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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)