Chords of echidna

Below are listed the dyadic chords of 11-limit echidna temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering by 540/539 are labeled swetismic, by 896/891 pentacircle, and by 176/175 valinorsmic.

The normal mapping for echidna is ech = [<2 1 9 2 12|, <0 3 -6 5 -7|]. From this we may derive a val v = ech[1] + 100 ech[2] = <2 301 -591 502 -688| which we may use to sort and normalize the chords of echidna. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v, and the next column gives a transversal which tempers to the chord by echidnic tempering. Finally, the Graham complexity is listed in the last column.

Echidna has MOS of size 6, 8, 14, 22, 36, 58 and 80. It may be seen that a modest number of chords are available to 14 notes, but that 22 has many more, and that may be a good choice as a scale with which to explore the harmonies of echidna. If for some reason you needed pentads and hexads, 36 notes would be fine.

 Number Chord Transversal Type Complexity 1 0-100-199 1-9/7-7/6 swetismic 4 2 0-101-199 1-20/11-7/6 swetismic 4 3 0-100-299 1-9/7-3/2 utonal 6 4 0-199-299 1-7/6-3/2 otonal 6 5 0-199-498 1-7/6-7/4 utonal 10 6 0-299-498 1-3/2-7/4 otonal 10 7 0-100-596 1-9/7-9/8 utonal 12 8 0-299-596 1-3/2-9/8 ambitonal 12 9 0-498-596 1-7/4-9/8 otonal 12 10 0-101-597 1-20/11-8/5 valinorsmic 12 11 0-498-597 1-7/4-8/5 valinorsmic 12 12 0-100-696 1-9/7-16/11 pentacircle 14 13 0-101-696 1-20/11-16/11 otonal 14 14 0-596-696 1-9/8-16/11 pentacircle 14 15 0-597-696 1-8/5-16/11 utonal 14 16 0-299-894 1-3/2-6/5 utonal 18 17 0-597-894 1-8/5-6/5 otonal 18 18 0-101-993 1-20/11-12/11 otonal 20 19 0-299-993 1-3/2-12/11 utonal 20 20 0-696-993 1-16/11-12/11 otonal 20 21 0-894-993 1-6/5-12/11 utonal 20 22 0-100-1093 1-9/7-7/5 swetismic 22 23 0-199-1093 1-7/6-7/5 utonal 22 24 0-498-1093 1-7/4-7/5 utonal 22 25 0-597-1093 1-8/5-7/5 otonal 22 26 0-894-1093 1-6/5-7/5 otonal 22 27 0-993-1093 1-12/11-7/5 swetismic 22 28 0-101-1192 1-20/11-14/11 otonal 24 29 0-199-1192 1-7/6-14/11 utonal 24 30 0-498-1192 1-7/4-14/11 utonal 24 31 0-596-1192 1-9/8-14/11 pentacircle 24 32 0-597-1192 1-8/5-14/11 valinorsmic 24 33 0-696-1192 1-16/11-14/11 otonal 24 34 0-993-1192 1-12/11-14/11 otonal 24 35 0-1093-1192 1-7/5-14/11 utonal 24 36 0-100-1193 1-9/7-9/5 utonal 24 37 0-299-1193 1-3/2-9/5 utonal 24 38 0-596-1193 1-9/8-9/5 utonal 24 39 0-597-1193 1-8/5-9/5 otonal 24 40 0-894-1193 1-6/5-9/5 otonal 24 41 0-1093-1193 1-7/5-9/5 otonal 24 42 0-100-1292 1-9/7-18/11 utonal 26 43 0-101-1292 1-20/11-18/11 otonal 26 44 0-199-1292 1-7/6-18/11 swetismic 26 45 0-299-1292 1-3/2-18/11 utonal 26 46 0-596-1292 1-9/8-18/11 utonal 26 47 0-696-1292 1-16/11-18/11 otonal 26 48 0-993-1292 1-12/11-18/11 otonal 26 49 0-1093-1292 1-7/5-18/11 swetismic 26 50 0-1192-1292 1-14/11-18/11 otonal 26 51 0-1193-1292 1-9/5-18/11 utonal 26

