Hemipyth
A hemipyth (or "hemipythagorean") interval is an interval in the √2.√3 subgroup i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.
Notable hemipyth intervals include the neutral third √(3/2) = √3/√2, semioctave √2 and the semifourth √(4/3) = 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))
| 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.
Other edos with hemipyth-supporting patent vals are 28, 30, 34, 38, 44, 48, 52, 54, 58, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo. You need to go all the way to 82edo in order to get an improvement in terms of relative error.
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.
A prototypical 5L 2s 5|1 (Ionian) scale would be spelled C, D, E, F, G, A, B, (C).
Simple otonal chords can be plucked out of the harmonic segment 1:2:3:4:6:8:9:12:16:18:24:27:32:36:48:54:64:72:81:96:108:128:... e.g. 6:8:9 is a sus4 chord.
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 (a.k.a. demisharps) etc.
A representative 3L 4s 4|2 (kleeth) scale would be spelled C, D, E 
, F, G, A , B , (C).
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 gets 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.
| 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 |
Where to put the greek notes on a staff is still being decided. Probably on the same lines as traditional notes but with distinct noteheads. E.g. a middle η would look like a middle C, but with an upwards pointing triangular notehead.
A representative 10L 2s 10|0(2) scale would be spelled C, η, D, α, E, β, γ, G, δ, A, ε, B, (C).
An alternative solution, although one which looses bijectivity, is to keep only the traditional nominals while having a dedicated accidental pair for √(256/243) (this was proposed by CompactStar).
Semifourths
Luckily we don't need to introduce any more generalizations to the notation to indicate √(4/3). It's a neutral 2½ or a α (alp semiflat) w.r.t middle C.
Nicknames are still assigned to make it easier to talk about the 5L 4s scale generated by √(4/3) against the octave.
| Nominal | Pronunciation | Meaning | Ratio with middle C | Cents |
|---|---|---|---|---|
| φ | phi | α |
√(4/3) | 249.022 |
| χ | chi | β |
√(27/16) | 452.933 |
| ψ | psi | ε |
√3 | 950.978 |
| ω | ome | ζ  |
√(243/64) | 1154.888 |
These particular definitions were chosen so that C, D, φ, χ, F, G, A, ψ, ω, (C) becomes the 6|2 (Stellerian) mode, all notated without accidentals.
Hemipyth
Putting it all together we can now spell a squashed Ionian scale, 10L 4s 10|2(2):
C, η, D, α , E , β , F , γ, G, δ, A , ε , B , ζ , (C)
Or equivalently with semiquartal nicknames:
C, η, D, φ, E , χ, F , γ, G, δ, A , ψ, B , ω, (C)
The 4L 6s 4|4(2) scale (called Pacific), can be spelled like so:
C, η, α , E , F, γ, G, A , ε , ζ, C
Cole prefers to spell it on D, giving:
D, α, β , F , G, δ, A, B , ζ , η, D
Simple hemipyth chords can be plucked out of the square root of the Pythagorean segment 1:√2:√3:2:√6:√8:3:√12:4:√18:√24:√27:√32:6:√48:√54:8:√72:9:√96:√108:√128:... e.g. 2:√6:3 is a neutral chord where spicy tension can be added by including the semioctave for 2:√6:√8:3 with no increase in complexity as far as the generator of the subgroup is concerned.
Here is a Xenpaper demo of all five representative scales listed above.
Musical significance
The semioctave sound is very familiar to westerners from repeated exposure of music tempered to 12-tone equal. It can provide a familiar dissonance in an otherwise xenharmonic piece.
The neutral third is featured in Arabic, Turkish, Persian music etc.
Close approximations to the semifourth can be found in many near-equal pentatonic scales around the world.
Their wide availability in edos makes it worthwhile to become familiar with their sound without having to make further interpretations of their meaning using fractions involving higher prime numbers.
Theoretical significance
The semioctave is always tuned pure when the octave is tuned pure.
