Harmony of 23edo

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If you take a look at the intervals of 23edo, you'll 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 trivialize 23edo by calling it "atonal" or "inharmonic", you should 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 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):

Degrees Armodue note Cents sizes Nearest Harmonic 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
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
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 64/33 1146.727 +1.099
23·· (or 0) 1 (9t) 1200.000 2/1 1200.000 none

You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. (<And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& 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. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Thus we produce ten triads, five tetrads, & one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & 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)

Pentad

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)