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The '''gamelismic family''' of temperaments are rank-3 temperaments tempering out [[1029/1024]].  
{{Technical data page}}
The '''gamelismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[gamelisma]], 1029/1024. The head of this family, gamelismic, tempers out 1029/1024 alone in the full 7-limit, so it has the same [[2.3.7 subgroup|2.3.7-subgroup]] structure as [[slendric]] but giving [[prime harmonic|prime]] [[5/1|5]] an independent generator.
 
See [[Gamelismic clan]] for the rank-2 temperament without the last generator of gamelismic, and its various extensions.
 
== Gamelismic ==
{{Main| Gamelismic and portent }}
 
[[Subgroup]]: 2.3.5.7


= Gamelan =
[[Comma list]]: 1029/1024
[[Comma list]]: 1029/1024


[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 0 -1 }},{{val| 0 0 1 0 }}]
{{Mapping|legend=1| 1 1 0 3 | 0 3 0 -1 | 0 0 1 0 }}
: mapping generators: ~2, ~8/7, ~5


Map to lattice: [{{val| 0 3 0 -1 }}, {{val| 0 0 1 0 }}]
[[Mapping to lattice]]: [{{val| 0 3 0 -1 }}, {{val| 0 0 1 0 }}]


[[Minkowski lattice basis]]:  
[[Minkowski lattice basis]]:  
Line 12: Line 20:
: Angle (8/7, 5/4) = 90 degrees
: Angle (8/7, 5/4) = 90 degrees


[[Minimax tuning]]: c = 1029/1024
[[Optimal tuning]]s:  
* 7-odd-limit: 3, 5, and 7 1/4c flat
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}, ~5/4 = 385.3412{{c}}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 5/2 -1/4 1 -3/4 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
: [[error map]]: {{val| +0.486 -0.123 -0.001 -1.151 }}
: Eigenmonzos: 2, 7/6, 6/5
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7474{{c}}, ~5/4 = 385.5205{{c}}
* 9-odd-limit: 3 1/7c flat, 5 and 7 2/7c flat
: error map: {{val| 0.000 -0.713 -0.793 -2.573 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 10/7 -1/7 1 -3/7 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: Eigenmonzos: 2, 6/5, 9/7


[[TE tuning]] map: {{val| 1200.486 1901.833 2786.314 3367.676 }}
[[Minimax tuning]]: ''c'' = 1029/1024
* [[7-odd-limit]]: 3, 5, and 7 (1/4)''c'' flat
: {{Monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 5/2 -1/4 1 -3/4 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3.5/3
* [[9-odd-limit]]: 3 1/7-comma flat, 5 and 7 2/7-comma flat
: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 10/7 -1/7 1 -3/7 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7


{{Val list|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 159, 190 }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 190 }}


[[Badness]]: 0.176 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.777


[[Projection pair]]: 3 1024/343 to 2.5.7
[[Projection pair]]: 3 1024/343 to 2.5.7


= Portent =
Scales: [[portent26]]
Portent has a [[Normal lists #normal interval list|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.


Comma list: 385/384, 441/440
== Portent ==
{{Main| Gamelismic and portent }}


Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 0 -1 4 }}, {{val| 0 0 1 0 -1 }}]
Portent tempers out [[385/384]] and [[441/440]] and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).


Mapping generators: ~2, ~8/7, ~5
[[Subgroup]]: 2.3.5.7.11


Map to lattice: [{{val| 0 3 1 -1 3 }}, {{val| 0 0 1 0 -1 }}]
[[Comma list]]: 385/384, 441/440
 
{{Mapping|legend=1| 1 1 0 3 5 | 0 3 0 -1 4 | 0 0 1 0 -1 }}
 
[[Mapping to lattice]]: [{{val| 0 3 1 -1 3 }}, {{val| 0 0 1 0 -1 }}]


Minkowski lattice basis:  
Minkowski lattice basis:  
Line 43: Line 59:
: Angle (8/7, 12/11) = 86.657 degrees
: Angle (8/7, 12/11) = 86.657 degrees


[[Minimax tuning]]: c = 1029/1024, e = 385/384
[[Optimal tuning]]s:
* [[11-odd-limit]]: 3 1/7c flat, 5 and 7 2/7c flat, 11 e-3/7c flat
* [[WE]]: ~2 = 1200.4902{{c}}, ~8/7 = 233.7839{{c}}, ~5/4 = 385.3191{{c}}
: [[error map]]: {{val| +0.490 -0.113 -0.014 -1.139 -0.031 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7616{{c}}, ~5/4 = 385.3149{{c}}
: error map: {{val| 0.000 -0.670 -0.999 -2.587 -1.586 }}
 
