Phoenix - Revision history

MisterShafXen: - redundant sentence

2026-05-28T10:58:41Z

- redundant sentence

← Older revision		Revision as of 10:58, 28 May 2026
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	== Instruments ==		== 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~~. Pianos are good instruments for phoenix because they have stretched octaves already~~.		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 ==		== Intervals ==

BudjarnLambeth at 01:45, 27 October 2024

2024-10-27T01:45:33Z

← Older revision		Revision as of 01:45, 27 October 2024
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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 [[Stretched ~~octave~~\|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 ==		== Etymology ==

BudjarnLambeth: Link

2024-10-27T01:39:52Z

Link

← Older revision		Revision as of 01:39, 27 October 2024
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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 octave\|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 ==		== Etymology ==

BudjarnLambeth: /* Harmonics and commas */ Add tables

2024-10-27T01:26:47Z

Harmonics and commas: Add tables

← Older revision		Revision as of 01:26, 27 October 2024
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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.		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 ==		== Fenghuang temperament ==

BudjarnLambeth: Add subheadings to improve readability

2024-10-27T01:10:01Z

Add subheadings to improve readability

← Older revision		Revision as of 01:10, 27 October 2024
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 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.		[[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].		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.		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.

			== 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.		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.		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.		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).		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).

			== 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. Pianos are good instruments for phoenix because they have stretched octaves already.		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.

			== Intervals ==
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-

BudjarnLambeth: Mention 16ed9/5

2024-10-27T01:02:19Z

Mention 16ed9/5

← Older revision		Revision as of 01:02, 27 October 2024
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), 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 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.

	[[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.		[[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.

BudjarnLambeth: Added links

2024-01-27T04:13:44Z

Added links

Cmloegcmluin: link to new page for Supports

2022-01-25T19:01:54Z

link to new page for Supports

← Older revision		Revision as of 19:01, 25 January 2022
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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.		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.		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.

	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).		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).

Moremajorthanmajor at 21:32, 30 May 2021

2021-05-30T21:32:26Z

Show changes

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

← Older revision		Revision as of 04:13, 27 January 2024
Line 1:		Line 1:
	The '''phoenix''' tuning continuum ranges consists of a range of equally-tempered scales ranging from 63.5998 cents (which divides the just 9:5 interval into 16 equal parts), 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 ~~harmonics~~ 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), 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.

	~~I (~~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.		[[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.

	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 [[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].

	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.

	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.		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 [[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.		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.

	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).		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 19:		Line 19:
	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.		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.

	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 ~~triads~~ considered incomplete. Unlike ~~tetrads~~ (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.		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).		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.		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.