Phoenix: Difference between revisions

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The '''phoenix''' tuning continuum ranges consists of a range of [[Equal-step Tuning|equally-tempered scales]] ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts, see [[16ed9/5]]), through 63.8141 cents (which divides the just perfect fifth into 11 equal parts, see [[11edf|11edf]]). All of these scales stretch the [[octave]] by around 8 to 12 cents. A distinctive feature of phoenix-tuned scales is that prime-numbered [[harmonic]]s are, on average, approximated more reliably than composite ones. Concentrating the error around composites provides greater overall benefit to tempering.

The '''phoenix''' tuning continuum ranges consists of a range of [[Equal-step Tuning|equally-tempered scales]] ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts, see [[16ed9/5]]), through 63.8141 cents (which divides the just perfect fifth into 11 equal parts, see [[11edf|11edf]]). All of these scales [[Stretched and compressed tuning|stretch]] the [[octave]] by around 8 to 12 cents. A distinctive feature of phoenix-tuned scales is that prime-numbered [[harmonic]]s are, on average, approximated more reliably than composite ones. Concentrating the error around composites provides greater overall benefit to tempering.

== Etymology ==

[[Mason Green]] chose the name phoenix because these scales approximate most small intervals reasonably well, but they have a noticeable weakness at the 8th harmonic ratio (8:1) which falls almost exactly between two scale degrees; the 9:1, 10:1 and 12:1 are also not approximated well. There is also a long stretch of missed harmonics from 24 (or 25) to 31. Figuratively, the scale, like a phoenix, "dies" at 24 and rises from the ashes again at 31 or 32.

== Harmonics and commas ==

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 [[Timbre|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].

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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.

'''Integers'''

{{Harmonics in equal|16|9|5|title=16ed9/5, lower bound of phoenix|intervals=integer|columns=13}}

{{Harmonics in equal|11|3|2|title=11edf, upper bound of phoenix|intervals=integer|columns=13}}

{{Harmonics in equal|12|2|1|title=12edo for comparison|intervals=integer|columns=13}}

'''Primes'''

{{Harmonics in equal|16|9|5|title=16ed9/5, lower bound of phoenix|intervals=prime|columns=13}}

{{Harmonics in equal|11|3|2|title=11edf, upper bound of phoenix|intervals=prime|columns=13}}

{{Harmonics in equal|12|2|1|title=12edo for comparison|intervals=prime|columns=13}}

== Fenghuang temperament ==

In addition to [[meantone]] temperament, phoenix also [[support]]s 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.

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12edo can be smoothly deformed into phoenix by gradually moving all pitches toward their nearest neighbors, and finally adding the new "interstitial" pitches at the end.

== Chords ==

In phoenix, the pentad 4:5:6:7:8 (represented by scale degrees 0, 6, 11, 15, 19) could be considered the basic chord, along with its utonal counterpart with [[triad]]s considered incomplete. Unlike [[tetrad]]s (which sound rather unstable since the top note is a seventh), but like triads, pentads have a great deal of stability and finality due to its highest note being a (tempered) octave, which is even more consonant than the perfect fifth of triads.

The [[otonal]] 4:5:6:7:8 has a happy sound, akin to a major triad (which it contains) but richer, whereas the utonal version sounds melancholy, like the minor triad (which it also contains). These are therefore the minor and major chords of phoenix. (Similar pentads can of course be voiced in [[19edo]] as well, but do not sound as good because the sum of squared errors is higher, the 7 in particular is way out of tune in 19edo).

Phoenix scales are best used over only a finite range of around five octaves; harmonics larger than this range are once again matched poorly. A phoenix piano could be built and would have around 97 keys on it (which is only slightly more than an ordinary piano); pianos are especially good instruments for phoenix because they have built-in octave stretching already. Another option though is to end at 87 keys (one fewer than an ordinary piano) since there the largest interval here is the 24th harmonic~~. Pianos are good instruments for phoenix because they have stretched octaves already~~.

== Instruments ==

Phoenix scales are best used over only a finite range of around five octaves; harmonics larger than this range are once again matched poorly. A phoenix piano could be built and would have around 97 keys on it (which is only slightly more than an ordinary piano); pianos are especially good instruments for phoenix because they have built-in octave stretching already. Another option though is to end at 87 keys (one fewer than an ordinary piano) since there the largest interval here is the 24th harmonic.

