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'''Blackwood''' may refer to:
: ''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]].''
* [[Easley Blackwood Jr.]], an American composer
* [[5L 5s]], a MOS pattern
* [[Limmic temperaments#5-limit (blackwood)|Blackwood]], a 5-limit temperament tempering out [[256/243]]


{{Disambiguation}}
'''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]]

Revision as of 15:09, 23 April 2025

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 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

5-limit prime-optimized 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 ¢
7-limit prime-optimized tunings
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 ¢

See also

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