Eratosthenes chords

Eratosthenes chords are essentially tempered chords tempered by the Eratosthenes comma, 1216/1215.

There are three triads, six tetrads and two pentads as 2.3.5.19 subgroup 19-odd-limit essentially tempered chords.

For triads, there are one palindromic chord and a pair of chords in inverse relationship. The palindromic chord is

• 1-9/8-19/15 chord with steps of 9/8-9/8-30/19.

The inversely related pair of chords is

• 1-16/15-9/8 with steps of 16/15-19/18-16/9 and its inverse 1-19/18-9/8 with steps of 19/18-16/15-16/9.

For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

• 1-16/15-9/8-6/5 chord with steps of 16/15-19/18-16/15-5/3;
• 1-19/18-9/8-19/16 chord with steps of 19/18-16/15-19/18-32/19.

The inversely related pairs of chords are

• 1-9/8-6/5-19/15 with steps of 9/8-16/15-19/18-30/19 and its inverse 1-19/18-9/8-19/15 with steps of 19/18-16/15-9/8-30/19;
• 1-9/8-19/16-19/15 with steps of 9/8-19/18-16/15-30/19 and its inverse 1-16/15-9/8-19/15 with steps of 16/15-19/18-9/8-30/19.

They can be extended to pentads:

• 1-16/15-9/8-6/5-19/15 with steps of 16/15-19/18-16/15-19/18-30/19 and its inverse 1-19/18-9/8-19/16-19/15 with steps of 19/18-16/15-19/18-16/15-30/19.

Equal temperaments with eratosthenes chords include 12, 41, 53, 65, 77, 87, 99, 118, 152, 164, 205, 217, 270 and 323.