Hemipyth

Hemipyth refers to the √2.√3 subgroup i.e. intervals that can be constructed by multiplying fractional powers of 2 and 3 where the exponents have a denominator at most 2.

Notable hemipyth intervals include the neutral third √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = (√2)²/√3.

Many temperaments naturally produce intervals that split ~3/2, ~2 or ~4/3 exactly in half and can thus be interpreted as neutral thirds, semioctaves or semifourths within the temperament.

An important property of edos > 1 is that they must by necessity include at least one of the notable hemipyth intervals:

• Either the edo is even and it features at least √2 (which is tuned "pure" when the octave is tuned pure).
• Or one of the following is true:
  • The closest approximation to 3/2 spans an even number of edosteps (leading to an approximation to √(3/2))
  • The closest approximation to 4/3 spans an even number of edosteps (leading to an approximation to √(4/3))

List of edo mappings with full or partial hemipyth support

Edo (warts)	Has √2	Has √(3/2)	Has √(4/3)
2	yes	no	no
3	no	yes	no
4	yes	yes	yes
5	no	no	yes
6	yes	yes	yes
7	no	yes	no
8	yes	no	no
9	no	no	yes
10	yes	yes	yes
11	no	yes	no
12	yes	no	no
13	no	yes	no
13b	no	no	yes