Xenharmonic Wiki

Septiennealimmal clan: Difference between revisions

VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
Created page with "This clan of temperaments tempers out the tritrizo comma, {{monzo| -11 -9 0 9 }} = 40353607/40310784, and includes these: * Niner * Marvel temper..."
 
Change several names, as planned
 
(48 intermediate revisions by 11 users not shown)
Line 1: Line 1:
This clan of temperaments tempers out the tritrizo comma, {{monzo| -11 -9 0 9 }} = 40353607/40310784, and includes these:
{{Technical data page}}
* [[Augmented family #Niner|Niner]]
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.
* [[Marvel temperaments #Enneaportent|Enneaportent]]
* [[Kleismic family #Novemkleismic|Novemkleismic]]
* [[Ragismic family #Ennealimmal|Ennealimmal]]


= No-five tritrizo =
Temperaments discussed elsewhere are:
Subgroup: 2.3.7
* ''[[Cobalt]]'' → [[Starling temperaments #Cobalt]]
* ''[[Niner]]'' → [[Augmented family #Niner]]
* ''[[Enneaportent]]'' → [[Marvel temperaments #Enneaportent]]
* ''[[Novemkleismic]]'' → [[Kleismic family #Novemkleismic]]
* ''[[Gamelstearn]]'' → [[Compton family #Gamelstearn]]
* ''[[Nonant]]'' → [[Schismatic family #Nonant]]
 
== 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.
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 9 0 11 24 | 0 2 2 1 }}
: mapping generators: ~121/112, ~343/198
 
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=0| 63, 72, 135, 342, 477, 1089, 1566 }}
 
Badness (Sintel): 0.161
 
== 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 [[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.
 
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.
 
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 4375/4374
 
{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
: mapping generators: ~27/25, ~5/3
 
[[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]]:
* 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|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
 
[[Badness]] (Sintel): 0.0914
 
=== Enneabiotic ===
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
 
Comma list: 2401/2400, 4375/4374, 5632/5625
 
Mapping: {{mapping| 9 1 1 12 -75 | 0 2 3 2 16 }}
 
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=0| 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: {{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
 
Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374
 
Mapping: {{mapping| 9 1 1 12 -75 93 | 0 2 3 2 16 -9 }}
 
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=0| 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: {{mapping| 9 1 1 12 -75 93 -3 | 0 2 3 2 16 -9 6 }}
 
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 }}
 
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: {{mapping| 9 1 1 12 -75 93 -3 -48 | 0 2 3 2 16 -9 6 13 }}
 
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=0| 99e, 171e, 270 }}
 
Badness (Sintel): 1.25
 
=== Ennealympic ===
Ennealympic ({{nowrap| 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: {{mapping| 9 1 1 12 124 | 0 2 3 2 -14 }}
 
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=0| 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: {{mapping| 9 1 1 12 124 93 | 0 2 3 2 -14 -9 }}
 
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=0| 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: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
 
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 }}
 
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: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
 
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}})


[[POTE generator]]: ~3/2 = 701.965
{{Optimal ET sequence|legend=0| 99, 171, 270, 441 }}


{{Val list|legend=1| 27, 36, 99, 135, 171, 306, 4419d, 4725d, ... , 8397dd, 8703dd }}
Badness (Sintel): 1.16
 
=== 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.


