12edt: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-03 16:45:31 UTC</tt>.<br>~~

~~: The original revision id was <tt>593933388</tt>.<br>~~

~~: The revision comment was: <tt></tt><br>~~

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~

~~<h4>Original Wikitext content:</h4>~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=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.

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.

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

== 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

!

Line 29:

Line 37:

1743.45875

3/1

</pre>

~~[[media type~~=~~"custom" key~~=~~"10532830"]]~~

== Compositions ==

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

=~~Exactly analogous to meantone~~=

In octave land, these simple temperaments, 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 [[tel:130321/130000|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.~~

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

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

[[Category:listen]]

[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]~~]</pre></div>~~

[[category:macrotonal]]

~~<h4>Original HTML content~~:~~</h4>~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>12edt</title></head><body><h1 id="toc0"><a name="Division of the tritave (3/1) into 12 equal parts"></a>Division of the tritave (3/1) into 12 equal parts</h1>

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 <a class="wiki_link" href="/Kleismic%20family#Hemikleismic">hemikleismic temperament</a>. 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.<br />

~~<br />~~

<h1 id="toc1"><a name="A scala formatted description of the tuning"></a>A scala formatted description of the tuning</h1>

~~<br />~~

~~! C:\Cakewalk\scales\tritave-in-12.scl<br />~~

~~!<br />~~

~~3/1 in 12<br />~~

~~12<br />~~

~~!<br />~~

~~158.49625<br />~~

~~316.99250<br />~~

~~475.48875<br />~~

~~633.98500<br />~~

~~792.48125<br />~~

~~950.97750<br />~~

~~1109.47375<br />~~

~~1267.97000<br />~~

~~1426.46625<br />~~

~~1584.96250<br />~~

~~1743.45875<br />~~

~~3/1<br />~~

~~<br />~~

<script type="text/javascript" src="http://mediaplayer.yahoo.com/js">

~~</script><br />~~

~~<br />~~

<h1 id="toc2"><a name="Exactly analogous to meantone"></a>Exactly analogous to meantone</h1>

In octave land, these simple temperaments, 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 [[~~tel~~:~~130321/130000|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.<br />~~

~~<br />~~

~~Another example of a macrodiatonic scale is <a class="wiki_link" href="/17ed5">hyperpyth</a> which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.<br />~~

~~<br />~~

~~<h1 id="toc3"><a name="Compositions"></a>Compositions</h1>~~

<a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow">Instant Gamelan</a> <a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a><br />

<a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow">Tritave in 12</a> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank">Chris Vaisvil</a></body></html></pre></div>

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-03 16:45:31 UTC</tt>.<br>
-: The original revision id was <tt>593933388</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Division of the tritave (3/1) into 12 equal parts=
-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.
+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.
-=A scala formatted description of the tuning=
+== 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
 !
@@ Line 29: / Line 37: @@
 .45875
 /1
+</pre>
-[[media type="custom" key="10532830"]]
+== Compositions ==
+[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]
-=Exactly analogous to meantone=
-In octave land, these simple temperaments, 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 [[tel:130321/130000|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.
-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://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]
-[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
+[[Category:listen]]
-[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]</pre></div>
+[[category:macrotonal]]
-<h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;12edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Division of the tritave (3/1) into 12 equal parts"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Division of the tritave (3/1) into 12 equal parts&lt;/h1&gt;
-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 &lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&gt;hemikleismic temperament&lt;/a&gt;. 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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="A scala formatted description of the tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;A scala formatted description of the tuning&lt;/h1&gt;
- &lt;br /&gt;
-! C:\Cakewalk\scales\tritave-in-12.scl&lt;br /&gt;
-!&lt;br /&gt;
-/1 in 12&lt;br /&gt;
-&lt;br /&gt;
-!&lt;br /&gt;
-.49625&lt;br /&gt;
-.99250&lt;br /&gt;
-.48875&lt;br /&gt;
-.98500&lt;br /&gt;
-.48125&lt;br /&gt;
-.97750&lt;br /&gt;
-.47375&lt;br /&gt;
-.97000&lt;br /&gt;
-.46625&lt;br /&gt;
-.96250&lt;br /&gt;
-.45875&lt;br /&gt;
-/1&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;10532830&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&gt;
-&lt;/script&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Exactly analogous to meantone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Exactly analogous to meantone&lt;/h1&gt;
- In octave land, these simple temperaments, 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 [[tel:130321/130000|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.&lt;br /&gt;
-&lt;br /&gt;
-Another example of a macrodiatonic scale is &lt;a class="wiki_link" href="/17ed5"&gt;hyperpyth&lt;/a&gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Compositions&lt;/h1&gt;
- &lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&gt;Instant Gamelan&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&gt;Tritave in 12&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&gt;Chris Vaisvil&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>