Phoenix: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-26 02:54:53 UTC</tt>.

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-26 10:05:04 UTC</tt>.

: The original revision id was <tt>~~567829659~~</tt>.

: The original revision id was <tt>567860657</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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The following harmonics are "split" (not matched): 8, 9, 10, 12, 15, 18. The fact that prime number harmonics are all approximated well, but composites often are not, implies that synthesized tones using the [[The Prime Harmonic Series|prime harmonic series]] should make a very good fit with phoenix. The generator of phoenix could be considered an analogue of prime number-generating functions such as [[https://en.wikipedia.org/wiki/Mills%27_constant|Mills' constant]].

There is another a very good reason to "split" the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the ~~septimal~~ (64:63) and ~~syntonic~~ (81:80) ~~commas~~ in the same tuning~~. One of~~ the ~~problems~~ with ~~extending beyond 12edo to higher scales is that 12edo tempers out both~~ of ~~these commas~~, ~~and many familiar melodies and chord progressions in 12edo depend on these [[comma pump|comma pumps]]~~. ~~Larger scales, with their finer~~ octave ~~divisions~~, ~~generally temper out either the syntonic comma (meantone, such as 19edo) or the septimal comma (superpythagorean, such as 17edo)~~ but ~~not both~~.

There is another a very good reason to "split" the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the syntonic comma (81:80) and the septimal comma (64:63) in the same tuning, if we do some fudging during modulation (for example, by representing the 8:7 with 3 instead of 4 steps, or by using the "blue" octave of 18 steps. The blue octave is not harmonically consonant, but is an interesting melodic interval.

~~By stretching~~ the ~~octave, though, it's possible~~ to ~~have a tuning~~ that ~~extends~~ 12edo ~~while maintaining most if not all~~ of ~~the same equivalencies found in it. Phoenix, like 12edo~~, ~~thus functions as both meantone //~~and~~// a superpythagorean tuning. Its whole tones (both of them) can each potentially function as 10:9, 9:8, //or// 8:7 (~~and ~~in this way are exactly analogous to the whole tone~~ in 12edo), ~~due to the "splitting"~~ of ~~the 8th and 9th harmonics~~.

One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these [[comma pump|comma pumps]]. Larger scales, with their finer octave divisions, generally temper out out one of these intervals but not both.

In addition to meantone ~~and superpythagorean temperaments~~, phoenix also supports the //fenghuang temperament.// A scale supports this temperament if it contains a tempered subminor //third// (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor //seventh// (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.

By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most of the same equivalencies found in it.

In addition to meantone temperament, phoenix also supports the //fenghuang temperament.// A scale supports this temperament if it contains a tempered subminor //third// (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor //seventh// (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.

Phoenix's approximations to undecimal and tridecimal intervals are dominated by the fact that the 12th harmonic is also split, meaning that 11:12 and 12:13 can //each// be approximated in two different ways (one corresponding to a whole tone and one to a diatonic semitone).

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|| Minor seventh || 16 || || 1021.025 || 9:5 ||

|| Major seventh || 17 || || 1084.840 || 15:8 ||

|| Diminished octave || 18 || || 1148.654 || 11:6 ||

|| Diminished octave, blue octave || 18 || || 1148.654 || 11:6 ||

|| Octave (stretched) || 19 || || 1212.468 || 2:1 ||

|| Chromatic minor ninth || 20 || || 1276.282 || ||

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The following harmonics are &quot;split&quot; (not matched): 8, 9, 10, 12, 15, 18. The fact that prime number harmonics are all approximated well, but composites often are not, implies that synthesized tones using the <a class="wiki_link" href="/The%20Prime%20Harmonic%20Series">prime harmonic series</a> should make a very good fit with phoenix. The generator of phoenix could be considered an analogue of prime number-generating functions such as <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Mills%27_constant" rel="nofollow">Mills' constant</a>.

There is another a very good reason to &quot;split&quot; the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the ~~septimal~~ (64:63) and ~~syntonic~~ (81:80) ~~commas~~ in the same tuning. One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these <a class="wiki_link" href="/comma%20pump">comma pumps</a>. Larger scales, with their finer octave divisions, generally temper out ~~either the syntonic comma (meantone, such as 19edo) or the septimal comma (superpythagorean, such as 17edo)~~ but not both.

