tempers out 131072/130977 (olympia).

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 131072/130977

Mapping: [⟨9 1 1 12 124], ⟨0 2 3 2 -14]]

Optimal tunings:

  • WE: ~27/25 = 133.3264 ¢, ~5/3 = 884.3631 ¢ (~36/35 = 48.9219 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4093 ¢ (~36/35 = 48.9240 ¢)

Optimal ET sequence: 99, 171, 270, 711, 981, 1251, 2232e

Badness (Sintel): 0.875

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 124 93], ⟨0 2 3 2 -14 -9]]

Optimal tunings:

  • WE: ~27/25 = 133.3281 ¢, ~5/3 = 884.3647 ¢ (~36/35 = 48.9317 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4006 ¢ (~36/35 = 48.9328 ¢)

Optimal ET sequence: 99, 171, 270, 711, 981, 1692e

Badness (Sintel): 0.686

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1225/1224, 1701/1700, 2401/2400, 4096/4095

Mapping: [⟨9 1 1 12 124 93 -3], ⟨0 2 3 2 -14 -9 6]]

Optimal tunings:

  • WE: ~27/25 = 133.3227 ¢, ~5/3 = 884.3102 ¢ (~36/35 = 48.9486 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3816 ¢ (~36/35 = 48.9518 ¢)

Optimal ET sequence: 99, 171, 270, 441, 711g

Badness (Sintel): 1.04

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1216/1215, 1225/1224, 1701/1700, 1729/1728, 2401/2400

Mapping: [⟨9 1 1 12 124 93 -3 -48], ⟨0 2 3 2 -14 -9 6 13]]

Optimal tunings:

  • WE: ~27/25 = 133.3255 ¢, ~5/3 = 884.3467 ¢ (~36/35 = 48.9320 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3982 ¢ (~36/35 = 48.9351 ¢)

Optimal ET sequence: 99, 171, 270, 441

Badness (Sintel): 1.16

Ennealimnic

Not to be confused with Ennealimmic.
See also: Chords of ennealimnic

Ennealimnic (72 & 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4375/4356

Mapping: [⟨9 1 1 12 -2], ⟨0 2 3 2 5]]

Optimal tunings:

  • WE: ~27/25 = 133.3514 ¢, ~5/3 = 884.0582 ¢ (~36/35 = 49.4015 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9977 ¢ (~36/35 = 49.3357 ¢)

Tuning ranges:

  • 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
  • 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]

Optimal ET sequence: 27e, 45e, 72, 171, 243

Badness (Sintel): 0.673

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 625/624

Mapping: [⟨9 1 1 12 -2 -33], ⟨0 2 3 2 5 10]]

Optimal tunings:

  • WE: ~27/25 = 133.3467 ¢, ~5/3 = 884.0809 ¢ (~36/35 = 49.3463 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0160 ¢ (~36/35 = 49.3173 ¢)

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
  • 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]

Optimal ET sequence: 72, 171, 243

Badness (Sintel): 0.961

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 375/374, 441/440, 595/594

Mapping: [⟨9 1 1 12 -2 -33 -3], ⟨0 2 3 2 5 10 6]]

Optimal tunings:

  • WE: ~27/25 = 133.3479 ¢, ~5/3 = 884.0943 ¢ (~36/35 = 49.3406 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0247 ¢ (~36/35 = 49.3087 ¢)

Tuning ranges:

  • 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
  • 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]

Optimal ET sequence: 72, 171, 243

Badness (Sintel): 0.744

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594

Mapping: [⟨9 1 1 12 -2 -33 -3 78], ⟨0 2 3 2 5 10 6 -6]]

Optimal tunings:

  • WE: ~27/25 = 133.3562 ¢, ~5/3 = 884.0991 ¢ (~36/35 = 49.3941 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9630 ¢ (~36/35 = 49.3703 ¢)

Optimal ET sequence: 72, 171, 243

Badness (Sintel): 1.18

Ennealim

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20], ⟨0 2 3 2 5 2]]

Optimal tunings:

  • WE: ~13/12 = 133.4086 ¢, ~5/3 = 884.1245 ¢ (~36/35 = 49.7357 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 883.8556 ¢ (~36/35 = 49.4777 ¢)

Optimal ET sequence: 27e, 45ef, 72

Badness (Sintel): 0.855

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20 -3], ⟨0 2 3 2 5 2 6]]

Optimal tunings:

