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{{Interwiki
| en = Whitewood
| de = Whitewood
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{{Infobox regtemp
{{Infobox regtemp
| Title = Whitewood
| Title = Whitewood
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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.
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]].
Whitewood was named by [[Mike Battaglia]] in 2010 to serve in contrast with the [[blackwood]] temperament, which tempers out 256/243, the [[Pythagorean limma]].<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_95296.html Yahoo! Tuning Group | ''7&14 temperament - 14 out of 35'']</ref> The [[2.3.7 subgroup|2.3.7-subgroup]] [[restriction]] of whitewood is sometimes known as '''purpleheart'''.
 
For technical data, see [[Whitewood family #Whitewood]] and [[No-fives subgroup temperaments #Purpleheart]].  


== Intervals ==
== Intervals ==
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|}
|}


<nowiki/>*in 7-limit CWE tuning
<nowiki/> * In 7-limit CWE tuning, octave reduced
 
== Scales ==
The [[7L 7s]] 14-note [[mos]] of whitewood, like the [[5L 5s]] 10-note mos of blackwood, shares a number of interesting properties which derive from the relatively small circle of fifths common to both. From any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the [[5L 2s|diatonic scale]], in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal", since resolutions that work in one key will work in all other keys in the scale, at least keys that share the same chord quality.
 
Another interesting property is that it becomes possible to construct "super-linked" 5-limit chords. In Whitewood[14], or Blackwood[10], if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that Whitewood[14] and Blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.
 
[[File:Whitewood14 21edo.mp3|14-note Whitewood scale (major, sLsLsLsLsLsLsL) in 21edo tuning]]
 
14-note Whitewood scale (major, sLsLsLsLsLsLsL) in 21edo tuning


== 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 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.
While blackwood fifths are sharp and thus necessitate the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning – targeting individually the [[5-limit|2.3.5-]] or [[2.3.7 subgroup|2.3.7-subgroup]]. Septimal whitewood entails a rather different tuning profile, as the vanishing of 36/35 means 5 and 7 should be tuned somewhat sharp.
 
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 the best it can be tuned harmonically, though the small step of Whitewood[14] shrinks to just 42.9 cents, thus becoming less melodically viable.


=== Norm-based tunings ===
=== Norm-based tunings ===
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! Comments
! Comments
|-
|-
| '''2\7'''
| '''[[7edo|2\7]]'''
|  
|  
| '''342.857'''
| '''342.857'''
| '''Lower bound of 5-odd-limit diamond monotone'''
| '''Lower bound of 5-odd-limit [[diamond monotone]]'''
|-
|-
|  
|  
| 9/5
| [[9/5]]
| 353.832
| 353.832
|  
|  
|-
|-
|  
|  
| 6/5
| [[6/5]]
| 370.073
| 370.073
|  
|  
|-
|-
| 11\35
| [[35edo|11\35]]
|  
|  
| 377.143
| 377.143
| 35d val
| 35d [[val]]
|-
|-
|  
|  
| 25/24
| [[25/24]]
| 378.193
| 378.193
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| '''9\28'''
| '''[[28edo|9\28]]'''
|  
|  
| '''385.714'''
| '''385.714'''
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|-
|-
|  
|  
| 5/4
| [[5/4]]
| 386.314
| 386.314
|  
| 5-limit CTE
|-
|-
|  
|  
| 21/20
| [[21/20]]
| 386.338
| 386.338
|  
|  
|-
|-
|  
|  
| 21/16
| [[21/16]]
| 386.362
| 386.362
|  
|  
|-
| [[49edo|16\49]]
|
| 391.837
| 49b val
|-
|-
|  
|  
| 7/5
| [[7/5]]
| 394.458
| 394.458
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
|-
|-
| 7\21
| [[21edo|7\21]]
|  
|  
| 400.000
| 400.000
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|-
|-
|  
|  
| 15/8
| [[15/8]]
| 402.554
| 402.554
|  
|  
|-
|-
|  
|  
| 15/14
| [[15/14]]
| 402.579
| 402.579
|  
|  
|-
|-
|  
|  
| 7/4
| [[7/4]]
| 402.603
| 402.603
|  
|  
|-
|-
|  
|  
| 49/48
| [[49/48]]
| 410.723
| 410.723
|  
|  
|-
| [[35edo|12\35]]
|
| 411.429
| 35c val
|-
|-
|  
|  
| 7/6
| [[7/6]]
| 418.843
| 418.843
|  
|  
|-
|-
| '''5\14'''
| '''[[14edo|5\14]]'''
|  
|  
| '''428.571'''
| '''428.571'''
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|-
|-
|  
|  
| 9/7
| [[9/7]]
| 435.084
| 435.084
|  
|  
|-
|-
| '''3\7'''
| '''[[7edo|3\7]]'''
|  
|  
| '''514.286'''
| '''514.286'''
| 7cd val, '''Upper bound of 5-odd-limit diamond monotone'''
| 7cd val, '''upper bound of 5-odd-limit diamond monotone'''
|}
|}
<nowiki/>* Besides the octave
<nowiki/>* Besides the octave


== References ==
[[Category:Whitewood| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Whitewood family]]
[[Category:Whitewood family]]
[[Category:Mint temperaments]]
[[Category:Mint temperaments]]
[[Category:Mirwomo temperaments]]
Retrieved from "https://en.xen.wiki/w/Whitewood"