Aberschismic

Hemifamity or Aberschismic, 2.3.5.7[41 & 46 & 53], is a 7-limit rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:

1. equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63
2. equates the limma 256/243 with 21/20
3. equates the apotome 2187/2048 with 15/14
4. equates the Pythagorean augmented fourth 729/512 with 10/7
5. splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63

It manifests in aberrismic theory where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.

Structural theory

Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.

Supporting edos

Comma resolution

The following edos map both 81/80 and 64/63 to one edostep: 41edo, 46edo, 48edo, 53edo, 58edo.

Half-comma resolution

The following edos map both 81/80 and 64/63 to two edosteps: 87edo, 94edo, 99edo, 111edo.

Third-comma resolution

The following edos map both 81/80 and 64/63 to three edosteps: 128edo, 133edo, 135edo, 140edo, 145edo, 147edo, 152edo, 157edo.

Supporting rank-2 temperaments

• 2.3.5.7[41 & 46]: Rodan
• 2.3.5.7[41 & 48]: Monkey
• 2.3.5.7[41 & 53]: Garibaldi
• 2.3.5.7[41 & 58]: Hemififths
• 2.3.5.7[46 & 53]: Amity
• 2.3.5.7[46 & 58]: Septimal Diaschismic
• 2.3.5.7[53 & 58]: Buzzard
• 2.3.5.7[87 & 99]: Misty

Extensions

• Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).
• Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as 140edo.

List of patent vals

Main article: Hemifamity/Patent vals