Whitewood family: Difference between revisions

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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.
The '''apotome family''' or '''whitewood family''' of temperaments tempers out the apotome, [[2187/2048]]. Consequently the fifths 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.


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


[[POTE generator]]: ~5/4 = 374.469
Mapping generators: ~9/8, ~5
 
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 374.469


{{Val list|legend=1| 7, 21, 28, 35, 77bb }}
{{Val list|legend=1| 7, 21, 28, 35, 77bb }}
Line 22: Line 24:
[[Badness]]: 0.154651
[[Badness]]: 0.154651


=== 7-limit ===
== Septimal whitewood ==
Subgroup: 2.3.5.7
[[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]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 -1 }}]


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


[[POTE generator]]: ~5/4 = 392.700
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 392.700


{{Val list|legend=1| 7, 14, 21, 28, 49b }}
{{Val list|legend=1| 7, 14, 21, 28, 49b }}
Line 42: Line 44:
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: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 -1 1 }}]


POTE generator: ~5/4 = 389.968
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 389.968


Optimal GPV sequence: {{Val list| 7, 14e, 21, 28, 49b }}
Optimal GPV sequence: {{Val list| 7, 14e, 21, 28, 49b }}
Line 55: Line 57:
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: [{{val| 7 11 16 20 24 26 }}, {{val| 0 0 1 -1 1 0 }}]


POTE generator: ~5/4 = 390.735
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 390.735


Optimal GPV sequence: {{Val list| 7, 14e, 21, 28, 49bf }}
Optimal GPV sequence: {{Val list| 7, 14e, 21, 28, 49bf }}
Line 64: Line 66:


== 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]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 -2 }}]


{{Multival|legend=1|0 7 -14 11 -22 -52}}
{{Multival|legend=1| 0 7 -14 11 -22 -52 }}


[[POTE generator]]: ~5/4 = 378.152
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 378.152


{{Val list|legend=1| 7, 21d, 28d, 35 }}
{{Val list|legend=1| 7, 21d, 28d, 35 }}
Line 83: Line 85:
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: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 -2 1 }}]


POTE generator: ~5/4 = 376.711
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 376.711


Optimal GPV sequence: {{Val list| 7, 21d, 28d, 35 }}
Optimal GPV sequence: {{Val list| 7, 21d, 28d, 35 }}
Line 92: Line 94:


== 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]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 0 }}]


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


[[POTE generator]]: ~5/4 = 395.187
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 395.187


{{Val list|legend=1| 7, 14d, 21dd }}
{{Val list|legend=1| 7, 14d, 21dd }}
Line 111: Line 113:
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: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 0 1 }}]


POTE generator: ~5/4 = 394.661
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 394.661


Optimal GPV sequence: {{Val list| 7, 14de, 21dd }}
Optimal GPV sequence: {{Val list| 7, 14de, 21dd }}
Line 124: Line 126:
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: [{{val| 7 11 16 20 24 26 }}, {{val| 0 0 1 0 1 0 }}]


POTE generator: ~5/4 = 395.071
Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.071


Optimal GPV sequence: {{Val list| 7, 14de, 21dd }}
Optimal GPV sequence: {{Val list| 7, 14de, 21dd }}
Line 133: Line 135:


== Greenwood ==
== Greenwood ==
{{see also|Greenwoodmic temperaments #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]]: [{{val| 7 11 1 12 }}, {{val| 0 0 2 1 }}]


{{Multival|legend=1|0 14 7 22 11 -23}}
{{Multival|legend=1| 0 14 7 22 11 -23 }}


[[POTE generator]]: ~8/7 = 241.490
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~8/7 = 241.490


{{Val list|legend=1| 14c, 21, 35 }}
{{Val list|legend=1| 14c, 21, 35 }}
Line 154: Line 156:
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: [{{val| 7 11 1 12 9 }}, {{val| 0 0 2 1 2 }}]


POTE generator: ~8/7 = 242.711
Optimal tuning (POTE): ~9/8 = 1\7, ~8/7 = 242.711


Optimal GPV sequence: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}
Optimal GPV sequence: {{Val list| 14c, 21, 35, 49bcde, 84bbccde }}

Revision as of 10:10, 21 December 2022

The apotome family or whitewood 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.

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 5n-edos, and whitewood is supported by 7n-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.

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.

Whitewood

Subgroup: 2.3.5

Comma list: 2187/2048

Mapping: [⟨7 11 16], ⟨0 0 1]]

Mapping generators: ~9/8, ~5

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 374.469

Template:Val list

Badness: 0.154651

Septimal whitewood

Subgroup: 2.3.5.7

Comma list: 36/35, 2187/2048

Mapping: [⟨7 11 16 20], ⟨0 0 1 -1]]

Wedgie: ⟨⟨ 0 7 -7 11 -11 -36 ]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 392.700

Template:Val list

Badness: 0.113987

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 11 16 20 24], ⟨0 0 1 -1 1]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 389.968

Optimal GPV sequence: Template:Val list

Badness: 0.060908

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 512/507

Mapping: [⟨7 11 16 20 24 26], ⟨0 0 1 -1 1 0]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 390.735

Optimal GPV sequence: Template:Val list

Badness: 0.039956

Redwood

Subgroup: 2.3.5.7

Comma list: 525/512, 729/700

Mapping: [⟨7 11 16 20], ⟨0 0 1 -2]]

Wedgie: ⟨⟨ 0 7 -14 11 -22 -52 ]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 378.152

Template:Val list

Badness: 0.165257

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 11 16 20 24], ⟨0 0 1 -2 1]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 376.711

Optimal GPV sequence: Template:Val list

Badness: 0.078193

Mujannab

Subgroup: 2.3.5.7

Comma list: 54/49, 64/63

Mapping: [⟨7 11 16 20], ⟨0 0 1 0]]

Wedgie: ⟨⟨ 0 7 0 11 0 -20 ]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.187

Template:Val list

Badness: 0.105820

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 54/49, 64/63

Mapping: [⟨7 11 16 20 24], ⟨0 0 1 0 1]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 394.661

Optimal GPV sequence: Template:Val list

Badness: 0.060985

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 52/49, 64/63

Mapping: [⟨7 11 16 20 24 26], ⟨0 0 1 0 1 0]]

Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.071

Optimal GPV sequence: Template:Val list

Badness: 0.042830

Greenwood

See also: Greenwoodmic temperaments § Greenwood

Subgroup: 2.3.5.7

Comma list: 405/392, 1323/1280

Mapping: [⟨7 11 1 12], ⟨0 0 2 1]]

Wedgie: ⟨⟨ 0 14 7 22 11 -23 ]]

Optimal tuning (POTE): ~9/8 = 1\7, ~8/7 = 241.490

Template:Val list

Badness: 0.121752

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 1323/1280

Mapping: [⟨7 11 1 12 9], ⟨0 0 2 1 2]]

Optimal tuning (POTE): ~9/8 = 1\7, ~8/7 = 242.711

Optimal GPV sequence: Template:Val list

Badness: 0.057471

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