Whitewood family

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.

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 0], ⟨0 0 1]]

mapping generators: ~9/8, ~5

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 386.314

error map: ⟨0.000 -16.241 0.000]

• POTE: ~9/8 = 171.429, ~5/4 = 374.469

error map: ⟨0.000 -16.241 -11.845]

Optimal ET sequence: 7, 21, 28, 35, 77bb

Badness (Smith): 0.154651

Septimal whitewood

Subgroup: 2.3.5.7

Comma list: 36/35, 2187/2048

Mapping: [⟨7 11 0 36], ⟨0 0 1 -1]]

Wedgie: ⟨⟨ 0 7 -7 11 -11 -36 ]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 392.930

error map: ⟨0.000 -16.241 +6.617 +9.672]

• POTE: ~9/8 = 171.429, ~5/4 = 392.700

error map: ⟨0.000 -16.241 +6.386 +9.903]

Optimal ET sequence: 7, 14, 21, 28, 49b

Badness (Smith): 0.113987

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:

• CTE: ~9/8 = 171.429, ~5/4 = 390.178
• POTE: ~9/8 = 171.429, ~5/4 = 389.968

Optimal ET sequence: 7, 14e, 21, 28, 49b

Badness (Smith): 0.060908

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:

• CTE: ~9/8 = 171.429, ~5/4 = 390.178
• POTE: ~9/8 = 171.429, ~5/4 = 390.735

Optimal ET sequence: 7, 14e, 21, 28, 49bf

Badness (Smith): 0.039956

Redwood

Subgroup: 2.3.5.7

Comma list: 525/512, 729/700

Mapping: [⟨7 11 0 52], ⟨0 0 1 -2]]

Wedgie: ⟨⟨ 0 7 -14 11 -22 -52 ]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 376.366

error map: ⟨0.000 -16.241 -9.948 -7.271]

• POTE: ~9/8 = 171.429, ~5/4 = 378.152

error map: ⟨0.000 -16.241 -8.162 -10.845]

Optimal ET sequence: 7, 28d, 35

Badness (Smith): 0.165257

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:

• CTE: ~9/8 = 171.429, ~5/4 = 376.745
• POTE: ~9/8 = 171.429, ~5/4 = 376.711

Optimal ET sequence: 7, 28d, 35

Badness (Smith): 0.078193

Mujannab

Subgroup: 2.3.5.7

Comma list: 54/49, 64/63

Mapping: [⟨7 11 0 20], ⟨0 0 1 0]]

Wedgie: ⟨⟨ 0 7 0 11 0 -20 ]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 386.314

error map: ⟨0.000 -16.241 0.000 +59.746]

• POTE: ~9/8 = 171.429, ~5/4 = 395.187

error map: ⟨0.000 -16.241 +8.873 +59.746]

Optimal ET sequence: 7, 14d

Badness (Smith): 0.105820

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 11 0 20 8], ⟨0 0 1 0 1]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 384.318
• POTE: ~9/8 = 171.429, ~5/4 = 394.661

Optimal ET sequence: 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: [⟨7 11 0 20 8 26], ⟨0 0 1 0 1 0]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~5/4 = 384.318
• POTE: ~9/8 = 171.429, ~5/4 = 395.071

Optimal ET sequence: 7, 14de

Badness (Smith): 0.042830

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

Wedgie: ⟨⟨ 0 14 7 22 11 -23 ]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~15/14 = 108.062

error map: ⟨0.000 -16.241 +1.239 -3.621]

• POTE: ~9/8 = 171.429, ~15/14 = 101.367

error map: ⟨0.000 -16.241 -12.152 -10.316]

Optimal ET sequence: 7c, 14c, 21, 35, 84bbccd

Badness (Smith): 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 tunings:

• CTE: ~9/8 = 171.429, ~15/14 = 106.997
• POTE: ~9/8 = 171.429, ~15/14 = 100.046

Optimal ET sequence: 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: [⟨7 11 1 12 9 26], ⟨0 0 2 1 2 0]]

Optimal tunings:

• CTE: ~9/8 = 171.429, ~15/14 = 106.997
• POTE: ~9/8 = 171.429, ~15/14 = 104.250

Optimal ET sequence: 7ce, 14c, 21, 35

Badness (Smith): 0.054009