  • WE: ~13/12 = 133.4072 ¢, ~5/3 = 884.1439 ¢ (~36/35 = 49.7066 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 883.8641 ¢ (~36/35 = 49.4692 ¢)

Optimal ET sequence: 27eg, 45efg, 72

Badness (Sintel): 0.774

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20 -3 25], ⟨0 2 3 2 5 2 6 2]]

Optimal tunings:

  • WE: ~13/12 = 133.3584 ¢, ~5/3 = 884.1121 ¢ (~36/35 = 49.3967 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 884.0107 ¢ (~36/35 = 49.3226 ¢)

Optimal ET sequence: 27eg, 45efg, 72

Badness (Sintel): 0.927

Ennealiminal

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 4375/4374

Mapping: [⟨9 1 1 12 51], ⟨0 2 3 2 -3]]

Optimal tunings:

  • WE: ~27/25 = 133.3883 ¢, ~5/3 = 884.1944 ¢ (~36/35 = 49.5240 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 883.8853 ¢ (~36/35 = 49.4480 ¢)

Optimal ET sequence: 27, 45, 72, 171e, 243e, 315e, 873bccdeeee

Badness (Sintel): 1.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20], ⟨0 2 3 2 -3 2]]

Optimal tunings:

  • WE: ~13/12 = 133.4091 ¢, ~5/3 = 884.3500 ¢ (~36/35 = 49.5139 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 883.9276 ¢ (~36/35 = 49.4057 ¢)

Optimal ET sequence: 27, 45f, 72, 171ef, 243eff

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20 50], ⟨0 2 3 2 -3 2 -2]]

Optimal tunings:

  • WE: ~13/12 = 133.4276 ¢, ~5/3 = 884.3160 ¢ (~36/35 = 49.6770 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 883.7517 ¢ (~36/35 = 49.5816 ¢)

Optimal ET sequence: 27, 45f, 72, 243effgg

Badness (Sintel): 1.26

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20 50 25], ⟨0 2 3 2 -3 2 -2 2]]

Optimal tunings:

  • WE: ~13/12 = 133.4067 ¢, ~5/3 = 884.1374 ¢ (~36/35 = 49.7094 ¢)
  • CWE: ~13/12 = 133.3333 ¢, ~5/3 = 883.7008 ¢ (~36/35 = 49.6326 ¢)

Optimal ET sequence: 27, 45f, 72

Badness (Sintel): 1.56

Hemiennealimmal

Hemiennealimmal (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 {(S22/S24 = S55 = S25/S27 × S99), S25/S27, S49, S33/S35 = S99}. Tempering out 9801/9800 leads to an octave split into two equal parts.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 4375/4374

Mapping: [⟨18 0 -1 22 48], ⟨0 2 3 2 1]]

mapping generators: ~80/77, ~400/231

Optimal tunings:

  • WE: ~80/77 = 66.6698 ¢, ~400/231 = 950.9982 ¢
  • CWE: ~80/77 = 66.6667 ¢, ~400/231 = 950.9736 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
  • 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]

Optimal ET sequence: 72, 198, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee

Badness (Sintel): 0.208

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨18 0 -1 22 48 -19], ⟨0 2 3 2 1 6]]

Optimal tunings:

  • WE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0838 ¢
  • CWE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0837 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
  • 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
  • 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
  • 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]

Optimal ET sequence: 72, 198, 270

Badness (Sintel): 0.517

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨18 0 -1 22 48 -19 -12], ⟨0 2 3 2 1 6 6]]

Optimal tunings:

  • WE: ~27/26 = 66.6681 ¢, ~26/15 = 951.0200 ¢
  • CWE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0063 ¢

Optimal ET sequence: 72, 198g, 270

Badness (Sintel): 0.664

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024

Mapping: [⟨18 0 -1 22 48 -19 -12 48 105], ⟨0 2 3 2 1 6 6 -2]]

Optimal tunings:

  • WE: ~27/26 = 66.6653 ¢, ~26/15 = 951.0226 ¢
  • CWE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0386 ¢

Optimal ET sequence: 72, 198g, 270

Badness (Sintel): 0.812

Semihemiennealimmal

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨18 0 -1 22 48 88], ⟨0 4 6 4 2 -3]]

mapping generators: ~80/77, ~1053/800

Optimal tunings:

  • WE: ~80/77 = 66.6702 ¢, ~1053/800 = 475.4979 ¢
  • CWE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4782 ¢

Optimal ET sequence: 126, 144, 270, 684, 954

Badness (Sintel): 0.541

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨18 0 -1 22 48 88 -119], ⟨0 4 6 4 2 -3 27]]

