Septiennealimmal clan: Difference between revisions

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The '''tritrizo clan''' of temperaments tempers out the tritrizo comma (no-five ennealimma), {{monzo| -11 -9 0 9 }} = [[40353607/40310784]], and includes these:
{{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.
 
Temperaments discussed elsewhere are:
* ''[[Cobalt]]'' → [[Starling temperaments #Cobalt]]
* ''[[Cobalt]]'' → [[Starling temperaments #Cobalt]]
* ''[[Niner]]'' → [[Augmented family #Niner]]
* ''[[Niner]]'' → [[Augmented family #Niner]]
* ''[[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]]


Primarily, this clan includes the 7-limit [[ennealimmal]] temperament and extensions of it.
== 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 [[1029/1024|gamelisma]], or [[63edo]], which in the 7-limit can be used for [[magic]] and in higher limits for [[parapyth]] among other things.


== No-five tritrizo ==
[[Subgroup]]: 2.3.7
Subgroup: 2.3.7


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


[[Sval]] [[mapping]]: [{{val| 9 0 11 }}, {{val| 0 1 1 }}]
{{Mapping|legend=2| 9 0 11 | 0 1 1 }}
: mapping generators: ~2592/2401, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3357{{c}}, ~3/2 = 701.9772{{c}}
: [[error map]]: {{val| +0.021 +0.043 -0.135 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}
: error map: {{val| 0.000 +0.028 -0.176 }}


[[POTE generator]]: ~3/2 = 701.965
{{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 }}
[[Badness]] (Sintel): 0.191


=== 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


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


[[POTE generator]]: ~99/98 = 17.626
Optimal tunings:  
* WE: ~121/112 = 133.3392{{c}}, 343/198 = 951.0013{{c}} (~99/98 = 17.6266{{c}})
* CWE: ~121/112 = 133.3333{{c}}, 343/198 = 950.9799{{c}} (~99/98 = 17.6466{{c}})


{{Optimal ET sequence|legend=1| 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 ==
{{Main| Ennealimmal }}
{{Main| 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 [[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). 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.  
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]]}.  


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 its hardly likely anyone could tell the difference.
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.  


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 2/3 to the octave mos.
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.


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]], [[Tritrizo clan #Undecentic|undecentic]], [[Tritrizo clan #Schisennealimmal|schisennealimmal]], and [[Tritrizo clan #Lunennealimmal|lunennealimmal]].
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.


7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. Interestingly, the [[landscape comma]] is equal to [[2401/2400|S49]]/([[4375/4374|S25/S27]]) while the [[wizma]] is equal to [[2401/2400|S49]]*[[4375/4374|S25/S27]].
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].  
 
''For the 5-limit temperament, see [[Ennealimma#Ennealimmal]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 53: 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]] ([[POTE]]): ~27/25 = 1\9, ~5/3 = 884.3129 (~36/35 = 49.0205)
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}} (~36/35 = 49.0214{{c}})
: [[error map]]: {{val| +0.022 +0.038 +0.009 -0.139 }}
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}} (~36/35 = 49.0118{{c}})
: error map: {{val| 0.000 +0.021 -0.016 -0.183 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 64: Line 77:
* 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
* 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]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~36/35 = [48.920, 49.179]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~36/35 = [48.920, 49.179]


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


=== 11-limit ===
=== Enneabiotic ===
The ennealimmal temperament can be described as 99e &amp; 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 79: Line 91:
Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4679 (~36/35 = 48.8654)
Optimal tunings:
* WE: ~27/25 = 133.3229{{c}}, ~5/3 = 884.3988 (~36/35 = 48.8616{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4596 (~36/35 = 48.8737{{c}})


{{Optimal ET sequence|legend=1| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}


Badness: 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 92: Line 121:
Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tunings:
* WE: ~27/25 = 133.3321{{c}}, ~5/3 = 884.4225{{c}} (~36/35 = 48.9025{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4301{{c}} (~36/35 = 48.9033{{c}})


{{Optimal ET sequence|legend=1| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}


