Blackwood: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
+music
No edit summary
 
(16 intermediate revisions by 5 users not shown)
Line 1: Line 1:
: ''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]].''
{{interwiki
 
| de = Blackwood-Limmisch
| en = 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]].''
{{Infobox regtemp
| Title = Blackwood
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[256/243]] (2.3.5); <br>[[28/27]], [[49/48]] (2.3.5.7)
| Edo join 1 = 5 | Edo join 2 = 10
| Mapping = 5; 0 1 0
| Generators = 5/4
| Generators tuning = 391.1
| Optimization method = CWE
| Pergen = (P8/5, ^1)
| Color name = Sawati
| MOS scales = [[5L&nbsp;5s]], [[10L&nbsp;5s]]
| Odd limit 1 = 5 | Mistuning 1 = 18.0 | Complexity 1 = 10
| Odd limit 2 = 9 | Mistuning 2 = 44.9 | Complexity 2 = 10
}}
'''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]] (sometimes known as ''blacksmith'') 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]].  
'''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]] (sometimes known as ''blacksmith'') 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]].  


Line 8: Line 26:


See [[Limmic temperaments #Blackwood]] for technical data.  
See [[Limmic temperaments #Blackwood]] for technical data.  
__TOC__
{{Clear}}


== Interval chain ==
== Interval chain ==
Line 66: Line 86:
== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
Line 92: Line 112:


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
Line 121: Line 141:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 138: Line 158:
| 15/8
| 15/8
| 368.269
| 368.269
|  
| -1/5-limma
|-
|11\35
|
| 377.143
|35b val
|-
|-
|  
|  
Line 153: Line 178:
| 5/4
| 5/4
| 386.314
| 386.314
|  
| Untempered, 5- and 7-limit CTE, etc.
|-
|-
|  
|  
| 25/24
| 25/24
| 395.336
| 395.336
|  
| 1/10-limma
|-
|-
|  
|  
Line 164: Line 189:
| 395.533
| 395.533
|  
|  
|-
|
|
|397.163
|DR-optimized 4:5:6 tuning
|-
|-
| 5\15
| 5\15
Line 173: Line 203:
| 5/3
| 5/3
| 404.359
| 404.359
|  
| 1/5-limma
|-
|-
| 7\20
| 7\20
Line 183: Line 213:
| 9/5
| 9/5
| 422.404
| 422.404
|  
| 2/5-limma
|-
|-
| 2\5
| 2\5
Line 206: Line 236:
[[Category:Blackwood| ]] <!-- main article -->
[[Category:Blackwood| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Trienstonic clan]]
[[Category:Trienstonic clan]]
[[Category:Semaphoresmic clan]]
[[Category:Semaphoresmic clan]]
[[Category:Archytas clan]]
[[Category:Archytas clan]]

Latest revision as of 05:18, 20 May 2026

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
Subgroups 2.3.5, 2.3.5.7
Comma basis 256/243 (2.3.5);
28/27, 49/48 (2.3.5.7)
Reduced mapping ⟨5; 0 1 0]
ET join 5 & 10
Generators (CWE) ~5/4 = 391.1 ¢
MOS scales 5L 5s, 10L 5s
Ploidacot pentaploid acot
Pergen (P8/5, ^1)
Color name Sawati
Minimax error 5-odd-limit: 18.0 ¢;
9-odd-limit: 44.9 ¢
Target scale size 5-odd-limit: 10 notes;
9-odd-limit: 10 notes

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

blacksmith10.jpg
Lattice of Blackwood[10]

Blackwood major scale in 15edo

Tunings

5-limit norm-based 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 norm-based 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 ¢

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 -1/5-limma
11\35 377.143 35b val
7/5 377.488
8\25 384.000
5/4 386.314 Untempered, 5- and 7-limit CTE, etc.
25/24 395.336 1/10-limma
21/20 395.533
397.163 DR-optimized 4:5:6 tuning
5\15 400.000
5/3 404.359 1/5-limma
7\20 420.000 20c val
9/5 422.404 2/5-limma
2\5 480.000 Upper bound of 7- and 9-odd-limit diamond monotone

* Besides the octave

Music

E8 Heterotic
  • Cyberfunk (2020) – in Blackwood[10], 15edo tuning
Sevish

See also

Retrieved from "https://en.xen.wiki/index.php?title=Blackwood&oldid=230643"