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 [[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 [[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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== Intervals ==

{| class="wikitable center-all"

|+ Intervals of 4edo

|+ style="font-size: 105%;" | Intervals of 4edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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| [[File:piano_1_1edo.mp3]]

|}

<nowiki />* Based on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

<nowiki>*~~</nowiki> based~~ on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==

{| class="wikitable center-all"

|+ Notation of 4edo

|+ style="font-size: 105%;" | Notation of 4edo

|-

! rowspan="2" | [[Degree]]

! rowspan="2" | [[Cent]]s

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In 4edo:

* [[ups and downs notation]] is identical to standard notation;

* ~~mixed~~ [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==

{| class="wikitable center-all"

|+ Solfege of 4edo

|+ style="font-size: 105%;" | Solfege of 4edo

|-

! [[Degree]]

! [[Cents]]

@@ 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 [[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.
+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 [[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.
@@ Line 24: / Line 24: @@
 == Intervals ==
 {| class="wikitable center-all"
-|+ Intervals of 4edo
+|+ style="font-size: 105%;" | Intervals of 4edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 81: / Line 82: @@
 | [[File:piano_1_1edo.mp3]]
 |}
+<nowiki />* Based on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
-<nowiki>*</nowiki> based on treating 4edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
 == Notation ==
 {| class="wikitable center-all"
-|+ Notation of 4edo
+|+ style="font-size: 105%;" | Notation of 4edo
+|-
 ! rowspan="2" | [[Degree]]
 ! rowspan="2" | [[Cent]]s
@@ Line 122: / Line 123: @@
 In 4edo:
 * [[ups and downs notation]] is identical to standard notation;
-* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
+* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
 == Solfege ==
 {| class="wikitable center-all"
-|+ Solfege of 4edo
+|+ style="font-size: 105%;" | Solfege of 4edo
+|-
 ! [[Degree]]
 ! [[Cents]]