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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:genewardsmith|genewardsmith]] and made on <tt>2012-05-29 18:57:27 UTC</tt>.<br>
 
: The original revision id was <tt>340713106</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
52edo has [[26edo]]'s very flat [[meantone]] [[perfect|fifth]] and a very sharp fifth close to 1/2-[[64/63|septimal-comma]] [[superpyth]]. The [[patent val]] has the same mapping for [[3/1|3]], [[7/1|7]], [[11/1|11]] and [[13/1|13]] as 26 does, but its [[5/1|5]] is sharp rather than flat. From this it tempers out [[648/625]] rather than [[81/80]] in the 5-limit, and [[225/224]] and [[1029/1024]] in the 7-limit, showing it [[support]]s [[miracle]], albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out [[99/98]] and [[176/175]] and in the 13-limit [[78/77]], [[144/143]] and [[169/168]]. It supplies the [[optimal patent val]] for then 12 &amp; 40 temperament of the diminished family in the 7- and 11-limit, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the {{nowrap|21 &amp; 52}} variant of miracle.
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">The 52 equal division divides the octave into 52 equal parts of 23.077 cents each. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out 648/625 rather than 81/80 in the 5-limit, and 225/224 and 1029/1024 in the 7-limit, showing it supports miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out 99/98 and 176/175 and in the 13-limit 78/77, 144/143 and 169/168. It supplies the optimal patent val for then 12&amp;40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&amp;52 variant of miracle.


Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[porcupine]] temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.
Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[porcupine]] temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.


=Music=
The 5\52 interval approximates [[31/29]] well, and when used as a generator produces [[tricesimoprimal miracloid]] temperament. The relationship is also preserved exactly in the period-52 [[french deck]] temperament.
[[http://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3|The Hidden Temple of Empathica III]] by [[Jon Lyle Smith]]</pre></div>
 
<h4>Original HTML content:</h4>
The 11\52 (253.846{{c}}) [[semifourth]] is a very accurate [[22/19]], with an error of only +0.041{{c}} and a closing error of only 9.3%.
<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;52edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 52 equal division divides the octave into 52 equal parts of 23.077 cents each. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out 648/625 rather than 81/80 in the 5-limit, and 225/224 and 1029/1024 in the 7-limit, showing it supports miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out 99/98 and 176/175 and in the 13-limit 78/77, 144/143 and 169/168. It supplies the optimal patent val for then 12&amp;amp;40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&amp;amp;52 variant of miracle.&lt;br /&gt;
 
&lt;br /&gt;
=== Odd harmonics ===
Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of &lt;a class="wiki_link" href="/porcupine"&gt;porcupine&lt;/a&gt; temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.&lt;br /&gt;
{{Harmonics in equal|52}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Music&lt;/h1&gt;
=== Subsets and supersets ===
&lt;a class="wiki_link_ext" href="http://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3" rel="nofollow"&gt;The Hidden Temple of Empathica III&lt;/a&gt; by &lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&gt;Jon Lyle Smith&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
Since 52 factors into {{factorization|52}}, 52edo contains subset edos {{EDOs| 2, 4, 13, and 26 }}.
 
== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! Degrees
! [[Cents]]s
! colspan="3" | [[Ups and downs notation]]
|-
| 0
| 0.000
| Perfect 1sn
| P1
| D
|-
| 1
| 23.077
| Up 1sn
| ^1
| ^D
|-
| 2
| 46.154
| Aug 1sn
| A1
| D#
|-
| 3
| 69.231
| Downdim 2nd, Upaug 1sn
| vd2, ^A1
| vEbb, ^D#
|-
| 4
| 92.308
| Dim 2nd
| d2
| Ebb
|-
| 5
| 115.358
| Downminor 2nd
| vm2
| vEb
|-
| 6
| 138.462
| Minor 2nd
| m2
| Eb
|-
| 7
| 161.538
| Mid 2nd
| ~2
| vE, ^Eb
|-
| 8
| 184.615
| Major 2nd
| M2
| E
|-
| 9
| 207.692
| Upmajor 2nd
| ^M2
| ^E
|-
| 10
| 230.769
| Aug 2nd
| A2
| E#
|-
| 11
| 253.846
| Downdim 3rd, Upaug 2nd
| vd3, ^A2
| vFb, ^E#
|-
| 12
| 276.923
| Dim 3rd
| d3
| Fb
|-
| 13
| 300.000
| Downminor 3rd
| vm3
| vF
|-
| 14
| 323.077
| Minor 3rd
| m3
| F
|-
| 15
| 346.154
| Mid 3rd
| ~3
| ^F, vF#
|-
| 16
| 369.231
| Major 3rd
| M3
| F#
|-
| 17
| 392.308
| Upmajor 3rd
| ^M3
| ^F#
|-
| 18
| 415.385
| Aug 3rd
| A3
| Fx
|-
| 19
| 438.462
| Downdim 4th, Upaug 3rd
| vd4, ^A4
| vGb, ^Fx
|-
| 20
| 461.538
| Dim 4th
| d4
| Gb
|-
| 21
| 484.615
| Down 4th
| v4
| vG
|-
| 22
| 507.692
| Perfect 4th
| P4
| G
|-
| 23
| 530.769
| Up 4th
| ^4
| ^G
|-
| 24
| 553.846
| Aug 4th
| A4
| G#
|-
| 25
| 576.293
| Upaug 4th
| ^A4
| ^G#
|-
| 26
|600.000
| Double-aug 4th, Double-dim 5th
| AA4, dd5
| Gx, Abb
|-
| 27
| 623.077
| Downdim 5th
| vd5
| vAb
|-
| 28
| 646.154
| Dim 5th
| d5
| Ab
|-
| 29
| 669.231
| Down 5th
| v5
| vA
|-
| 30
| 692.308
| Perfect 5th
| P5
| A
|-
| 31
| 715.385
| Up 5th
| ^5
| ^A
|-
| 32
| 738.462
| Aug 5th
| A5
| A#
|-
| 33
| 761.538
| Downdim 6th, Upaug 5th
| vd6, ^A5
| vBbb, ^A#
|-
| 34
| 784.615
| Dim 6th
| d6
| Bbb
|-
| 35
| 807.692
| Downminor 6th
| vm6
| vBb
|-
| 36
| 830.769
| Minor 6th
| m6
| Bb
|-
| 37
| 853.846
| Mid 6th
| ~6
| vB, ^Bb
|-
| 38
| 876.923
| Major 6th
| M6
| B
|-
| 39
| 900.000
| Upmajor 6th
| ^M6
| ^B
|-
| 40
| 923.077
| Aug 6th
| A6
| B#
|-
| 41
| 946.154
| Downdim 7th, Upaug 6th
| vd7, ^A6
| vCb, ^B#
|-
| 42
| 969.231
| Dim 7th
| d7
| Cb
|-
| 43
| 992.308
| Downminor 7th
| vm7
| vC
|-
| 44
| 1015.385
| Minor 7th
| m7
| C
|-
| 45
| 1038.462
| Mid 7th
| ~7
| ^C, vC#
|-
| 46
| 1061.538
| Major 7th
| M7
| C#
|-
| 47
| 1084.615
| Upmajor 7th
| ^M7
| ^C#
|-
| 48
| 1107.692
| Aug 7th
| A7
| Cx
|-
| 49
| 1130.769
| Downdim 8ve, Upaug 7th
| vd8, ^A7
| vDb, ^Cx
|-
| 50
| 1153.846
| Dim 8ve
| d8
| Db
|-
| 51
| 1176.923
| Down 8ve
| v8
| vD
|-
| 52
| 1200.000
| Perfect 8ve
| P8
| D
|}
 
== Notation ==
[[13edo#Notation|13edo notation]] can be used together with +/- eighth-tone accidentals.
 
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals, 52edo can also be notated using [[ups and downs notation]] or Stein–Zimmerman [[24edo#Notation|quarter tone]] accidentals:
{{Sharpness-sharp2a}}
 
{{sharpness-sharp2}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[59edo#Second-best fifth notation|59b]], and is a superset of the notations for EDOs [[26edo#Sagittal notation|26]] and [[13edo#Sagittal notation|13]].
 
==== Evo flavor ====
<imagemap>
File:52-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:52-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:52-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:52-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:52-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:52-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
 
== Instruments ==
'''Lumatone'''
 
See [[Lumatone mapping for 52edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/lNJCZz7EjL0 ''microtonal improvisation in 52edo''] (2025)
* [https://www.youtube.com/watch?v=3uo24YpEN0E ''Waltz in 52edo''] (2025)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=5HkEM0ZchP0 ''5-in-1 Canon on Happy Birthday''] (2020)
 
; [[Jon Lyle Smith]]
* [https://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 ''The Hidden Temple of Empathica III'']{{dead link}}
 
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/52edo"