4edo: Difference between revisions

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{{interwiki

| de = ~~4edo~~

| de = 4-EDO

| en = 4edo

| es =

| ja =

| ja = 4平均律

}}

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, 9/8 is mapped to the unison also.

~~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 ==

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.

~~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~~.

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.

~~==Music==~~

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.

[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~~)

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.

=== Odd harmonics ===

=== Subsets and supersets ===

4edo is the first composite edo, containing [[2edo]] as the only nontrivial subset edo.

== Intervals ==

{| class="wikitable center-all"

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

|-

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

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

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])

! rowspan="2" | Audio

|-

! [[3-limit]]

! [[5-limit]]

! [[7-limit]]

! Other

|-

| 0

| Unison (prime)

| [[1/1]] (just)

|

| [[File:piano_0_1edo.mp3]]

|-

| 1

| 300

| Minor third

| [[32/27]] (+5.865)

| [[6/5]] (-15.641)

| [[7/6]] (+33.129) [[25/21]] (-1.847)

| [[19/16]] (+2.487)

| [[File:piano_1_4edo.mp3]]

|-

| 2

| 600

| [[Tritone]]

|

| [[7/5]] (+17.488) [[10/7]] (-17.488)

| [[24/17]] (+3.000) [[99/70]] (-0.088) [[17/12]] (-3.000)

| [[File:piano_1_2edo.mp3]]

|-

| 3

| 900

| Major sixth

| [[27/16]] (-5.865)

| [[5/3]] (+15.641)

| [[42/25]] (+1.847) [[12/7]] (-33.129)

| [[32/19]] (-2.487)

| [[File:piano_3_4edo.mp3]]

|-

| 4

| 1200

| Octave

| [[2/1]] (just)

|

| [[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.

== Notation ==

{| class="wikitable center-all"

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

|-

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

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

! colspan="2" | [[12edo]] [[subset notation]]

|-

! [[5L 2s|Diatonic]] interval names

! Note names (on D)

|-

| 0

| '''Perfect unison (P1)'''

| '''D'''

|-

| 1

| 300

| Augmented second (A2) '''Minor third (m3)'''

| E# '''F'''

|-

| 2

| 600

| Augmented fourth (A4) Diminished fifth (d5)

| G# Ab

|-

| 3

| 900

| '''Major sixth (M6)''' Diminished seventh (d7)

| '''B''' Cb

|-

| 4

| 1200

| '''Perfect octave (P8)'''

| '''D'''

|}

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.

== Solfege ==

{| class="wikitable center-all"

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

|-

! [[Degree]]

! [[Cents]]

! 12edo subset standard [[solfege]] (movable do)

! 12edo subset [[Uniform solfege]] (2-3 vowels)

|-

| 0

| Do (P1)

| Da (P1)

|-

| 1

| 300

| Ri (A2) Me (m3)

| Ru (A2) Na (m3)

|-

| 2

| 600

| Fi (A4) Se (d5)

| Pa (A4) Sha (d5)

|-

| 3

| 900

| La (M6)

| La (M6) Tho (d7)

|-

| 4

| 1200

| Do (P8)

| Da (P8)

|}

== Music ==

; [[Aeterna]]

* [https://www.youtube.com/watch?v=catncLv5oSk "Mimetism"], from [https://youtube.com/playlist?list=OLAK5uy_meMItS3Zalh3jNeQ-4XVwEp0JUPO3A9rQ ''Tribute to Armodue''] (2008)

; [[No Clue Music]]

* [https://www.youtube.com/watch?v=JBGqDbbZ0Fw ''Dimpulse''] (2024)

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=rCfWHwrEaA0 "Entering"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]

* [https://www.youtube.com/watch?v=T0J1D_E94L4 ''Diminished''] (2024)

; [[Rozencrantz|Rozencrantz the Sane]]

* ''Nothing of any importance'' (2006) – his contribution to the [[MMMday06|MMM day 2006]][''where?'']

; [[Gene Ward Smith]]

* [https://web.archive.org/web/20201127012143/http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 ''A simple 4EDO piece''] (2011?) – see also [[Composing with tablets]]

; [[STC_1003]]

* "Neainaz Antithetica, Variation II", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/4-stc-s1003-neainaz SoundCloud] | [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/4-neainaz-antithetica-variation-ii 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:Listen]]

~~[[Category:macrotonal~~]]

