Whitewood: Difference between revisions
→Intervals: add for symmetry |
Major and minor -> up and down |
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| Generators tuning = 392.7 | | Generators tuning = 392.7 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[7L 7s]], [[7L 14s]], | | MOS scales = [[7L 7s]], [[7L 14s]], … | ||
| Pergen = (P8/7, ^1) | | Pergen = (P8/7, ^1) | ||
| Color name = Lawati | | Color name = Lawati | ||
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| Odd limit 2 = 9 | Mistuning 2 = 40.6 | Complexity 2 = 21 | | Odd limit 2 = 9 | Mistuning 2 = 40.6 | Complexity 2 = 21 | ||
}} | }} | ||
'''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 (interval quality)|neutral]] in quality. The whitewood temperament adds prime [[5/1|5]] as an independent [[generator]], adding | '''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 (interval quality)|neutral]] in quality. The whitewood temperament adds prime [[5/1|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 [[7/1|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 | The canonical [[extension]] to prime [[7/1|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. | ||
For technical data, see [[Whitewood family #Whitewood]]. | For technical data, see [[Whitewood family #Whitewood]]. | ||
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== Intervals == | == Intervals == | ||
In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''. | In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''. | ||
{| class="wikitable center-1 right-2 right-4 right-6 right-8" | {| class="wikitable center-1 right-2 right-4 right-6 right-8" | ||
! rowspan="2" | Period | ! rowspan="2" | Period | ||
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== Tunings == | == Tunings == | ||
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 | 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 [[14:18:21|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{{c}} 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 === | === Norm-based tunings === | ||
Revision as of 08:39, 30 April 2026
| 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.
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
Tunings
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 | ||
| 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