[[Minimax tuning]]: ''c''<sub>1</sub> = 1029/1024, ''c''<sub>2</sub> = 385/384
* [[11-odd-limit]]: 3 (1/7)''c''<sub>1</sub> flat, 5 and 7 (2/7)''c''<sub>1</sub> flat, 11 (''c''<sub>2</sub> - (3/7)''c''<sub>1</sub>) flat
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 39/14 4/7 1/2 -2/7 -1/2 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 39/14 4/7 -1/2 -2/7 1/2 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 39/14 4/7 1/2 -2/7 -1/2 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 39/14 4/7 -1/2 -2/7 1/2 }}]
: [[Eigenmonzo]]s: 2, 11/10, 9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5
 
{{Optimal ET sequence|legend=1| 15, 26, 31, 41, 72, 118, 159, 190 }}
 
[[Badness]] (Sintel): 0.281


{{Val list|legend=1| 10, 15, 26, 31, 41, 46, 57, 72, 77, 103, 118, 149, 159, 190 }}
[[Projection pair]]s: 3 1024/343 11 131072/12005 to 2.5.7


Badness: 0.234 × 10<sup>-3</sup>
Scales: [[portent26]]


Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7
=== Portending ===
Subgroup: 2.3.5.7.11.13


== Portending ==
Comma list: 325/324, 364/363, 385/384
Comma list: 325/324, 364/363, 441/440


Mapping: [{{val| 1 1 0 3 5 6 }}, {{val| 0 3 0 -1 4 12 }}, {{val| 0 0 1 0 -1 -2 }}]
Mapping: {{mapping| 1 1 0 3 5 6 | 0 3 0 -1 4 12 | 0 0 1 0 -1 -2 }}


{{Val list|legend=1| 15, 26, 41, 46, 72, 87, 159, 477cdf }}
Optimal tunings:
* WE: ~2 = 1200.4540{{c}}, ~8/7 = 234.0013{{c}}, ~5/4 = 384.8733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9748{{c}}, ~5/4 = 384.8812{{c}}


Badness: 0.627 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 15, 26, 31f, 41, 46, 72, 87, 159 }}
 
Badness (Sintel): 0.587
 
Complexity spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13
 
=== Portentous ===
Subgroup: 2.3.5.7.11.13


== Portentous ==
Comma list: 385/384, 441/440, 625/624
Comma list: 385/384, 441/440, 625/624


Mapping: [{{val| 1 1 0 3 5 -5 }}, {{val| 0 3 0 -1 4 -3 }}, {{val| 0 0 1 0 -1 4 }}]
Mapping: {{mapping| 1 1 0 3 5 -5 | 0 3 0 -1 4 -3 | 0 0 1 0 -1 4 }}
 
Optimal tunings:
* WE: ~2 = 1200.4888{{c}}, ~8/7 = 233.7795{{c}}, ~5/4 = 385.1398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.7575{{c}}, ~5/4 = 385.1447{{c}}
 
{{Optimal ET sequence|legend=0| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef }}


{{Val list|legend=1| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 349f, 452cdef, 611cdef }}
Badness (Sintel): 0.618


Badness: 0.662 × 10<sup>-3</sup>
=== Ominous ===
Subgroup: 2.3.5.7.11.13


== Ominous ==
Comma list: 351/350, 385/384, 441/440
Comma list: 351/350, 385/384, 441/440


Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 -10 }}, {{val| 0 0 1 0 -1 2 }}]
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 -10 | 0 0 1 0 -1 2 }}
 
Optimal tunings:
* WE: ~2 = 1200.7019{{c}}, ~8/7 = 233.5453{{c}}, ~5/4 = 385.6079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4510{{c}}, ~5/4 = 385.6739{{c}}


{{Val list|legend=1| 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdef }}
{{Optimal ET sequence|legend=0| 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}


Badness: 0.751 × 10<sup>-3</sup>
Badness (Sintel): 0.702


=== 17-limit ===
==== 17-limit ====
Comma list: 273/272, 351/350, 441/440, 561/560
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 1 0 3 5 1 1 }}, {{val| 0 3 0 -1 4 -10 -8 }}, {{val| 0 0 1 0 -1 2 2 }}]
Comma list: 273/272, 351/350, 385/384, 441/440


Mapping generators: 2, 8/7, 5
Mapping: {{mapping| 1 1 0 3 5 1 1 | 0 3 0 -1 4 -10 -8 | 0 0 1 0 -1 2 2 }}


