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'' | : ''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 [[regular temperament|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 [[tempering out|tempers out]] the [[Pythagorean limma]], and it extends to the [[7-limit]] by recognizing that 4\5 is a good [[7/4|harmonic seventh]], thus tempering out [[28/27]], [[49/48]], and [[64/63]], making it a member of [[trienstonic clan]], [[semaphoresmic clan]], and [[archytas clan]]. | |||
[[Category:Blackwood]] | The main interest in this temperament is in its [[mos scale]]s, featuring [[5L 5s|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'''. | |||
{| class="wikitable center-1 right-2 right-4" | |||
! rowspan="2" | Period | |||
! colspan="2" | Generator 0 | |||
! colspan="2" | 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 | |||
|} | |||
<nowiki/>* In 7-limit CWE tuning, octave reduced | |||
== Scales == | |||
[[File:BlackwoodMajor 15edo.mp3]] [[:File:BlackwoodMajor 15edo.mp3|Blackwood major scale in 15edo]] | |||
== Tunings == | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Equilateral | |||
| CEE: ~5/4 = 386.3137{{c}} | |||
| CSEE: ~5/4 = 392.3287{{c}} | |||
| POEE: ~5/4 = 405.2729{{c}} | |||
|- | |||
! Tenney | |||
| CTE: ~5/4 = 386.3137{{c}} | |||
| CWE: ~5/4 = 395.1256{{c}} | |||
| POTE: ~5/4 = 399.5938{{c}} | |||
|- | |||
! Benedetti, <br>Wilson | |||
| CBE: ~5/4 = 386.3137{{c}} | |||
| CSBE: ~5/4 = 396.3386{{c}} | |||
| POBE: ~5/4 = 400.3211{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Equilateral | |||
| CEE: ~5/4 = 386.3137{{c}} | |||
| CSEE: ~5/4 = 388.6185{{c}} | |||
| POEE: ~5/4 = 387.1612{{c}} | |||
|- | |||
! Tenney | |||
| CTE: ~5/4 = 386.3137{{c}} | |||
| CWE: ~5/4 = 391.0976{{c}} | |||
| POTE: ~5/4 = 392.7675{{c}} | |||
|- | |||
! Benedetti, <br>Wilson | |||
| CBE: ~5/4 = 386.3137{{c}} | |||
| CSBE: ~5/4 = 392.2565{{c}} | |||
| POBE: ~5/4 = 395.3830{{c}} | |||
|} | |||
== See also == | |||
* [[Blackwood temperament modal harmony (in 15edo)]] | |||
[[Category:Blackwood| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | |||
[[Category:Trienstonic clan]] | |||
[[Category:Semaphoresmic clan]] | |||
[[Category:Archytas clan]] | |||