52edo: Difference between revisions

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== 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 21 & 52 variant of miracle.

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.

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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.

The 11\52 (253.~~846¢~~) [[semifourth]] is a very accurate [[22/19]], with an error of only +0.~~041¢~~ and a closing error of only 9.3%.

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 ===

=== Subsets and supersets ===

Since 52 factors into {{factorization|52}}, 52edo contains subset edos {{EDOs| 2, 4, 13, and 26 }}.

== Intervals ==

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! Degrees

! [[Cents]]s

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

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

|-

| 0

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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 ==

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== 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]

* [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}}

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

[[Category:Listen]]

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|52}}
+{{ED intro}}
 == 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 21 &amp; 52 variant of miracle.
+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.
@@ Line 9: / Line 9: @@
 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.
-The 11\52 (253.846¢) [[semifourth]] is a very accurate [[22/19]], with an error of only +0.041¢ and a closing error of only 9.3%.
+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 ===
 {{Harmonics in equal|52}}
+=== Subsets and supersets ===
+Since 52 factors into {{factorization|52}}, 52edo contains subset edos {{EDOs| 2, 4, 13, and 26 }}.
 == Intervals ==
@@ Line 19: / Line 22: @@
 ! Degrees
 ! [[Cents]]s
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 339: / 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 ==
@@ Line 346: / Line 393: @@
 == 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]
+* [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}}
+* [https://archive.org/download/TheHiddenTempleOfEmpathicaIii/TheHiddenTempleOfEmpathicaIii.mp3 ''The Hidden Temple of Empathica III'']{{dead link}}
 [[Category:Listen]]
 [[Category:Todo:add rank 2 temperaments table]]