Map to lattice: [{{val| 0 1 1 0 0 -1 0 }}, {{val| 0 -1 -1 0 -1 2 1 }}]
Mapping to lattice: [{{val| 0 1 1 0 0 -1 0 }}, {{val| 0 -1 -1 0 -1 2 1 }}]


Lattice basis:  
Lattice basis:  
: 8/7 length = 0.3859, 6/5 length = 1.1303
: 8/7 length = 0.3859, 6/5 length = 1.1303
: Angle (8/7, 6/5) = 98.6015
: Angle (8/7, 6/5) = 98.6015
Optimal tunings:
* WE: ~2 = 1200.6745{{c}}, ~8/7 = 233.5625{{c}}, ~5/4 = 385.5056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4679{{c}}, ~5/4 = 385.5892{{c}}


Minimax tuning:  
Minimax tuning:  
* [[17-odd-limit]]:
* 17-odd-limit
: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 7/4 9/10 0 0 -3/10 -3/20 0 }}, {{monzo| 5/2 7/5 0 0 -4/5 1/10 0 }}, {{monzo| 11/4 -3/10 0 0 1/10 1/20 0 }}, {{monzo| 7/2 -1/5 0 0 2/5 -3/10 0 }}, {{monzo| 7/2 -1/5 0 0 -3/5 7/10 0 }}, {{monzo| 4 2/5 0 0 -4/5 3/5 0 }}]
: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 7/4 9/10 0 0 -3/10 -3/20 0 }}, {{monzo| 5/2 7/5 0 0 -4/5 1/10 0 }}, {{monzo| 11/4 -3/10 0 0 1/10 1/20 0 }}, {{monzo| 7/2 -1/5 0 0 2/5 -3/10 0 }}, {{monzo| 7/2 -1/5 0 0 -3/5 7/10 0 }}, {{monzo| 4 2/5 0 0 -4/5 3/5 0 }}]
: Eigenmonzos: 2, 13/11, 11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
 
{{Optimal ET sequence|legend=0| 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef }}


{{Val list|legend=1| 26, 31, 46, 72, 103, 149, 175f, 221ef }}
Badness (Sintel): 0.582


Badness: 0.612 × 10<sup>-3</sup>
=== Momentous ===
Subgroup: 2.3.5.7.11.13


== Momentous ==
Comma list: 196/195, 352/351, 385/384
Comma list: 196/195, 352/351, 385/384


Mapping: [{{val| 1 1 0 3 5 7 }}, {{val| 0 3 0 -4 1 -5 }}, {{val| 0 0 1 0 -1 -1 }}]
Mapping: {{mapping| 1 1 0 3 5 7 | 0 3 0 -4 1 -5 | 0 0 1 0 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0652{{c}}, ~8/7 = 234.1856{{c}}, ~5/4 = 386.6199{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1748{{c}}, ~5/4 = 386.5951{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d }}


{{Val list|legend=1| 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def }}
Badness (Sintel): 0.778


Badness: 0.832 × 10<sup>-3</sup>
=== Foreboding ===
Subgroup: 2.3.5.7.11.13


== Foreboding ==
Comma list: 105/104, 144/143, 275/273
Comma list: 105/104, 144/143, 275/273


Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 0 -1 4 2 }}, {{val| 0 0 1 0 -1 1 }}]
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 2 | 0 0 1 0 -1 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.2251{{c}}, ~8/7 = 233.4102{{c}}, ~5/4 = 382.4142{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.4017{{c}}, ~5/4 = 382.4261{{c}}
 
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 26, 31, 41, 72f }}
 
Badness (Sintel): 0.816
 
=== Portannic ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 441/440, 10985/10976
 
Mapping: {{mapping| 1 1 2 3 3 4 | 0 3 0 -1 4 -1 | 0 0 3 0 -3 -1 }}
: mapping generators: ~2, ~8/7, ~14/13
 
Optimal tunings:
* WE: ~2 = 1200.5451{{c}}, ~8/7 = 233.7495{{c}}, ~14/13 = 128.4023{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6930{{c}}, ~14/13 = 128.3684{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}
 
Badness (Sintel): 1.67
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 385/384, 441/440, 8624/8619
 
Mapping: {{mapping| 1 1 2 3 3 4 4 | 0 3 0 -1 4 -1 1 | 0 0 3 0 -3 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.4416{{c}}, ~8/7 = 233.7663{{c}}, ~14/13 = 128.4622{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.7173{{c}}, ~14/13 = 128.4269{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}
 
Badness (Sintel): 1.24
 
== Gamel ==
This esoteric alternative extension tempers out [[540/539]], and sometimes comes up in temperament searches. In practice however, it is almost always desirable to further temper it to [[miracle]], in which case is is identical to portent anyway. This enables many more [[essentially tempered chord]]s while introducing virtually no additional error.


