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 major and minor intervals 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 minor seventh, 7/6 at the minor third, and 9/7 at the major third.
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 | 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
Tunings
Any multiple of 7edo, up to 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 major seconds and minor thirds are equated, and every interval is either a 7edo interval or halfway between two 7edo intervals. While the 14edo tuning 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.
Target tunings
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Todo: complete section |
Tuning spectrum
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Todo: complete section |