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'''4 equal divisions of the octave''' ('''4edo''') is the [[tuning system]] derived by dividing the [[octave]] into 4 equal steps of 300 [[cent]]s each.
'''4 equal divisions of the octave''' ('''4EDO''') is the [[tuning system]] derived by dividing the [[octave]] into 4 equal steps of 300 [[cent]]s each.


== Theory ==
== Theory ==
Like [[3edo|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 tuning map, or [[Vals and_Tuning Space|val]], for 4EDO goes {{val|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]] [[consistent]]ly, 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 [[Very low accuracy temperaments #Antitonic|antitonic]], though this can be traced to both 3/2 and 4/3 being mapped to 2\4.


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 tuning map, or [[Vals_and_Tuning_Space|val]], for 4edo goes {{val|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|7-limit]] [[consistent]]ly, 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 [[Very low accuracy temperaments#Antitonic|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 [[The_Seven_Limit_Symmetrical_Lattices|7-limit cubic lattice of tetrads]] with the number of 4EDO steps returned by the {{val|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.


By putting together the triples of integers which uniquely represent 7-limit tetrads in the [[The_Seven_Limit_Symmetrical_Lattices|7-limit cubic lattice of tetrads]] with the number of 4edo steps returned by the {{val|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.


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 family #Dimipent|dimipent temperament]], since it tempers [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave.


When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent_family#Dimipent|dimipent temperament]], since it tempers [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave.
=== Differences between distributionally-even scales and smaller EDOs ===
 
=== Differences between distributionally-even scales and smaller edos ===
{| class="wikitable"
{| class="wikitable"
|+
|+
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!s-Nedo
!s-Nedo
|-
|-
|3
| 3
|200¢
| 200¢
| -100¢
| -100¢
|}
|}
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|His contribution to the [[MMMday06|MMM day 2006]]
|His contribution to the [[MMMday06|MMM day 2006]]
|-
|-
|[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 ''A simple 4edo piece'']
|[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 ''A simple 4EDO piece'']
|[[Gene Ward Smith]]
|[[Gene Ward Smith]]
|2011 (?)
|2011 (?)
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