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.

Background

"Hyper-" is the Greek prefix for high or above and "pent" is Greek for five. The term is loosely associated with "superpythagorean," but specifically refers to equal division of the octave that result in any fifth tempered sharp, while "hypo-" is the Greek prefix for low or below and "pent" is Greek for five. The term is loosely associated with "meantone," but specifically refers to equal division of the octave that result in any fifth tempered flat.

Large edo's are generally "amphipent," meaning both hyperpent and hypopent, because they contain multiple representations of the fifth that may be sharp and flat. Some edo's 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."

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

Amphipent (both hyperpent and hypopent)

35edo, 37edo, 40edo, 42edo, 44edo, 45edo, 47edo

Analysis

If 5edo and 7edo are taken as the smallest hyperpent and hypopent edo's, respectively, other edo tuning systems (let's call it "Xedo" with "X" equal divisions of the octave) can be determined to be hyperpent, hypopen, anpent, or amphipent by addition as follows (with a few exceptions):

If X can be represented as a sum of 5's, Xedo is hyperpent.

If X can be represented as a sum of 7's, Xedo is hypopent.

If X can be represented as a sum of 5's and 7's, with more 5's than 7's, Xedo is hyperpent.

If X can be represented as a sum of 5's and 7's, with more 7's than 5's, or an equal number of 7's and 5's, Xedo is hypopent.

If X cannot be represented as a sum of 5's and 7's, Xedo is anpent. If X can be represented as a sum of 5's and 7's in multiple ways (for example, 35 can be represented as five sevens or as seven fives), Xedo is amphipent.

Some exceptions are 37edo and 44edo, which are amphipent, even though 37 cannot be written as multiple sums of 5's and 7's, nor can 44.