Whitewood family

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 Pythagorean 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

Main article: 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

Optimal tunings:

• 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:

• Sept → Very low accuracy temperaments

Considered below are septimal whitewood, redwood, greenwood, and jamesbond.

Septimal whitewood

Main article: Whitewood

Subgroup: 2.3.5.7

Comma list: 36/35, 2187/2048

Mapping: [⟨7 11 0 36], ⟨0 0 1 -1]]

Optimal tunings:

• 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]]

Optimal tunings:

• 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

Optimal tunings:

• 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

Jamesbond

This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its "wedgie" (a kind of mathematical object representing the temperament) starts with ⟨⟨ 0 0 7 … ]] (in fact, it is ⟨⟨ 0 0 7 0 11 16 ]])

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]

mapping generators: ~10/9, ~7

Optimal tunings:

• WE: ~10/9 = 172.790 ¢, ~7/4 = 949.343 ¢

error map: ⟨+9.533 -1.261 -21.668 -0.418]

• CWE: ~10/9 = 171.429 ¢, ~7/4 = 948.499 ¢

error map: ⟨-0.000 -16.241 -43.457 -20.327]

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 1.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 45/44

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

Optimal tunings:

• WE: ~10/9 = 172.830 ¢, ~7/4 = 948.784 ¢
• CWE: ~10/9 = 171.429 ¢, ~7/4 = 946.554 ¢

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 0.778

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 27/26, 33/32, 40/39

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

Optimal tunings:

• WE: ~10/9 = 172.390 ¢, ~7/4 = 954.559 ¢
• CWE: ~10/9 = 171.429 ¢, ~7/4 = 952.367 ¢

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 0.951

Austinpowers

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 33/32, 45/44, 65/63

Mapping: [⟨7 11 16 0 24 6], ⟨0 0 0 1 0 1]]

Optimal tunings:

• WE: ~10/9 = 172.873 ¢, ~7/4 = 960.581 ¢
• CWE: ~10/9 = 171.429 ¢, ~7/4 = 958.793 ¢

Optimal ET sequence: 7(df), 14cf

Badness (Sintel): 0.933