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| <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:guest|guest]] and made on <tt>2011-12-30 00:47:58 UTC</tt>.<br>
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| : The original revision id was <tt>288802419</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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=
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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.
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| | 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. |
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| =A scala formatted description of the tuning= | | == Prime harmonics == |
| | {{Harmonics in equal|12|3|1|intervals=prime}} |
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| | == 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]]). |
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| | 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 | | ! C:\Cakewalk\scales\tritave-in-12.scl |
| ! | | ! |
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| 1743.45875 | | 1743.45875 |
| 3/1 | | 3/1 |
| | </pre> |
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| [[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 3.13.17 -- 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=13/1. Tempering out the 85293/83521 comma. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 chords.
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| 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.
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| =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>
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| <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><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Division of the tritave (3/1) into 12 equal parts"></a><!-- ws:end:WikiTextHeadingRule:1 -->Division of the tritave (3/1) into 12 equal parts</h1>
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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 <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 />
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| <br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="A scala formatted description of the tuning"></a><!-- ws:end:WikiTextHeadingRule:3 -->A scala formatted description of the tuning</h1>
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| <br />
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| ! C:\Cakewalk\scales\tritave-in-12.scl<br />
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| !<br />
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| 3/1 in 12<br />
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| 12<br />
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| !<br />
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| 158.49625<br />
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| 316.99250<br />
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| 475.48875<br />
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| 633.98500<br />
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| 792.48125<br />
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| 950.97750<br />
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| 1109.47375<br />
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| 1267.97000<br />
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| 1426.46625<br />
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| 1584.96250<br />
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| 1743.45875<br />
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| 3/1<br />
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| <br />
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| <!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;w=0&quot; class=&quot;WikiMedia WikiMediaCustom&quot; id=&quot;wikitext@@media@@type=&amp;quot;custom&amp;quot; key=&amp;quot;10532830&amp;quot;&quot; title=&quot;Custom Media&quot;/&gt; --><script type="text/javascript" src="http://mediaplayer.yahoo.com/js">
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| </script><!-- ws:end:WikiTextMediaRule:0 --><br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc2"><a name="Exactly analogous to meantone"></a><!-- ws:end:WikiTextHeadingRule:5 -->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 3.13.17 -- 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=13/1. Tempering out the 85293/83521 comma. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 chords.<br />
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| <br />
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| 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 />
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| <br />
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| <br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc3"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:7 -->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 />
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| <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>
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| Prime factorization
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22 × 3 (highly composite)
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| Step size
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158.496 ¢
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| Octave
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8\12edt (1267.97 ¢) (→ 2\3edt)
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| Consistency limit
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3
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| Distinct consistency limit
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3
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12 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 12edt or 12ed3), is a nonoctave tuning system that divides the interval of 3/1 into 12 equal parts of about 158 ¢ each. Each step represents a frequency ratio of 31/12, or the 12th root of 3.
12edt corresponds to 7.571 edo, and can be used as a generator chain for 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
Approximation of prime harmonics in 12edt
| Harmonic
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2
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3
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5
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7
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11
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13
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17
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19
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23
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29
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31
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| Error
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Absolute (¢)
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+68.0
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+0.0
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+66.6
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-40.4
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-30.4
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-2.6
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+8.4
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-25.6
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-39.4
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+34.8
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+77.8
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| Relative (%)
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+42.9
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+0.0
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+42.0
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-25.5
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-19.2
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-1.7
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+5.3
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-16.2
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-24.9
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+21.9
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+49.1
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Steps
(reduced)
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8
(8)
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12
(0)
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18
(6)
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21
(9)
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26
(2)
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28
(4)
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31
(7)
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32
(8)
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34
(10)
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37
(1)
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38
(2)
|
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 tritaves, so it contains an extremely stretched diatonic scale or macrodiatonic 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 hyperpyth which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.
Interval table
Scala file
! 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
Compositions
Instant Gamelan by Carlo Serafini
Tritave in 12 by Chris Vaisvil