4edo

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← 3edo4edo5edo →
Prime factorization 22
Step size 300¢
Fifth 2\4 (600¢) (→1\2)
Semitones (A1:m2) -2:2 (-600¢ : 600¢)
Dual sharp fifth 3\4 (900¢)
Dual flat fifth 2\4 (600¢) (→1\2)
Dual major 2nd 1\4 (300¢)
Consistency limit 7
Distinct consistency limit 1
Special properties

4 equal divisions of the octave (4edo) is the tuning system derived by dividing the octave into 4 equal steps of 300 cents each.

Theory

Like 3edo, 4edo is already familiar as a chord of 12edo. Not only that, but 4edo establishes tonality in much the same ways that 3edo does — with only two notes at a time as opposed to three aside from octave reduplications of the tonic, though the Tonic-Antitonic contrast from 2edo also works. Also like with 3edo, it has a theoretical interest in that it preserves a kind of outline, or skeleton, of melodic movement while erasing key distinctions concerning harmony. The 7-limit mapping, or val, for 4edo goes 4 6 9 11], all of which are distinct modulo 4. It therefore goes with tetradic harmony in much the same way that 3edo goes with triadic harmony, mapping the 7-limit consistently, and sending 15/14, 21/20, 25/24, and 36/35 to the unison. Somewhat confusingly, the patent mapping of 4edo sees 9/8 mapped to the unison also, leading to antitonic, though this can be traced to both 3/2 and 4/3 being mapped to 2\4.

By putting together the triples of integers which uniquely represent 7-limit tetrads in the 7-limit cubic lattice of tetrads with the number of 4edo steps returned by the 4 6 9 11] we obtain a representation of the 7-limit in terms of four integers, which differs from the usual (monzo) representation in that the triple representing the chord can be swapped for another such triple, resulting in a similar note tuned to a different chord. It is even possible under some circumstances to create a sort of recombinant merging of two pieces of music by using the chords of one with the 4edo skeletons of another.

We can also add more kinds of chords, for instance the subminor (1-7/6-3/2-5/3) and supermajor (1-9/7-3/2-9/5) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.

When viewed from a regular temperament perspective, 4edo can be seen as a tuning of the dimipent temperament, since it tempers 648/625 (the major diesis) by equating four minor thirds (6/5) to an octave. Alternately, it can be viewed as a critically flat hanson or myna scale, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves, 4edo is the first EDO that is zeta peak but not zeta peak integer, which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two EDOs. If you compress the octaves instead, it can be interpreted as critically sharp Gariberttet.

Harmonics

Approximation of odd harmonics in 4edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -102 -86 -69 +96 +49 +59 +112 -105 +2 +129 -28
relative (%) -34 -29 -23 +32 +16 +20 +37 -35 +1 +43 -9
Steps
(reduced)
6
(2)
9
(1)
11
(3)
13
(1)
14
(2)
15
(3)
16
(0)
16
(0)
17
(1)
18
(2)
18
(2)

Music

NullPointerException Music

Aeterna

Rozencrantz the Sane
  • Nothing of any importance (2006) (his contribution to the MMM day 2006)
Gene Ward Smith
STC_1003