Xenharmonic Wiki

Gamelismic family: Difference between revisions

← Older edit
VisualWikitext
m Units & misc. cleanup
m Recategorize
 
(5 intermediate revisions by 2 users not shown)
Line 20: Line 20:
: Angle (8/7, 5/4) = 90 degrees
: Angle (8/7, 5/4) = 90 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 233.6875{{c}}, ~5/4 = 385.1853{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}, ~5/4 = 385.3412{{c}}
: [[error map]]: {{val| +0.486 -0.123 -0.001 -1.151 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.7474{{c}}, ~5/4 = 385.5205{{c}}
: error map: {{val| 0.000 -0.713 -0.793 -2.573 }}


[[Minimax tuning]]: ''c'' = 1029/1024
[[Minimax tuning]]: ''c'' = 1029/1024
Line 30: Line 34:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7


[[TE tuning|TE]] [[tuning map]]: {{val| 1200.486 1901.833 2786.314 3367.676 }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 190 }}


{{Optimal ET sequence|legend=1| 5, 10, 15, 26, 31, 41, 72, 118, 159, 190 }}
[[Badness]] (Sintel): 0.777
 
[[Badness]] (Smith): 0.176 × 10<sup>-3</sup>


[[Projection pair]]: 3 1024/343 to 2.5.7
[[Projection pair]]: 3 1024/343 to 2.5.7
Line 43: Line 45:
{{Main| Gamelismic and 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
Line 57: Line 59:
: Angle (8/7, 12/11) = 86.657 degrees
: Angle (8/7, 12/11) = 86.657 degrees


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 233.6884{{c}}, ~5/4 = 385.1618{{c}}
[[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''<sub>1</sub> = 1029/1024, ''c''<sub>2</sub> = 385/384
[[Minimax tuning]]: ''c''<sub>1</sub> = 1029/1024, ''c''<sub>2</sub> = 385/384
Line 64: Line 70:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5
: [[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]] (Smith): 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 79: Line 85:
Mapping: {{mapping| 1 1 0 3 5 6 | 0 3 0 -1 4 12 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.9128{{c}}, ~5/4 = 384.7278{{c}}
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}}


{{Optimal ET sequence|legend=0| 15, 26, 41, 46, 72, 87, 159, 477cdf }}
{{Optimal ET sequence|legend=0| 15, 26, 31f, 41, 46, 72, 87, 159 }}


Badness (Smith): 0.627 × 10<sup>-3</sup>
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
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
Line 94: Line 102:
Mapping: {{mapping| 1 1 0 3 5 -5 | 0 3 0 -1 4 -3 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.6843{{c}}, ~5/4 = 384.9830{{c}}
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, 349f, 452cdef, 611cdef }}
{{Optimal ET sequence|legend=0| 15, 31, 56, 72, 87, 103, 159, 190, 262df, 452cdef, 611cddef }}


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


=== Ominous ===
=== Ominous ===
Line 107: Line 117:
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 -10 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.4088{{c}}, ~5/4 = 385.3825{{c}}
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=0| 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 (Smith): 0.751 × 10<sup>-3</sup>
Badness (Sintel): 0.702


==== 17-limit ====
==== 17-limit ====
Line 126: Line 138:
: Angle (8/7, 6/5) = 98.6015
: Angle (8/7, 6/5) = 98.6015


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.4313{{c}}, ~5/4 = 385.2890{{c}}
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:  
Line 133: Line 147:
: unchanged-interval (eigenmonzo) basis: 2.11/9.13/9
: unchanged-interval (eigenmonzo) basis: 2.11/9.13/9


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


Badness (Smith): 0.612 × 10<sup>-3</sup>
Badness (Sintel): 0.582


=== Momentous ===
=== Momentous ===
Line 144: Line 158:
Mapping: {{mapping| 1 1 0 3 5 7 | 0 3 0 -4 1 -5 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.4088{{c}}, ~5/4 = 385.3825{{c}}
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| 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 (Smith): 0.832 × 10<sup>-3</sup>
Badness (Sintel): 0.778


=== Foreboding ===
=== Foreboding ===
Line 157: Line 173:
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 0 -1 4 2 | 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 tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.3664{{c}}, ~5/4 = 382.3425{{c}}
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, 26, 31, 41, 72f, 185cf }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 26, 31, 41, 72f }}


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


=== Portannic ===
=== Portannic ===
Line 169: Line 187:


Mapping: {{mapping| 1 1 2 3 3 4 | 0 3 0 -1 4 -1 | 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


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.6434{{c}}, ~14/13 = 128.3440{{c}}
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 }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 570ddeff, 832bcdddeefff }}


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


==== 17-limit ====
==== 17-limit ====
Line 183: Line 204:
Mapping: {{mapping| 1 1 2 3 3 4 4 | 0 3 0 -1 4 -1 1 | 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 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~8/7 = 233.6803{{c}}, ~14/13 = 128.4149{{c}}
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 }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 93e, 102, 103, 149, 159, 262df, 308def }}


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


== Gamel ==
== 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.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 196: Line 221:
{{Mapping|legend=1| 1 1 0 3 -1 | 0 3 0 -1 11 | 0 0 1 0 1 }}
{{Mapping|legend=1| 1 1 0 3 -1 | 0 3 0 -1 11 | 0 0 1 0 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~8/7 = 233.2909{{c}}, ~5/4 = 384.2120{{c}}
[[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]] (Smith): 0.853 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.02


[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Gamelismic family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Gamelismic_family"