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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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== 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}}


[[POTE generator]]: ~5/4 = 392.700
[[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 }}


{{Val list|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 ===
Line 42: Line 51:
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)


Optimal GPV sequence: {{Val list| 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 ===
Line 55: Line 66:
Comma list: 27/26, 36/35, 45/44, 512/507
Comma list: 27/26, 36/35, 45/44, 512/507


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


POTE generator: ~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 GPV sequence: {{Val list| 7, 14e, 21, 28, 49bf }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28, 49bf }}


Badness: 0.039956
Badness (Smith): 0.039956


== Redwood ==
== Redwood ==
Subgroup: 2.3.5.7
[[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 ===
Line 83: Line 98:
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)


Optimal GPV sequence: {{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 }}


{{Multival|legend=1|0 7 0 11 0 -20}}
[[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 }}


[[POTE generator]]: ~5/4 = 395.187
{{Optimal ET sequence|legend=1| 7, 14d }}


{{Val list|legend=1| 7, 14d, 21dd }}
[[Badness]] (Smith): 0.105820
 
[[Badness]]: 0.105820


=== 11-limit ===
=== 11-limit ===
Line 111: Line 130:
Comma list: 45/44, 54/49, 64/63
Comma list: 45/44, 54/49, 64/63


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


POTE generator: ~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 GPV sequence: {{Val list| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7, 14de }}


Badness: 0.060985
Badness (Smith): 0.060985


=== 13-limit ===
=== 13-limit ===
Line 124: Line 145:
Comma list: 27/26, 45/44, 52/49, 64/63
Comma list: 27/26, 45/44, 52/49, 64/63


Mapping: [{{val|7 11 16 20 24 26}}, {{val|0 0 1 0 1 0}}]
Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}


POTE generator: ~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 GPV sequence: {{Val list| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7, 14de }}


Badness: 0.042830
Badness (Smith): 0.042830


== Greenwood ==
== Greenwood ==
{{see also|Greenwoodmic temperaments #Greenwood}}
[[Subgroup]]: 2.3.5.7
 
Subgroup: 2.3.5.7


[[Comma list]]: 405/392, 1323/1280
[[Comma list]]: 405/392, 1323/1280


[[Mapping]]: [{{val|7 11 1 12}}, {{val|0 0 2 1}}]
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}


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


[[POTE generator]]: ~8/7 = 241.490
[[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 }}


{{Val list|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 ===
Line 154: Line 179:
Comma list: 45/44, 99/98, 1323/1280
Comma list: 45/44, 99/98, 1323/1280


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


POTE generator: ~8/7 = 242.711
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 GPV sequence: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}


Badness: 0.057471
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"