The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of √(3/2) also makes it more tolerant of imprecise tuning.
The same goes for the semifourth. A poorly tuned ~4/3 still results in a decent ~√(4/3) (assuming it's featured in the tuning in the first place).
Signposts
Due to their low damage in supporting temperaments octave 2/1, semioctave √2, fifth 3/2, fourth 4/3, neutral third √(3/2), neural sixth √(8/3), semifourth √(4/3), semitwelfth √3, "hemitone" √(9/8) and "contrahemitone" √(32/9) all provide good signposts for navigating around otherwise unfamiliar scales.
While untempered semitones usually come as unequal pairs consisting of an augmented unison and a minor second, the "hemitone" is always exactly the geometric half of a 9/8 whole tone. The "contrahemitone" is its octave-complement.
Temperament interpretations
Under ploidacot classification diploid temperaments feature ~√2, dicot temperaments have ~√(3/2) and alpha-dicot temperaments feature ~√(4/3) (by virtue of having a ~√3).
Full hemipyth support is indicated by at least "diploid dicot". Examples include:
| Temperament | ~√2 | ~√(3/2) | ~√(4/3) | contorted | rank-2 |
|---|---|---|---|---|---|
| decimal | ~7/5 | ~5/4 | ~7/6 | no | yes |
| anguirus | ~45/32 | ~56/45 | ~7/6 | no | yes |
| sruti | ~45/32 | ~175/144 | ~81/70 | no | yes |
| pakkanian hemipyth | ~17/12 | ~11/9 | ~15/13 | no | yes |
| harry | ~17/12 | ~11/9 | ~15/13 | yes | yes |
| semimiracle | ~91/64 | ~11/9 | ~15/13 | yes | yes |
| hemidim | ~36/25 | ~25/21 | ~7/6 | yes | yes |
| greenland | ~99/70 | ~49/40 | ~15/13~231/200 | no | no |
| semisema | ~108/77 | ~11/9 | ~7/6 | no | yes |
| quadritikleismic | ~625/441 | ~49/40 | ~125/108 | yes | yes |
| decoid | ~99/70 | ~49/40 | ~4725/4096 | yes | yes |
Above contorted tunings don't have a ~√2 period with a ~√3 generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.
Some possible interpretations for ~√2 are:
| Temperament | ~√2 | contorted | rank-2 |
|---|---|---|---|
| jubilic | ~7/5 | no | yes (2.5.7) |
| diaschismic | ~45/32 | no | yes (2.3.5) |
| semitonic | ~17/12 | no | yes (2.3.17) |
| kalismic | ~99/70 | no | no |
Some possible interpretations for ~√3 are:
| Temperament | ~√3 | contorted | rank-2 |
|---|---|---|---|
| semaphore | ~7/4 | no | yes (2.3.7) |
| barbados | ~26/15 | no | yes (2.3.13/5) |
Some possible interpretations for ~√(3/2) are:
| Temperament | ~√(3/2) | contorted | rank-2 |
|---|---|---|---|
| dicot | ~5/4 | no | yes (2.3.5) |
| neutral | ~11/9 | no | yes (2.3.11) |
| jove | ~11/9~49/40 | no | no |
MOS patterns
By default hemipyth[n] refers to a MOS pattern of size n inside the octave i.e. there are always two periods per octave. The generator is √3 unless otherwise stated.
| hemipyth[n] | MOS pattern | hardness (untempered) |
|---|---|---|
| hemipyth[4] | 2L 2s | 1.41 |
| hemipyth[6] | 4L 2s | 2.44 |
| hemipyth[10] | 4L 6s | 1.44 |
| hemipyth[14] | 10L 4s | 2.26 |
| hemipyth[24] | 10L 14s | 1.26 |
Music
The Hymn of Pergele, a short piece in Hemipyth[10] 4|4(2) (Pacific mode of 4L 6s), written by Cole.