Hyperpent and hypopent

Hyperpent or hyperpental edo is an equal division of the octave which has a just fifth that is less than a syntonic comma sharp. It is juxtaposed by hypopent or hypopental edo tuning systems, which have a just fifth that is less than a syntonic comma flat. In particular, a hyperpent edo has a fifth ranging from 3/2 to 243/160 and a hypopent edo has a fifth ranging from 40/27 to 3/2.

Background

"Hyperpental" (about 702 to 723c) loosely corresponds to tunings of the 5L 7s sharp chromatic scale (700 to 720 cents) while "hypopental" (about 680-702c) loosely corresponds to tunings of the 7L 5s mellow chromatic scale (686 to 700 cents).

Some edos may contain no intervals in the span of a just fifth plus or minus a syntonic comma, and thus are neither hyperpent nor hypopent, and are refered to as "anpent." By necessity, these edos have no diatonic fifth, and are a finite set (the largest being 23edo, which is also the largest edo to lack a diatonic scale).

Some edos are hypopental but generate 5L 7s. One example is 53edo, which tempers the fifth a fraction of a cent flat.

Partial List

Anpent (neither hypopent nor hyperpent)

2edo, 3edo, 4edo, 6edo, 8edo, 9edo, 11edo, 13edo, 16edo, 18edo, 23edo

Hyperpent

5edo, 10edo, 15edo, 17edo, 20edo, 22edo, 25edo, 27edo, 29edo, 30edo, 32edo, 34edo, 39edo, 41edo, 46edo

Hypopent

7edo, 12edo, 14edo, 19edo, 21edo, 24edo, 26edo, 28edo, 31edo, 33edo, 36edo, 38edo, 43edo