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Septiennealimmal clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
Switch to Sintel's badness, WE & CWE tunings (2/)
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: [[error map]]: {{val| +0.021 +0.043 -0.135 }}
: [[error map]]: {{val| +0.021 +0.043 -0.135 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}
: error map: {{val| 0.0000 +0.028 -0.176 }}
: error map: {{val| 0.000 +0.028 -0.176 }}


{{Optimal ET sequence|legend=1| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}
{{Optimal ET sequence|legend=1| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}
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: ''For the 5-limit version, see [[Ennealimma #Ennealimmal]].''
: ''For the 5-limit version, see [[Ennealimma #Ennealimmal]].''


Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, [[2401/2400]] and [[4375/4374]], leading to a temperament of unusual [[efficiency]]. It also tempers out the [[landscape comma]], which is (2401/2400)/(4375/4374), and the [[wizma]], which is (2401/2400)⋅(4375/4374).
Ennealimmal tempers out the two smallest 7-limit [[superparticular]] commas, [[2401/2400]] and [[4375/4374]], leading to a temperament of unusual [[efficiency]]. It also tempers out the [[landscape comma]], which is (2401/2400)/(4375/4374), and the [[wizma]], which is (2401/2400)⋅(4375/4374). 7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}.  


In the 5-limit, it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2), and [[ploidacot]] enneaploid dicot. While [[27/25]] is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.  
In the 5-limit, it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, which leads to the identification of (27/25)<sup>9</sup> with the [[octave]], and gives ennealimmal a [[period]] of 1/9 octave. Its [[pergen]] is (P8/9, P5/2), and [[ploidacot]] enneaploid dicot. While [[27/25]] is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.  


Aside from 10/9 which has already been mentioned, possible generators include [[36/35]], [[21/20]], [[6/5]], [[7/5]] and the neutral thirds pair [[49/40]][[~]][[60/49]], all of which have their own interesting advantages. Possible tunings are [[441edo|441-]], [[612edo|612-]], or [[3600edo]], though its hardly likely anyone could tell the difference.
Aside from 10/9 which has already been mentioned, possible generators include [[36/35]], [[21/20]], [[6/5]], [[7/5]] and the neutral thirds pair [[49/40]][[~]][[60/49]], all of which have their own interesting advantages. Possible tunings are [[441edo|441-]], [[612edo|612-]], or [[3600edo]], though it is hardly likely anyone could tell the difference.


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28- or 43-note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17{{frac|2|3}} to the octave mos.
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "[[tritave]]s" as analogous to octaves might consider the 28- or 43-note [[mos]] with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 17{{frac|2|3}} to the octave mos.


Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].  
Ennealimmal extensions discussed elsewhere include [[Compton family #Omicronbeta|omicronbeta]].  
7-limit ennealimmal's [[S-expression]]-based comma list is {[[4375/4374|S25/S27]], [[2401/2400|S49]]}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3317{{c}} (~36/35 = 49.0016{{c}})
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}} (~36/35 = 49.0214{{c}})
: [[error map]]: {{val| 0.0000 +0.0416 +0.0146 -0.1626 }}
: [[error map]]: {{val| +0.022 +0.038 +0.009 -0.139 }}
* [[POTE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3129{{c}} (~36/35 = 49.0205{{c}})
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}} (~36/35 = 49.0118{{c}})
: error map: {{val| 0.0000 +0.0040 -0.0418 -0.2002 }}
: error map: {{val| 0.000 +0.021 -0.016 -0.183 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
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{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}
{{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612 }}


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


=== 11-limit ===
=== 11-limit ===
The ennealimmal temperament can be described as {{nowrap| 99e & 171e }}, which tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).
The undecimal 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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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4385 (~36/35 = 48.8948{{c}})
* WE: ~27/25 = 133.3229{{c}}, ~5/3 = 884.3988 (~36/35 = 48.8616{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4679 (~36/35 = 48.8654{{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 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}


Badness (Smith): 0.027332
Badness (Sintel): 0.904


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4285{{c}} (~36/35 = 48.9048{{c}})
* CTE: ~27/25 = 133.3321{{c}}, ~5/3 = 884.4225{{c}} (~36/35 = 48.9025{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4304{{c}} (~36/35 = 48.9030{{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 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}


Badness (Smith): 0.029404
Badness (Sintel): 1.22


===== 17-limit =====
===== 17-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4110{{c}} (~36/35 = 48.9223{{c}})
* WE: ~27/25 = 133.3268{{c}}, ~5/3 = 884.3797{{c}} (~36/35 = 48.9076{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4234{{c}} (~36/35 = 48.9099{{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=0| 99e, 171e, 270 }}
Badness (Sintel): 1.44


===== 19-limit =====
===== 19-limit =====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4139{{c}} (~36/35 = 48.9194{{c}})
* WE: ~27/25 = 133.3271{{c}}, ~5/3 = 884.3856{{c}} (~36/35 = 48.9040{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4270{{c}} (~36/35 = 48.9063{{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 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
Badness (Sintel): 1.25


==== Ennealimmalis ====
==== Ennealimmalis ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4560{{c}} (~36/35 = 48.8773{{c}})
* 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 = 48.8588{{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 }}
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}


Badness (Smith): 0.022068
Badness (Sintel): 0.912


=== Ennealimmia ===
=== Ennealimmia ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4113{{c}} (~36/35 = 48.9220{{c}})
* WE: ~27/25 = 133.3264{{c}}, ~5/3 = 884.3631{{c}} (~36/35 = 48.9219{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4089{{c}} (~36/35 = 48.9244{{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 }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}


Badness (Smith): 0.026463
Badness (Sintel): 0.875


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4055{{c}} (~36/35 = 48.9278{{c}})
* WE: ~27/25 = 133.3281{{c}}, ~5/3 = 884.3647{{c}} (~36/35 = 48.9317{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3997{{c}} (~36/35 = 48.9336{{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 }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}


Badness (Smith): 0.016607
Badness (Sintel): 0.686


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


Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374
Comma list: 936/935, 1225/1224, 1701/1700, 2401/2400, 4096/4095


Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}
Mapping: {{mapping| 9 1 1 12 124 93 -3 | 0 2 3 2 -14 -9 6 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3867{{c}} (~36/35 = 48.9466{{c}})
* WE: ~27/25 = 133.3227{{c}}, ~5/3 = 884.3102{{c}} (~36/35 = 48.9486{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3808{{c}} (~36/35 = 48.9525{{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 }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 441, 711g }}
 
Badness (Sintel): 1.04


===== 19-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374
Comma list: 936/935, 1216/1215, 1225/1224, 1701/1700, 1729/1728, 2401/2400


Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}
Mapping: {{mapping| 9 1 1 12 124 93 -3 -48 | 0 2 3 2 -14 -9 6 13 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3960{{c}} (~36/35 = 48.9373{{c}})
* WE: ~27/25 = 133.3255{{c}}, ~5/3 = 884.3467{{c}} (~36/35 = 48.9320{{c}})
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3985{{c}} (~36/35 = 48.9348{{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=0| 99, 171, 270 }}
Badness (Sintel): 1.16


=== Ennealimnic ===
=== Ennealimnic ===
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