Ramanujanismic chords

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Ramanujanismic chords are essentially tempered 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.

These chords feature the octave-reduced 19th harmonic:

  • 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.

These chords feature the octave-reduced 7th harmonic:

  • 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;
  • 1-18/13-7/4 with steps 18/13-24/19-8/7, and its inverse
  • 1-24/19-7/4 with steps 24/19-18/13-8/7.

These chords feature the octave-reduced 13th harmonic:

  • 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;
  • 1-9/7-13/8 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.

These chords put an otonal 19 against a utonal 7, or the other way around:

  • 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;
  • 1-3/2-7/4-36/19 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;
  • 1-24/19-3/2-7/4 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;
  • 1-9/7-3/2-19/12 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;

These chords put an otonal 19 against a utonal 13, or the other way around:

  • 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;
  • 1-3/2-13/8-36/19 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;
  • 1-24/19-3/2-13/8 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.
  • 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;

These chords put an otonal 7 against a utonal 13, or the other way around:

  • 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;
  • 1-18/13-3/2-7/4 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;
  • 1-9/7-3/2-13/8 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;
  • 1-7/6-18/13-3/2 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;

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.

These chords put an otonal 19 against utonal 7 and 13, or the other way around, without the fifth:

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

These chords put an otonal 7 against utonal 13 and 19, or the other way around, without the fifth:

  • 1-18/13-7/4-36/19 with steps 18/13-24/19-13/12-19/18, and its inverse
  • 1-24/19-7/4-24/13 with steps 24/19-18/13-19/18-13/12;
  • 1-19/16-13/8-12/7 with steps 19/16-26/19-19/18-7/6, and its inverse
  • 1-19/18-13/9-12/7 with steps 19/18-26/19-19/16-7/6;
  • 1-7/6-24/19-24/13 with steps 7/6-13/12-19/13-13/12.

These chords put an otonal 13 against utonal 7 and 19, or the other way around, without the fifth:

  • 1-9/7-13/8-36/19 with steps 9/7-24/19-7/6-19/18, and its inverse
  • 1-24/19-13/8-12/7 with steps 24/19-9/7-19/18-7/6;
  • 1-19/16-7/4-24/13 with steps 19/16-28/19-19/18-13/12, and its inverse
  • 1-19/18-14/9-24/13 with steps 19/18-28/19-19/16-13/12;
  • 1-13/12-24/19-12/7 with steps 13/12-7/6-19/14-7/6.

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.

For pentads, the following chords put an otonal 19 against utonal 7 and 13, or the other way around:

  • 1-19/16-9/7-18/13-3/2 with steps 19/16-13/12-14/13-13/12-4/3, and its inverse
  • 1-13/12-7/6-24/19-3/2 with steps 13/12-14/13-13/12-19/16-4/3;
  • 1-3/2-13/8-7/4-36/19 with steps 3/2-13/12-14/13-13/12-19/18, and its inverse
  • 1-3/2-19/12-12/7-24/13 with steps 3/2-19/18-13/12-14/13-13/12.

These chords put an otonal 7 against utonal 13 and 19, or the other way around:

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

These chords put an otonal 13 against utonal 7 and 19, or the other way around:

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

These chords put two otonal 19's against a utonal 7 or 13, or the other way around:

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

These chords put two otonal 7's against a utonal 13 or 19, or the other way around:

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

These chords put two otonal 13's against a utonal 7 or 19, or the other way around:

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

For hexads, the following chords put two otonal 19's against utonal 7 and 13, or the other way around:

  • 1-19/16-3/2-19/12-12/7-24/13 with steps 19/16-24/19-19/18-13/12-14/13-13/12, and its inverse
  • 1-24/19-3/2-13/8-7/4-36/19 with steps 24/19-19/16-13/12-14/13-13/12-19/18.

These chords put two otonal 7's against utonal 13 and 19, or the other way around:

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

These chords put two otonal 13's against utonal 7 and 19, or the other way around:

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

These chords put otonal 9 and 19 against utonal 7 and 13, or the other way around:

  • 1-9/8-19/16-9/7-18/13-3/2 with steps 9/8-19/18-13/12-14/13-13/12-4/3, and its inverse
  • 1-13/12-7/6-24/19-4/3-3/2 with steps 13/12-14/13-13/12-19/18-9/8-4/3.

These chords put otonal 9 and 7 against utonal 13 and 19, or the other way around:

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

These chords put otonal 9 and 13 against utonal 7 and 19, or the other way around:

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

Equal temperaments with ramanujanismic chords include 17, 27, 36, 41, 53, 77, 94, 104, 113, 130, 140, 166, 207, 234, 243, 270, 347, 400 and 477.

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