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), 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.