{{Val list|legend=1| 5, 10, 15, 26, 31, 41, 72f, 185cf }}
[[Subgroup]]: 2.3.5.7.11


Badness: 0.873 × 10<sup>-3</sup>
[[Comma list]]: 540/539, 1029/1024


= Gamel =
{{Mapping|legend=1| 1 1 0 3 -1 | 0 3 0 -1 11 | 0 0 1 0 1 }}
Comma list: 1029/1024, 540/539


Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 0 -1 11 }}, {{val| 0 0 1 0 1 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6462{{c}}, ~8/7 = 233.4166{{c}}, ~5/4 = 384.4189{{c}}
: [[error map]]: {{val| +0.646 -1.059 -0.602 -0.304 +1.329 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.3289{{c}}, ~5/4 = 384.5627{{c}}
: error map: {{val| 0.000 -1.968 -1.751 -2.155 -0.137 }}


{{Val list|legend=1| 10, 31, 41, 72 }}
{{Optimal ET sequence|legend=1| 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}


Badness: 0.853 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.02


[[Category:Theory]]
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Gamelan]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Gamelismic]]
[[Category:Rank 3]]

Latest revision as of 11:46, 16 August 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.

The gamelismic family of rank-3 temperaments tempers out the gamelisma, 1029/1024. The head of this family, gamelismic, tempers out 1029/1024 alone in the full 7-limit, so it has the same 2.3.7-subgroup structure as slendric but giving prime 5 an independent generator.

See Gamelismic clan for the rank-2 temperament without the last generator of gamelismic, and its various extensions.

Gamelismic

Main article: Gamelismic and portent

Subgroup: 2.3.5.7

Comma list: 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 0 -1], ⟨0 0 1 0]]

mapping generators: ~2, ~8/7, ~5

Mapping to lattice: [⟨0 3 0 -1], ⟨0 0 1 0]]

Minkowski lattice basis:

8/7 length = 0.5192, 5/4 length = log25
Angle (8/7, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1200.4859 ¢, ~8/7 = 233.7822 ¢, ~5/4 = 385.3412 ¢
error map: ⟨+0.486 -0.123 -0.001 -1.151]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7474 ¢, ~5/4 = 385.5205 ¢
error map: ⟨0.000 -0.713 -0.793 -2.573]

Minimax tuning: c = 1029/1024

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [5/2 -1/4 1 -3/4⟩, [5/2 -1/4 0 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3.5/3
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [10/7 -1/7 1 -3/7⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

Optimal ET sequence: 5, 10, 15, 26, 31, 41, 72, 118, 190

Badness (Sintel): 0.777

Projection pair: 3 1024/343 to 2.5.7

Scales: portent26

Portent

Main article: Gamelismic and portent

Portent tempers out 385/384 and 441/440 and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440

Mapping: [⟨1 1 0 3 5], ⟨0 3 0 -1 4], ⟨0 0 1 0 -1]]

Mapping to lattice: [⟨0 3 1 -1 3], ⟨0 0 1 0 -1]]

Minkowski lattice basis:

8/7 length = 0.46467, 12/11 length = 1.931
Angle (8/7, 12/11) = 86.657 degrees

Optimal tunings:

  • WE: ~2 = 1200.4902 ¢, ~8/7 = 233.7839 ¢, ~5/4 = 385.3191 ¢
error map: ⟨+0.490 -0.113 -0.014 -1.139 -0.031]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7616 ¢, ~5/4 = 385.3149 ¢
error map: ⟨0.000 -0.670 -0.999 -2.587 -1.586]

Minimax tuning: c1 = 1029/1024, c2 = 385/384

  • 11-odd-limit: 3 (1/7)c1 flat, 5 and 7 (2/7)c1 flat, 11 (c2 - (3/7)c1) flat
[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [39/14 4/7 1/2 -2/7 -1/2⟩, [20/7 -2/7 0 1/7 0⟩, [39/14 4/7 -1/2 -2/7 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7.11/5

Optimal ET sequence: 15, 26, 31, 41, 72, 118, 159, 190

Badness (Sintel): 0.281

Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7

Scales: portent26

Portending

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384

Mapping: [⟨1 1 0 3 5 6], ⟨0 3 0 -1 4 12], ⟨0 0 1 0 -1 -2]]

Optimal tunings:

  • WE: ~2 = 1200.4540 ¢, ~8/7 = 234.0013 ¢, ~5/4 = 384.8733 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.9748 ¢, ~5/4 = 384.8812 ¢

