Harmony of 23edo
If you take a look at the intervals of 23edo, you will find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than it atonal or inharmonic, we could consider the higher-limit harmonies that could serve as useful sonorities, perhaps even consonances, in the context of careful composition. 23edo contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 and 117. Let us compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):
| Degrees | Armodue note | Cents sizes | Just interval | Cents | Error |
|---|---|---|---|---|---|
| 0 | 1 | 0 | 1/1 | 0.000 | none |
| 1 | 1t (2b) | 52.174 | 33/32 | 53.273 | -1.099 |
| 2 | 2v (1#) | 104.348 | 17/16 | 104.955 | -0.607 |
| 3 | 2 | 156.522 | 35/32 | 155.140 | +1.382 |
| 4· | 2t (3b) | 208.696 | 9/8 | 203.910 | +4.786 |
| 5 | 3v (2#) | 260.869 | 50/43 | 261.110 | -0.241 |
| 6 | 3 | 313.043 | 6/5 | 315.641 | -2.598 |
| 7· | 3t (4b) | 365.217 | 79/64 | 364.537 | +0.68 |
| 8 | 4v (3#) | 417.391 | 14/11 | 417.508 | -0.117 |
| 9 | 4 (5v) | 469.565 | 21/16 | 470.781 | -1.216 |
| 10· | 5 (4t) | 521.739 | 23/17 | 523.319 | -1.58 |
| 11 | 5t (6b) | 573.913 | 32/23 | 571.726 | +2.187 |
| 12 | 6v (5#) | 626.087 | 23/16 | 628.274 | -2.187 |
| 13· | 6 | 678.261 | 34/23 | 676.681 | +1.58 |
| 14 | 6t (7b) | 730.435 | 32/21 | 729.219 | +1.216 |
| 15 | 7v (6#) | 782.609 | 11/7 | 782.492 | +0.117 |
| 16· | 7 | 834.783 | 34/21 | 834.175 | +0.608 |
| 17 | 7t (8b) | 886.957 | 5/3 | 884.359 | +2.598 |
| 18 | 8v (7#) | 939.130 | 55/32 | 937.632 | +1.498 |
| 19· | 8 | 991.304 | 39/22 | 991.165 | +0.139 |
| 20 | 8t (9b) | 1043.478 | 117/64 | 1044.438 | -0.96 |
| 21 | 9v (8#) | 1095.652 | 32/17 | 1095.045 | +0.607 |
| 22 | 9 (1v) | 1147.826 | 31/16 | 1145.036 | +2.791 |
| 23·· (or 0) | 1 (9t) | 1200.000 | 2/1 | 1200.000 | none |
You will see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, and 55/32; and 1 cent of 17/16, 79/64, and 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8; some can hear a distinct sound in spite of being only 6 cents sharper than 9/8. Let us also note the excellent representations of 14/11 and its inverse, 11/7. In fact they might be considered good enough that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament. Of course, it also has perfect unisons and octaves, by definition. This means we could potentially build a very strange (and slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117:124. This cluster can be a little hard to listen to, whether tuned to JI or 23edo, so consider smaller chords, triads and tetrads, as a starting point.
Here we set the 23rd harmonic as an arbitrary limit on how high up the harmonic series we will go, so harmonics 1, 9, 17, 21, and 23 are considered, and 33, 35, 55, 79, and 117 are excluded. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, we will leave them out.
Thus we produce ten triads, five tetrads, and one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. The chords below are written in closed position (within one octave), but trying different voicings is recommended. Moving chord tones up and down by octaves, you can unmuddy a muddy chord.
Triads
16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
18/17 (98.955, error: +5.393)
16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).
17/16 (104.955, error -0.607)
21/16 (470.781, error -1.216)
21/17 (365.825, error: -0.608)
16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).
17/16 (104.955, error -0.607)
23/16 (628.274, error -2.187)
23/17 (523.319, error: -1.578)
16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
21/18 = 7/6 (266.871, error: -6.001)
16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).
18/16 = 9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
23/18 (424.364, error: -6.973)
16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
23/21 (157.493, error: -0.971)
17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
21/18 = 7/6 (266.871, error: -6.001)
17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)
17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -0.971)
18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)
Tetrads
16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
21/18 = 7/6 (266.871, error: -6.001)
16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
23/17 (523.319, error: -1.578)
23/18 (424.364, error: -6.973)
16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).
17/16 (104.955, error -0.607)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
23/21 (157.493, error: -0.971)
16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)
17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)
Pentads
16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).
17/16 (104.955, error -0.607)
18/16 = 9/8 (203.910, error +4.786)
21/16 (470.781, error -1.216)
23/16 (628.274, error -2.187)
18/17 (98.955, error: +5.393)
21/17 (365.825, error: -0.608)
23/17 (523.319, error: -1.578)
21/18 = 7/6 (266.871, error: -6.001)
23/18 (424.364, error: -6.973)
23/21 (157.493, error: -0.971)