12edt: Difference between revisions

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~~=Division of the tritave (3/1) into 12 equal parts=~~

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. [[category:macrotonal]]

~~=A scala formatted description~~ of the ~~tuning=~~

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.

== 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]] {{mos scalesig|5L 2s<3/1>}} 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 ==

<pre>

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

!

3/1 in 12

12

!

158.49625

316.99250

475.48875

633.98500

792.48125

950.97750

1109.47375

1267.97000

1426.46625

1584.96250

1743.45875

3/1

</pre>

=~~Exactly analogous to meantone~~=

== Compositions ==

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. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

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

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

~~=Compositions~~=

[~~http~~://~~www~~.~~seraph.it~~/~~XenoTunes3.html~~ Instant Gamelan~~] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play~~] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil] ~~[[Category:edonoi]]~~

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]

~~[[Category:edt]]~~

~~[[Category:equal]]~~

[[Category:listen]]

[[category:macrotonal]]

@@ Line 1: / Line 1: @@
-=Division of the tritave (3/1) into 12 equal parts=
+{{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. [[category:macrotonal]]
+{{ED intro}}
-=A scala formatted description of the tuning=
+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.
+== 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]] {{mos scalesig|5L 2s<3/1>}} 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 ==
+<pre>
 ! C:\Cakewalk\scales\tritave-in-12.scl
 !
 /1 in 12
 !
 .49625
 .99250
 .48875
 .98500
 .48125
 .97750
 .47375
 .97000
 .46625
 .96250
 .45875
 /1
+</pre>
-=Exactly analogous to meantone=
+== Compositions ==
-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. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.
+[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]
-Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.
-=Compositions=
-[http://www.seraph.it/XenoTunes3.html Instant Gamelan] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play] by [[Carlo_Serafini|Carlo Serafini]]
-[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      [[Category:edonoi]]
+[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]
-[[Category:edt]]
-[[Category:equal]]
 [[Category:listen]]
+[[category:macrotonal]]