@@ Line 1: / Line 1: @@
 {{interwiki
-| de = 4edo
+| de = 4-EDO
 | en = 4edo
 | es =
-| ja =
+| ja = 4平均律
 }}
-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, 9/8 is mapped to the unison also.
+{{Infobox ET}}
+{{ED intro}}
-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 ==
+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.
-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.
+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.
-==Music==
+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.
-[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)
+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.
+=== Odd harmonics ===
+{{Harmonics in equal|4}}
+=== Subsets and supersets ===
+edo is the first composite edo, containing [[2edo]] as the only nontrivial subset edo.
+== Intervals ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Intervals of 4edo
+|-
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! rowspan="2" | [[Interval region]]
+! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
+! rowspan="2" | Audio
+|-
+! [[3-limit]]
+! [[5-limit]]
+! [[7-limit]]
+! Other
+|-
+| 0
+| 0
+| Unison (prime)
+| [[1/1]] (just)
+|
+|
+|
+| [[File:piano_0_1edo.mp3]]
+|-
+| 1
+| 300
+| Minor third
+| [[32/27]] (+5.865)
+| [[6/5]] (-15.641)
+| [[7/6]] (+33.129)<br>[[25/21]] (-1.847)
+| [[19/16]] (+2.487)
+| [[File:piano_1_4edo.mp3]]
+|-
+| 2
+| 600
+| [[Tritone]]
+|
+|
+| [[7/5]] (+17.488)<br>[[10/7]] (-17.488)
+| [[24/17]] (+3.000)<br>[[99/70]] (-0.088)<br>[[17/12]] (-3.000)
+| [[File:piano_1_2edo.mp3]]
+|-
+| 3
+| 900
+| Major sixth
+| [[27/16]] (-5.865)
+| [[5/3]] (+15.641)
+| [[42/25]] (+1.847)<br>[[12/7]] (-33.129)
+| [[32/19]] (-2.487)
+| [[File:piano_3_4edo.mp3]]
+|-
+| 4
+| 1200
+| Octave
+| [[2/1]] (just)
+|
+|
+|
+| [[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.
+== Notation ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Notation of 4edo
+|-
+! rowspan="2" | [[Degree]]
+! rowspan="2" | [[Cent]]s
+! colspan="2" | [[12edo]] [[subset notation]]
+|-
+! [[5L 2s|Diatonic]] interval names
+! Note names (on D)
+|-
+| 0
+| 0
+| '''Perfect unison (P1)'''
+| '''D'''
+|-
+| 1
+| 300
+| Augmented second (A2)<br>'''Minor third (m3)'''
+| E#<br>'''F'''
+|-
+| 2
+| 600
+| Augmented fourth (A4)<br>Diminished fifth (d5)
+| G#<br>Ab
+|-
+| 3
+| 900
+| '''Major sixth (M6)'''<br>Diminished seventh (d7)
+| '''B'''<br>Cb
+|-
+| 4
+| 1200
+| '''Perfect octave (P8)'''
+| '''D'''
+|}
+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.
+== Solfege ==
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Solfege of 4edo
+|-
+! [[Degree]]
+! [[Cents]]
+! 12edo subset<br>standard [[solfege]]<br>(movable do)
+! 12edo subset<br>[[Uniform solfege]]<br>(2-3 vowels)
+|-
+| 0
+| 0
+| Do (P1)
+| Da (P1)
+|-
+| 1
+| 300
+| Ri (A2)<br>Me (m3)
+| Ru (A2)<br>Na (m3)
+|-
+| 2
+| 600
+| Fi (A4)<br>Se (d5)
+| Pa (A4)<br>Sha (d5)
+|-
+| 3
+| 900
+| La (M6)
+| La (M6)<br>Tho (d7)
+|-
+| 4
+| 1200
+| Do (P8)
+| Da (P8)
+|}
+== Music ==
+; [[Aeterna]]
+* [https://www.youtube.com/watch?v=catncLv5oSk "Mimetism"], from [https://youtube.com/playlist?list=OLAK5uy_meMItS3Zalh3jNeQ-4XVwEp0JUPO3A9rQ ''Tribute to Armodue''] (2008)
+; [[No Clue Music]]
+* [https://www.youtube.com/watch?v=JBGqDbbZ0Fw ''Dimpulse''] (2024)
+; [[NullPointerException Music]]
+* [https://www.youtube.com/watch?v=rCfWHwrEaA0 "Entering"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
+; [[User:Phanomium|Phanomium]]
+* [https://www.youtube.com/watch?v=T0J1D_E94L4 ''Diminished''] (2024)
+; [[Rozencrantz|Rozencrantz the Sane]]
+* ''Nothing of any importance'' (2006) – his contribution to the [[MMMday06|MMM day 2006]]<sup>[''where?'']</sup>
+; [[Gene Ward Smith]]
+* [https://web.archive.org/web/20201127012143/http://clones.soonlabel.com/public/micro/gene_ward_smith/transformers/fouredo.mp3 ''A simple 4EDO piece''] (2011?) – see also [[Composing with tablets]]
+; [[STC_1003]]
+* "Neainaz Antithetica, Variation II", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/4-stc-s1003-neainaz SoundCloud] | [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/4-neainaz-antithetica-variation-ii 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:Listen]]
-[[Category:macrotonal]]