Supermagic chords

A supermagic chord is an 11-odd-limit essentially tempered chord in the supermagic temperament. Since 100/99 is tempered out, ptolemismic chords are also supermagic chords; since 385/384 is tempered out, keenanismic chords are also supermagic chords. Aside from these, there are also essentially supermagic tempered chords.

The most basic of these is the magical seventh chord, consisting of three sharp classical minor thirds and a septimal whole tone, which closes at the octave since both 100/99 and 385/384 (and therefore 875/864) are tempered out. This means the chord is the tempering of

• 1–6/5–16/11–7/4 with steps 6/5, 6/5, 6/5, 8/7.

In an optimized tuning for the supermagic temperament, the marvel comma 225/224 shrinks in size and may reverse direction, and adding it to the list of commas does little tuning damage; this results in 11-limit magic temperament, which has the same optimal patent val (104edo). Hence magic (19&22) temperament is practically the most accurate temperament that include this chord. Magic, however, does give it a Graham complexity of 12, so it does not appear that often.

Other temperaments that feature this chord prominently include 11-limit keemun, superkleismic, porcupine and doublewide.

For other tetrads, there are

• 1–5/4–16/11–9/5 with steps 5/4, 7/6, 5/4, 10/9;
• 1–12/11–5/4–9/5 with steps 12/11, 8/7, 16/11, 10/9, and its inverse
• 1–12/11–6/5–7/4 with steps 12/11, 10/9, 16/11, 8/7.

For pentads, there are

• 1–12/11–5/4–3/2–9/5 with steps 12/11, 8/7, 6/5, 6/5, 10/9, and its inverse
• 1–6/5–11/8–3/2–5/3 with steps 6/5, 8/7, 12/11, 10/9, 6/5;
• 1–5/4–11/8–3/2–12/7 with steps 5/4, 11/10, 12/11, 8/7, 7/6, and its inverse
• 1–12/11–6/5–3/2–7/4 with steps 12/11, 11/10, 5/4, 7/6, 8/7.

The count of supermagic chords is therefore tetrads: 4, and pentads: 4, for a total of 8.