12edt: Difference between revisions

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12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.

~~== Interval table ==~~

12edt corresponds to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.

~~{{Interval table}}~~

== Prime harmonics ==

== Theory ==

In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===

12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{sl|5L 2s}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==

== Scala file ==

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3/1

</pre>

~~== Theory ==~~

In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

~~=== Macrodiatonic and macromeantone ===~~

12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s<3/1>]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable—for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

~~Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.~~

== Compositions ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.
+{{ED intro}}
-== Interval table ==
+edt corresponds to 7.571&nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.
-{{Interval table}}
 == Prime harmonics ==
 {{Harmonics in equal|12|3|1|intervals=prime}}
+== Theory ==
+In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.
+=== Macrodiatonic and macromeantone ===
+edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{sl|5L 2s}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).
+Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.
+== Interval table ==
+{{Interval table}}
 == Scala file ==
@@ Line 28: / Line 38: @@
 /1
 </pre>
-== Theory ==
-In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.
-=== Macrodiatonic and macromeantone ===
-edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s<3/1>]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable—for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).
-Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.
 == Compositions ==