Whitewood family: Difference between revisions
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{{Technical data page}} | {{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. | ||
== Whitewood == | == Whitewood == | ||
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Scales: [[7L 7s in 140edo]] | Scales: [[7L 7s in 140edo]] | ||
=== Overview to extensions === | |||
Temperaments discussed elsewhere include: | |||
* ''[[Jamesbond]]'' → [[7th-octave temperaments#Jamesbond|7th-octave temperaments]] | |||
* ''[[Sept]]'' → [[Very low accuracy temperaments #Sept|Very low accuracy temperaments]] | |||
Considered below are septimal whitewood, redwood, and greenwood. | |||
== Septimal whitewood == | == Septimal whitewood == | ||
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Badness (Sintel): 2.59 | Badness (Sintel): 2.59 | ||
== Greenwood == | == Greenwood == | ||
Revision as of 10:55, 29 May 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The 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.
Whitewood
Whitewood is the natural counterpart of blackwood: whereas blackwood can be thought of as a closed chain of five fifths and a 5/4 major third generator, whitewood is a closed chain of seven fifths and a 5/4 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.
Subgroup: 2.3.5
Comma list: 2187/2048
Mapping: [⟨7 11 0], ⟨0 0 1]]
- mapping generators: ~9/8, ~5
- WE: ~9/8 = 172.1541 ¢, ~5/4 = 376.0535 ¢ (~80/81 = 31.7453 ¢)
- error map: ⟨+5.079 -8.260 -0.102]
- CWE: ~9/8 = 171.4286 ¢, ~5/4 = 378.3830 ¢ (~80/81 = 35.5258 ¢)
- error map: ⟨0.000 -16.241 -7.931]
Optimal ET sequence: 7, 21, 28, 35, 77bbc
Badness (Sintel): 3.63
Scales: 7L 7s in 140edo
Overview to extensions
Temperaments discussed elsewhere include:
Considered below are septimal whitewood, redwood, and greenwood.
Septimal whitewood
Subgroup: 2.3.5.7
Comma list: 36/35, 2187/2048
Mapping: [⟨7 11 0 36], ⟨0 0 1 -1]]
- WE: ~9/8 = 171.5524 ¢, ~5/4 = 392.9834 ¢ (~64/63 = 49.8786 ¢)
- error map: ⟨+0.867 -14.879 +8.403 +12.343]
- CWE: ~9/8 = 171.4286 ¢, ~5/4 = 392.7412 ¢ (~64/63 = 49.8841 ¢)
- error map: ⟨0.000 -16.241 +6.428 +9.861]
Optimal ET sequence: 7, 14, 21, 28, 49b
Badness (Sintel): 2.88
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 2079/2048
Mapping: [⟨7 11 0 36 8], ⟨0 0 1 -1 1]]
Optimal tunings:
- WE: ~11/10 = 171.4451 ¢, ~5/4 = 390.0053 ¢ (~64/63 = 47.1151 ¢)
- CWE: ~11/10 = 171.4286 ¢, ~5/4 = 389.9864 ¢ (~64/63 = 47.1293 ¢)
Optimal ET sequence: 7, 14e, 21, 28
Badness (Sintel): 2.01
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 512/507
Mapping: [⟨7 11 0 36 8 26], ⟨0 0 1 -1 1 0]]
Optimal tunings:
- WE: ~11/10 = 171.3236 ¢, ~5/4 = 390.4957 ¢ (~64/63 = 47.8484 ¢)
- CWE: ~11/10 = 171.4286 ¢, ~5/4 = 390.6336 ¢ (~64/63 = 47.7765 ¢)
Optimal ET sequence: 7, 14e, 21, 28
Badness (Sintel): 1.65
Redwood
Subgroup: 2.3.5.7
Comma list: 525/512, 729/700
Mapping: [⟨7 11 0 52], ⟨0 0 1 -2]]
- WE: ~9/8 = 172.0521 ¢, ~5/4 = 379.5277 ¢ (~36/35 = 35.4234 ¢)
- error map: ⟨+4.365 -9.382 +1.944 +1.370]
- CWE: ~9/8 = 171.4286 ¢, ~5/4 = 377.7903 ¢ (~36/35 = 34.9331 ¢)
- error map: ⟨0.000 -16.241 -8.523 -10.121]
Optimal ET sequence: 7, 28d, 35
Badness (Sintel): 4.18
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 385/384, 729/700
Mapping: [⟨7 11 0 52 8], ⟨0 0 1 -2 1]]
Optimal tunings:
- WE: ~11/10 = 171.9390 ¢, ~5/4 = 377.8321 ¢ (~36/35 = 33.9542 ¢)
- CWE: ~11/10 = 171.4286 ¢, ~5/4 = 376.7162 ¢ (~36/35 = 33.8590 ¢)
Optimal ET sequence: 7, 28d, 35
Badness (Sintel): 2.59
Greenwood
Subgroup: 2.3.5.7
Comma list: 405/392, 1323/1280
Mapping: [⟨7 11 1 12], ⟨0 0 2 1]]
- mapping generators: ~9/8, ~15/7
- WE: ~9/8 = 172.1073 ¢, ~15/14 = 101.7681 ¢ (~21/20 = 70.3391 ¢)
- error map: ⟨+4.751 -8.775 -1.169 +2.980]
- CWE: ~9/8 = 171.4286 ¢, ~15/14 = 103.3802 ¢ (~21/20 = 68.0484 ¢)
- error map: ⟨0.000 -16.241 -8.125 -8.303]
Optimal ET sequence: 7c, 14c, 21, 35, 84bbccd
Badness (Sintel): 3.08
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 tunings:
- WE: ~11/10 = 172.0795 ¢, ~15/14 = 100.5259 ¢ (~21/20 = 71.5536 ¢)
- CWE: ~11/10 = 171.4286 ¢, ~15/14 = 102.1866 ¢ (~21/20 = 69.2419 ¢)
Optimal ET sequence: 7ce, 14c, 21, 35, 49bcde
Badness (Sintel): 1.90
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 45/44, 99/98, 640/637
Mapping: [⟨7 11 1 12 9 26], ⟨0 0 2 1 2 0]]
Optimal tunings:
- WE: ~11/10 = 171.6777 ¢, ~15/14 = 104.4016 ¢ (~21/20 = 67.2761 ¢)
- CWE: ~11/10 = 171.4286 ¢, ~15/14 = 104.8518 ¢ (~21/20 = 66.5768 ¢)
Optimal ET sequence: 7ce, 14c, 21, 35
Badness (Sintel): 2.23