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Septiennealimmal clan: Difference between revisions

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{{Technical data page}}
The '''septiennealimmal clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septimal ennealimma]] ({{monzo|legend=1| -11 -9 0 9 }}, [[ratio]]: 40353607/40310784). Primarily, this clan includes the 7-limit [[ennealimmal]] temperament and extensions of it.
The '''septiennealimmal clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[septimal ennealimma]] ({{monzo|legend=1| -11 -9 0 9 }}, [[ratio]]: 40353607/40310784). Primarily, this clan includes the 7-limit [[ennealimmal]] temperament and extensions of it.


Line 6: Line 7:
* ''[[Enneaportent]]'' → [[Marvel temperaments #Enneaportent]]
* ''[[Enneaportent]]'' → [[Marvel temperaments #Enneaportent]]
* ''[[Novemkleismic]]'' → [[Kleismic family #Novemkleismic]]
* ''[[Novemkleismic]]'' → [[Kleismic family #Novemkleismic]]
* ''[[Decades]]'' → [[Compton family #Decades]]
* ''[[Gamelstearn]]'' → [[Compton family #Gamelstearn]]
* ''[[Nonant]]'' → [[Schismatic family #Nonant]]
* ''[[Nonant]]'' → [[Schismatic family #Nonant]]


== 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 17: 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 = 133.33... = 1\9, ~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 = 133.33... = 1\9, ~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/intuitive mapping of prime 13 via explaining ~7/6 = 2\9 as a consequence of ~14/13~13/12 = 1\9.
 
A notable tuning of septiennealic not appearing in the optimal ET sequence is [[63edo]].
 
[[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 = 133.33... = 1\9, ~3/2 = 702.638
 
{{Optimal ET sequence|legend=1| 27, 36, 99, 135f, 171f }}
 
[[Badness]] (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 = 133.333, 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 = 133.333, 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| 63, 72, 135, 342, 477, 1089, 1566 }}


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


== Ennealimmal ==
== Ennealimmal ==
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: ''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 81: 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 99: 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 111: 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 126: 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 =====
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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 =====
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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


Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
=== Ennealympic ===
 
Ennealympic ({{nowrap| 99 & 171 }}, formerly ''ennealimmia'') is an alternative extension which tempers out [[131072/130977]] (olympia).  
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 184: 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 199: 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 242: 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 249: 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 263: 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 272: 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 282: 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 291: 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 301: 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 314: 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 329: 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 342: 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 355: 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 370: 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 385: 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, 243effgg }}


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


===== 19-limit =====
===== 19-limit =====
Line 398: 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 413: 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 423: 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 435: 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 445: 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 455: 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 467: 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 477: 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 493: 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 509: 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 527: 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 545: 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 557: 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 573: 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 593: 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 612: 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 627: 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 643: 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 657: 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 673: 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 687: 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 701: 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 715: 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 729: 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 745: 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 759: 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]]
Retrieved from "https://en.xen.wiki/w/Septiennealimmal_clan"