Phoenix: Difference between revisions
Wikispaces>MasonGreen1 **Imported revision 567860657 - Original comment: ** |
Wikispaces>MasonGreen1 **Imported revision 567860713 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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 10:05: | : This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-26 10:05:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>567860713</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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]]. | 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 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. | 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). | ||
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. | 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. | ||
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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>.<br /> | 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>.<br /> | ||
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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.<br /> | 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).<br /> | ||
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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.<br /> | 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.<br /> | ||