Badness: 0.029404
Badness (Sintel): 1.22


===== 17-limit =====
===== 17-limit =====
Line 105: Line 136:
Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
Mapping: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tunings:
* WE: ~27/25 = 133.3268{{c}}, ~5/3 = 884.3797{{c}} (~36/35 = 48.9076{{c}})
* 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=1| 99e, 171e, 270 }}
Badness (Sintel): 1.44


===== 19-limit =====
===== 19-limit =====
Line 116: Line 151:
Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
Mapping: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4304 (~36/35 = 48.9030)
Optimal tunings:
* WE: ~27/25 = 133.3271{{c}}, ~5/3 = 884.3856{{c}} (~36/35 = 48.9040{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4251{{c}} (~36/35 = 48.9083{{c}})


{{Optimal ET sequence|legend=1| 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 tuning (CTE): ~27/25 = 1\9, ~5/3 = 884.4560 (~36/35 = 48.8773)
 
{{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
 
Badness: 0.022068
 
=== Ennealimmia ===
The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 142: Line 168:
Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
Mapping: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.4089 (~36/35 = 48.9244)
Optimal tunings:
* WE: ~27/25 = 133.3264{{c}}, ~5/3 = 884.3631{{c}} (~36/35 = 48.9219{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4093{{c}} (~36/35 = 48.9240{{c}})


{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}


Badness: 0.026463
Badness (Sintel): 0.875


==== 13-limit ====
==== 13-limit ====
Line 155: Line 183:
Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
Mapping: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tunings:
* WE: ~27/25 = 133.3281{{c}}, ~5/3 = 884.3647{{c}} (~36/35 = 48.9317{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4006{{c}} (~36/35 = 48.9328{{c}})


{{Optimal ET sequence|legend=1| 99, 171, 270, 711, 981, 1692e, 2673e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}


Badness: 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 tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tunings:
* WE: ~27/25 = 133.3227{{c}}, ~5/3 = 884.3102{{c}} (~36/35 = 48.9486{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3816{{c}} (~36/35 = 48.9518{{c}})
 
{{Optimal ET sequence|legend=0| 99, 171, 270, 441, 711g }}


{{Optimal ET sequence|legend=1| 99, 171, 270 }}
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 tuning (POTE): ~27/25 = 1\9, ~5/3 = 884.3997 (~36/35 = 48.9336)
Optimal tunings:
* WE: ~27/25 = 133.3255{{c}}, ~5/3 = 884.3467{{c}} (~36/35 = 48.9320{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3982{{c}} (~36/35 = 48.9351{{c}})
 
{{Optimal ET sequence|legend=0| 99, 171, 270, 441 }}


{{Optimal ET sequence|legend=1| 99, 171, 270 }}
Badness (Sintel): 1.16


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 192: Line 233:
Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)
Optimal tunings:
* WE: ~27/25 = 133.3514{{c}}, ~5/3 = 884.0582{{c}} (~36/35 = 49.4015{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9977{{c}} (~36/35 = 49.3357{{c}})


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


{{Optimal ET sequence|legend=1| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 27e, 45e, 72, 171, 243 }}


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


==== 13-limit ====
==== 13-limit ====
Line 212: Line 252:
Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
Mapping: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9920 (~36/35 = 49.3414)
Optimal tunings:
* WE: ~27/25 = 133.3467{{c}}, ~5/3 = 884.0809{{c}} (~36/35 = 49.3463{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0160{{c}} (~36/35 = 49.3173{{c}})


Tuning ranges:  
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 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]
* 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~36/35 = [48.825, 50.000]


{{Optimal ET sequence|legend=1| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}


Badness: 0.023250
Badness (Sintel): 0.961


===== 17-limit =====
===== 17-limit =====
Line 230: Line 271:
Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
Mapping: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9981 (~36/35 = 49.3353)
Optimal tunings:
* WE: ~27/25 = 133.3479{{c}}, ~5/3 = 884.0943{{c}} (~36/35 = 49.3406{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0247{{c}} (~36/35 = 49.3087{{c}})


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


{{Optimal ET sequence|legend=1| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}


Badness: 0.014602
Badness (Sintel): 0.744


===== 19-limit =====
===== 19-limit =====
Line 248: Line 290:
Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}
Mapping: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}


{{Optimal ET sequence|legend=1| 72, 171, 243 }}
Optimal tunings:
* WE: ~27/25 = 133.3562{{c}}, ~5/3 = 884.0991{{c}} (~36/35 = 49.3941{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9630{{c}} (~36/35 = 49.3703{{c}})
 
