4edo: Difference between revisions

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~~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 <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~~.

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

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.

== Theory ==

~~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~~.

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.

~~==Music==~~

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.

[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 A simple 4edo piece] by [[~~Gene_Ward_Smith|~~Gene Ward Smith]] (~~see~~ [[Composing_with_tablets|Composing with tablets]] ~~for explanation~~)

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.

== Music ==

{| class="wikitable sortable"

!Title

!Composer

!Year

!Genre

!Additional links

|-

|''Nothing of any importance''

|[[Rozencrantz|Rozencrantz the Sane]]

|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'']

|[[Gene Ward Smith]]

|2011 (?)

|Classical

|Explanation: [[Composing_with_tablets|Composing with tablets]]

|-

|data-sort-value="Entering (from Edolian)"|[https://www.youtube.com/watch?v=rCfWHwrEaA0 "Entering"] (from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''])

|NullPointerException Music

|2020

|Classical

|

|-

|data-sort-value="Neainaz Antithetica, Variation II (from STAFFcirc vol. 7)"|"[https://soundcloud.com/sexytoadsandfrogsfriendcircle/4-stc-s1003-neainaz Neainaz Antithetica, Variation II]" (from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''])

|STC_1003

|2021

|Electronic

|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}

~~"Nothing of any importance" by [[Rozencrantz|Rozencrantz the Sane]] (his contribution to the [[MMMday06|MMM day 2006]])~~

[[Category:7-limit]]

[[Category:Equal divisions of the octave]]

[[Category:macrotonal]]

@@ Line 5: / Line 5: @@
 | ja =
 }}
-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 &lt;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.
+'''4 equal divisions of the octave''' (or '''4edo''') is the [[tuning system]] derived by dividing the [[octave]] into 4 equal steps of 300 [[cent]]s each.
-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 &lt;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.
+== Theory ==
-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.
+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.
-==Music==
+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.
-[http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 A simple 4edo piece] by [[Gene_Ward_Smith|Gene Ward Smith]] (see [[Composing_with_tablets|Composing with tablets]] for explanation)
+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.
+== Music ==
+{| class="wikitable sortable"
+!Title
+!Composer
+!Year
+!Genre
+!Additional links
+|-
+|''Nothing of any importance''
+|[[Rozencrantz|Rozencrantz the Sane]]
+|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'']
+|[[Gene Ward Smith]]
+|2011 (?)
+|Classical
+|Explanation: [[Composing_with_tablets|Composing with tablets]]
+|-
+|data-sort-value="Entering (from Edolian)"|[https://www.youtube.com/watch?v=rCfWHwrEaA0 "Entering"] (from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''])
+|NullPointerException Music
+|2020
+|Classical
+|
+|-
+|data-sort-value="Neainaz Antithetica, Variation II (from STAFFcirc vol. 7)"|"[https://soundcloud.com/sexytoadsandfrogsfriendcircle/4-stc-s1003-neainaz Neainaz Antithetica, Variation II]" (from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''])
+|STC_1003
+|2021
+|Electronic
+|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]
+|}
-"Nothing of any importance" by [[Rozencrantz|Rozencrantz the Sane]] (his contribution to the [[MMMday06|MMM day 2006]])
 [[Category:7-limit]]
 [[Category:Equal divisions of the octave]]
 [[Category:macrotonal]]