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 30edt width cents 16ed9:5 width cents (min) 11edf width cents (max) Equivalencies (closest listed first)
Chromatic semitone 1 63.3985 63.5998 63.8141 28:27, 25:24, 33:32
Diatonic semitone 2 126.797 127.1995 127.6282 15:14, 16:15
Whole tone 3 190.1955 190.7993 191.4423 10:9, 9:8, 8:7(functional)
Subminor third 4 253.594 254.3991 255.2564 7:6, 8:7
Minor third 5 316.9925 317.9998 319.07045 6:5
Major third 6 380.391 381.5986 382.88455 5:4
Supermajor third 7 443.7895 445.1984 446.6986 9:7
Perfect fourth 8 507.188 508.7981 510.5127 4:3
Augmented fourth 9 570.5865 572.3979 574.3268 7:5
Diminished fifth 10 633.985 635.9977 638.1409
Perfect fifth 11 697.3835 699.59745 701.955 3:2
Subminor sixth 12 760.782 763.1972 765.7691 14:9
Minor sixth 13 824.1805 826.797 829.5832 8:5
Major sixth 14 887.579 890.39675 893.3973 5:3
Subminor seventh 15 950.9775 953.9965 957.2114 7:4
Minor seventh 16 1014.376 1017.5963 1021.02545 9:5
Major seventh 17 1077.7745 1081.1961 1084.83955 15:8
Diminished octave, blue octave 18 1141.173 1144.7958 1148.6536 11:6
Octave (stretched) 19 1204.5715 1208.3956 1212.4678 2:1
Chromatic minor ninth 20 1267.97 1271.9953 1276.2818
Diatonic minor ninth 21 1331.3685 1335.5951 1340.0959 13:6
Major ninth 22 1394.767 1399.1945 1403.910 9:4
Subminor tenth 23 1458.1655 1462.7947 1467.7241 7:3
Minor tenth 24 1521.564 1526.3944 1531.5382 17:7, 12:5
Major tenth 25 1584.9625 1589.9942 1595.3523 5:2, 19:8
Minor eleventh 26 1648.361 1653.594 1659.1674 13:5
Perfect eleventh 27 1711.7595 1717.1937 1722.9805 8:3
Augmented eleventh, diminished twelfth 28 1775.158 1780.7935 1786.79455 14:15
Minor twelfth 29 1838.5565 1844.3933 1850.6086
Perfect twelfth, tritave, 3rd harmonic 30 1901.955 1907.993 1914.4227 3:1
Minor thirteenth 31 1965.3535 1971.5928 1978.2368
Neutral thirteenth 32 2028.752 2035.1926 2042.0509 13:4
Major thirteenth 33 2092.1505 2098.7923 2105.865 10:3, 27:8
Subminor fourteenth 34 2155.549 2162.3921 2169.6791 7:2
Minor fourteenth 35 2218.9475 2225.9919 2233.4932 18:5, 11:3
Major fourteenth 36 2282.346 2289.59165 2297.3073 15:4
Double octave, fifteenth (flat) 37 2345.7445 2353.1914 2361.1214
Double octave, fifteenth (sharp) 38 2409.143 2416.7912 2424.9355 4:1
Minor sixteenth 39 2472.5415 2480.39095 2488.74955 19:9
Neutral sixteenth 40 2535.94 2543.9907 2552.5636 13:4
Major sixteenth 41 2599.3405 2607.5905 2616.3777 9:2
Minor seventeenth 42 2662.737 2671.1903 2680.1918 14:3
Neutral seventeenth 43 2726.1355 2734.79 2744.0059
Major seventeenth; fifth harmonic 44 2789.534 2798.3898 2807.82 5:1
Minor eighteenth 45 2852.9325 2861.9896 2871.6341 21:4
Major eighteenth 46 2916.331 2925.5893 2935.4482 11:2 (flat)
Augmented eighteenth; diminished nineteenth 47 2979.7395 2989.1891 2999.2623 11:2 (sharp)
Minor nineteenth 48 3043.138 3052.7889 3063.0764
Perfect nineteenth, sixth harmonic 49 3106.5265 3116.3886 3126.8905 6:1
Minor twentieth 50 3169.925 3179.9884 3190.70455 19:3
Neutral twentieth 51 3233.3235 3243.5882 3254.5196 13:2
Major twentieth 52 3296.722 3307.1879 3318.3327 27:4
