Septiennealimmal clan: Difference between revisions

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


Temperaments discussed elsewhere are:
Temperaments discussed elsewhere are:
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== 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 [[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 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.
Aside from 10/9 which has already been mentioned, possible generators include [[36/35]], [[21/20]], [[6/5]], [[7/5]] and the neutral thirds pair [[49/40]][[~]][[60/49]], all of which have their own interesting advantages. Possible tunings are [[441edo|441-]], [[612edo|612-]], or [[3600edo]], though its 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 2/3 to the octave mos.
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28- or 43-note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17{{frac|2|3}} to the octave mos.


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


7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}. 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]].
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]].
''For the 5-limit temperament, see [[Ennealimma#Ennealimmal]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: 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:
* [[POTE]]: ~27/25 = 133.3333, ~5/3 = 884.3129 (~36/35 = 49.0205)


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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* 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 }}
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=== 11-limit ===
=== 11-limit ===
The ennealimmal temperament can be described as 99e &amp; 171e, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
The ennealimmal temperament can be described as {{nowrap| 99e & 171e }}, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4679 (~36/35 = 48.8654)


{{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: 0.027332
Line 96: Line 96:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)


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


Badness: 0.029404
Badness: 0.029404
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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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)


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


===== 19-limit =====
===== 19-limit =====
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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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)


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


==== Ennealimmalis ====
==== Ennealimmalis ====
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Mapping: {{mapping| 9 1 1 12 -75 -106 | 0 2 3 2 16 21 }}
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 tunings:
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4560 (~36/35 = 48.8773)


{{Optimal ET sequence|legend=1| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}


Badness: 0.022068
Badness: 0.022068


=== Ennealimmia ===
=== Ennealimmia ===
The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out [[131072/130977]] (olympia).  
The ennealimmia temperament is an alternative extension and can be described as {{nowrap| 99 & 171 }}, which tempers out [[131072/130977]] (olympia).  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4089 (~36/35 = 48.9244)


{{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: 0.026463
Line 159: Line 164:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)


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


Badness: 0.016607
Badness: 0.016607
Line 172: Line 178:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)


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


===== 19-limit =====
===== 19-limit =====
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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:
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)


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


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 201: Line 209:
* 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| 72, 171, 243 }}


Badness: 0.020347
Badness: 0.020347
Line 216: Line 223:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9920 (~36/35 = 49.3414)


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: 0.023250
Line 234: Line 241:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9981 (~36/35 = 49.3353)


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: 0.014602
Line 252: Line 259:
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 ET sequence|legend=0| 72, 171, 243 }}


==== Ennealim ====
==== Ennealim ====
Line 261: Line 268:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)


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


Badness: 0.020697
Badness: 0.020697
Line 274: Line 282:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)


{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}


===== 19-limit =====
===== 19-limit =====
Line 285: Line 294:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)


{{Optimal ET sequence|legend=1| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}


=== Ennealiminal ===
=== Ennealiminal ===
Line 296: Line 306:
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:
* POTE: ~27/25 = 133.3333, ~5/3 = 883.8298 (~36/35 = 49.5036)


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


Badness: 0.031123
Badness: 0.031123
Line 309: Line 320:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)


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


Badness: 0.030325
Badness: 0.030325
Line 322: Line 334:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)


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


===== 19-limit =====
===== 19-limit =====
Line 333: Line 346:
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:
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)


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


=== Hemiennealimmal ===
=== Hemiennealimmal ===

Revision as of 09:31, 6 March 2025

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 tritrizo

Subgroup: 2.3.7

Comma list: 40353607/40310784

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

sval mapping generators: ~2592/2401, ~3

Optimal tunings:

  • POTE: ~2592/2401 = 133.333, ~3/2 = 701.965

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

Ennea

See also: No-fives subgroup temperaments

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

Optimal tunings:

  • POTE: ~99/98 = 17.626

Optimal ET sequence: 54, 63, 72, 135, 342, 477, 1089, 1566

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 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 its 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.

7-limit ennealimmal's S-expression-based comma list is {S25/S27, S49}. Interestingly, the landscape comma is equal to S49/(S25/S27) while the wizma is equal to S49*S25/S27.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12], ⟨0 2 3 2]]

Wedgie: ⟨⟨ 18 27 18 1 -22 -34 ]]

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

Optimal tunings:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.3129 (~36/35 = 49.0205)

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: 0.003610

11-limit

The ennealimmal temperament can be described as 99e & 171e, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.4679 (~36/35 = 48.8654)

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

Badness: 0.027332

13-limit

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)

Optimal ET sequence: 99e, 171e, 270

Badness: 0.029404

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)

Optimal ET sequence: 99e, 171e, 270

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)

Optimal ET sequence: 99e, 171e, 270

Ennealimmalis

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:

  • CTE: ~27/25 = 133.3333, ~5/3 = 884.4560 (~36/35 = 48.8773)

Optimal ET sequence: 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

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

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

Optimal tunings:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.4089 (~36/35 = 48.9244)

