Xenharmonic Wiki

Whitewood: Difference between revisions

← Older edit
VisualWikitext
Overthink (talk | contribs)
autopatrolled
2,671 edits
→Target tunings: add norm-based tunings
Overthink (talk | contribs)
autopatrolled
2,671 edits
→Tuning spectrum: more edo tunings; links
 
(13 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Interwiki
| en = Whitewood
| de = Whitewood
}}
{{Infobox regtemp
{{Infobox regtemp
| Title = Whitewood
| Title = Whitewood
Line 8: Line 12:
| Generators tuning = 392.7
| Generators tuning = 392.7
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[7L 7s]], [[7L 14s]], ...
| MOS scales = [[7L 7s]], [[7L 14s]], …
| Pergen = (P8/7, ^1)
| Pergen = (P8/7, ^1)
| Color name = Lawati
| Color name = Lawati
Line 14: Line 18:
| Odd limit 2 = 9 | Mistuning 2 = 40.6 | Complexity 2 = 21
| Odd limit 2 = 9 | Mistuning 2 = 40.6 | Complexity 2 = 21
}}
}}
'''Whitewood''' is the [[rank-2 temperament]] tempering out [[2187/2048]], the Pythagorean chromatic semitone. As a result, the [[circle of fifths]] is the same as that of [[7edo]], and every interval on the chain of fifths is [[neutral (interval quality)|neutral]] in quality. The whitewood temperament adds prime [[5/1|5]] as an independent [[generator]], adding major and minor intervals on either side of the neutral ones.
'''Whitewood''' is the [[rank-2 temperament]] tempering out [[2187/2048]], the Pythagorean chromatic semitone. As a result, the [[circle of fifths]] is the same as that of [[7edo]], and every interval on the chain of fifths is [[neutral (interval quality)|neutral]] in quality. The whitewood temperament adds prime [[5/1|5]] as an independent [[generator]], adding subchromatically inflected intervals (notated with ups and downs below) on either side of the neutral ones.


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 minor seventh, [[7/6]] at the minor third, and [[9/7]] at the major 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 ==
In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''.
In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''.
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
! rowspan="2" | Period
! rowspan="2" | Period
Line 79: Line 86:
| '''8/5''', 14/9
| '''8/5''', 14/9
| 857.1
| 857.1
| 27/16, 128/81
| 27/16, 128/81, 105/64
| 907.0
| 907.0
| 5/3, 12/7
| 5/3, 12/7
Line 100: Line 107:
|}
|}


<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 major seconds and minor thirds are equated, and every interval is either a 7edo interval or halfway between two 7edo intervals. While the 14edo tuning 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.  


=== Target 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 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 ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
Line 135: Line 153:
| CWE: ~5/4 = 392.741{{C}}
| CWE: ~5/4 = 392.741{{C}}
| POTE: ~5/4 = 392.699{{C}}
| POTE: ~5/4 = 392.699{{C}}
|}
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~5/4 = 378.193{{C}}
| 25/24
|-
| 7-odd-limit
| ~5/4 = 394.458{{C}}
| 7/5
|-
| 9-odd-limit
| ~5/4 = 394.458{{C}}
| 7/5
|}
|}


=== Tuning spectrum ===
=== Tuning spectrum ===
{{Todo|inline=1|complete section}}
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged&nbsp;interval<br>(eigenmonzo)]]*
! Generator&nbsp;(¢)
! Comments
|-
| '''[[7edo|2\7]]'''
|
| '''342.857'''
| '''Lower bound of 5-odd-limit [[diamond monotone]]'''
|-
|
| [[9/5]]
| 353.832
|
|-
|
| [[6/5]]
| 370.073
|
|-
| [[35edo|11\35]]
|
| 377.143
| 35d [[val]]
|-
|
| [[25/24]]
| 378.193
| 5-odd-limit minimax
|-
| '''[[28edo|9\28]]'''
|
| '''385.714'''
| '''Lower bound of 7-odd-limit diamond monotone'''
|-
|
| [[5/4]]
| 386.314
| 5-limit CTE
|-
|
| [[21/20]]
| 386.338
|
|-
|
| [[21/16]]
| 386.362
|
|-
| [[49edo|16\49]]
|
| 391.837
| 49b val
|-
|
| [[7/5]]
| 394.458
| 7- and 9-odd-limit minimax
|-
| [[21edo|7\21]]
|
| 400.000
|
|-
|
| [[15/8]]
| 402.554
|
|-
|
| [[15/14]]
| 402.579
|
|-
|
| [[7/4]]
| 402.603
|
|-
|
| [[49/48]]
| 410.723
|
|-
| [[35edo|12\35]]
|
| 411.429
| 35c val
|-
|
| [[7/6]]
| 418.843
|
|-
| '''[[14edo|5\14]]'''
|
| '''428.571'''
| '''Upper bound of 7-odd-limit diamond monotone'''
|-
|
| [[9/7]]
| 435.084
|
|-
| '''[[7edo|3\7]]'''
|
| '''514.286'''
| 7cd val, '''upper bound of 5-odd-limit diamond monotone'''
|}
<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"