Phoenix

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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). 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.

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.

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 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 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).

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 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.

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).

Fenghuang tempers out the 49:48 diesis. Fenghuang offers unexpected new possibilities for melody and harmony since it violates the usual rule that the octave inversions of sixths and sevenths must always be thirds and seconds, respectively. It is also a good temperament for increasing overall consonance, since 7:6 is more consonant than 8:7, while 7:4 is more consonant than 12:7. This temperament provides a defining aspect of phoenix's sound, particularly in how it handles the 7-limit and in how its "new" intervals (those not shared with 12edo) behave.

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.

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.

Interval Width in steps 16ed9:5 width cents (min) 11edf width cents (max) Equivalencies (closest listed first)
Chromatic semitone 1 63.814 28:27, 25:24, 33:32
Diatonic semitone 2 127.628 15:14, 16:15
Whole tone 3 191.442 10:9, 9:8, 8:7(functional)
Subminor third 4 255.256 7:6, 8:7
Minor third 5 319.070 6:5
Major third 6 382.884 5:4
Supermajor third 7 446.699 9:7
Perfect fourth 8 510.513 4:3
Augmented fourth 9 574.327 7:5
Diminished fifth 10 638.141
Perfect fifth 11 701.955 3:2
Subminor sixth 12 765.769 14:9
Minor sixth 13 829.583 8:5
Major sixth 14 893.397 5:3
Subminor seventh 15 957.211 7:4
Minor seventh 16 1021.025 9:5
Major seventh 17 1084.840 15:8
Diminished octave, blue octave 18 1148.654 11:6
Octave (stretched) 19 1212.468 2:1
Chromatic minor ninth 20 1276.282
Diatonic minor ninth 21 1340.096 13:6
Major ninth 22 1403.910 9:4
Subminor tenth 23 1467.724 7:3
Minor tenth 24 1531.538 17:7, 12:5
Major tenth 25 1595.352 5:2, 19:8
Minor eleventh 26 1659.167 13:5
Perfect eleventh 27 1722.980 8:3
Augmented eleventh, diminished twelfth 28 1786.795 14:15
Minor twelfth 29 1850.609
Perfect twelfth, tritave, 3rd harmonic 30 1914.423 3:1
Minor 31 1978.237
Neutral thirteenth 32 2042.051 13:4
Major thirteenth 33 2105.865 10:3, 27:8
Subminor fourteenth 34 2169.680 7:2
Minor fourteenth 35 2233.493 18:5, 11:3
Major fourteenth 36 2297.307 15:4
Double octave, fifteenth (flat) 37 2361.121
Double octave, fifteenth (sharp) 38 2424.935 4:1
Minor sixteenth 39 2488.750 19:9
Neutral sixteenth 40 2552.564 13:4
Major sixteenth 41 2616.378 9:2
Minor seventeenth 42 2680.192 14:3
Neutral seventeenth 43 2744.006
Major seventeenth; fifth harmonic 44 2807.82 5:1
Minor eighteenth 45 2871.634 21:4
Major eighteenth 46 2935.448 11:2 (flat)
Augmented eighteenth; diminished nineteenth 47 2999.262 11:2 (sharp)
Minor nineteenth 48 3063.076
Perfect nineteenth, sixth harmonic 49 3126.890 6:1
Minor twentieth 50 3190.705 19:3
Neutral twentieth 51 3254.519 13:2
Major twentieth 52 3318.333 27:4
(Sub)minor twenty-first; 7th harmonic 53 3382.147 7:1
Minor/neutral twenty-first 54 3445.961 22:3
Major twenty-first 55 3509.775 15:2
Twenty-second, triple octave, 8th harmonic (flat) 56 3573.589 8:1 (flat)
Twenty-second, triple octave, 8th harmonic (sharp) 57 3637.403 8:1 (sharp)
Minor twenty-third 58 3701.217 17:2
Neutral twenty-third 59 3765.031
Major twenty-third; 9th harmonic 60 3828.845 9:1
Minor twenty-fourth 61 3892.659 19:2
Neutral twenty-fourth 62 3956.474 29:3
Major twenty-fourth, decade 63 4020.288 10:1 (sharp), 31:3 (flat)
Twenty-fifth 64 4084.102 21:2
Half-augmented twenty-fifth; 11th harmonic 65 4147.916 11:1
Diminished twenty-sixth 66 4211.730 34:3
Minor twenty-sixth 67 4275.544 12:1 (flat)
Major twenty-sixth 68 4339.358 12:1 (sharp)
Minor twenty-seventh 69 4403.172
Neutral twenty-seventh, thirteenth harmonic 70 4466.986 13:1
Major twenty-seventh 71 4530.800 27:2
Minor twenty-eighth, 14th harmonic 72 4594.615 14:1
Neutral twenty-eighth 73 4658.429 15:1 (flat)
Major twenty-eighth 74 4722.243 15:1 (sharp)
Twenty-ninth; quadruple octave, 16th harmonic 75 4786.057 16:1
Half-augmented twenty-ninth 76 4849.871 33:2
Minor thirtieth, 17th harmonic 77 4913.685 17:1
Chromatic major thirtieth 78 4977.499 35:2
Diatonic major thirtieth 79 5041.313 37:2
Minor thirty-first, 19th harmonic 80 5105.127 19:1
Neutral thirty-first 81 5168.941
Major thirty-first 82 5232.755 41:2
Thirty-second, 21st harmonic 83 5296.570 21:1
Half-augmented thirty-second, 22nd harmonic 84 5360.384 22:1
23rd harmonic 85 5424.198 23:1
Thirty-third 86 24:1, 47:2 *
Half-augmented thirty-third 87 5551.826 49:2
Minor thirty-fourth 88 5615.640 51:2
Major thirty-fourth 89 5679.454 53:2
Supermajor thirty-fourth 90 5743.268 55:2
Minor thirty-fifth 91 5807.082 57:2
Neutral/major thirty-fifth 92 5870.896
31st harmonic 93 5934.710 31:1 (wide-end only)
32nd harmonic 94 5998.52 32:1
33rd harmonic 95 6062.339 33:1
34th harmonic 96 6126.153 34:1 (narrow-end only)

Harmonics beyond the 34th are not matched well.

The wide end of the phoenix continuum matches 86 scale degrees to the 24th harmonic, whereas the narrow end matches 86 degrees to 47:2. The wide end of the continuum matches the 31st harmonic, the narrow end does not. On the other hand, the narrow end of the continuum matches the 34th harmonic, unlike the wide end.