Optimal tunings:

  • WE: ~80/77 = 66.6698 ¢, ~1053/800 = 475.5039 ¢
  • CWE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4837 ¢

Optimal ET sequence: 270, 684g, 954, 1224, 2178ef

Badness (Sintel): 0.994

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨18 0 -1 22 48 88 -119 -2], ⟨0 4 6 4 2 -3 27 11]]

Optimal tunings:

  • WE: ~80/77 = 66.6702 ¢, ~1053/800 = 475.5078 ¢
  • CWE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4854 ¢

Optimal ET sequence: 270, 684gh, 954h, 1224, 2178efh

Badness (Sintel): 0.927

Ennealimmapine

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

Comma list: 2401/2400, 4000/3993, 4375/4374

Mapping: [⟨9 3 4 14 18], ⟨0 6 9 6 7]]

mapping generators: ~27/25, ~140/121

Optimal tunings:

  • WE: ~27/25 = 133.3264 ¢, ~140/121 = 250.3236 ¢
  • CWE: ~27/25 = 133.3333 ¢, ~140/121 = 250.3283 ¢

Optimal ET sequence: 72, …, 297e, 369, 441

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨9 3 4 14 18 -8], ⟨0 6 9 6 7 22]]

Optimal tunings:

  • WE: ~27/25 = 133.3262 ¢, ~140/121 = 250.3241 ¢
  • CWE: ~27/25 = 133.3333 ¢, ~140/121 = 250.3317 ¢

Optimal ET sequence: 72, …, 297ef, 369f, 441

Badness (Sintel): 1.08

Quadraennealimmal

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 234375/234256

Mapping: [⟨9 1 1 12 -7], ⟨0 8 12 8 23]]

mapping generators: ~27/25, ~25/22

Optimal tunings:

  • WE: ~27/25 = 133.3372 ¢, ~25/22 = 221.0781 ¢
  • CWE: ~27/25 = 133.3333 ¢, ~25/22 = 221.0746 ¢

Optimal ET sequence: 27e, …, 342, 1053, 1395, 1737

Badness (Sintel): 0.705

Trinealimmal

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 2097152/2096325

Mapping: [⟨27 1 0 34 177], ⟨0 2 3 2 -4]]

mapping generators: ~2744/2673, ~2352/1375

Optimal tunings:

  • WE: ~2744/2673 = 44.4437 ¢, ~2352/1375 = 928.7852 ¢
  • CWE: ~2744/2673 = 44.4444 ¢, ~2352/1375 = 928.7985 ¢

Optimal ET sequence: 27, 243, 270, 783, 1053, 1323

Badness (Sintel): 0.986

Rhodium

Main article: Rhodium

Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45. Thus the name is given after the 45th element.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 117440512/117406179

Mapping: [⟨45 1 -1 56 226], ⟨0 2 3 2 -2]]

mapping generators: ~3072/3025, ~55/32

Optimal tunings:

  • WE: ~3072/3025 = 26.6668 ¢, ~55/32 = 937.6664 ¢ (~385/384 = 4.3288 ¢)
  • CWE: ~3072/3025 = 26.6667 ¢, ~55/32 = 937.6630 ¢ (~385/384 = 4.3297 ¢)

Optimal ET sequence: 45, 225c, 270, 1125, 1395, 1665, 5265d

Badness (Sintel): 1.26

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨45 1 -1 56 226 272], ⟨0 2 3 2 -2 -3]]

Optimal tunings:

  • WE: ~66/65 = 26.6670 ¢, ~55/32 = 937.6633 ¢ (~385/384 = 4.3172 ¢)
  • CWE: ~66/65 = 26.6667 ¢, ~55/32 = 937.6515 ¢ (~385/384 = 4.3182 ¢)

Optimal ET sequence: 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df

Badness (Sintel): 0.936

Undecentic

Not to be confused with Undecental.

