Hemipyth: Difference between revisions

Newer edit →

VisualWikitext

@@ Line 1: / Line 1: @@
 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/√3.
+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))
+{| class="wikitable"
+|+ 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
+|}