There is another a very good reason to &quot;split&quot; the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the syntonic comma (81:80) and the septimal comma (64:63) in the same tuning, if we do some fudging during modulation (for example, by representing the 8:7 with 3 instead of 4 steps, or by using the &quot;blue&quot; octave of 18 steps. The blue octave is not harmonically consonant, but is an interesting melodic interval.

One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these <a class="wiki_link" href="/comma%20pump">comma pumps</a>. Larger scales, with their finer octave divisions, generally temper out out one of these intervals but not both.

By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most ~~if not all~~ of the same equivalencies found in it. Phoenix, like 12edo, thus functions as both meantone and a superpythagorean tuning. Its whole tones (both of them) can each potentially function as 10:9, 9:8, or 8:7 (and in this way are exactly analogous to the whole tone in 12edo), due to the &quot;splitting&quot; of the 8th and 9th harmonics.

By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most of the same equivalencies found in it.

In addition to meantone ~~and superpythagorean temperaments~~, phoenix also supports the fenghuang temperament. A scale supports this temperament if it contains a tempered subminor third (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor seventh (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.

In addition to meantone temperament, phoenix also supports the fenghuang temperament. A scale supports this temperament if it contains a tempered subminor third (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor seventh (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.

Phoenix's approximations to undecimal and tridecimal intervals are dominated by the fact that the 12th harmonic is also split, meaning that 11:12 and 12:13 can each be approximated in two different ways (one corresponding to a whole tone and one to a diatonic semitone).

