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.

Equal temperaments

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
14	yes	yes	yes
15	no	no	yes
16	yes	no	no
17	no	yes	no
18	yes	no	no
18b	yes	yes	yes
19	no	yes	no
20*	yes	yes	yes
20b	yes	no	no
21	no	yes	no
22	yes	no	no
23	no	no	yes
24	yes	yes	yes

*) Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.

Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.

Notation

The Pythagorean 2.3 part of hemipyth can be notated using traditional notation where octaves represent multiples of 2/1, chain of fifths denotes multiples of 3/2, the sharp sign is equal to 2187/2048 etc.

Neutral thirds

The 2.√(3/2) part can be notated using neutral chain-of-fifths notation. This introduces a neutral interval quality between major and minor, semisharps etc.

Semioctaves

In traditional notation the octave spans 7 diasteps which means that it splits into two interordinal 3½ diasteps or two perfect 4.5ths if we wish to remain backwards compatible with the 1-indexed traditional notation.

Intervals retain their quality when the frequency ratio is multiplied by the perfect semioctave √2 e.g. M6 - P4.5 = M2.5 = (9/8)^(3/2).

Relative interordinal intervals are either called by their double i.e. M2.5 is a major semifourth due to being exactly the half of an augmented fourth (Aug4), or by simply adding the suffix "-and-a-halfth" i.e. "major second-and-a-halfth". The semisecond get the special nickname "sesquith".

The nominals for absolute pitches are denoted using lowercase Greek nominals (uppercase often looks identical to pre-existing Latin nominals). The logic being that Latin and Greek notes differ by a multiple of √2 when paired up alphabetically. The direction is determined by octaves starting from the middle C.

Semioctave nominals

Nominal	Pronuciation	Meaning	Ratio with middle C	Cents
γ	gam	C + P4.5	√2	600.000
δ	del	D + P4.5	√(81/32)	803.910
ε	eps	E + P4.5	√(6561/2048)	1007.820
ζ	zet	F + P4.5	√(32/9)	1098.045
η	eta	G - P4.5	√(9/8)	101.955
α	alp	A - P4.5	(9/8)^(3/2)	305.865
β	bet	B - P4.5	(9/8)^(5/2)	509.775