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Gamelismic family: Difference between revisions

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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.
{{Main| Portent }}


Subgroup: 2.3.5.7
== Gamelismic ==
{{Main| Gamelismic and portent }}
 
[[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


[[Mapping 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 }}]
Line 16: Line 20:
: Angle (8/7, 5/4) = 90 degrees
: Angle (8/7, 5/4) = 90 degrees


[[POTE generator]]s: ~8/7 = 233.6875, ~5/4 = 385.1853
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}, ~5/4 = 385.3412{{c}}
[[Minimax tuning]]: c = 1029/1024
: [[error map]]: {{val| +0.486 -0.123 -0.001 -1.151 }}
* [[7-odd-limit]]: 3, 5, and 7 1/4c flat
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7474{{c}}, ~5/4 = 385.5205{{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| 0.000 -0.713 -0.793 -2.573 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 6/5
* [[9-odd-limit]]: 3 1/7c flat, 5 and 7 2/7c flat
: [{{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 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 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


{{Optimal ET sequence|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
Line 37: Line 43:


== Portent ==
== Portent ==
{{Main| Portent }}
{{Main| Gamelismic and portent }}


Portent has a [[Normal lists #normal interval list|normal comma list]] [1029/1024, 385/384] and also tempers out 441/440.
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
[[Subgroup]]: 2.3.5.7.11


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


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


Mapping 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 55: Line 59:
: Angle (8/7, 12/11) = 86.657 degrees
: Angle (8/7, 12/11) = 86.657 degrees


[[POTE generator]]s: ~8/7 = 233.6884, ~5/4 = 385.1618
[[Optimal tuning]]s:  
* [[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 = 1029/1024, e = 385/384
[[Minimax tuning]]: ''c''<sub>1</sub> = 1029/1024, ''c''<sub>2</sub> = 385/384
* [[11-odd-limit]]: 3 1/7c flat, 5 and 7 2/7c flat, 11 e-3/7c flat
* [[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 (unchanged-intervals): 2, 11/10, 9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5


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


[[Badness]]: 0.234 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.281


[[Projection pair]]s: 3 1024/343 11 131072/12005 to 2.5.7
[[Projection pair]]s: 3 1024/343 11 131072/12005 to 2.5.7
Line 75: Line 83:
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 }}
 
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}}


POTE generators: ~8/7 = 233.9128, ~5/4 = 384.7278
{{Optimal ET sequence|legend=0| 15, 26, 31f, 41, 46, 72, 87, 159 }}


{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 159, 477cdf }}
Badness (Sintel): 0.587


Badness: 0.627 × 10<sup>-3</sup>
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 ===
=== Portentous ===
Line 88: Line 100:
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 }}


POTE generators: ~8/7 = 233.6843, ~5/4 = 384.9830
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=1| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 349f, 452cdef, 611cdef }}
{{Optimal ET sequence|legend=0| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef }}


Badness: 0.662 × 10<sup>-3</sup>
Badness (Sintel): 0.618


=== Ominous ===
=== Ominous ===
Line 101: Line 115:
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 }}


POTE generators: ~8/7 = 233.4088, ~5/4 = 385.3825
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}}


{{Optimal ET sequence|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 ====
Line 114: Line 130:
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


Mapping 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 }}]
Line 124: Line 138:
: Angle (8/7, 6/5) = 98.6015
: Angle (8/7, 6/5) = 98.6015


POTE generators: ~8/7 = 233.4313, ~5/4 = 385.2890
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 (unchanged-intervals): 2, 13/11, 11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9.13/9


{{Optimal ET sequence|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


=== Momentous ===
=== Momentous ===
Line 140: Line 156:
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 }}


POTE generators: ~8/7 = 233.4088, ~5/4 = 385.3825
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=1| 10, 31, 41, 46, 77, 87, 118, 164, 205d, 574def }}
{{Optimal ET sequence|legend=0| 15f, 21e, 31, 41, 46, 72f, 77, 87, 118, 164, 205d }}


Badness: 0.832 × 10<sup>-3</sup>
Badness (Sintel): 0.778


=== Foreboding ===
=== Foreboding ===
Line 153: Line 171:
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 }}


POTE generators: ~8/7 = 233.3664, ~5/4 = 382.3425
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=1| 5, 10, 15, 26, 31, 41, 72f, 185cf }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 26, 31, 41, 72f }}


Badness: 0.873 × 10<sup>-3</sup>
Badness (Sintel): 0.816


=== Portannic ===
=== Portannic ===
Line 166: Line 186:
Comma list: 385/384, 441/440, 10985/10976
Comma list: 385/384, 441/440, 10985/10976


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


POTE generators: ~8/7 = 233.6434, ~14/13 = 128.3440
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=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}


Badness: 1.783 × 10<sup>-3</sup>
Badness (Sintel): 1.67


==== 17-limit ====
==== 17-limit ====
Line 179: Line 202:
Comma list: 273/272, 385/384, 441/440, 8624/8619
Comma list: 273/272, 385/384, 441/440, 8624/8619


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


POTE generators: ~8/7 = 233.6803, ~14/13 = 128.4149
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=1| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}


Badness: 1.303 × 10<sup>-3</sup>
Badness (Sintel): 1.24


== Gamel ==
== Gamel ==
Subgroup: 2.3.5.7.11
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.
 
[[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 }}


[[POTE generator]]s: ~8/7 = 233.2909, ~5/4 = 384.2120
[[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 }}


{{Optimal ET sequence|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:Gamelismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
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