Whitewood
| Whitewood |
36/35, 2187/2048 (2.3.5.7)
9-odd-limit: 40.6 ¢
9-odd-limit: 21 notes
Whitewood is the rank-2 temperament tempering out 2187/2048, the Pythagorean chromatic semitone. As a result, the circle of fifths is the same as that of 7edo, and every interval on the chain of fifths is neutral in quality. The whitewood temperament adds prime 5 as an independent generator, adding subchromatically inflected intervals (notated with ups and downs below) on either side of the neutral ones.
The canonical extension to prime 7 adds 36/35 to the commas, thus equating 5-limit major and minor intervals with 7-limit subminor and supermajor ones. It finds 7/4 at the down seventh, 7/6 at the down third, and 9/7 at the up third.
Whitewood was named by Mike Battaglia in 2010 to serve in contrast with the blackwood temperament, which tempers out 256/243, the Pythagorean limma.[1]
For technical data, see Whitewood family #Whitewood.
Intervals
In the following table, odd harmonics and subharmonics 1–9 are in bold.
| Period | Generator -1 | Generator 0 | Generator 1 | |||
|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | 49.9 | 64/63, 135/128 | ||
| 1 | 121.5 | 16/15, 28/27 | 171.4 | 9/8, 35/32 | 221.3 | 8/7, 10/9 |
| 2 | 293.0 | 6/5, 7/6 | 342.9 | 32/27, 81/64, 128/105 | 392.7 | 5/4 |
| 3 | 464.4 | 21/16 | 514.3 | 4/3 | 564.2 | 45/32 |
| 4 | 635.8 | 64/45 | 685.7 | 3/2 | 735.6 | 32/21 |
| 5 | 807.3 | 8/5, 14/9 | 857.1 | 27/16, 128/81, 105/64 | 907.0 | 5/3, 12/7 |
| 6 | 978.7 | 7/4, 9/5 | 1028.6 | 16/9, 64/35 | 1078.5 | 15/8, 27/14 |
| 7 | 1150.1 | 63/32, 256/135 | 1200.0 | 2/1 | ||
* In 7-limit CWE tuning, octave reduced
Scales
The 7L 7s 14-note mos of whitewood, like the 5L 5s 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.
14-note Whitewood scale (major, sLsLsLsLsLsLsL) in 21edo tuning
Tunings
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 – targeting individually the 2.3.5- or 2.3.7-subgroup. Septimal whitewood entails a rather different tuning profile, as the vanishing of 36/35 means 5 and 7 should be tuned somewhat sharp.
Any multiple of 7edo, up until 35edo, contains 7edo's perfect fifth, and thus supports whitewood, with all but 35edo supporting the canonical 7-limit extension by patent val. The most extreme tuning is 14edo, where up seconds and down thirds are equated, and every interval is either a 7edo interval or halfway between two 7edo intervals. While the 14edo tuning poorly approximates 5-limit intervals, it does approximate the 6:7:9 subminor and 1/(9:7:6) supermajor triads fairly well. A less extreme tuning is 21edo, tuning 7/4 close to just and tuning 5/4 to the same 400 ¢ major third as in 12edo, though 6/5 is still about 30 cents flat. The 28edo tuning has a near-just 5/4, and tunes whitewood about as best as it can be tuned.
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~5/4 = 386.314 ¢ | CWE: ~5/4 = 376.383 ¢ | POTE: ~5/4 = 374.469 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~5/4 = 392.930 ¢ | CWE: ~5/4 = 392.741 ¢ | POTE: ~5/4 = 392.699 ¢ |
Target tunings
| Target | Minimax | |
|---|---|---|
| Generator | Eigenmonzo* | |
| 5-odd-limit | ~5/4 = 378.193 ¢ | 25/24 |
| 7-odd-limit | ~5/4 = 394.458 ¢ | 7/5 |
| 9-odd-limit | ~5/4 = 394.458 ¢ | 7/5 |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 2\7 | 342.857 | Lower bound of 5-odd-limit diamond monotone | |
| 9/5 | 353.832 | ||
| 6/5 | 370.073 | ||
| 11\35 | 377.143 | 35d val | |
| 25/24 | 378.193 | 5-odd-limit minimax | |
| 9\28 | 385.714 | Lower bound of 7-odd-limit diamond monotone | |
| 5/4 | 386.314 | 5-limit CTE | |
| 21/20 | 386.338 | ||
| 21/16 | 386.362 | ||
| 7/5 | 394.458 | 7- and 9-odd-limit minimax | |
| 7\21 | 400.000 | ||
| 15/8 | 402.554 | ||
| 15/14 | 402.579 | ||
| 7/4 | 402.603 | ||
| 49/48 | 410.723 | ||
| 7/6 | 418.843 | ||
| 5\14 | 428.571 | Upper bound of 7-odd-limit diamond monotone | |
| 9/7 | 435.084 | ||
| 3\7 | 514.286 | 7cd val, upper bound of 5-odd-limit diamond monotone |
* Besides the octave