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Whitewood: Difference between revisions

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{{interwiki
{{Interwiki
| en = Whitewood
| de = Whitewood
| de = Whitewood
| en = Whitewood
}}
}}{{Infobox regtemp
{{Infobox regtemp
| Title = Whitewood
| Title = Whitewood
| Subgroups = 2.3.5, 2.3.5.7
| Subgroups = 2.3.5, 2.3.5.7
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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.


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>  
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]].
For technical data, see [[Whitewood family #Whitewood]] and [[No-fives subgroup temperaments #Purpleheart]].  


== Intervals ==
== Intervals ==
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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.  
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 as best as it can be tuned.
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
| 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'''
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