Blackwood
- This article is about the regular temperament. For the musician, see Easley Blackwood Jr. For the scale structure sometimes associated with it, see 5L 5s.
Blackwood is a temperament that takes 5edo's circle of fifths for the 3-limit, but adds multiple copies to improve the tuning of the 5-limit. In the fundamental sense, it is the 5-limit temperament that tempers out the Pythagorean limma, and it extends to the 7-limit (sometimes known as blacksmith) by recognizing that 4\5 is a good harmonic seventh, thus tempering out 28/27, 49/48, and 64/63, making it a member of trienstonic clan, semaphoresmic clan, and archytas clan.
The main interest in this temperament is in its mos scales, featuring pentawood (5L 5s). 15edo provides an excellent tuning for this temperament as well as for pentawood.
Blackwood was named in honor of Easley Blackwood Jr.
See Limmic temperaments #Blackwood for technical data.
Interval chain
In the following table, odd harmonics 1–9 are in bold.
| Period | Generator 0 | Generator 1 | ||
|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | ||
| 1 | 240.0 | 7/6, 8/7, 9/8 | 151.1 | 10/9, 15/14 |
| 2 | 480.0 | 4/3 | 391.1 | 5/4 |
| 3 | 720.0 | 3/2 | 631.1 | 10/7 |
| 4 | 960.0 | 7/4, 12/7, 16/9 | 871.1 | 5/3 |
| 5 | 1200.0 | 2/1 | 1111.1 | 15/8, 40/21 |
* In 7-limit CWE tuning, octave reduced
Scales
Blackwood major scale in 15edo
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~5/4 = 386.3137 ¢ | CSEE: ~5/4 = 392.3287 ¢ | POEE: ~5/4 = 405.2729 ¢ |
| Tenney | CTE: ~5/4 = 386.3137 ¢ | CWE: ~5/4 = 395.1256 ¢ | POTE: ~5/4 = 399.5938 ¢ |
| Benedetti, Wilson |
CBE: ~5/4 = 386.3137 ¢ | CSBE: ~5/4 = 396.3386 ¢ | POBE: ~5/4 = 400.3211 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~5/4 = 386.3137 ¢ | CSEE: ~5/4 = 388.6185 ¢ | POEE: ~5/4 = 387.1612 ¢ |
| Tenney | CTE: ~5/4 = 386.3137 ¢ | CWE: ~5/4 = 391.0976 ¢ | POTE: ~5/4 = 392.7675 ¢ |
| Benedetti, Wilson |
CBE: ~5/4 = 386.3137 ¢ | CSBE: ~5/4 = 392.2565 ¢ | POBE: ~5/4 = 395.3830 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 15/14 | 359.443 | ||
| 3\10 | 360.000 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 15/8 | 368.269 | ||
| 7/5 | 377.488 | ||
| 8\25 | 384.000 | ||
| 5/4 | 386.314 | ||
| 25/24 | 395.336 | ||
| 21/20 | 395.533 | ||
| 5\15 | 400.000 | ||
| 5/3 | 404.359 | ||
| 7\20 | 420.000 | 20c val | |
| 9/5 | 422.404 | ||
| 2\5 | 480.000 | Upper bound of 7- and 9-odd-limit diamond monotone |
* Besides the octave
Music
- Cyberfunk (2020) – in Blackwood[10], 15edo tuning
- From Harmony Hacker (2017)
- "Freathy" – in Blackwood[10], TOP tuning
- "So Thankful" – in Blackwood[10], 15edo tuning