4edo: Difference between revisions

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== 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|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 [[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.

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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 == 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|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 [[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.
 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.