Hemipyth: Difference between revisions

A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3.

Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.

! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>

| 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

| 14 || yes || yes || yes

| 15 || no || no || yes

| 16 || yes || no || no

| 17 || no || yes || no

| 18 || yes || no || no

| 19 || no || no || yes

| 20* || yes || yes || yes

| 21 || no || yes || no

| 22 || yes || no || no

| 23 || no || no || yes

| 24 || yes || yes || yes

{{asterisk}} 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.

Nicknames are still assigned to make it easier to talk about the [[5L&nbsp;4s]] scale generated by <math>\sim\vsp\sqrt{\frac{4}{3}}</math> against the octave.

The same goes for the semifourth. A poorly tuned ~{{sfrac|4|3}} still results in a decent <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

Under [[ploidacot]] classification diploid temperaments feature <math>\sim\vsp\sqrt{2}</math>, dicot temperaments have <math>\sim\vsp\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\vsp\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\vsp\sqrt{3}</math>).

! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! <math>\sim\vsp\sqrt{\frac{3}{2}}</math> !! <math>\sim\vsp\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2

| [[greenland]] || ~{{sfrac|99|70}} || ~{{sfrac|49|40}} || ~{{sfrac|15|13}}~{{sfrac|231|200}} || no || no

Above contorted tunings don't have a <math>\sim\vsp\sqrt{2}</math> period with a <math>\sim\vsp\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

Some possible interpretations for <math>\sim\vsp\sqrt{2}</math> are:

! Temperament !! <math>\sim\vsp\sqrt{2}</math> !! contorted !! rank-2

| [[kalismic temperaments|kalismic]] || ~{{sfrac|99|70}} || no || no

Some possible interpretations for <math>\sim\vsp\sqrt{3}</math> are:

! Temperament !! <math>\sim\vsp\sqrt{3}</math> !! contorted !! rank-2

Some possible interpretations for <math>\sim\vsp\frac{3}{2}</math> are:

! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2

| [[jove]] || ~{{sfrac|11|9}}~{{sfrac|49|40}} || no || no

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