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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 16:59:43 UTC</tt>.<br>
: The original revision id was <tt>340674494</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">Rare, but adds approximations of the 5th and 9th harmonics to [[26edo]] which it doubles.


Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[porcupine]] temperament.
== Theory ==
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.


=Music=
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.
[[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 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.
<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;Rare, but adds approximations of the 5th and 9th harmonics to &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt; which it doubles.&lt;br /&gt;
 
&lt;br /&gt;
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%.
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.&lt;br /&gt;
 
&lt;br /&gt;
=== Odd harmonics ===
&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;
{{Harmonics in equal|52}}
&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>
 
=== Subsets and supersets ===
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"