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Whitewood family: Difference between revisions

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+CTE tunings and error maps; restore POTE tunings for greenwood. Spell 1\7 etc. in decimal. Note the badness metric being used. Misc. cleanup
 
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{{Technical data page}}
The '''whitewood family''' of [[temperament]]s [[tempering out|tempers out]] the apotome, [[2187/2048]]. Consequently the [[3/2|fifth]]s are always 4/7 of an [[octave]], a distinctly flat 685.714 [[cent]]s. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
The '''whitewood family''' of [[temperament]]s [[tempering out|tempers out]] the apotome, [[2187/2048]]. Consequently the [[3/2|fifth]]s are always 4/7 of an [[octave]], a distinctly flat 685.714 [[cent]]s. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314 (~80/81 = 43.457)
: [[error map]]: {{val| 0.000 -16.241 0.000 }}
: [[error map]]: {{val| 0.000 -16.241 0.000 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 374.469
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 374.469 (~80/81 = 31.612)
: error map: {{val| 0.000 -16.241 -11.845 }}
: error map: {{val| 0.000 -16.241 -11.845 }}


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{{Mapping|legend=1| 7 11 0 36 | 0 0 1 -1 }}
{{Mapping|legend=1| 7 11 0 36 | 0 0 1 -1 }}
{{Multival|legend=1| 0 7 -7 11 -11 -36 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 392.930
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 392.930 (~64/63 = 50.073)
: [[error map]]: {{val| 0.000 -16.241 +6.617 +9.672 }}
: [[error map]]: {{val| 0.000 -16.241 +6.617 +9.672 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 392.700
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 392.700 (~64/63 = 49.843)
: error map: {{val| 0.000 -16.241 +6.386 +9.903 }}
: error map: {{val| 0.000 -16.241 +6.386 +9.903 }}


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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~5/4 = 390.178
* CTE: ~9/8 = 171.429, ~5/4 = 390.178 (~64/63 = 47.321)
* POTE: ~9/8 = 171.429, ~5/4 = 389.968
* POTE: ~9/8 = 171.429, ~5/4 = 389.968 (~64/63 = 47.111)


{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49b }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49b }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~5/4 = 390.178
* CTE: ~9/8 = 171.429, ~5/4 = 390.178 (~64/63 = 47.321)
* POTE: ~9/8 = 171.429, ~5/4 = 390.735
* POTE: ~9/8 = 171.429, ~5/4 = 390.735 (~64/63 = 47.878)


{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49bf }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49bf }}
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{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
{{Multival|legend=1| 0 7 -14 11 -22 -52 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 376.366
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 376.366 (~36/35 = 33.509)
: [[error map]]: {{val| 0.000 -16.241 -9.948 -7.271 }}
: [[error map]]: {{val| 0.000 -16.241 -9.948 -7.271 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 378.152
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 378.152 (~36/35 = 35.295)
: error map: {{val| 0.000 -16.241 -8.162 -10.845 }}
: error map: {{val| 0.000 -16.241 -8.162 -10.845 }}


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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~5/4 = 376.745
* CTE: ~9/8 = 171.429, ~5/4 = 376.745 (~36/35 = 33.888)
* POTE: ~9/8 = 171.429, ~5/4 = 376.711
* POTE: ~9/8 = 171.429, ~5/4 = 376.711 (~36/35 = 33.854)


{{Optimal ET sequence|legend=0| 7, 28d, 35 }}
{{Optimal ET sequence|legend=0| 7, 28d, 35 }}
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{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
{{Multival|legend=1| 0 7 0 11 0 -20 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314 (~80/81 = 43.457)
: [[error map]]: {{val| 0.000 -16.241 0.000 +59.746 }}
: [[error map]]: {{val| 0.000 -16.241 0.000 +59.746 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 395.187
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 395.187 (~15/14 = 52.330)
: error map: {{val| 0.000 -16.241 +8.873 +59.746 }}
: error map: {{val| 0.000 -16.241 +8.873 +59.746 }}


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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~5/4 = 384.318
* CTE: ~9/8 = 171.429, ~5/4 = 384.318 (~33/32 = 41.461)
* POTE: ~9/8 = 171.429, ~5/4 = 394.661
* POTE: ~9/8 = 171.429, ~5/4 = 394.661 (~33/32 = 51.804)


{{Optimal ET sequence|legend=0| 7, 14de }}
{{Optimal ET sequence|legend=0| 7, 14de }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~5/4 = 384.318
* CTE: ~9/8 = 171.429, ~5/4 = 384.318 (~33/32 = 41.461)
* POTE: ~9/8 = 171.429, ~5/4 = 395.071
* POTE: ~9/8 = 171.429, ~5/4 = 395.071 (~33/32 = 52.214)


{{Optimal ET sequence|legend=0| 7, 14de }}
{{Optimal ET sequence|legend=0| 7, 14de }}
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: mapping generators: ~9/8, ~15/7
: mapping generators: ~9/8, ~15/7
{{Multival|legend=1| 0 14 7 22 11 -23 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 171.429, ~15/14 = 108.062
* [[CTE]]: ~9/8 = 171.429, ~15/14 = 108.062 (~21/20 = 63.367)
: [[error map]]: {{val| 0.000 -16.241 +1.239 -3.621 }}
: [[error map]]: {{val| 0.000 -16.241 +1.239 -3.621 }}
* [[POTE]]: ~9/8 = 171.429, ~15/14 = 101.367
* [[POTE]]: ~9/8 = 171.429, ~15/14 = 101.367 (~21/20 = 70.062)
: error map: {{val| 0.000 -16.241 -12.152 -10.316 }}
: error map: {{val| 0.000 -16.241 -12.152 -10.316 }}


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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~15/14 = 106.997
* CTE: ~9/8 = 171.429, ~15/14 = 106.997 (~21/20 = 64.432)
* POTE: ~9/8 = 171.429, ~15/14 = 100.046
* POTE: ~9/8 = 171.429, ~15/14 = 100.046 (~21/20 = 71.383)


{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~9/8 = 171.429, ~15/14 = 106.997
* CTE: ~9/8 = 171.429, ~15/14 = 106.997 (~21/20 = 64.432)
* POTE: ~9/8 = 171.429, ~15/14 = 104.250
* POTE: ~9/8 = 171.429, ~15/14 = 104.250 (~21/20 = 67.179)


{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Whitewood| ]] <!-- key article -->
[[Category:Whitewood| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Whitewood_family"