Avicennmic temperaments: Difference between revisions

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These temper out the avicennma, [[525/512]], also known as Avicenna's enharmonic diesis. Avicennmic temperaments include [[negri]], [[Dicot family|dichotic]], [[flattone]], [[hexadecimal]], [[muggles]], [[lemba]], [[Augmented family|deflated]], and [[pycnic]].
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the avicennma, [[525/512]], also known as Avicenna's enharmonic diesis.  


== Roman ==
Tempereaments discussed elsewhere are:
{{see also|Wesley family #Roman}}
* ''[[Dichotic]]'' (+25/24) → [[Dicot family #Dichotic|Dicot family]]
* ''[[Armodue (temperament)|Armodue]]'' (+36/35) → [[Mavila family #Pelogic|Mavila family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Muggles]] (+126/125) → [[Magic family #Muggles|Magic family]]
* ''[[Deflated]]'' (+21/20) → [[Augmented family #Deflated|Augmented family]]
* ''[[Pycnic]]'' (+245/243) → [[Sensamagic clan #Pycnic|Sensamagic clan]]
* ''[[Secund]]'' (+405/392) → [[Greenwoodmic temperaments #Secund|Greenwoodmic temperaments]]
* ''[[Roman]]'' (+3125/3024) → [[Wesley family #Roman|Wesley family]]


=== 7-limit (Crusher) ===
Considered below is submerged.  
Subgroup: 2.3.5.7


[[Comma list]]: 525/512, 3125/3024
== Submerged ==
{{Main| Submerged }}


[[Mapping]]: [{{Val|1 4 3 -1}}, {{Val|0 -7 -2 11}}]
Named by [[Fitzgerald Lee]] in 2025, submerged tempers out 3125/3087, the [[gariboh comma]]. It may be described as the {{nowrap| 16 & 29 }} temperament, generated by a sharply tuned ~8/5, eleven of which minus seven octaves make a perfect fifth; its [[ploidacot]] is therefore zeta-hendecacot. [[45edo]] makes for an excellent tuning.


{{Multival|legend=1|7 2 -11 -13 -37 -31}}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~63/50 = 414.552
[[Comma list]]: 525/512, 3125/3087


{{Val list|legend=1| 3, 23, 26, 29, 55c }}
{{Mapping|legend=1| 1 -6 3 9 | 0 11 -1 -9 }}
: mapping generators: ~2, ~8/5


[[Badness]]: 0.113386
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.3342{{c}}, ~8/5 = 829.2070{{c}}
: [[error map]]: {{val| +3.334 -0.683 -5.518 -1.681 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 826.9483{{c}}
: error map: {{val| 0.000 -5.523 -13.262 -11.361 }}


=== 11-limit ===
{{Optimal ET sequence|legend=1| 16, 29, 45, 74cd, 119bccdd }}
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 525/512
 
Mapping: [{{Val|1 4 3 -1 0}}, {{Val|0 -7 -2 11 10}}]


POTE generator: ~14/11 = 414.471
[[Badness]] (Sintel): 3.98
 
Vals: {{Val list| 3, 23, 26, 29, 55c }}
 
Badness: 0.052841
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 65/64, 100/99, 105/104, 245/242
 
Mapping: [{{Val|1 4 3 -1 0 3}}, {{Val|0 -7 -2 11 10 2}}]
 
POTE generator: ~14/11 = 414.472
 
Vals: {{Val list| 3, 23, 26, 29, 55cf }}
 
Badness: 0.030043
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 35/34, 52/51, 65/64, 100/99, 121/119
 
Mapping: [{{Val|1 4 3 -1 0 3 1}}, {{Val|0 -7 -2 11 10 2 9}}]
 
POTE generator: ~14/11 = 414.164
 
Vals: {{Val list| 3, 26g, 29 }}
 
Badness: 0.041555
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 35/34, 52/51, 65/64, 100/99, 121/119, 133/132
 
Mapping: [{{Val|1 4 3 -1 0 3 1 7}}, {{Val|0 -7 -2 11 10 2 9 -8}}]
 
POTE generator: ~14/11 = 414.130
 
Vals: {{Val list| 3, 26g, 29 }}
 
Badness: 0.033730
 
== Secund ==
{{see also|Greenwoodmic temperaments #Secund}}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 405/392, 525/512
 
