Swetismic chords

A swetismic chord is an essentially tempered chord tempered by the swetisma, 540/539.

Swetismic chords are of pattern 2 in the 11-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

There are six swetismic triads, consisting of three pairs of chords in inverse relationship:

• 1–7/5–12/7 with steps 7/5, 11/9, 7/6, and its inverse
• 1–7/6–10/7 with steps 7/6, 11/9, 7/5;
• 1–7/6–9/7 with steps 7/6, 11/10, 14/9, and its inverse
• 1–14/9–12/7 with steps 14/9, 11/10, 7/6;
• 1–9/7–7/5 with steps 9/7, 12/11, 10/7, and its inverse
• 1–10/7–14/9 with steps 10/7, 12/11, 9/7.

There are fifteen swetismic tetrads, consisting of three palindromic (self-inverse) chords and six pairs of chords in inverse relationship. The palindromic tetrads are:

• 1–7/6–9/7–3/2 with steps 7/6, 11/10, 7/6, 4/3;
• 1–6/5–7/5–12/7 with steps 6/5, 7/6, 11/9, 7/6;
• 1–9/7–7/5–9/5 with steps 9/7, 12/11, 9/7, 10/9.

The six pairs are:

• 1–9/7–3/2–11/6 with steps 9/7, 7/6, 11/9, 12/11, and its inverse
• 1–7/6–3/2–18/11 with steps 7/6, 9/7, 12/11, 11/9;
• 1–7/6–9/7–10/7 with steps 7/6, 11/10, 10/9, 7/5, and its inverse
• 1–7/5–14/9–12/7 with steps 7/5, 10/9, 11/10, 7/6;
• 1–7/6–9/7–18/11 with steps 7/6, 11/10, 14/11, 11/9, and its inverse
• 1–14/11–7/5–18/11 with steps 14/11, 11/10, 7/6, 11/9;
• 1–7/6–10/7–11/6 with steps 7/6, 11/9, 9/7, 12/11, and its inverse
• 1–9/7–11/7–11/6 with steps 9/7, 11/9, 7/6, 12/11;
• 1–7/6–9/7–11/6 with steps 7/6, 11/10, 10/7, 12/11, and its inverse
• 1–10/7–11/7–11/6 with steps 10/7, 11/10, 7/6, 12/11;
• 1–10/7–14/9–12/7 with steps 10/7, 12/11, 11/10, 7/6, and its inverse
• 1–7/6–9/7–7/5 with steps 7/6, 11/10, 12/11, 10/7.

Finally, there are six swetismic pentads coming in three pairs:

• 1–7/6–9/7–3/2–11/6 with steps 7/6, 11/10, 7/6, 11/9, 12/11, and its inverse
• 1–7/6–9/7–3/2–18/11 with steps 7/6, 11/10, 7/6, 12/11, 11/9;
• 1–7/6–10/7–5/3–11/6 with steps 7/6, 11/9, 7/6, 11/10, 12/11, and its inverse
• 1–7/6–10/7–5/3–20/11 with steps 7/6, 11/9, 7/6, 12/11, 11/10;
• 1–7/6–9/7–10/7–11/6 with steps 7/6, 11/10, 10/9, 9/7, 12/11, and its inverse
• 1–9/7–10/7–11/7–11/6 with steps 9/7, 10/9, 11/10, 7/6, 12/11.

If we are willing to consider the 15-odd-limit, there are also 15-odd-limit swetismic tetrads, including something important for functional harmony. The swetismic dominant seventh chord is a tempering of

• 1–9/7–3/2–7/4 with steps 9/7, 7/6, 7/6, 8/7.

Its inversion might be called the swetismic half-diminished chord, a tempering of

• 1–7/6–3/2–12/7 with steps 7/6, 9/7, 8/7, 7/6.

We also have

• 1–11/9–10/7–5/3 with steps 11/9, 7/6, 7/6, 6/5; and its inverse
• 1–7/6–15/11–5/3 with steps 7/6, 7/6, 11/9, 6/5.

Equal temperaments with swetismic chords include 19, 22, 31, 41, 53, 58, 72, 80, 94, 103, 111, 121, 130, 152, 183, 205, 224, 354, 537, 578, 761d, 1115de, 1339de, 1491de, 1715de and 1845de.