User:Frostburn/Hemiptol

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Oops! Misguided article. The name is technically correct, but I have no clue why this would be more interesting than √2.√3.5.

Hemiptol is to hemipyth what 5-limit (i.e. Ptolemaic tuning) is to 3-limit (i.e. Pythagorean tuning). It is the √2.√3.√5 subgroup i.e. the set of intervals that can be constructed by multiplying half-integer powers of 2, 3 and 5.

Supporting edos

To support hemiptol an equal temperament must map all 2, 3 and 5 to even numbers of edosteps.

Edos (patent vals) under 100 that achieve this are: 6, 20, 24, 30, 38, 44, 62, 68, 76, 82, 86, 92 (and 100 itself).

Of these, 82 being the double of 41edo is arguably the most convincing/accurate tuning for hemiptol.

Higher prime fudging

Motivation for the system can be found from temperaments that are accurate enough to fudging most of the higher primes with the simple action of squashing the 5-limit to 50% of its size in cents.

Prime fudges
Prime Hemiptol interval Associated comma
7 √(4000/81) octagar comma
11 √(243/2) rastma
13 √(675/4) island comma
17 √288 semitonisma
19 √(729/2) 729/722
23 √(1600/3) 1600/1587
29 √(3375/4) 3375/3364
31 √960 961/960

Most of these don't even need the square root of 5, smh.