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Whitewood family: Difference between revisions

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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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The 14-note mos of whitewood, like the 10-note mos of blackwood, shares a number of interesting properties which derive from the relatively small circle of fifths common to both. From any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal", since resolutions that work in one key will work in all other keys in the scale, at least keys that share the same chord quality.
The 14-note mos of whitewood, like the 10-note mos of blackwood, shares a number of interesting properties which derive from the relatively small circle of fifths common to both. From any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal", since resolutions that work in one key will work in all other keys in the scale, at least keys that share the same chord quality.


Another interesting property is that it becomes possible to construct "super-linked" 5-limit chords. In whitewood[14], or blackwood[10], if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that whitewood[14] and blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.
Another interesting property is that it becomes possible to construct "super-linked" 5-limit chords. In Whitewood[14], or Blackwood[10], if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that Whitewood[14] and Blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.


Lastly, while blackwood fifths are sharp and thus necessitate the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.
Lastly, while blackwood fifths are sharp and thus necessitate the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.
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[[Comma list]]: 2187/2048
[[Comma list]]: 2187/2048


{{Mapping|legend=1| 7 11 16 | 0 0 1 }}
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}


: mapping generators: ~9/8, ~5
: mapping generators: ~9/8, ~5


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 374.469
[[Optimal tuning]]s:
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314 (~80/81 = 43.457)
: [[error map]]: {{val| 0.000 -16.241 0.000 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 374.469 (~80/81 = 31.612)
: error map: {{val| 0.000 -16.241 -11.845 }}


{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bb }}
{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bb }}


[[Badness]]: 0.154651
[[Badness]] (Smith): 0.154651


== Septimal whitewood ==
== Septimal whitewood ==
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[[Comma list]]: 36/35, 2187/2048
[[Comma list]]: 36/35, 2187/2048


{{Mapping|legend=1| 7 11 16 20 | 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:
 
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 392.930 (~64/63 = 50.073)
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 392.700
: [[error map]]: {{val| 0.000 -16.241 +6.617 +9.672 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 392.700 (~64/63 = 49.843)
: error map: {{val| 0.000 -16.241 +6.386 +9.903 }}


{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}
{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}


[[Badness]]: 0.113987
[[Badness]] (Smith): 0.113987


=== 11-limit ===
=== 11-limit ===
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Comma list: 36/35, 45/44, 2079/2048
Comma list: 36/35, 45/44, 2079/2048


Mapping: {{mapping| 7 11 16 20 24 | 0 0 1 -1 1 }}
Mapping: {{mapping| 7 11 0 36 8 | 0 0 1 -1 1 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 389.968
Optimal tunings:
* CTE: ~9/8 = 171.429, ~5/4 = 390.178 (~64/63 = 47.321)
* POTE: ~9/8 = 171.429, ~5/4 = 389.968 (~64/63 = 47.111)


{{Optimal ET sequence|legend=1| 7, 14e, 21, 28, 49b }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49b }}


Badness: 0.060908
Badness (Smith): 0.060908


=== 13-limit ===
=== 13-limit ===
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Comma list: 27/26, 36/35, 45/44, 512/507
Comma list: 27/26, 36/35, 45/44, 512/507


Mapping: {{mapping| 7 11 16 20 24 26 | 0 0 1 -1 1 0 }}
Mapping: {{mapping| 7 11 0 36 8 26 | 0 0 1 -1 1 0 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 390.735
Optimal tunings:
* CTE: ~9/8 = 171.429, ~5/4 = 390.178 (~64/63 = 47.321)
* POTE: ~9/8 = 171.429, ~5/4 = 390.735 (~64/63 = 47.878)


{{Optimal ET sequence|legend=1| 7, 14e, 21, 28, 49bf }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49bf }}


Badness: 0.039956
Badness (Smith): 0.039956


== Redwood ==
== Redwood ==
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[[Comma list]]: 525/512, 729/700
[[Comma list]]: 525/512, 729/700


{{Mapping|legend=1| 7 11 16 20 | 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]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 378.152
[[Optimal tuning]]s:
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 376.366 (~36/35 = 33.509)
: [[error map]]: {{val| 0.000 -16.241 -9.948 -7.271 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 378.152 (~36/35 = 35.295)
: error map: {{val| 0.000 -16.241 -8.162 -10.845 }}


{{Optimal ET sequence|legend=1| 7, 21d, 28d, 35 }}
{{Optimal ET sequence|legend=1| 7, 28d, 35 }}


