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This family of temperaments tempers out the apotome, [[2187/2048]]. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
{{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 5-limit version of this temperament is called "whitewood" temperament, to serve in contrast with the "blackwood" temperament which tempers out [[256/243]], the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5''n''-EDOs, and whitewood is supported by 7''n''-EDOs, and the MOS of both scales follow a similar pattern.
The 5-limit version of this temperament is called ''whitewood'', to serve in contrast with the [[blackwood]] temperament which tempers out [[256/243]], the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5''n''-edos, and whitewood is supported by 7''n''-edos, and the [[mos]] of both scales follow a similar pattern.


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.
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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.


= Whitewood =
== Whitewood ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: 2187/2048
[[Comma list]]: 2187/2048


[[Mapping]]: [{{val|7 11 16}}, {{val|0 0 1}}]
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}


[[POTE generator]]: ~5/4 = 374.469
: mapping generators: ~9/8, ~5


{{Val list|legend=1| 7, 21, 28, 35, 77bb }}
[[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 }}


[[Badness]]: 0.154651
{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bb }}


== 7-limit ==
[[Badness]] (Smith): 0.154651
Subgroup: 2.3.5.7
 
== Septimal whitewood ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2187/2048
[[Comma list]]: 36/35, 2187/2048


[[Mapping]]: [{{val|7 11 16 20}}, {{val|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)
: [[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 }}


[[POTE generator]]: ~5/4 = 392.700
{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}


{{Val list|legend=1| 7, 14, 21, 28, 49b }}
[[Badness]] (Smith): 0.113987


[[Badness]]: 0.113987
=== 11-limit ===
 
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 36/35, 45/44, 2079/2048
Comma list: 36/35, 45/44, 2079/2048


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


POTE generator: ~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)


Vals: {{Val list| 7, 14e, 21, 28, 49b }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49b }}


Badness: 0.060908
Badness (Smith): 0.060908


= Redwood =
=== 13-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 512/507
 
Mapping: {{mapping| 7 11 0 36 8 26 | 0 0 1 -1 1 0 }}
 
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=0| 7, 14e, 21, 28, 49bf }}
 
Badness (Smith): 0.039956
 
== Redwood ==
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|7 11 16 20}}, {{val|0 0 1 -2}}]
{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
 
{{Multival|legend=1|0 7 -14 11 -22 -52}}


[[POTE generator]]: ~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 }}


{{Val list|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 ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 385/384, 729/700
Comma list: 45/44, 385/384, 729/700


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


POTE generator: ~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)


Vals: {{Val list| 7, 21d, 28d, 35 }}
{{Optimal ET sequence|legend=0| 7, 28d, 35 }}


Badness: 0.078193
Badness (Smith): 0.078193


= Mujannab =
== Mujannab ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|7 11 16 20}}, {{val|0 0 1 0}}]
{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
 
[[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 }}
 
[[Badness]] (Smith): 0.105820
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 54/49, 64/63
 
Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}
 
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=0| 7, 14de }}
 
Badness (Smith): 0.060985
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 52/49, 64/63
 
Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
 
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=0| 7, 14de }}
 
Badness (Smith): 0.042830
 
== Greenwood ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 405/392, 1323/1280


{{Multival|legend=1|0 7 0 11 0 -20}}
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}


[[POTE generator]]: ~5/4 = 395.187
: mapping generators: ~9/8, ~15/7


{{Val list|legend=1| 7, 14d, 21dd }}
[[Optimal tuning]]s:
* [[CTE]]: ~9/8 = 171.429, ~15/14 = 108.062 (~21/20 = 63.367)
: [[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 }}


[[Badness]]: 0.105820
{{Optimal ET sequence|legend=1| 7c, 14c, 21, 35, 84bbccd }}


== 11-limit ==
[[Badness]] (Smith): 0.121752
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 54/49, 64/63
Comma list: 45/44, 99/98, 1323/1280
 
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}
 
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=0| 7ce, 14c, 21, 35, 49bcde }}
 
Badness (Smith): 0.057471
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 99/98, 640/637


Mapping: [{{val|7 11 16 20 24}}, {{val|0 0 1 0 1}}]
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}


POTE generator: ~5/4 = 394.661
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)


Vals: {{Val list| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}


Badness: 0.060985
Badness (Smith): 0.054009


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Apotomic]]
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Whitewood| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Whitewood_family"