[[Mapping]]: [{{val|1 1 3 2}}, {{val|0 5 -6 7}}]
 
{{Multival|legend=1|5 -6 7 -21 -3 33}}
 
[[POTE generator]]: ~16/15 = 137.958
 
{{Val list|legend=1| 9, 17, 26, 61bc, 87bcc }}
 
[[Badness]]: 0.089840


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 99/98, 385/384
Comma list: 121/120, 441/440, 525/512


Mapping: [{{val|1 1 3 2 3}}, {{val|0 5 -6 7 4}}]
Mapping: {{mapping| 1 -6 3 9 0 | 0 11 -1 -9 5 }}


POTE generator: ~12/11 = 137.900
Optimal tunings:  
* WE: ~2 = 1203.6231{{c}}, ~8/5 = 829.4195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 826.9730{{c}}


Vals: {{Val list| 9, 17, 26, 61bc, 87bcc }}
{{Optimal ET sequence|legend=0| 16, 29, 45e, 74cde, 119bccddeee }}


Badness: 0.042562
Badness (Sintel): 1.93


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


Comma list: 27/26, 45/44, 99/98, 385/384
Comma list: 65/64, 105/104, 121/120, 441/440
 
Mapping: [{{val|1 1 3 2 3 2}}, {{val|0 5 -6 7 4 15}}]
 
POTE generator: ~12/11 = 136.937
 
Vals: {{Val list| 9, 26f, 35 }}
 
Badness: 0.049684
 
=== Secundly ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 65/64, 78/77, 99/98


Mapping: [{{val|1 1 3 2 3 3}}, {{val|0 5 -6 7 4 6}}]
Mapping: {{mapping| 1 -6 3 9 0 3 | 0 11 -1 -9 5 1 }}


POTE generator: ~12/11 = 137.917
Optimal tunings:  
* WE: ~2 = 1203.6316{{c}}, ~8/5 = 829.4249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 826.9677{{c}}


Vals: {{Val list| 9, 17, 26, 61bcf, 87bccf }}
{{Optimal ET sequence|legend=0| 16, 29, 45ef, 74cdef, 119bccddeeeff }}


Badness: 0.026212
Badness (Sintel): 1.35


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Avicennmic]]
[[Category:Avicennmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 18:46, 24 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.

This is a collection of rank-2 temperaments tempering out the avicennma, 525/512, also known as Avicenna's enharmonic diesis.

Tempereaments discussed elsewhere are:

Considered below is submerged.

Submerged

Main article: Submerged

Named by Fitzgerald Lee in 2025, submerged tempers out 3125/3087, the gariboh comma. It may be described as the 16 & 29 temperament, generated by a sharply tuned ~8/5, eleven of which minus seven octaves make a perfect fifth; its ploidacot is therefore zeta-hendecacot. 45edo makes for an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 525/512, 3125/3087

Mapping: [⟨1 -6 3 9], ⟨0 11 -1 -9]]

mapping generators: ~2, ~8/5

Optimal tunings:

  • WE: ~2 = 1203.3342 ¢, ~8/5 = 829.2070 ¢
error map: ⟨+3.334 -0.683 -5.518 -1.681]
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 826.9483 ¢
error map: ⟨0.000 -5.523 -13.262 -11.361]

Optimal ET sequence: 16, 29, 45, 74cd, 119bccdd

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 525/512

Mapping: [⟨1 -6 3 9 0], ⟨0 11 -1 -9 5]]

Optimal tunings:

  • WE: ~2 = 1203.6231 ¢, ~8/5 = 829.4195 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 826.9730 ¢

Optimal ET sequence: 16, 29, 45e, 74cde, 119bccddeee

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 105/104, 121/120, 441/440

Mapping: [⟨1 -6 3 9 0 3], ⟨0 11 -1 -9 5 1]]

Optimal tunings:

  • WE: ~2 = 1203.6316 ¢, ~8/5 = 829.4249 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/5 = 826.9677 ¢

Optimal ET sequence: 16, 29, 45ef, 74cdef, 119bccddeeeff

Badness (Sintel): 1.35

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