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

Badness: 0.026463

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)

Optimal ET sequence: 99, 171, 270, 711, 981, 1692e, 2673e

Badness: 0.016607

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374

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

Optimal tunings:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)

Optimal ET sequence: 99, 171, 270

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374

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

Optimal tunings:

  • POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)

Optimal ET sequence: 99, 171, 270

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 tuning (POTE): ~27/25 = 1\9, ~5/3 = 883.9386 (~36/35 = 49.3948)

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: 72, 171, 243

Badness: 0.020347

See also: Chords of ennealimnic

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 883.9920 (~36/35 = 49.3414)

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: 0.023250

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 883.9981 (~36/35 = 49.3353)

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: 0.014602

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 ET sequence: 72, 171, 243

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)

Optimal ET sequence: 27e, 45ef, 72

Badness: 0.020697

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)

Optimal ET sequence: 27eg, 45efg, 72

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)

Optimal ET sequence: 27eg, 45efg, 72

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:

  • POTE: ~27/25 = 133.3333, ~5/3 = 883.8298 (~36/35 = 49.5036)

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

Badness: 0.031123

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)

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

Badness: 0.030325

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)

Optimal ET sequence: 27, 45f, 72

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:

  • POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)

Optimal ET sequence: 27, 45f, 72

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. 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.

Its S-expression-based comma list is {(S22/S24 = S55 = S25/S27 * S99,) S25/S27, S49, S33/S35 = S99}.

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 tuning (POTE): ~80/77 = 1\18, ~400/231 = 950.9553

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]
  • 11-odd-limit diamond monotone and tradeoff: ~99/98 = [17.304, 17.985]

Optimal ET sequence: 72, 198, 270, 342, 612, 954, 1566

Badness: 0.006283

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 tuning (POTE): ~27/26 = 1\18, ~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]
  • 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: 72, 198, 270

Badness: 0.012505

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 tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

Optimal ET sequence: 72, 198g, 270

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 tuning (POTE): ~27/26 = 1\18, ~26/15 = 951.0837

Optimal ET sequence: 72, 198g, 270

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 tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

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

Badness: 0.013104

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]]

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

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

Optimal ET sequence: 270, 684, 954

Badness: 0.013104

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]]

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

Optimal tuning (POTE): ~80/77 = 1\18, ~1053/800 = 475.4727

Optimal ET sequence: 270, 684h, 954h, 1224

Badness: 0.013104

Semiennealimmal

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.

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 tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3367

Optimal ET sequence: 72, 369, 441

Badness: 0.034196

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 tuning (POTE): ~27/25 = 1\9, ~140/121 = 250.3375

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

Badness: 0.026122

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 tuning (POTE): ~27/25 = 1\9, ~25/22 = 221.0717

Optimal ET sequence: 342, 1053, 1395, 1737, 4869dd, 6606cdd

Badness: 0.021320

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 tuning (POTE): ~2744/2673 = 1\27, ~2352/1375 = 928.8000

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

Badness: 0.029812

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:

  • CTE: ~3072/3025 = 1\45, ~55/32 = 937.6658 (~385/384 = 4.3325)
  • CWE: ~3072/3025 = 1\45, ~55/32 = 937.6630 (~385/384 = 4.3397)

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

Badness: 0.0381

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:

  • CTE: ~66/65 = 1\45, ~55/32 = 937.6569 (~385/384 = 4.3236)
  • CWE: ~66/65 = 1\45, ~55/32 = 937.6515 (~385/384 = 4.3182)

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

Badness: 0.0226

Undecentic

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]]

POTE generator: ~11/8 = 552.756

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

Badness: 0.058801

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]]

POTE generator: ~11/8 = 552.024

Optimal ET sequence: 99ef, 198

Badness: 0.042547

Schisennealimmal

Schisennealimmal (171&342) has a period of 1/171 octave. 171EDO and its multiples are members of both schismic and ennealimmal, and from this it derives its name.

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]]

POTE generator: ~11/8 = 550.954

Optimal ET sequence: 171, 342

Badness: 0.031739

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]]

POTE generator: ~11/8 = 551.322

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

Badness: 0.054029

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]]

POTE generator: ~11/8 = 551.365

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

Badness: 0.031323

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]]

POTE generator: ~11/8 = 551.625

Optimal ET sequence: 171, 342f, 513, 855f

Badness: 0.046843

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]]

POTE generator: ~11/8 = 551.756

Optimal ET sequence: 171, 342f, 513, 855f

Badness: 0.030622

Lunennealimmal

Lunennealimmal (441&882) has has a period of 1/441 octave. 441EDO and its multiples are members of both luna and ennealimmal, and from this it derives its name.

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]]

POTE generator: ~11/8 = 551.3584

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

Badness: 0.091939

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]]

POTE generator: ~11/8 = 551.4043

Optimal ET sequence: 441, 882, 1323, 3528f, 4851ff, 6174dff

Badness: 0.042975

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]]

POTE generator: ~11/8 = 551.3688

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

Badness: 0.029334

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