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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
Line 109: Line 110:
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
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* 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 ===
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