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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:xenwolf|xenwolf]] and made on <tt>2016-12-30 15:56:09 UTC</tt>.<br>
: The original revision id was <tt>602918480</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">=43 EDT=


This tuning is related to [[27edo]] having compressed octaves of 1194.251 cents, a small but significant deviation. This is particularly relevant because 27edo is a "sharp tending" system, and flattening its octaves has been suggested before as an improvement (I think by no less than Ivor Darreg, but I'll have to check that).
== Theory ==
43edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. Like 27edo, it is consistent to the [[9-odd-limit|10-integer-limit]]. It has octaves compressed by about 5.7492{{c}}, a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.


However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, Bohlen-Pierce harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The 4L+5s MOS has L=7 s=3.
However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, [[Bohlen–Pierce]] harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.


||~ Degrees ||~ Cents ||
=== Harmonics ===
|| 1 || 44.2315 ||
{{Harmonics in equal|43|3|1}}
|| 2 || 88.463 ||
{{Harmonics in equal|43|3|1|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 43edt (continued)}}
|| 3 || 132.6945 ||
|| 4 || 176.926 ||
|| 5 || 221.158 ||
|| 6 || 265.389 ||
|| 7 || 309.621 ||
|| 8 || 353.852 ||
|| 9 || 398.084 ||
|| 10 || 442.315 ||
|| 11 || 486.547 ||
|| 12 || 530.778 ||
|| 13 || 575.01 ||
|| 14 || 619.241 ||
|| 15 || 663.473 ||
|| 16 || 707.704 ||
|| 17 || 751.936 ||
|| 18 || 796.167 ||
|| 19 || 840.399 ||
|| 20 || 884.63 ||
|| 21 || 928.862 ||
|| 22 || 973.093 ||
|| 23 || 1017.325 ||
|| 24 || 1061.556 ||
|| 25 || 1105.788 ||
|| 26 || 1150.019 ||
|| 27 || 1194.251 ||
|| 28 || 1238.482 ||
|| 29 || 1282.713 ||
|| 30 || 1326.946 ||
|| 31 || 1371.177 ||
|| 32 || 1415.408 ||
|| 33 || 1459.64 ||
|| 34 || 1503.871 ||
|| 35 || 1548.193 ||
|| 36 || 1592.334 ||
|| 37 || 1636.566 ||
|| 38 || 1680.797 ||
|| 39 || 1725.029 ||
|| 40 || 1769.2605 ||
|| 41 || 1813.492 ||
|| 42 || 1857.7235 ||
|| 43 || 1901.955 ||</pre></div>
<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;43edt&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x43 EDT"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;43 EDT&lt;/h1&gt;
&lt;br /&gt;
This tuning is related to &lt;a class="wiki_link" href="/27edo"&gt;27edo&lt;/a&gt; having compressed octaves of 1194.251 cents, a small but significant deviation. This is particularly relevant because 27edo is a &amp;quot;sharp tending&amp;quot; system, and flattening its octaves has been suggested before as an improvement (I think by no less than Ivor Darreg, but I'll have to check that).&lt;br /&gt;
&lt;br /&gt;
However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, Bohlen-Pierce harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt; is not. The 4L+5s MOS has L=7 s=3.&lt;br /&gt;
&lt;br /&gt;


