Ptolemismic chords

Ptolemismic chords are essentially tempered chords tempered by the ptolemisma, 100/99.

There are one triad, six tetrads and four pentads as 2.3.5.11 subgroup 11-odd-limit essentially tempered chords.

The ptolemismic triad consists of 11/8 and a stack of two 6/5's closing at the octave:

• 1-11/8-5/3 with steps of 11/8-6/5-6/5.

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

• 1-9/8-5/4-9/5 with steps 9/8-11/10-16/11-11/10;
• 1-3/2-5/3-9/5 with steps 3/2-11/10-12/11-11/10.

The inversely related pairs of chords are

• 1-11/8-3/2-5/3 with steps 11/8-12/11-11/10-6/5, and its inverse
• 1-11/8-5/3-11/6 with steps 11/8-6/5-11/10-12/11;
• 1-5/4-11/8-5/3 with steps 5/4-11/10-6/5-6/5, and its inverse
• 1-11/10-11/8-5/3 with steps 11/10-5/4-6/5-6/5.

For pentads, there are two palindromic chords and a pair of chords in inverse relationship. The palindromic chords are

• 1-11/8-3/2-5/3-11/6 with steps 11/8-12/11-11/10-11/10-12/11;
• 1-9/8-5/4-3/2-9/5 with steps 9/8-11/10-6/5-6/5-11/10.

The inversely related pair of chords is

• 1-5/4-11/8-3/2-5/3 with steps 5/4-11/10-12/11-11/10-6/5, and its inverse
• 1-11/10-11/8-5/3-11/6 with steps 11/10-5/4-6/5-11/10-12/11.

Equal divisions with ptolemismic chords include 19edo, 22edo, 37edo, 41edo and 104edo.