{{Optimal ET sequence|legend=0| 72, 171, 243 }}
 
Badness (Sintel): 1.18


==== Ennealim ====
==== Ennealim ====
Line 257: Line 305:
Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tunings:
* WE: ~13/12 = 133.4086{{c}}, ~5/3 = 884.1245{{c}} (~36/35 = 49.7357{{c}})
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8556{{c}} (~36/35 = 49.4777{{c}})


{{Optimal ET sequence|legend=1| 27e, 45ef, 72 }}
{{Optimal ET sequence|legend=0| 27e, 45ef, 72 }}


Badness: 0.020697
Badness (Sintel): 0.855


===== 17-limit =====
===== 17-limit =====
Line 270: Line 320:
Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
Mapping: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tunings:
* WE: ~13/12 = 133.4072{{c}}, ~5/3 = 884.1439{{c}} (~36/35 = 49.7066{{c}})
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8641{{c}} (~36/35 = 49.4692{{c}})


{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
 
Badness (Sintel): 0.774


===== 19-limit =====
===== 19-limit =====
Line 281: Line 335:
Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
Mapping: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.6257 (~36/35 = 49.7076)
Optimal tunings:
* WE: ~13/12 = 133.3584{{c}}, ~5/3 = 884.1121{{c}} (~36/35 = 49.3967{{c}})
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.0107{{c}} (~36/35 = 49.3226{{c}})


{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
 
Badness (Sintel): 0.927


=== Ennealiminal ===
=== Ennealiminal ===
Line 292: Line 350:
Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.8298 (~36/35 = 49.5036)
Optimal tunings:
* WE: ~27/25 = 133.3883{{c}}, ~5/3 = 884.1944{{c}} (~36/35 = 49.5240{{c}})
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.8853{{c}} (~36/35 = 49.4480{{c}})


{{Optimal ET sequence|legend=1| 27, 45, 72, 171e, 243e, 315e }}
{{Optimal ET sequence|legend=0| 27, 45, 72, 171e, 243e, 315e, 873bccdeeee }}


Badness: 0.031123
Badness (Sintel): 1.03


==== 13-limit ====
==== 13-limit ====
Line 305: Line 365:
Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
Mapping: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tunings:
* WE: ~13/12 = 133.4091{{c}}, ~5/3 = 884.3500{{c}} (~36/35 = 49.5139{{c}})
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.9276{{c}} (~36/35 = 49.4057{{c}})


{{Optimal ET sequence|legend=1| 27, 45f, 72, 171ef, 243eff }}
{{Optimal ET sequence|legend=0| 27, 45f, 72, 171ef, 243eff }}


Badness: 0.030325
Badness (Sintel): 1.25


===== 17-limit =====
===== 17-limit =====
Line 318: Line 380:
Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
Mapping: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tunings:
* WE: ~13/12 = 133.4276{{c}}, ~5/3 = 884.3160{{c}} (~36/35 = 49.6770{{c}})
* 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=1| 27, 45f, 72 }}
Badness (Sintel): 1.26


===== 19-limit =====
===== 19-limit =====
Line 329: Line 395:
Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
Mapping: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}


Optimal tuning (POTE): ~13/12 = 1\9, ~5/3 = 883.8476 (~36/35 = 49.4857)
Optimal tunings:
* WE: ~13/12 = 133.4067{{c}}, ~5/3 = 884.1374{{c}} (~36/35 = 49.7094{{c}})
* CWE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.7008{{c}} (~36/35 = 49.6326{{c}})


{{Optimal ET sequence|legend=1| 27, 45f, 72 }}
{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
 
Badness (Sintel): 1.56


=== Hemiennealimmal ===
=== Hemiennealimmal ===
Hemiennealimmal (72 &amp; 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 343: 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 tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553
Optimal tunings:
* 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:  
* 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
* 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]
* 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]
* 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]


{{Optimal ET sequence|legend=1| 72, 198, 270, 342, 612, 954, 1566 }}
{{Optimal ET sequence|legend=0| 72, 198, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee }}