=== Subsets and supersets ===
43edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{| class="wikitable center-1 right-2 right-3"
        &lt;th&gt;Degrees&lt;br /&gt;
|-
&lt;/th&gt;
! #
        &lt;th&gt;Cents&lt;br /&gt;
! Cents
&lt;/th&gt;
! [[Hekt]]s
    &lt;/tr&gt;
! Approximate ratios
    &lt;tr&gt;
|-
        &lt;td&gt;1&lt;br /&gt;
| 1
&lt;/td&gt;
| 44.2
        &lt;td&gt;44.2315&lt;br /&gt;
| 30.2
&lt;/td&gt;
| 39/38, 40/39
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 2
        &lt;td&gt;2&lt;br /&gt;
| 88.5
&lt;/td&gt;
| 60.5
        &lt;td&gt;88.463&lt;br /&gt;
| [[20/19]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 3
    &lt;tr&gt;
| 132.7
        &lt;td&gt;3&lt;br /&gt;
| 90.7
&lt;/td&gt;
| [[27/25]]
        &lt;td&gt;132.6945&lt;br /&gt;
|-
&lt;/td&gt;
| 4
    &lt;/tr&gt;
| 176.9
    &lt;tr&gt;
| 120.9
        &lt;td&gt;4&lt;br /&gt;
| [[10/9]]
&lt;/td&gt;
|-
        &lt;td&gt;176.926&lt;br /&gt;
| 5
&lt;/td&gt;
| 221.2
    &lt;/tr&gt;
| 151.2
    &lt;tr&gt;
| [[25/22]]
        &lt;td&gt;5&lt;br /&gt;
|-
&lt;/td&gt;
| 6
        &lt;td&gt;221.158&lt;br /&gt;
| 265.4
&lt;/td&gt;
| 181.4
    &lt;/tr&gt;
| [[7/6]]
    &lt;tr&gt;
|-
        &lt;td&gt;6&lt;br /&gt;
| 7
&lt;/td&gt;
| 309.6
        &lt;td&gt;265.389&lt;br /&gt;
| 211.6
&lt;/td&gt;
| [[6/5]]
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 8
        &lt;td&gt;7&lt;br /&gt;
| 353.9
&lt;/td&gt;
| 241.9
        &lt;td&gt;309.621&lt;br /&gt;
| [[27/22]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 9
    &lt;tr&gt;
| 398.1
        &lt;td&gt;8&lt;br /&gt;
| 272.1
&lt;/td&gt;
| [[24/19]]
        &lt;td&gt;353.852&lt;br /&gt;
|-
&lt;/td&gt;
| 10
    &lt;/tr&gt;
| 442.3
    &lt;tr&gt;
| 302.3
        &lt;td&gt;9&lt;br /&gt;
| [[9/7]]
&lt;/td&gt;
|-
        &lt;td&gt;398.084&lt;br /&gt;
| 11
&lt;/td&gt;
| 486.5
    &lt;/tr&gt;
| 332.6
    &lt;tr&gt;
| [[45/34]]
        &lt;td&gt;10&lt;br /&gt;
|-
&lt;/td&gt;
| 12
        &lt;td&gt;442.315&lt;br /&gt;
| 530.8
&lt;/td&gt;
| 362.8
    &lt;/tr&gt;
| [[34/25]]
    &lt;tr&gt;
|-
        &lt;td&gt;11&lt;br /&gt;
| 13
&lt;/td&gt;
| 575.0
        &lt;td&gt;486.547&lt;br /&gt;
| 393.0
&lt;/td&gt;
| [[39/28]]
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 14
        &lt;td&gt;12&lt;br /&gt;
| 619.2
&lt;/td&gt;
| 423.3
        &lt;td&gt;530.778&lt;br /&gt;
| [[10/7]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 15
    &lt;tr&gt;
| 663.5
        &lt;td&gt;13&lt;br /&gt;
| 453.5
&lt;/td&gt;
| [[22/15]]
        &lt;td&gt;575.01&lt;br /&gt;
|-
&lt;/td&gt;
| 16
    &lt;/tr&gt;
| 707.7
    &lt;tr&gt;
| 483.7
        &lt;td&gt;14&lt;br /&gt;
| [[3/2]]
&lt;/td&gt;
|-
        &lt;td&gt;619.241&lt;br /&gt;
| 17
&lt;/td&gt;
| 751.9
    &lt;/tr&gt;
| 514.0
    &lt;tr&gt;
| [[20/13]], 105/68
        &lt;td&gt;15&lt;br /&gt;
|-
&lt;/td&gt;
| 18
        &lt;td&gt;663.473&lt;br /&gt;
| 796.2
&lt;/td&gt;
| 544.2
    &lt;/tr&gt;
| [[19/12]]
    &lt;tr&gt;
|-
        &lt;td&gt;16&lt;br /&gt;
| 19
&lt;/td&gt;
| 840.4
        &lt;td&gt;707.704&lt;br /&gt;
| 574.4
&lt;/td&gt;
| [[13/8]]
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 20
        &lt;td&gt;17&lt;br /&gt;
| 884.6
&lt;/td&gt;
| 604.7
        &lt;td&gt;751.936&lt;br /&gt;
| [[5/3]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 21
    &lt;tr&gt;
| 928.9
        &lt;td&gt;18&lt;br /&gt;
| 634.9
&lt;/td&gt;
| [[12/7]]
        &lt;td&gt;796.167&lt;br /&gt;
|-
&lt;/td&gt;
| 22
    &lt;/tr&gt;
| 973.1
    &lt;tr&gt;
| 665.1
        &lt;td&gt;19&lt;br /&gt;
| [[7/4]]
&lt;/td&gt;
|-
        &lt;td&gt;840.399&lt;br /&gt;
| 23
&lt;/td&gt;
| 1017.3
    &lt;/tr&gt;
| 695.3
    &lt;tr&gt;
| [[9/5]]
        &lt;td&gt;20&lt;br /&gt;
|-
