Whitewood: Difference between revisions

Line 8:

| Generators tuning = 392.7

| Optimization method = CWE

| MOS scales = [[7L 7s]], [[7L 14s]], ~~...~~

| MOS scales = [[7L 7s]], [[7L 14s]], …

| Pergen = (P8/7, ^1)

| Color name = Lawati

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| 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]].

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== Intervals ==

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"

! rowspan="2" | Period

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== 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.

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.

=== Norm-based tunings ===

@@ Line 8: / Line 8: @@
 | Generators tuning = 392.7
 | Optimization method = CWE
-| MOS scales = [[7L 7s]], [[7L 14s]], ...
+| MOS scales = [[7L 7s]], [[7L 14s]], …
 | Pergen = (P8/7, ^1)
 | Color name = Lawati
@@ Line 14: / Line 14: @@
 | 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]].
@@ Line 22: / Line 22: @@
 == Intervals ==
 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"
 ! rowspan="2" | Period
@@ Line 103: / Line 104: @@
 == 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.
+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.
 === Norm-based tunings ===