Named by Xenllium in 2021, undecentic (99 & 198) has a period of 1/99 octave.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 4375/4374

Mapping: [⟨99 157 230 278 0], ⟨0 0 0 0 1]]

mapping generators: ~126/125, ~11

Optimal tunings:

  • WE: ~126/125 = 12.1170 ¢, ~11/8 = 552.5647 ¢
  • CWE: ~126/125 = 12.1212 ¢, ~11/8 = 552.4684 ¢

Optimal ET sequence: 99e, 198, 297e, 495ce

Badness (Sintel): 1.94

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3136/3125

Mapping: [⟨99 157 230 278 0 24], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~144/143 = 12.1170 ¢, ~11/8 = 551.8308 ¢
  • CWE: ~144/143 = 12.1212 ¢, ~11/8 = 551.7241 ¢

Optimal ET sequence: 99ef, 198, 693bcdefff

Badness (Sintel): 1.76

Schisennealimmal

Schisennealimmal (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

Comma list: 2401/2400, 4375/4374, 32805/32768

Mapping: [⟨171 271 397 480 0], ⟨0 0 0 0 1]]

mapping generators: ~225/224, ~11

Optimal tunings:

  • WE: ~225/224 = 7.0182 ¢, ~11/8 = 551.0022 ¢
  • CWE: ~225/224 = 7.0175 ¢, ~11/8 = 551.0267 ¢

Optimal ET sequence: 171, 342

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 2205/2197, 2401/2400

Mapping: [⟨171 271 397 480 0 633], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~225/224 = 7.0175 ¢, ~11/8 = 551.3212 ¢
  • CWE: ~225/224 = 7.0175 ¢, ~11/8 = 551.3210 ¢

Optimal ET sequence: 171, 342

Badness (Sintel): 2.23

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 833/832, 1225/1224, 2205/2197

Mapping: [⟨171 271 397 480 0 633 699], ⟨0 0 0 0 1 0 0]]

Optimal tunings:

  • WE: ~225/224 = 7.0175 ¢, ~11/8 = 551.3583 ¢
  • CWE: ~225/224 = 7.0175 ¢, ~11/8 = 551.3578 ¢

Optimal ET sequence: 171, 342, 855ff, 1197fff

Badness (Sintel): 1.60

Schisennealimmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374, 32805/32768

Mapping: [⟨171 271 397 480 0 41], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~225/224 = 7.0182 ¢, ~11/8 = 551.6748 ¢
  • CWE: ~225/224 = 7.0175 ¢, ~11/8 = 551.7024 ¢

Optimal ET sequence: 171, 342f, 513

Badness (Sintel): 1.94

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1225/1224, 1701/1700, 2025/2023, 11271/11264

Mapping: [⟨171 271 397 480 0 41 699], ⟨0 0 0 0 1 1 0]]

Optimal tunings:

  • WE: ~225/224 = 7.0180 ¢, ~11/8 = 551.7893 ¢
  • CWE: ~225/224 = 7.0175 ¢, ~11/8 = 551.7990 ¢

Optimal ET sequence: 171, 342f, 513

Badness (Sintel): 1.56

Lunennealimmal

Lunennealimmal (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

Comma list: 2401/2400, 4375/4374, 274877906944/274658203125

Mapping: [⟨441 699 1024 1238 1526], ⟨0 0 0 0 -1]]

mapping generators: ~32805/32768, ~11

Optimal tunings:

  • WE: ~32805/32768 = 2.7211 ¢, ~11/8 = 551.3530 ¢
  • CWE: ~32805/32768 = 2.7211 ¢, ~11/8 = 551.3503 ¢

Optimal ET sequence: 441, 882, 1323, 2205, 3528

Badness (Sintel): 3.04

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4096/4095, 4375/4374, 85750/85683

Mapping: [⟨441 699 1024 1238 1526 1632], ⟨0 0 0 0 -1 0]]

Optimal tunings:

  • WE: ~729/728 = 2.7210 ¢, ~11/8 = 551.3928 ¢
  • CWE: ~729/728 = 2.7211 ¢, ~11/8 = 551.3899 ¢

Optimal ET sequence: 441, 882, 1323

Badness (Sintel): 1.78

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 4096/4095, 4375/4374, 8624/8619, 14161/14157

Mapping: [⟨441 699 1024 1238 1526 1632 1803], ⟨0 0 0 0 -1 0 -1]]

Optimal tunings:

  • WE: ~729/728 = 2.7210 ¢, ~11/8 = 551.3572 ¢
  • CWE: ~729/728 = 2.7211 ¢, ~11/8 = 551.3532 ¢

Optimal ET sequence: 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.

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 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: [⟨9 0 11 19], ⟨0 1 1 1]]

Optimal tunings:

  • WE: ~13/12 = 133.3847 ¢, ~3/2 = 701.9342 ¢
  • CWE: ~13/12 = 133.3333 ¢, ~3/2 = 702.0763 ¢

Optimal ET sequence: 27, 36, 99, 135f, 171f

Badness (Sintel): 0.540

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