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 [[mapping]], or [[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 [[mapping]], or [[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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We can also add more kinds of chords, for instance the subminor ([[6:7:9:10|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126|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.

4edo can be viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone ~~is the~~ the [[patent val]] while the major sixth is in 4b~~-edo~~ (using [[wart notation]]). 4b~~-edo~~ has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

4edo can be viewed as a [[dual-fifth]] system (the smallest in fact, besides the trivial [[1edo]]), with the tritone and major sixth as the flat and sharp "fifths". The tritone represents 3/2 in the [[patent val]], while the major sixth represents 3/2 in the 4b val (using [[wart notation]]). The 4b val has one of the sharpest mappings of 3/2 of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

4edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[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]] tuning, 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 [[The Riemann zeta function and tuning #Zeta edo lists|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 a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.

4edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[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]] tuning, 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 [[The Riemann zeta function and tuning #Zeta edo lists|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 a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.

=== Odd harmonics ===

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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 [[mapping]], or [[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 [[mapping]], or [[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.
@@ Line 15: / Line 15: @@
 We can also add more kinds of chords, for instance the subminor ([[6:7:9:10|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126|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.
-edo can be viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].
+edo can be viewed as a [[dual-fifth]] system (the smallest in fact, besides the trivial [[1edo]]), with the tritone and major sixth as the flat and sharp "fifths". The tritone represents 3/2 in the [[patent val]], while the major sixth represents 3/2 in the 4b val (using [[wart notation]]). The 4b val has one of the sharpest mappings of 3/2 of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].
-edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[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]] tuning, 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 [[The Riemann zeta function and tuning #Zeta edo lists|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 a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.
+edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[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]] tuning, 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 [[The Riemann zeta function and tuning #Zeta edo lists|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 a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.
 === Odd harmonics ===