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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.


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


Period: 1\1
== Gamelismic ==
{{Main| Gamelismic and portent }}


Optimal ([[POTE]]) generators: ~8/7 = 233.6875, ~5/4 = 385.1853
[[Subgroup]]: 2.3.5.7
 
EDO generator: [[31edo|(6, 10)\31]], [[41edo|(8, 13)\41]], [[46edo|(9, 15)\46]]
 
Scales (Scala files): [[portent26]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Subgroup: 2.3.5.7


[[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 27: 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


</div></div>
Scales: [[portent26]]
 
= Portent =
Portent has a [[Normal lists #normal interval list|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~8/7 = 233.6884, ~5/4 = 385.1618
 
EDO generator: [[31edo|(6, 10)\31]], [[41edo|(8, 13)\41]], [[46edo|(9, 15)\46]]
 
Scales (Scala files): [[portent26]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Portent ==
<div style="line-height:1.6;">Technical data</div>
{{Main| Gamelismic and portent }}
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
Portent tempers out [[385/384]] and [[441/440]] and is the main extension of gamelismic. Notice the identity 1029/1024 = (385/384)⋅(441/440).  


Comma list: 385/384, 441/440
[[Subgroup]]: 2.3.5.7.11


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


Mapping generators: ~2, ~8/7, ~5
{{Mapping|legend=1| 1 1 0 3 5 | 0 3 0 -1 4 | 0 0 1 0 -1 }}


Map to lattice: [{{val| 0 3 1 -1 3 }}, {{val| 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 74: 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
 
{{Val list|legend=1| 10, 15, 26, 31, 41, 72, 118, 159, 190 }}
 
Badness: 0.234 × 10<sup>-3</sup>
 
Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7
 
</div></div>
 
== Portending ==
 
Period: 1\1


Optimal ([[POTE]]) generators: ~8/7 = 233.9128, ~5/4 = 384.7278
{{Optimal ET sequence|legend=1| 15, 26, 31, 41, 72, 118, 159, 190 }}


EDO generator: [[41edo|(8, 13)\41]], [[46edo|(9, 15)\46]], [[72edo|(14, 23)\72]]
[[Badness]] (Sintel): 0.281


Scales (Scala files):  
[[Projection pair]]s: 3 1024/343 11 131072/12005 to 2.5.7


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Scales: [[portent26]]
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Portending ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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 }}
 
Badness: 0.627 × 10<sup>-3</sup>
 
</div></div>
 
== Portentous ==
 
Period: 1\1


Optimal ([[POTE]]) generators: ~8/7 = 233.6843, ~5/4 = 384.9830
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}}


EDO generator: [[31edo|(6, 10)\31]], [[72edo|(14, 23)\72]], [[87edo|(17, 28)\87]]
{{Optimal ET sequence|legend=0| 15, 26, 31f, 41, 46, 72, 87, 159 }}


Scales (Scala files):  
Badness (Sintel): 0.587


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Portentous ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Badness: 0.662 × 10<sup>-3</sup>
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}}


</div></div>
{{Optimal ET sequence|legend=0| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef }}


== Ominous ==
Badness (Sintel): 0.618
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~8/7 = 233.4088, ~5/4 = 385.3825
 
EDO generator: [[31edo|(6, 10)\31]], [[46edo|(9, 15)\46]], [[72edo|(14, 23)\72]]
 
Scales (Scala files):  
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Ominous ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}
 
{{Val list|legend=1| 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdef }}
 
Badness: 0.751 × 10<sup>-3</sup>
 
</div></div>
 
=== 17-limit ===
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~8/7 = 233.4313, ~5/4 = 385.2890


EDO generator: [[31edo|(6, 10)\31]], [[46edo|(9, 15)\46]], [[72edo|(14, 23)\72]]
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}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 15f, 26, 31, 46, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.702
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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


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


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


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


{{Val list|legend=1| 26, 31, 46, 72, 103, 149, 175f, 221ef }}
{{Optimal ET sequence|legend=0| 15f, 20c, 26, 31, 46, 72, 103, 149, 221ef }}


Badness: 0.612 × 10<sup>-3</sup>
Badness (Sintel): 0.582
 
</div></div>
 
== Momentous ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~8/7 = 233.4088, ~5/4 = 385.3825
 
EDO generator: [[31edo|(6, 10)\31]], [[41edo|(8, 13)\41]], [[46edo|(9, 15)\46]]
 
Scales (Scala files):
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Momentous ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}


{{Val list|legend=1| 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def }}
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}}


Badness: 0.832 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d }}


</div></div>
Badness (Sintel): 0.778


== Foreboding ==
=== Foreboding ===
Subgroup: 2.3.5.7.11.13


Period: 1\1
Comma list: 105/104, 144/143, 275/273


Optimal ([[POTE]]) generators: ~8/7 = 233.3664, ~5/4 = 382.3425
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 2 | 0 0 1 0 -1 1 }}


EDO generator: [[10edo|(3, 6)\10]], [[15edo|(5, 9)\15]], [[31edo|(6, 10)\31]]
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}}


Scales (Scala files):
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 26, 31, 41, 72f }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 0.816
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


=== Portannic ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


{{Val list|legend=1| 5, 10, 15, 26, 31, 41, 72f, 185cf }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}


Badness: 0.873 × 10<sup>-3</sup>
Badness (Sintel): 1.67


</div></div>
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


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


Period: 1\1
Mapping: {{mapping| 1 1 2 3 3 4 4 | 0 3 0 -1 4 -1 1 | 0 0 3 0 -3 -1 -1 }}


Optimal ([[POTE]]) generators: ~8/7 = 233.2909, ~5/4 = 384.2120
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}}


EDO generator: [[31edo|(6, 10)\31]], [[36edo|(7, 12)\36]], [[72edo|(14, 23)\72]]
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}


Scales (Scala files):  
Badness (Sintel): 1.24


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
== Gamel ==
<div style="line-height:1.6;">Technical data</div>
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.  
<div class="mw-collapsible-content">


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Comma list: 540/539, 1029/1024
[[Comma list]]: 540/539, 1029/1024


Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 0 -1 11 }}, {{val| 0 0 1 0 1 }}]
{{Mapping|legend=1| 1 1 0 3 -1 | 0 3 0 -1 11 | 0 0 1 0 1 }}


{{Val list|legend=1| 10, 31, 41, 72 }}
[[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 }}


Badness: 0.853 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 5e, 10, 21e, 26e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}


</div></div>
[[Badness]] (Sintel): 1.02


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

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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