12edt: Difference between revisions

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==Scala file==

== Scala file ==

<pre>

! C:\Cakewalk\scales\tritave-in-12.scl

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</pre>

==Theory==

== 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.

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===

=== 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]]).

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==

== Compositions ==

[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

@@ Line 8: / Line 8: @@
 {{Harmonics in equal|12|3|1|intervals=prime}}
-==Scala file==
+== Scala file ==
 <pre>
 ! C:\Cakewalk\scales\tritave-in-12.scl
@@ Line 30: / Line 29: @@
 </pre>
-==Theory==
+== 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.
+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===
+=== 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]]).
+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==
+== Compositions ==
 [https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]