Boethius chords

Boethius chords are essentially tempered chords tempered by the Boethius comma, 513/512.

There are three triads, seven tetrads, five pentads and one hexad as 2.3.19 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

• 1-9/8-24/19 chord with steps of 9/8-9/8-19/12.

The inversely related pair of chords is

• 1-9/8-4/3 with steps of 9/8-19/16-3/2 and its inverse 1-19/16-4/3 with steps of 19/16-9/8-3/2.

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

• 1-9/8-4/3-3/2 chord with steps of 9/8-19/16-9/8-4/3;
• 1-19/16-4/3-19/12 chord with steps of 19/16-9/8-19/16-24/19;
• 1-9/8-19/16-4/3 chord with steps of 9/8-19/18-9/8-3/2.

The inversely related pairs of chords are

• 1-9/8-24/19-3/2 with steps of 9/8-9/8-19/16-4/3 and its inverse 1-19/16-4/3-3/2 with steps of 19/16-9/8-9/8-4/3;
• 1-9/8-24/19-4/3 with steps of 9/8-9/8-19/18-3/2 and its inverse 1-19/18-19/16-4/3 with steps of 19/18-9/8-9/8-3/2.

For pentads, there are one palindromic chord and two pairs of chords in inverse relationship. The palindromic chord is

• 1-9/8-4/3-3/2-16/9 chord with steps of 9/8-19/16-9/8-19/16-9/8.

The inversely related pairs of chords are

• 1-9/8-24/19-4/3-3/2 with steps of 9/8-9/8-19/18-9/8-4/3 and its inverse 1-9/8-19/16-4/3-3/2 with steps of 9/8-19/18-9/8-9/8-4/3;
• 1-9/8-24/19-4/3-32/19 with steps of 9/8-9/8-19/18-24/19-19/16 and its inverse 1-19/18-19/16-4/3-19/12 with steps of 19/18-9/8-9/8-19/16-24/19.

They can be extended to a hexad:

• 1-9/8-24/19-4/3-3/2-32/19 chord with steps of 9/8-9/8-19/18-9/8-9/8-19/16.

Equal temperaments with boethius chords include 5, 7, 12, 17, 19, 29, 41, 53, 65, 77, 89 and 101.