Tetlist

 Number Chord Transversal Type Complexity 1 0-100-199-299 1-9/7-7/6-3/2 swetismic 6 2 0-199-299-498 1-7/6-3/2-7/4 ambitonal 10 3 0-100-299-596 1-9/7-3/2-9/8 utonal 12 4 0-299-498-596 1-3/2-7/4-9/8 otonal 12 5 0-100-596-696 1-9/7-9/8-16/11 pentacircle 14 6 0-101-597-696 1-20/11-8/5-16/11 valinorsmic 14 7 0-101-696-993 1-20/11-16/11-12/11 otonal 20 8 0-299-894-993 1-3/2-6/5-12/11 utonal 20 9 0-100-199-1093 1-9/7-7/6-7/5 swetismic 22 10 0-199-498-1093 1-7/6-7/4-7/5 utonal 22 11 0-498-597-1093 1-7/4-8/5-7/5 valinorsmic 22 12 0-597-894-1093 1-8/5-6/5-7/5 otonal 22 13 0-894-993-1093 1-6/5-12/11-7/5 swetismic 22 14 0-101-199-1192 1-20/11-7/6-14/11 swetismic 24 15 0-199-498-1192 1-7/6-7/4-14/11 utonal 24 16 0-498-596-1192 1-7/4-9/8-14/11 pentacircle 24 17 0-101-597-1192 1-20/11-8/5-14/11 valinorsmic 24 18 0-498-597-1192 1-7/4-8/5-14/11 valinorsmic 24 19 0-101-696-1192 1-20/11-16/11-14/11 otonal 24 20 0-596-696-1192 1-9/8-16/11-14/11 pentacircle 24 21 0-597-696-1192 1-8/5-16/11-14/11 valinorsmic 24 22 0-101-993-1192 1-20/11-12/11-14/11 otonal 24 23 0-696-993-1192 1-16/11-12/11-14/11 otonal 24 24 0-199-1093-1192 1-7/6-7/5-14/11 utonal 24 25 0-498-1093-1192 1-7/4-7/5-14/11 utonal 24 26 0-597-1093-1192 1-8/5-7/5-14/11 valinorsmic 24 27 0-993-1093-1192 1-12/11-7/5-14/11 swetismic 24 28 0-100-299-1193 1-9/7-3/2-9/5 utonal 24 29 0-100-596-1193 1-9/7-9/8-9/5 utonal 24 30 0-299-596-1193 1-3/2-9/8-9/5 utonal 24 31 0-299-894-1193 1-3/2-6/5-9/5 ambitonal 24 32 0-597-894-1193 1-8/5-6/5-9/5 otonal 24 33 0-100-1093-1193 1-9/7-7/5-9/5 swetismic 24 34 0-597-1093-1193 1-8/5-7/5-9/5 otonal 24 35 0-894-1093-1193 1-6/5-7/5-9/5 otonal 24 36 0-100-199-1292 1-9/7-7/6-18/11 swetismic 26 37 0-101-199-1292 1-20/11-7/6-18/11 swetismic 26 38 0-100-299-1292 1-9/7-3/2-18/11 utonal 26 39 0-199-299-1292 1-7/6-3/2-18/11 swetismic 26 40 0-100-596-1292 1-9/7-9/8-18/11 utonal 26 41 0-299-596-1292 1-3/2-9/8-18/11 utonal 26 42 0-100-696-1292 1-9/7-16/11-18/11 pentacircle 26 43 0-101-696-1292 1-20/11-16/11-18/11 otonal 26 44 0-596-696-1292 1-9/8-16/11-18/11 pentacircle 26 45 0-101-993-1292 1-20/11-12/11-18/11 otonal 26 46 0-299-993-1292 1-3/2-12/11-18/11 ambitonal 26 47 0-696-993-1292 1-16/11-12/11-18/11 otonal 26 48 0-100-1093-1292 1-9/7-7/5-18/11 swetismic 26 49 0-199-1093-1292 1-7/6-7/5-18/11 swetismic 26 50 0-993-1093-1292 1-12/11-7/5-18/11 swetismic 26 51 0-101-1192-1292 1-20/11-14/11-18/11 otonal 26 52 0-199-1192-1292 1-7/6-14/11-18/11 swetismic 26 53 0-596-1192-1292 1-9/8-14/11-18/11 pentacircle 26 54 0-696-1192-1292 1-16/11-14/11-18/11 otonal 26 55 0-993-1192-1292 1-12/11-14/11-18/11 otonal 26 56 0-1093-1192-1292 1-7/5-14/11-18/11 swetismic 26 57 0-100-1193-1292 1-9/7-9/5-18/11 utonal 26 58 0-299-1193-1292 1-3/2-9/5-18/11 utonal 26 59 0-596-1193-1292 1-9/8-9/5-18/11 utonal 26 60 0-1093-1193-1292 1-7/5-9/5-18/11 swetismic 26

Pentlist

 Number Chord Transversal Type Complexity 1 0-101-597-696-1192 1-20/11-8/5-16/11-14/11 valinorsmic 24 2 0-101-696-993-1192 1-20/11-16/11-12/11-14/11 otonal 24 3 0-199-498-1093-1192 1-7/6-7/4-7/5-14/11 utonal 24 4 0-498-597-1093-1192 1-7/4-8/5-7/5-14/11 valinorsmic 24 5 0-100-299-596-1193 1-9/7-3/2-9/8-9/5 utonal 24 6 0-597-894-1093-1193 1-8/5-6/5-7/5-9/5 otonal 24 7 0-100-199-299-1292 1-9/7-7/6-3/2-18/11 swetismic 26 8 0-100-299-596-1292 1-9/7-3/2-9/8-18/11 utonal 26 9 0-100-596-696-1292 1-9/7-9/8-16/11-18/11 pentacircle 26 10 0-101-696-993-1292 1-20/11-16/11-12/11-18/11 otonal 26 11 0-100-199-1093-1292 1-9/7-7/6-7/5-18/11 swetismic 26 12 0-101-199-1192-1292 1-20/11-7/6-14/11-18/11 swetismic 26 13 0-101-696-1192-1292 1-20/11-16/11-14/11-18/11 otonal 26 14 0-596-696-1192-1292 1-9/8-16/11-14/11-18/11 pentacircle 26 15 0-101-993-1192-1292 1-20/11-12/11-14/11-18/11 otonal 26 16 0-696-993-1192-1292 1-16/11-12/11-14/11-18/11 otonal 26 17 0-199-1093-1192-1292 1-7/6-7/5-14/11-18/11 swetismic 26 18 0-993-1093-1192-1292 1-12/11-7/5-14/11-18/11 swetismic 26 19 0-100-299-1193-1292 1-9/7-3/2-9/5-18/11 utonal 26 20 0-100-596-1193-1292 1-9/7-9/8-9/5-18/11 utonal 26 21 0-299-596-1193-1292 1-3/2-9/8-9/5-18/11 utonal 26 22 0-100-1093-1193-1292 1-9/7-7/5-9/5-18/11 swetismic 26