&lt;/td&gt;
| 24
        &lt;td&gt;884.63&lt;br /&gt;
| 1061.6
&lt;/td&gt;
| 725.6
    &lt;/tr&gt;
| [[24/13]]
    &lt;tr&gt;
|-
        &lt;td&gt;21&lt;br /&gt;
| 25
&lt;/td&gt;
| 1105.8
        &lt;td&gt;928.862&lt;br /&gt;
| 755.8
&lt;/td&gt;
| [[36/19]]
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 26
        &lt;td&gt;22&lt;br /&gt;
| 1150.0
&lt;/td&gt;
| 786.0
        &lt;td&gt;973.093&lt;br /&gt;
| [[39/20]], [[68/35]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 27
    &lt;tr&gt;
| 1194.3
        &lt;td&gt;23&lt;br /&gt;
| 816.3
&lt;/td&gt;
| [[2/1]]
        &lt;td&gt;1017.325&lt;br /&gt;
|-
&lt;/td&gt;
| 28
    &lt;/tr&gt;
| 1238.5
    &lt;tr&gt;
| 846.5
        &lt;td&gt;24&lt;br /&gt;
| [[45/22]]
&lt;/td&gt;
|-
        &lt;td&gt;1061.556&lt;br /&gt;
| 29
&lt;/td&gt;
| 1282.7
    &lt;/tr&gt;
| 876.7
    &lt;tr&gt;
| [[21/10]]
        &lt;td&gt;25&lt;br /&gt;
|-
&lt;/td&gt;
| 30
        &lt;td&gt;1105.788&lt;br /&gt;
| 1326.9
&lt;/td&gt;
| 907.0
    &lt;/tr&gt;
| [[28/13]]
    &lt;tr&gt;
|-
        &lt;td&gt;26&lt;br /&gt;
| 31
&lt;/td&gt;
| 1371.2
        &lt;td&gt;1150.019&lt;br /&gt;
| 937.2
&lt;/td&gt;
| 75/34
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 32
        &lt;td&gt;27&lt;br /&gt;
| 1415.4
&lt;/td&gt;
| 967.4
        &lt;td&gt;1194.251&lt;br /&gt;
| [[34/15]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 33
    &lt;tr&gt;
| 1459.6
        &lt;td&gt;28&lt;br /&gt;
| 997.7
&lt;/td&gt;
| [[7/3]]
        &lt;td&gt;1238.482&lt;br /&gt;
|-
&lt;/td&gt;
| 34
    &lt;/tr&gt;
| 1503.9
    &lt;tr&gt;
| 1027.9
        &lt;td&gt;29&lt;br /&gt;
| [[19/8]]
&lt;/td&gt;
|-
        &lt;td&gt;1282.713&lt;br /&gt;
| 35
&lt;/td&gt;
| 1548.1
    &lt;/tr&gt;
| 1058.1
    &lt;tr&gt;
| [[22/9]]
        &lt;td&gt;30&lt;br /&gt;
|-
&lt;/td&gt;
| 36
        &lt;td&gt;1326.946&lt;br /&gt;
| 1592.3
&lt;/td&gt;
| 1088.3
    &lt;/tr&gt;
| [[5/2]]
    &lt;tr&gt;
|-
        &lt;td&gt;31&lt;br /&gt;
| 37
&lt;/td&gt;
| 1636.6
        &lt;td&gt;1371.177&lt;br /&gt;
| 1118.6
&lt;/td&gt;
| [[18/7]]
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 38
        &lt;td&gt;32&lt;br /&gt;
| 1680.8
&lt;/td&gt;
| 1148.8
        &lt;td&gt;1415.408&lt;br /&gt;
| [[66/25]]
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 39
    &lt;tr&gt;
| 1725.0
        &lt;td&gt;33&lt;br /&gt;
| 1179.1
&lt;/td&gt;
| [[27/10]]
        &lt;td&gt;1459.64&lt;br /&gt;
|-
&lt;/td&gt;
| 40
    &lt;/tr&gt;
| 1769.3
    &lt;tr&gt;
| 1209.3
        &lt;td&gt;34&lt;br /&gt;
| [[25/9]]
&lt;/td&gt;
|-
        &lt;td&gt;1503.871&lt;br /&gt;
| 41
&lt;/td&gt;
| 1813.5
    &lt;/tr&gt;
| 1239.5
    &lt;tr&gt;
| 57/20
        &lt;td&gt;35&lt;br /&gt;
|-
&lt;/td&gt;
| 42
        &lt;td&gt;1548.193&lt;br /&gt;
| 1857.7
&lt;/td&gt;
| 1269.8
    &lt;/tr&gt;
| [[38/13]], 117/40
    &lt;tr&gt;
|-
        &lt;td&gt;36&lt;br /&gt;
| 43
&lt;/td&gt;
| 1902.0
        &lt;td&gt;1592.334&lt;br /&gt;
| 1300.0
&lt;/td&gt;
| [[3/1]]
    &lt;/tr&gt;
|}
    &lt;tr&gt;
        &lt;td&gt;37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1636.566&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1680.797&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1725.029&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1769.2605&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1813.492&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1857.7235&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1901.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
== Related regular temperaments ==
43edt tempers out the no-twos comma of {{monzo| 0 63 -43 }}, leading to the regular temperament [[support]]ed by [[27edo|27-]], [[190edo|190-]], and [[217edo]].
 
