Wizardharry chords

Wizardharry chords are essentially tempered chords tempered by the wizardharry comma, 4000/3993.

The wizardharry temperament has a seven-tone scale which consists of a chain of six submajor seconds, of size approximately 11/10, and a whole tone of size approximately 9/8, which closes at the octave since the wizardharry comma is tempered out. It contains two triads, four tetrads and two pentads as 2.3.11/5 subgroup 33-odd-limit essentially tempered chords.

The triads are

• 1-11/10-40/33 chord with steps of 11/10-11/10-33/20;
• 1-40/33-22/15 chord with steps of 40/33-40/33-15/11.

The tetrads are

• 1-11/10-40/33-4/3 chord with steps of 11/10-11/10-11/10-3/2;
• 1-11/10-40/33-22/15 chord with steps of 11/10-11/10-40/33-15/11, and its inverse
• 1-40/33-4/3-22/15 chord with steps of 40/33-11/10-11/10-15/11;
• 1-40/33-22/15-16/9 chord with steps of 40/33-40/33-40/33-9/8.

The pentads are

• 1-11/10-40/33-4/3-22/15 chord with steps of 11/10-11/10-11/10-11/10-15/11;
• 1-40/33-4/3-22/15-16/9 chord with steps of 40/33-11/10-11/10-40/33-9/8.

In the 2.3.5.11 subgroup, the wizardharry temperament contains many possibilities for essentially tempered chords. It includes 12 triads, and even nonads, as 33-odd-limit essentially tempered chords.

For triads, there are five additional pairs of chords in inverse relationship:

• 1-11/10-25/22 with steps of 11/10-33/32-44/25 and its inverse 1-33/32-25/22 with steps of 33/32-11/10-44/25;
• 1-40/33-11/8 with steps of 40/33-25/22-16/11 and its inverse 1-25/22-11/8 with steps of 25/22-40/33-16/11;
• 1-11/8-50/33 with steps of 11/8-11/10-33/25 and its inverse 1-11/10-50/33 with steps of 11/10-11/8-33/25;
• 1-22/15-50/33 with steps of 22/15-33/32-33/25 and its inverse 1-33/32-50/33 with steps of 33/32-22/15-33/25;
• 1-50/33-11/6 with steps of 50/33-40/33-12/11 and its inverse 1-40/33-11/6 with steps of 40/33-50/33-12/11.

Remarkable nonads include a pair of chords in inverse relationship:

• 1-33/32-11/10-25/22-5/4-11/8-3/2-33/20-20/11 with steps of 33/32-16/15-33/32-11/10-11/10-12/11-11/10-11/10-11/10, and its inverse
• 1-33/32-25/22-5/4-11/8-3/2-33/20-20/11-15/8 with steps of 33/32-11/10-11/10-11/10-12/11-11/10-11/10-33/32-16/15.

Equal temperaments with wizardharry chords include 7, 15, 22, 29, 43, 50, 58, 65, 72, 80, 87, 94, 137, 152, 159, 217, 224 and 289.