Badness: 0.006283
Badness (Sintel): 0.208


==== 13-limit ====
==== 13-limit ====
Line 364: Line 432:
Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
Mapping: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}


Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
Optimal tunings:
* 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 371: Line 441:
* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
* 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
* 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]
* 13-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.309]
* 15-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 18.926]


{{Optimal ET sequence|legend=1| 72, 198, 270 }}
{{Optimal ET sequence|legend=0| 72, 198, 270 }}


Badness: 0.012505
Badness (Sintel): 0.517


===== 17-limit =====
===== 17-limit =====
Line 385: Line 453:
Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
Mapping: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}


Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
Optimal tunings:
* 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=1| 72, 198g, 270 }}
Badness (Sintel): 0.664


===== 19-limit =====
===== 19-limit =====
Line 396: Line 468:
Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}
Mapping: {{mapping| 18 0 -1 22 48 -19 -12 48 105 | 0 2 3 2 1 6 6 -2 }}


Optimal tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837
Optimal tunings:
* WE: ~27/26 = 66.6653{{c}}, ~26/15 = 951.0226{{c}}
* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0386{{c}}
 
{{Optimal ET sequence|legend=0| 72, 198g, 270 }}


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


==== Semihemiennealimmal ====
==== Semihemiennealimmal ====
Line 406: 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 tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
Optimal tunings:
* 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=1| 126, 144, 270, 684, 954 }}
{{Optimal ET sequence|legend=0| 126, 144, 270, 684, 954 }}


Badness: 0.013104
Badness (Sintel): 0.541


===== 17-limit =====
===== 17-limit =====
Line 422: Line 499:
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:  
* WE: ~80/77 = 66.6698{{c}}, ~1053/800 = 475.5039{{c}}
* CWE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4837{{c}}


Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
{{Optimal ET sequence|legend=0| 270, 684g, 954, 1224, 2178ef }}


{{Optimal ET sequence|legend=1| 270, 684, 954 }}
Badness (Sintel): 0.994
 
Badness: 0.013104


===== 19-limit =====
===== 19-limit =====
Line 437: Line 514:
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:  
 
* WE: ~80/77 = 66.6702{{c}}, ~1053/800 = 475.5078{{c}}
Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727
* CWE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4854{{c}}


{{Optimal ET sequence|legend=1| 270, 684h, 954h, 1224 }}
{{Optimal ET sequence|legend=0| 270, 684gh, 954h, 1224, 2178efh }}


Badness: 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 453: 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 tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367
Optimal tunings:
* 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=1| 72, 369, 441 }}
{{Optimal ET sequence|legend=0| 72, …, 297e, 369, 441 }}


Badness: 0.034196
Badness (Sintel): 1.13


==== 13-limit ====
==== 13-limit ====
Line 469: Line 547:
Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
Mapping: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}


Optimal tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375
Optimal tunings:
* 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=1| 72, 297ef, 369f, 441 }}
{{Optimal ET sequence|legend=0| 72, …, 297ef, 369f, 441 }}


Badness: 0.026122
Badness (Sintel): 1.08


=== Quadraennealimmal ===
=== Quadraennealimmal ===
Line 481: 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 tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717
Optimal tunings:
* 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=1| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
{{Optimal ET sequence|legend=0| 27e, …, 342, 1053, 1395, 1737 }}


Badness: 0.021320
Badness (Sintel): 0.705


=== Trinealimmal ===
=== Trinealimmal ===
Line 496: 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 tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000
Optimal tunings:
* 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=1| 27, 243, 270, 783, 1053, 1323 }}
{{Optimal ET sequence|legend=0| 27, 243, 270, 783, 1053, 1323 }}


Badness: 0.029812
Badness (Sintel): 0.986


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 514: 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 = 1\45, ~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 = 1\45, ~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: {{Optimal ET sequence| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
{{Optimal ET sequence|legend=0| 45, 225c, 270, 1125, 1395, 1665, 5265d }}


Badness: 0.0381
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 533: Line 615:


Optimal tunings:  
Optimal tunings:  
* CTE: ~66/65 = 1\45, ~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 = 1\45, ~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: {{Optimal ET sequence| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
Badness (Sintel): 0.936


Badness: 0.0226
== Undecentic ==
{{Distinguish| Undecental }}


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


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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