(Sub)minor twenty-first; 7th harmonic 53 3360.1205 3370.7877 3382.1468 7:1
Minor/neutral twenty-first 54 3423.519 3434.3875 3445.9608 22:3
Major twenty-first 55 3486.9175 3497.9872 3509.775 15:2
Twenty-second, triple octave, 8th harmonic (flat) 56 3550.316 3561.587 3573.5891 8:1 (flat)
Twenty-second, triple octave, 8th harmonic (sharp) 57 3613.7145 3625.1868 3637.4032 8:1 (sharp)
Minor twenty-third 58 3677.113 3688.7865 3701.2173 17:2
Neutral twenty-third 59 3740.5115 3752.3863 3765.0314
Major twenty-third; 9th harmonic 60 3803.91 3815.9861 3828.8455 9:1
Minor twenty-fourth 61 3867.3085 3879.58585 3892.65955 19:2
Neutral twenty-fourth 62 3930.707 3943.1856 3956.4736 29:3
Major twenty-fourth, decade 63 3994.1055 4006.7854 4020.2877 10:1 (sharp), 31:3 (flat)
Twenty-fifth 64 4057.504 4070.38515 4084.1018 21:2
Half-augmented twenty-fifth; 11th harmonic 65 4120.9025 4133.9849 4147.9159 11:1
Diminished twenty-sixth 66 4184.301 4197.5847 4211.730 34:3
Minor twenty-sixth 67 4247.6995 4261.1845 4275.5441 12:1 (flat)
Major twenty-sixth 68 4311.098 4324.7842 4339.3582 12:1 (sharp)
Minor twenty-seventh 69 4374.4965 4388.384 4403.1723 25:2
Neutral twenty-seventh, thirteenth harmonic 70 4437.895 4451.9838 4466.9864 13:1
Major twenty-seventh 71 4501.2955 4515.5835 4530.8005 27:2
Minor twenty-eighth, 14th harmonic 72 4564.692 4579.1833 4594.61455 14:1
Neutral twenty-eighth 73 4628.0905 4642.7831 4658.4286 15:1 (flat)
Major twenty-eighth 74 4691.489 4706.3828 4722.2427 15:1 (sharp)
Twenty-ninth; quadruple octave, 16th harmonic 75 4754.8875 4769.9826 4786.0568 16:1
Half-augmented twenty-ninth 76 4818.286 4833.5824 4849.8709 33:2
Minor thirtieth, 17th harmonic 77 4881.6945 4897.1821 4913.685 17:1
Chromatic major thirtieth 78 4945.093 4960.7819 4977.4991 35:2
Diatonic major thirtieth 79 5008.4815 5024.3817 5041.3132 37:2
Minor thirty-first, 19th harmonic 80 5071.87 5087.9814 5105.1273 19:1
Neutral thirty-first 81 5135.528 5151.5812 5168.9414
Major thirty-first 82 5198.677 5215.181 5232.7555 41:2
Thirty-second, 21st harmonic 83 5262.0755 5278.7807 5296.56955 21:1
Half-augmented thirty-second, 22nd harmonic 84 5325.474 5342.8051 5360.3836 22:1
23rd harmonic 85 5388.8725 5405.9803 5424.1977 23:1
Thirty-third 86 5452.271 5469.58005 5488.0118 24:1, 47:2 *
Half-augmented thirty-third 87 5515.6695 5533.1798 5551.8259 49:2
Minor thirty-fourth 88 5579.068 5596.7796 5615.640 51:2
Major thirty-fourth 89 5642.4665 5660.37935 5679.4541 53:2
Supermajor thirty-fourth 90 5705.865 5723.9791 5743.2682 55:2
Minor thirty-fifth 91 5769.2635 5787.5789 5807.0823 57:2
Neutral/major thirty-fifth 92 5832.662 5851.1787 5870.8964
31st harmonic 93 5896.0605 5914.7784 5934.7105 31:1 (wide-end only)
32nd harmonic, Thirty-sixth 94 5959.459 5978.3782 5998.52455 32:1
33rd harmonic 95 6022.8575 6041.978 6062.3386 33:1
34th harmonic 96 6086.256 6105.5777 6126.1527 34:1 (narrow-end only)
35th~36th harmonic 97 6149.6545 6169.1775 6189.9688 35:1, 36:1
37th harmonic 98 6213.053 6232,7773 6253.7809 37:1
Minor thirty-eighth, 38th harmonic 99 6276.4515 6296.377 6317.595 192:5, 38:1

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.