=== 27 &amp; 190 temperament ===
==== 5-limit ====
Subgroup: 2.3.5
 
Comma list: {{monzo| 0 63 -43 }}
 
Mapping: {{mapping| 1 0 0 | 0 43 63 }}
 
Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294
 
{{Optimal ET sequence|legend=0| 27, 190, 217, 407, 597, 624, 841 }}
 
==== 7-limit ====
Subgroup: 2.3.5.7
 
Comma list: 4375/4374, 40353607/40000000
 
Mapping: {{mapping| 1 0 0 1 | 0 43 63 49 }}
 
Optimal tuning (POTE): ~1029/1000 = 44.2288
 
{{Optimal ET sequence|legend=0| 27, 190, 217 }}
 
Badness: 0.1659
 
=== 217 &amp; 407 temperament ===
==== 7-limit ====
Subgroup: 2.3.5.7
 
Comma list: 134217728/133984375, 512557306947/512000000000
 
Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}
 
Optimal tuning (POTE): ~525/512 = 44.2320
 
{{Optimal ET sequence|legend=0| 217, 407, 624, 841, 1058, 1465 }}
 
Badness: 0.3544
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
 
Comma list: 46656/46585, 131072/130977, 234375/234256
 
Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}
 
Optimal tuning (POTE): ~525/512 = 44.2312
 
{{Optimal ET sequence|legend=0| 217, 407, 624 }}
 
Badness: 0.1129
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375
 
Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}
 
Optimal tuning (POTE): ~40/39 = 44.2312
 
{{Optimal ET sequence|legend=0| 217, 407, 624 }}
 
Badness: 0.0503
 
== See also ==
* [[16edf]] – relative edf
* [[27edo]] – relative edo
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12
 
[[Category:27edo]]
Retrieved from "https://en.xen.wiki/w/43edt"