Phoenix
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, see 16ed9/5), 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.
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 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.
Integers
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.4 | +6.0 | +16.8 | +12.1 | +14.4 | +2.0 | +25.2 | +12.1 | +20.5 | -17.3 | +22.8 | +11.5 | +10.4 |
| Relative (%) | +13.2 | +9.5 | +26.4 | +19.0 | +22.7 | +3.1 | +39.6 | +19.0 | +32.2 | -27.3 | +35.9 | +18.0 | +16.3 | |
| Steps (reduced) |
19 (3) |
30 (14) |
38 (6) |
44 (12) |
49 (1) |
53 (5) |
57 (9) |
60 (12) |
63 (15) |
65 (1) |
68 (4) |
70 (6) |
72 (8) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.5 | +12.5 | +24.9 | +21.5 | +24.9 | +13.3 | -26.4 | +24.9 | -29.8 | -3.4 | -26.4 | +26.5 | +25.8 |
| Relative (%) | +19.5 | +19.5 | +39.1 | +33.7 | +39.1 | +20.9 | -41.4 | +39.1 | -46.8 | -5.3 | -41.4 | +41.5 | +40.4 | |
| Steps (reduced) |
19 (8) |
30 (8) |
38 (5) |
44 (0) |
49 (5) |
53 (9) |
56 (1) |
60 (5) |
62 (7) |
65 (10) |
67 (1) |
70 (4) |
72 (6) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -2.0 | +0.0 | +13.7 | -2.0 | +31.2 | +0.0 | -3.9 | +13.7 | +48.7 | -2.0 | -40.5 | +31.2 |
| Relative (%) | +0.0 | -2.0 | +0.0 | +13.7 | -2.0 | +31.2 | +0.0 | -3.9 | +13.7 | +48.7 | -2.0 | -40.5 | +31.2 | |
| Steps (reduced) |
12 (0) |
19 (7) |
24 (0) |
28 (4) |
31 (7) |
34 (10) |
36 (0) |
38 (2) |
40 (4) |
42 (6) |
43 (7) |
44 (8) |
46 (10) | |
Primes
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.4 | +6.0 | +12.1 | +2.0 | -17.3 | +11.5 | -7.8 | -9.5 | -22.3 | +21.6 | -30.3 | -18.6 | -5.5 |
| Relative (%) | +13.2 | +9.5 | +19.0 | +3.1 | -27.3 | +18.0 | -12.2 | -15.0 | -35.1 | +34.0 | -47.6 | -29.2 | -8.6 | |
| Steps (reduced) |
19 (3) |
30 (14) |
44 (12) |
53 (5) |
65 (1) |
70 (6) |
77 (13) |
80 (0) |
85 (5) |
92 (12) |
93 (13) |
98 (2) |
101 (5) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.5 | +12.5 | +21.5 | +13.3 | -3.4 | +26.5 | +8.7 | +7.6 | -4.1 | -22.5 | -10.3 | +2.4 | +16.2 |
| Relative (%) | +19.5 | +19.5 | +33.7 | +20.9 | -5.3 | +41.5 | +13.7 | +11.9 | -6.4 | -35.3 | -16.2 | +3.8 | +25.3 | |
| Steps (reduced) |
19 (8) |
30 (8) |
44 (0) |
53 (9) |
65 (10) |
70 (4) |
77 (0) |
80 (3) |
85 (8) |
91 (3) |
93 (5) |
98 (10) |
101 (2) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -2.0 | +13.7 | +31.2 | +48.7 | -40.5 | -5.0 | +2.5 | -28.3 | -29.6 | -45.0 | +48.7 | -29.1 |
| Relative (%) | +0.0 | -2.0 | +13.7 | +31.2 | +48.7 | -40.5 | -5.0 | +2.5 | -28.3 | -29.6 | -45.0 | +48.7 | -29.1 | |
| Steps (reduced) |
12 (0) |
19 (7) |
28 (4) |
34 (10) |
42 (6) |
44 (8) |
49 (1) |
51 (3) |
54 (6) |
58 (10) |
59 (11) |
63 (3) |
64 (4) | |
Fenghuang temperament
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.
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 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).
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.
Intervals
| 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.