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
• 1–12/11–3/2–9/5 with steps 12/11, 11/8, 6/5, 11/10;
• 1–5/4–3/2–9/5 with steps 5/4, 6/5, 6/5, 11/10, and its inverse
• 1–6/5–3/2–5/3 with steps 6/5, 5/4, 11/10, 6/5.

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

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

The inversely related pair of chords is

• 1–5/4–3/2–5/3–9/5 with steps 5/4, 6/5, 11/10, 12/11, 11/10, and its inverse
• 1–6/5–3/2–5/3–9/5 with steps 6/5, 5/4, 11/10, 12/11, 11/10.

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