12edo: Difference between revisions

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| ja = 12平均律

}}

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today.

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

== Theory ==

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

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<nowiki>*</nowiki> based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==

=== Selected just intervals by error ===

{| class="wikitable" style="text-align:center;"

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[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===

Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et.

{| class="wikitable"

! colspan="2" |

! 3-limit

! 5-limit

! 7-limit

! 2.3.5.7.17.19

|-

! colspan="2" |Octave stretch (¢)

| 0.617

| -1.56

| -3.95

| -2.53

|-

! rowspan="2" |Error

! [[TE error|absolute]] (¢)

| 0.617

| 3.11

| 4.92

| 4.52

|-

! [[TE simple badness|relative]] (%)

| 0.617

| 3.11

| 4.94

| 4.53

|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively.

* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==

@@ Line 5: / Line 5: @@
 | ja = 12平均律
 }}
-'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].
+'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today.
-It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
+== Theory ==
+edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
 The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].
@@ Line 113: / Line 114: @@
 <nowiki>*</nowiki> based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
+== Just approximation ==
 === Selected just intervals by error ===
 {| class="wikitable" style="text-align:center;"
@@ Line 229: / Line 231: @@
 [[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]
+=== Temperament measures ===
+Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et.
+{| class="wikitable"
+! colspan="2" |
+! 3-limit
+! 5-limit
+! 7-limit
+! 2.3.5.7.17.19
+|-
+! colspan="2" |Octave stretch (¢)
+| 0.617
+| -1.56
+| -3.95
+| -2.53
+|-
+! rowspan="2" |Error
+! [[TE error|absolute]] (¢)
+| 0.617
+| 3.11
+| 4.92
+| 4.52
+|-
+! [[TE simple badness|relative]] (%)
+| 0.617
+| 3.11
+| 4.94
+| 4.53
+|}
+* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively.
+* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.
 == Rank two temperaments ==