= 171&amp;342 temperament =
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 4375/4356
Mapping: {{mapping| 9 1 1 12 -2 | 0 2 3 2 5 }}
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:
* 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|legend=0| 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: {{mapping| 9 1 1 12 -2 -33 | 0 2 3 2 5 10 }}
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:
* 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|legend=0| 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: {{mapping| 9 1 1 12 -2 -33 -3 | 0 2 3 2 5 10 6 }}
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:
* 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|legend=0| 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: {{mapping| 9 1 1 12 -2 -33 -3 78  | 0 2 3 2 5 10 6 -6 }}
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 ====
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 243/242, 325/324, 441/440
Mapping: {{mapping| 9 1 1 12 -2 20 | 0 2 3 2 5 2 }}
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=0| 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: {{mapping| 9 1 1 12 -2 20 -3 | 0 2 3 2 5 2 6 }}
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=0| 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: {{mapping| 9 1 1 12 -2 20 -3 25 | 0 2 3 2 5 2 6 2 }}
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=0| 27eg, 45efg, 72 }}
Badness (Sintel): 0.927
=== Ennealiminal ===
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1375/1372, 4375/4374
Mapping: {{mapping| 9 1 1 12 51 | 0 2 3 2 -3 }}
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=0| 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: {{mapping| 9 1 1 12 51 20 | 0 2 3 2 -3 2 }}
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=0| 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: {{mapping| 9 1 1 12 51 20 50 | 0 2 3 2 -3 2 -2 }}
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 }}
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: {{mapping| 9 1 1 12 51 20 50 25 | 0 2 3 2 -3 2 -2 2 }}
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=0| 27, 45f, 72 }}
Badness (Sintel): 1.56
=== 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]]. 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.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 4375/4374
Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
: mapping generators: ~80/77, ~400/231
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:
* 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|legend=0| 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: {{mapping| 18 0 -1 22 48 -19 | 0 2 3 2 1 6 }}
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:
* 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|legend=0| 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: {{mapping| 18 0 -1 22 48 -19 -12 | 0 2 3 2 1 6 6 }}
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 }}
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: {{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{{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 }}
Badness (Sintel): 0.812
==== Semihemiennealimmal ====
Subgroup: 2.3.5.7.11.13
Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374
Mapping: {{mapping| 18 0 -1 22 48 88 | 0 4 6 4 2 -3 }}
: mapping generators: ~80/77, ~1053/800
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=0| 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: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
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 ET sequence|legend=0| 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: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
Optimal tunings:
* 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, 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: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
: mapping generators: ~27/25, ~140/121
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=0| 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: {{mapping| 9 3 4 14 18 -8 | 0 6 9 6 7 22 }}
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=0| 72, …, 297ef, 369f, 441 }}
Badness (Sintel): 1.08
=== Quadraennealimmal ===
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 4375/4374, 234375/234256
Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
: mapping generators: ~27/25, ~25/22
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=0| 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: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
: mapping generators: ~2744/2673, ~2352/1375
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=0| 27, 243, 270, 783, 1053, 1323 }}
Badness (Sintel): 0.986
=== Rhodium ===
{{Main| Rhodium }}
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
Comma list: 2401/2400, 4375/4374, 117440512/117406179
Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
: mapping generators: ~3072/3025, ~55/32
Optimal tunings:
* WE: ~3072/3025 = 26.6668{{c}}, ~55/32 = 937.6664{{c}} (~385/384 = 4.3288{{c}})
* CWE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6630{{c}} (~385/384 = 4.3297{{c}})
{{Optimal ET sequence|legend=0| 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: {{mapping| 45 1 -1 56 226 272 | 0 2 3 2 -2 -3 }}
Optimal tunings:
* WE: ~66/65 = 26.6670{{c}}, ~55/32 = 937.6633{{c}} (~385/384 = 4.3172{{c}})
* CWE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6515{{c}} (~385/384 = 4.3182{{c}})
{{Optimal ET sequence|legend=0| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
Badness (Sintel): 0.936
== Undecentic ==
{{Distinguish| Undecental }}
Named by [[Xenllium]] in 2021, undecentic ({{nowrap| 99 & 198 }}) has a period of 1/99 octave.
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 2401/2400, 3136/3125, 4375/4374
{{Mapping|legend=1| 99 157 230 278 0 | 0 0 0 0 1 }}
: mapping generators: ~126/125, ~11
[[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 }}
[[Badness]] (Sintel): 1.94
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 847/845, 2401/2400, 3136/3125
Mapping: {{mapping| 99 157 230 278 0 24 | 0 0 0 0 1 1 }}
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=0| 99ef, 198, 693bcdefff }}
Badness (Sintel): 1.76
== Schisennealimmal ==
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


[[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}}


{{Val list|legend=1| 171, 342 }}
{{Optimal ET sequence|legend=1| 171, 342 }}


[[Badness]]: 0.031739
[[Badness]] (Sintel): 1.05


== 13-limit 171&amp;342 ==
=== 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 }}
 
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=0| 171, 342 }}
 
Badness (Sintel): 2.23


POTE generator: ~11/8 = 551.322
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Vals: {{Val list| 171, 342 }}
Comma list: 625/624, 729/728, 833/832, 1225/1224, 2205/2197


Badness: 0.054029
Mapping: {{mapping| 171 271 397 480 0 633 699 | 0 0 0 0 1 0 0 }}


== 13-limit 171&amp;513 ==
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=0| 171, 342, 855ff, 1197fff }}
 
Badness (Sintel): 1.60
 
=== 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 }}
 
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 }}
 
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: {{mapping| 171 271 397 480 0 41 699 | 0 0 0 0 1 1 0 }}
 
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=0| 171, 342f, 513 }}
 
Badness (Sintel): 1.56
 
== Lunennealimmal ==
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
 
[[Comma list]]: 2401/2400, 4375/4374, 274877906944/274658203125
 
{{Mapping|legend=1| 441 699 1024 1238 1526 | 0 0 0 0 -1 }}
: mapping generators: ~32805/32768, ~11
 
[[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 }}
 
[[Badness]] (Sintel): 3.04
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2401/2400, 4096/4095, 4375/4374, 85750/85683
 
Mapping: {{mapping| 441 699 1024 1238 1526 1632 | 0 0 0 0 -1 0 }}
 
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=0| 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: {{mapping| 441 699 1024 1238 1526 1632 1803 | 0 0 0 0 -1 0 -1 }}
 
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 }}


POTE generator: ~11/8 = 551.625
Optimal tunings:  
* WE: ~13/12 = 133.3847{{c}}, ~3/2 = 701.9342{{c}}
* CWE: ~13/12 = 133.3333{{c}}, ~3/2 = 702.0763{{c}}


Vals: {{Val list| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=0| 27, 36, 99, 135f, 171f }}


Badness: 0.046843
Badness (Sintel): 0.540


[[Category:Regular temperament theory]]
[[Category:Septiennealimmal clan| ]] <!-- main article -->
[[Category:Temperament clan]]
[[Category:Temperament clans]]
Retrieved from "https://en.xen.wiki/w/Septiennealimmal_clan"