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


[[POTE generator]]: ~11/8 = 552.756
[[Optimal tuning]]s:  
* [[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]]: 0.058801
[[Badness]] (Sintel): 1.94


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val|99 157 230 278 0 24}}, {{val|0 0 0 0 1 1}}]
Mapping: {{mapping| 99 157 230 278 0 24 | 0 0 0 0 1 1 }}


POTE generator: ~11/8 = 552.024
Optimal tunings:  
* 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=1| 99ef, 198 }}
{{Optimal ET sequence|legend=0| 99ef, 198, 693bcdefff }}


Badness: 0.042547
Badness (Sintel): 1.76


=== Schisennealimmal ===
== Schisennealimmal ==
Schisennealimmal (171&amp;342) has a period of 1/171 octave. [[171edo|171EDO]] and its multiples are members of both [[Schismatic family|schismic]] and [[Ragismic microtemperaments #Ennealimmal|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


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


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


[[POTE generator]]: ~11/8 = 550.954
[[Optimal tuning]]s:  
* [[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]]: 0.031739
[[Badness]] (Sintel): 1.05


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 171 271 397 480 0 633 }}, {{val| 0 0 0 0 1 0 }}]
Mapping: {{mapping| 171 271 397 480 0 633 | 0 0 0 0 1 0 }}


POTE generator: ~11/8 = 551.322
Optimal tunings:  
* 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=1| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342 }}


Badness: 0.054029
Badness (Sintel): 2.23


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


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


Mapping: [{{val| 171 271 397 480 0 633 699 }}, {{val| 0 0 0 0 1 0 0 }}]
Mapping: {{mapping| 171 271 397 480 0 633 699 | 0 0 0 0 1 0 0 }}


POTE generator: ~11/8 = 551.365
Optimal tunings:  
* 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=1| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}


Badness: 0.031323
Badness (Sintel): 1.60


==== Schisennealimmic ====
=== Schisennealimmic ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 171 271 397 480 0 41 }}, {{val| 0 0 0 0 1 1 }}]
Mapping: {{mapping| 171 271 397 480 0 41 | 0 0 0 0 1 1 }}


POTE generator: ~11/8 = 551.625
Optimal tunings:  
* 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: 0.046843
Badness (Sintel): 1.94


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


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


Mapping: [{{val|171 271 397 480 0 41 699 }}, {{val|0 0 0 0 1 1 0 }}]
Mapping: {{mapping| 171 271 397 480 0 41 699 | 0 0 0 0 1 1 0 }}


POTE generator: ~11/8 = 551.756
Optimal tunings:  
* 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=1| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=0| 171, 342f, 513 }}


Badness: 0.030622
Badness (Sintel): 1.56


=== Lunennealimmal ===
== Lunennealimmal ==
Lunennealimmal (441&amp;882) has has a period of 1/441 octave. [[441edo|441EDO]] and its multiples are members of both [[Luna family|luna]] and [[Ragismic microtemperaments #Ennealimmal|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


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


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


[[POTE generator]]: ~11/8 = 551.3584
[[Optimal tuning]]s:  
* [[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]]: 0.091939
[[Badness]] (Sintel): 3.04


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val|441 699 1024 1238 1526 1632}}, {{val|0 0 0 0 -1 0}}]
Mapping: {{mapping| 441 699 1024 1238 1526 1632 | 0 0 0 0 -1 0 }}


POTE generator: ~11/8 = 551.4043
Optimal tunings:  
* 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=1| 441, 882, 1323, 3528f, 4851ff, 6174dff }}
{{Optimal ET sequence|legend=0| 441, 882, 1323 }}


Badness: 0.042975
Badness (Sintel): 1.78


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


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


Mapping: [{{val|441 699 1024 1238 1526 1632 1803}}, {{val|0 0 0 0 -1 0 -1}}]
Mapping: {{mapping| 441 699 1024 1238 1526 1632 1803 | 0 0 0 0 -1 0 -1 }}


POTE generator: ~11/8 = 551.3688
Optimal tunings:  
* 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=1| 441, 882, 1323, 2205f, 3528f }}
{{Optimal ET sequence|legend=0| 27, 36, 99, 135f, 171f }}


Badness: 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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