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</tr>

<tr>

<td>Diminished octave

<td>Diminished octave, blue octave

</td>

<td>18

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-26 02:54:53 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-26 10:05:04 UTC</tt>.<br>
-: The original revision id was <tt>567829659</tt>.<br>
+: The original revision id was <tt>567860657</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 12: / Line 12: @@
 The following harmonics are "split" (not matched): 8, 9, 10, 12, 15, 18. The fact that prime number harmonics are all approximated well, but composites often are not, implies that synthesized tones using the [[The Prime Harmonic Series|prime harmonic series]] should make a very good fit with phoenix. The generator of phoenix could be considered an analogue of prime number-generating functions such as [[https://en.wikipedia.org/wiki/Mills%27_constant|Mills' constant]].
-There is another a very good reason to "split" the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the septimal (64:63) and syntonic (81:80) commas in the same tuning. One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these [[comma pump|comma pumps]]. Larger scales, with their finer octave divisions, generally temper out either the syntonic comma (meantone, such as 19edo) or the septimal comma (superpythagorean, such as 17edo) but not both.
+There is another a very good reason to "split" the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the syntonic comma (81:80) and the septimal comma (64:63) in the same tuning, if we do some fudging during modulation (for example, by representing the 8:7 with 3 instead of 4 steps, or by using the "blue" octave of 18 steps. The blue octave is not harmonically consonant, but is an interesting melodic interval.
-By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most if not all of the same equivalencies found in it. Phoenix, like 12edo, thus functions as both meantone //and// a superpythagorean tuning. Its whole tones (both of them) can each potentially function as 10:9, 9:8, //or// 8:7 (and in this way are exactly analogous to the whole tone in 12edo), due to the "splitting" of the 8th and 9th harmonics.
+One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these [[comma pump|comma pumps]]. Larger scales, with their finer octave divisions, generally temper out out one of these intervals but not both.
-In addition to meantone and superpythagorean temperaments, phoenix also supports the //fenghuang temperament.// A scale supports this temperament if it contains a tempered subminor //third// (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor //seventh// (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.
+By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most of the same equivalencies found in it.
+In addition to meantone temperament, phoenix also supports the //fenghuang temperament.// A scale supports this temperament if it contains a tempered subminor //third// (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor //seventh// (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.
 Phoenix's approximations to undecimal and tridecimal intervals are dominated by the fact that the 12th harmonic is also split, meaning that 11:12 and 12:13 can //each// be approximated in two different ways (one corresponding to a whole tone and one to a diatonic semitone).
@@ Line 44: / Line 46: @@
 || Minor seventh || 16 ||   || 1021.025 || 9:5 ||
 || Major seventh || 17 ||   || 1084.840 || 15:8 ||
-|| Diminished octave || 18 ||   || 1148.654 || 11:6 ||
+|| Diminished octave, blue octave || 18 ||   || 1148.654 || 11:6 ||
 || Octave (stretched) || 19 ||   || 1212.468 || 2:1 ||
 || Chromatic minor ninth || 20 ||   || 1276.282 ||   ||
@@ Line 133: / Line 135: @@
 The following harmonics are &amp;quot;split&amp;quot; (not matched): 8, 9, 10, 12, 15, 18. The fact that prime number harmonics are all approximated well, but composites often are not, implies that synthesized tones using the &lt;a class="wiki_link" href="/The%20Prime%20Harmonic%20Series"&gt;prime harmonic series&lt;/a&gt; should make a very good fit with phoenix. The generator of phoenix could be considered an analogue of prime number-generating functions such as &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Mills%27_constant" rel="nofollow"&gt;Mills' constant&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-There is another a very good reason to &amp;quot;split&amp;quot; the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the septimal (64:63) and syntonic (81:80) commas in the same tuning. One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt;. Larger scales, with their finer octave divisions, generally temper out either the syntonic comma (meantone, such as 19edo) or the septimal comma (superpythagorean, such as 17edo) but not both.&lt;br /&gt;
+There is another a very good reason to &amp;quot;split&amp;quot; the eighth harmonic. Having two approximations (one sharp and one flat) for 8 makes it possible to temper out both the syntonic comma (81:80) and the septimal comma (64:63) in the same tuning, if we do some fudging during modulation (for example, by representing the 8:7 with 3 instead of 4 steps, or by using the &amp;quot;blue&amp;quot; octave of 18 steps. The blue octave is not harmonically consonant, but is an interesting melodic interval.&lt;br /&gt;
+&lt;br /&gt;
+One of the problems with extending beyond 12edo to higher scales is that 12edo tempers out both of these commas, and many familiar melodies and chord progressions in 12edo depend on these &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pumps&lt;/a&gt;. Larger scales, with their finer octave divisions, generally temper out out one of these intervals but not both.&lt;br /&gt;
 &lt;br /&gt;
-By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most if not all of the same equivalencies found in it. Phoenix, like 12edo, thus functions as both meantone &lt;em&gt;and&lt;/em&gt; a superpythagorean tuning. Its whole tones (both of them) can each potentially function as 10:9, 9:8, &lt;em&gt;or&lt;/em&gt; 8:7 (and in this way are exactly analogous to the whole tone in 12edo), due to the &amp;quot;splitting&amp;quot; of the 8th and 9th harmonics.&lt;br /&gt;
+By stretching the octave, though, it's possible to have a tuning that extends 12edo while maintaining most of the same equivalencies found in it.&lt;br /&gt;
 &lt;br /&gt;
-In addition to meantone and superpythagorean temperaments, phoenix also supports the &lt;em&gt;fenghuang temperament.&lt;/em&gt; A scale supports this temperament if it contains a tempered subminor &lt;em&gt;third&lt;/em&gt; (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor &lt;em&gt;seventh&lt;/em&gt; (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.&lt;br /&gt;
+In addition to meantone temperament, phoenix also supports the &lt;em&gt;fenghuang temperament.&lt;/em&gt; A scale supports this temperament if it contains a tempered subminor &lt;em&gt;third&lt;/em&gt; (which must be closer to 7:6 than to 8:7) that is the octave inversion of a tempered subminor &lt;em&gt;seventh&lt;/em&gt; (which must be closer to 7:4 than 12:7). Only stretched-octave temperaments can accomplish this.&lt;br /&gt;
 &lt;br /&gt;
 Phoenix's approximations to undecimal and tridecimal intervals are dominated by the fact that the 12th harmonic is also split, meaning that 11:12 and 12:13 can &lt;em&gt;each&lt;/em&gt; be approximated in two different ways (one corresponding to a whole tone and one to a diatonic semitone).&lt;br /&gt;
@@ Line 367: / Line 371: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Diminished octave&lt;br /&gt;
+         &lt;td&gt;Diminished octave, blue octave&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;18&lt;br /&gt;