[[Badness]]: 0.165257
[[Badness]] (Smith): 0.165257


=== 11-limit ===
=== 11-limit ===
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Comma list: 45/44, 385/384, 729/700
Comma list: 45/44, 385/384, 729/700


Mapping: {{mapping| 7 11 16 20 24 | 0 0 1 -2 1 }}
Mapping: {{mapping| 7 11 0 52 8 | 0 0 1 -2 1 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 376.711
Optimal tunings:
* CTE: ~9/8 = 171.429, ~5/4 = 376.745 (~36/35 = 33.888)
* POTE: ~9/8 = 171.429, ~5/4 = 376.711 (~36/35 = 33.854)


{{Optimal ET sequence|legend=1| 7, 21d, 28d, 35 }}
{{Optimal ET sequence|legend=0| 7, 28d, 35 }}


Badness: 0.078193
Badness (Smith): 0.078193


== Mujannab ==
== Mujannab ==
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[[Comma list]]: 54/49, 64/63
[[Comma list]]: 54/49, 64/63


{{Mapping|legend=1| 7 11 16 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]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 395.187
[[Optimal tuning]]s:
* [[CTE]]: ~9/8 = 171.429, ~5/4 = 386.314 (~80/81 = 43.457)
: [[error map]]: {{val| 0.000 -16.241 0.000 +59.746 }}
* [[POTE]]: ~9/8 = 171.429, ~5/4 = 395.187 (~15/14 = 52.330)
: error map: {{val| 0.000 -16.241 +8.873 +59.746 }}


{{Optimal ET sequence|legend=1| 7, 14d, 21dd }}
{{Optimal ET sequence|legend=1| 7, 14d }}


[[Badness]]: 0.105820
[[Badness]] (Smith): 0.105820


=== 11-limit ===
=== 11-limit ===
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Comma list: 45/44, 54/49, 64/63
Comma list: 45/44, 54/49, 64/63


Mapping: {{mapping| 7 11 16 20 24 | 0 0 1 0 1 }}
Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 394.661
Optimal tunings:
* CTE: ~9/8 = 171.429, ~5/4 = 384.318 (~33/32 = 41.461)
* POTE: ~9/8 = 171.429, ~5/4 = 394.661 (~33/32 = 51.804)


{{Optimal ET sequence|legend=1| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7, 14de }}


Badness: 0.060985
Badness (Smith): 0.060985


=== 13-limit ===
=== 13-limit ===
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Comma list: 27/26, 45/44, 52/49, 64/63
Comma list: 27/26, 45/44, 52/49, 64/63


Mapping: {{mapping| 7 11 16 20 24 26 | 0 0 1 0 1 0 }}
Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.071
Optimal tunings:
* CTE: ~9/8 = 171.429, ~5/4 = 384.318 (~33/32 = 41.461)
* POTE: ~9/8 = 171.429, ~5/4 = 395.071 (~33/32 = 52.214)


{{Optimal ET sequence|legend=1| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7, 14de }}


Badness: 0.042830
Badness (Smith): 0.042830


== Greenwood ==
== Greenwood ==
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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:
 
* [[CTE]]: ~9/8 = 171.429, ~15/14 = 108.062 (~21/20 = 63.367)
[[Optimal tuning]] ([[CTE]]): ~9/8 = 1\7, ~15/14 = 108.062
: [[error map]]: {{val| 0.000 -16.241 +1.239 -3.621 }}
* [[POTE]]: ~9/8 = 171.429, ~15/14 = 101.367 (~21/20 = 70.062)
: error map: {{val| 0.000 -16.241 -12.152 -10.316 }}


{{Optimal ET sequence|legend=1| 14c, 21, 35 }}
{{Optimal ET sequence|legend=1| 7c, 14c, 21, 35, 84bbccd }}


[[Badness]]: 0.121752
[[Badness]] (Smith): 0.121752


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}


Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
Optimal tunings:
* CTE: ~9/8 = 171.429, ~15/14 = 106.997 (~21/20 = 64.432)
* POTE: ~9/8 = 171.429, ~15/14 = 100.046 (~21/20 = 71.383)


{{Optimal ET sequence|legend=1| 14c, 21, 35, 49bcde, 84bbccde }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}


Badness: 0.057471
Badness (Smith): 0.057471


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}


Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
Optimal tunings:
* CTE: ~9/8 = 171.429, ~15/14 = 106.997 (~21/20 = 64.432)
* POTE: ~9/8 = 171.429, ~15/14 = 104.250 (~21/20 = 67.179)


{{Optimal ET sequence|legend=1| 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}


Badness: 0.054009
Badness (Smith): 0.054009


[[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"