4edo: Difference between revisions

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== 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 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|~~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.

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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 ===

{| class="wikitable"

|+

!N

!L-Nedo

!s-Nedo

|-

|3

|200¢

| -100¢

|}

== Music ==

@@ Line 9: / Line 9: @@
 == 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 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|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.
@@ Line 16: / Line 16: @@
 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 ===
+{| class="wikitable"
+|+
+!N
+!L-Nedo
+!s-Nedo
+|-
+|3
+|200¢
+| -100¢
+|}
 == Music ==