52edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 2<sup>2</sup> × 13~~

~~| Step size = 23.077¢~~

~~| Fifth = 30\52 = 692.31¢~~

~~| Major 2nd = 8\52 = 185¢~~

~~| Minor 2nd = 6\52 = 138¢~~

~~| Augmented 1sn = 2\52 = 46¢~~

}}

~~'''~~52edo'~~'' divides the~~ [[~~octave~~]] ~~into 52 equal parts of 23.077~~ [[~~Cent~~|~~cents~~]] ~~each. It has 26edo's very flat meantone fifth~~ and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth.

== 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 & 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 & 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.

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.

~~== Theory ==~~

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%.

=== Odd harmonics ===

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&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&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~~|~~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~~.

=== 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|~~Cents]]

! [[Cents]]s

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! colspan="3" | [[Ups and downs notation]]

|-

| 0

Line 343:

Line 342:

| 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:

=== 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 ====

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 ====

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 ====

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)

* [~~http~~://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 The Hidden Temple of Empathica III~~] by [[Jon Lyle Smith]~~] {{dead link}}

; [[Jon Lyle Smith]]

* [https://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 ''The Hidden Temple of Empathica III'']{{dead link}}

[[Category:~~Equal divisions of the octave~~]]

[[Category:Listen]]

[[Category:Todo:add rank 2 temperaments table]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>2</sup> × 13
+{{ED intro}}
-| Step size = 23.077¢
-| Fifth = 30\52 = 692.31¢
-| Major 2nd = 8\52 = 185¢
-| Minor 2nd = 6\52 = 138¢
-| Augmented 1sn = 2\52 = 46¢
-}}
-'''52edo''' divides the [[octave]] into 52 equal parts of 23.077 [[Cent|cents]] each. It has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth.
+== Theory ==
+edo 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.
+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.
+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.
-== Theory ==
+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%.
-{{primes in edo|52}}
+=== Odd harmonics ===
-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.
+{{Harmonics in equal|52}}
-Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[Porcupine|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.
+=== 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|Cents]]
+! [[Cents]]s
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 343: / Line 342: @@
 | 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)
-* [http://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 The Hidden Temple of Empathica III] by [[Jon Lyle Smith]] {{dead link}}
+; [[Jon Lyle Smith]]
+* [https://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 ''The Hidden Temple of Empathica III'']{{dead link}}
-[[Category:Equal divisions of the octave]]
+[[Category:Listen]]
+[[Category:Todo:add rank 2 temperaments table]]