Ramanujanismic chords

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Ramanujanismic chords are essentially tempered dyadic chords tempered by the ramanujanisma, 1729/1728.

Ramanujanismic chords are numerous. A basic ramanujanismic triad in the 2.3.7.13.19 subgroup 19-odd-limit consists of 7/6, 13/12, and 19/12 closing at the octave:

  • 1-7/6-24/19 with steps 7/6-13/12-19/12, and its inverse
  • 1-7/6-24/13 with steps 7/6-19/12-13/12;

Other ramanujanismic triads are

  • 1-19/16-9/7 with steps 19/16-13/12-14/9, and its inverse
  • 1-19/16-24/13 with steps 19/16-14/9-13/12;
  • 1-19/16-18/13 with steps 19/16-7/6-13/9, and its inverse
  • 1-19/16-12/7 with steps 19/16-13/9-7/6.
  • 1-7/4-36/19 with steps 7/4-13/12-19/18, and its inverse
  • 1-7/4-24/13 with steps 7/4-19/18-13/12;
  • 1-13/8-36/19 with steps 13/8-7/6-19/18, and its inverse
  • 1-13/8-12/7 with steps 13/8-19/18-7/6;
  • 1-18/13-7/4 with steps 18/13-24/19-8/7, and its inverse
  • 1-24/19-7/4 with steps 24/19-18/13-8/7;
  • 1-9/7-13/8 with steps 9/7-24/19-16/13, and its inverse
  • 1-24/19-13/8 with steps 24/19-9/7-16/13;

If we allow the 21-odd-limit, we have these additional triads:

  • 1-21/16-36/19 with steps 21/16-13/9-19/18, and its inverse
  • 1-21/16-18/13 with steps 21/16-19/18-13/9.

All the triads listed above work well with a perfect fifth, so we have the following tetrads:

  • 1-19/16-9/7-3/2 with steps 19/16-13/12-7/6-4/3, and its inverse
  • 1-7/6-24/19-3/2 with steps 7/6-13/12-19/16-4/3;
  • 1-19/16-18/13-3/2 with steps 19/16-7/6-13/12-4/3, and its inverse
  • 1-13/12-24/19-3/2 with steps 13/12-7/6-19/16-4/3;
  • 1-7/6-18/13-3/2 with steps 7/6-19/16-13/12-4/3, and its inverse
  • 1-13/12-9/7-3/2 with steps 13/12-19/16-7/6;
  • 1-3/2-7/4-36/19 with steps 3/2-7/6-13/12-19/18, and its inverse
  • 1-3/2-19/12-12/7 with steps 3/2-19/18-13/12-7/6;
  • 1-3/2-7/4-24/13 with steps 3/2-7/6-19/18-13/12, and its inverse
  • 1-3/2-13/8-12/7 with steps 3/2-13/12-19/18-7/6;
  • 1-3/2-13/8-36/19 with steps 3/2-13/12-7/6-19/18, and its inverse
  • 1-3/2-19/12-24/13 with steps 3/2-19/18-7/6-13/12;
  • 1-9/7-3/2-13/8 with steps 9/7-7/6-13/12-16/13, and its inverse
  • 1-7/6-3/2-24/13 with steps 7/6-9/7-16/13-13/12;
  • 1-9/7-3/2-19/12 with steps 9/7-7/6-19/18-24/19, and its inverse
  • 1-7/6-3/2-36/19 with steps 7/6-9/7-24/19-19/18;
  • 1-18/13-3/2-7/4 with steps 18/13-13/12-7/6-8/7, and its inverse
  • 1-13/12-3/2-12/7 with steps 13/12-18/13-8/7-7/6;
  • 1-18/13-3/2-19/12 with steps 18/13-13/12-19/18-24/19, and its inverse
  • 1-13/12-3/2-36/19 with steps 13/12-18/13-24/19-19/18;
  • 1-24/19-3/2-7/4 with steps 24/19-19/16-7/6-8/7, and its inverse
  • 1-19/16-3/2-12/7 with steps 19/16-24/19-8/7-7/6;
  • 1-24/19-3/2-13/8 with steps 24/19-19/16-13/12-16/13, and its inverse
  • 1-19/16-3/2-24/13 with steps 19/16-24/19-16/13-13/12.

Otherwise they work with a major second:

  • 1-9/8-19/16-9/7 with steps 9/8-19/18-13/12-14/9, and its inverse
  • 1-9/8-7/4-36/19 with steps 9/8-14/9-13/12-19/18;
  • 1-9/8-19/16-18/13 with steps 9/8-19/18-7/6-13/9, and its inverse
  • 1-9/8-13/8-36/19 with steps 9/8-13/9-7/6-19/18;
  • 1-9/8-18/13-7/4 with steps 9/8-16/13-24/19-8/7, and its inverse
  • 1-9/8-9/7-13/8 with steps 9/8-8/7-24/19-16/13.

If we allow the 21-odd-limit, we have these additional tetrads:

  • 1-21/16-18/13-3/2 with steps 21/16-19/18-13/12-4/3, and its inverse
  • 1-13/12-8/7-3/2 with steps 13/12-19/18-21/16-4/3;
  • 1-21/16-3/2-36/19 with steps 21/16-8/7-24/19-19/18, and its inverse
  • 1-8/7-3/2-19/12 with steps 8/7-21/16-19/18-24/19;
  • 1-9/8-21/16-18/13 with steps 9/8-7/6-19/18-13/9, and its inverse
  • 1-9/8-13/8-12/7 with steps 9/8-13/9-19/18-7/6;
  • 1-9/8-21/16-36/19 with steps 9/8-7/6-13/9-19/18, and its inverse
  • 1-9/8-19/16-12/7 with steps 9/8-19/18-13/9-7/6.