User:Frostburn/Hemiptol
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 | 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.