Hemipyth: Difference between revisions

Line 74:

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.

{| class="wikitable"

|+ 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

|}

@@ Line 74: / Line 74: @@
 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.
+{| class="wikitable"
+|+ 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
+|}