== Intervals ==

{| class="wikitable"

|-

@@ Line 1: / Line 1: @@
-The '''phoenix''' tuning continuum ranges consists of a range of [[Equal-step Tuning|equally-tempered scales]] ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts, see [[16ed9/5]]), through 63.8141 cents (which divides the just perfect fifth into 11 equal parts, see [[11edf|11edf]]). All of these scales stretch the [[octave]] by around 8 to 12 cents. A distinctive feature of phoenix-tuned scales is that prime-numbered [[harmonic]]s are, on average, approximated more reliably than composite ones. Concentrating the error around composites provides greater overall benefit to tempering.
+The '''phoenix''' tuning continuum ranges consists of a range of [[Equal-step Tuning|equally-tempered scales]] ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts, see [[16ed9/5]]), through 63.8141 cents (which divides the just perfect fifth into 11 equal parts, see [[11edf|11edf]]). All of these scales [[Stretched and compressed tuning|stretch]] the [[octave]] by around 8 to 12 cents. A distinctive feature of phoenix-tuned scales is that prime-numbered [[harmonic]]s are, on average, approximated more reliably than composite ones. Concentrating the error around composites provides greater overall benefit to tempering.
+== Etymology ==
 [[Mason Green]] chose the name phoenix because these scales approximate most small intervals reasonably well, but they have a noticeable weakness at the 8th harmonic ratio (8:1) which falls almost exactly between two scale degrees; the 9:1, 10:1 and 12:1 are also not approximated well. There is also a long stretch of missed harmonics from 24 (or 25) to 31. Figuratively, the scale, like a phoenix, "dies" at 24 and rises from the ashes again at 31 or 32.
+== Harmonics and commas ==
 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 [[Timbre|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].
@@ Line 11: / Line 13: @@
 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.
+'''Integers'''
+{{Harmonics in equal|16|9|5|title=16ed9/5, lower bound of phoenix|intervals=integer|columns=13}}
+{{Harmonics in equal|11|3|2|title=11edf, upper bound of phoenix|intervals=integer|columns=13}}
+{{Harmonics in equal|12|2|1|title=12edo for comparison|intervals=integer|columns=13}}
+'''Primes'''
+{{Harmonics in equal|16|9|5|title=16ed9/5, lower bound of phoenix|intervals=prime|columns=13}}
+{{Harmonics in equal|11|3|2|title=11edf, upper bound of phoenix|intervals=prime|columns=13}}
+{{Harmonics in equal|12|2|1|title=12edo for comparison|intervals=prime|columns=13}}
+== Fenghuang temperament ==
 In addition to [[meantone]] temperament, phoenix also [[support]]s 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.
@@ Line 19: / Line 33: @@
 edo can be smoothly deformed into phoenix by gradually moving all pitches toward their nearest neighbors, and finally adding the new "interstitial" pitches at the end.
+== Chords ==
 In phoenix, the pentad 4:5:6:7:8 (represented by scale degrees 0, 6, 11, 15, 19) could be considered the basic chord, along with its utonal counterpart with [[triad]]s considered incomplete. Unlike [[tetrad]]s (which sound rather unstable since the top note is a seventh), but like triads, pentads have a great deal of stability and finality due to its highest note being a (tempered) octave, which is even more consonant than the perfect fifth of triads.
 The [[otonal]] 4:5:6:7:8 has a happy sound, akin to a major triad (which it contains) but richer, whereas the utonal version sounds melancholy, like the minor triad (which it also contains). These are therefore the minor and major chords of phoenix. (Similar pentads can of course be voiced in [[19edo]] as well, but do not sound as good because the sum of squared errors is higher, the 7 in particular is way out of tune in 19edo).
-Phoenix scales are best used over only a finite range of around five octaves; harmonics larger than this range are once again matched poorly. A phoenix piano could be built and would have around 97 keys on it (which is only slightly more than an ordinary piano); pianos are especially good instruments for phoenix because they have built-in octave stretching already. Another option though is to end at 87 keys (one fewer than an ordinary piano) since there the largest interval here is the 24th harmonic. Pianos are good instruments for phoenix because they have stretched octaves already.
+== Instruments ==
+Phoenix scales are best used over only a finite range of around five octaves; harmonics larger than this range are once again matched poorly. A phoenix piano could be built and would have around 97 keys on it (which is only slightly more than an ordinary piano); pianos are especially good instruments for phoenix because they have built-in octave stretching already. Another option though is to end at 87 keys (one fewer than an ordinary piano) since there the largest interval here is the 24th harmonic.
+== Intervals ==
 {| class="wikitable"
 |-