Optimal ET sequence: 15, 26, 31f, 41, 46, 72, 87, 159

Badness (Sintel): 0.587

Complexity spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13

Portentous

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624

Mapping: [⟨1 1 0 3 5 -5], ⟨0 3 0 -1 4 -3], ⟨0 0 1 0 -1 4]]

Optimal tunings:

  • WE: ~2 = 1200.4888 ¢, ~8/7 = 233.7795 ¢, ~5/4 = 385.1398 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7575 ¢, ~5/4 = 385.1447 ¢

Optimal ET sequence: 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef

Badness (Sintel): 0.618

Ominous

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 -10], ⟨0 0 1 0 -1 2]]

Optimal tunings:

  • WE: ~2 = 1200.7019 ¢, ~8/7 = 233.5453 ¢, ~5/4 = 385.6079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4510 ¢, ~5/4 = 385.6739 ¢

Optimal ET sequence: 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff

Badness (Sintel): 0.702

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1 1], ⟨0 3 0 -1 4 -10 -8], ⟨0 0 1 0 -1 2 2]]

Mapping to lattice: [⟨0 1 1 0 0 -1 0], ⟨0 -1 -1 0 -1 2 1]]

Lattice basis:

8/7 length = 0.3859, 6/5 length = 1.1303
Angle (8/7, 6/5) = 98.6015

Optimal tunings:

  • WE: ~2 = 1200.6745 ¢, ~8/7 = 233.5625 ¢, ~5/4 = 385.5056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4679 ¢, ~5/4 = 385.5892 ¢

Minimax tuning:

  • 17-odd-limit
[[1 0 0 0 0 0 0⟩, [7/4 9/10 0 0 -3/10 -3/20 0⟩, [5/2 7/5 0 0 -4/5 1/10 0⟩, [11/4 -3/10 0 0 1/10 1/20 0⟩, [7/2 -1/5 0 0 2/5 -3/10 0⟩, [7/2 -1/5 0 0 -3/5 7/10 0⟩, [4 2/5 0 0 -4/5 3/5 0⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9.13/9

Optimal ET sequence: 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef

Badness (Sintel): 0.582

Momentous

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384

Mapping: [⟨1 1 0 3 5 7], ⟨0 3 0 -4 1 -5], ⟨0 0 1 0 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0652 ¢, ~8/7 = 234.1856 ¢, ~5/4 = 386.6199 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 234.1748 ¢, ~5/4 = 386.5951 ¢

Optimal ET sequence: 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d

Badness (Sintel): 0.778

Foreboding

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 275/273

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 2], ⟨0 0 1 0 -1 1]]

Optimal tunings:

  • WE: ~2 = 1200.2251 ¢, ~8/7 = 233.4102 ¢, ~5/4 = 382.4142 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.4017 ¢, ~5/4 = 382.4261 ¢

Optimal ET sequence: 5, 10, 15, 25e, 26, 31, 41, 72f

Badness (Sintel): 0.816

Portannic

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 10985/10976

Mapping: [⟨1 1 2 3 3 4], ⟨0 3 0 -1 4 -1], ⟨0 0 3 0 -3 -1]]

mapping generators: ~2, ~8/7, ~14/13

Optimal tunings:

  • WE: ~2 = 1200.5451 ¢, ~8/7 = 233.7495 ¢, ~14/13 = 128.4023 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.6930 ¢, ~14/13 = 128.3684 ¢

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff

Badness (Sintel): 1.67

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 385/384, 441/440, 8624/8619

Mapping: [⟨1 1 2 3 3 4 4], ⟨0 3 0 -1 4 -1 1], ⟨0 0 3 0 -3 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.4416 ¢, ~8/7 = 233.7663 ¢, ~14/13 = 128.4622 ¢
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.7173 ¢, ~14/13 = 128.4269 ¢

Optimal ET sequence: 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def

Badness (Sintel): 1.24

Gamel

This esoteric alternative extension tempers out 540/539, and sometimes comes up in temperament searches. In practice however, it is almost always desirable to further temper it to miracle, in which case is is identical to portent anyway. This enables many more essentially tempered chords while introducing virtually no additional error.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024

Mapping: [⟨1 1 0 3 -1], ⟨0 3 0 -1 11], ⟨0 0 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1200.6462 ¢, ~8/7 = 233.4166 ¢, ~5/4 = 384.4189 ¢
error map: ⟨+0.646 -1.059 -0.602 -0.304 +1.329]
  • CWE: ~2 = 1200.0000 ¢, ~8/7 = 233.3289 ¢, ~5/4 = 384.5627 ¢
error map: ⟨0.000 -1.968 -1.751 -2.155 -0.137]

Optimal ET sequence: 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde

Badness (Sintel): 1.02

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