Harmony of 23edo: Difference between revisions

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You'll 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, & 55/32; & 1 cent of 17/16, 79/64, & 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, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's 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''' -- 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.

You'll 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, & 55/32; & 1 cent of 17/16, 79/64, & 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, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's 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''' -- 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 64:66:68:70:72:79:84:92:110:117:124. 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.

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-You'll 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, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 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, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (&lt;'''And let's 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''' -- AKJ) Of course, it also has perfect unisons &amp; octaves, by definition. This means we could potentially build a very strange (&amp; 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 &amp; tetrads, as a starting point.
+You'll 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, &amp; 55/32; &amp; 1 cent of 17/16, 79/64, &amp; 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, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (&lt;'''And let's 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''' -- AKJ) Of course, it also has perfect unisons &amp; octaves, by definition. This means we could potentially build a very strange (&amp; 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. 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 &amp; 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, &amp; 23, excluding (at least for now) 33, 35, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.