50edo: Difference between revisions

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== Theory ==

As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 11/9 and 18/11.

As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 11/9 and 18/11.

It tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.

=== Odd harmonics ===

=== Relations ===

Line 20:

! Ratios<ref group="note">{{sg|13-limit}}</ref>

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

([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and vvd2)

|-

| 0

Line 379:

Line 380:

|}

== JI ~~approximation~~ ==

== Notation ==

=== Ups and downs notation ===

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

Using [[Helmholtz–Ellis]] accidentals, 50edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[57edo#Sagittal notation|57]], [[64edo#Sagittal notation|64]], and [[71edo#Second-best fifth notation|71b]].

==== Evo flavor ====

File:50-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 160 106 [[1053/1024]]

default [[File:50-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:50-EDO_Revo_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 80 160 106 [[1053/1024]]

default [[File:50-EDO_Revo_Sagittal.svg]]

</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

== Approximation to JI ==

=== 15-odd-limit mappings ===

=== 15-odd-limit interval mappings ===

== Regular temperament properties ==

=== Temperament measures ===

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

Line 422:

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| 1.57

| 6.54

~~{{comma basis end}~~}

|}

=== Commas ===

~~50edo~~ [[tempers out]] the following [[comma]]s. ~~(Note:~~ This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

50et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places. This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

{| class="commatable wikitable center-all left-3 right-4 left-5"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

Line 493:

Line 538:

| {{monzo| 6 0 -5 2 }}

| 6.08

| Hemimean

| Hemimean comma

|-

| 7

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Line 544:

| {{monzo| 11 -10 -10 10 }}

| 5.57

| [[Linus]]

| [[Linus comma]]

|-

| 7

Line 517:

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| {{monzo| -1 0 1 2 -2 }}

| 21.33

| ~~Cassacot~~

| Frostma

|-

| 11

Line 535:

Line 580:

| {{monzo| 5 -1 3 0 -3 }}

| 3.03

| Wizardharry

| Wizardharry comma

|-

| 11

Line 565:

Line 610:

| {{monzo| 2 3 0 -1 1 -2 }}

| 7.30

| Kestrel ~~Comma~~

| Kestrel comma

|-

| 13

Line 577:

Line 622:

| {{monzo| 2 -1 0 1 -2 1 }}

| 4.76

| ~~Gentle comma~~

| Minor minthma

|-

| 13

Line 669:

Line 714:

| Triaphonisma

|}

~~<references/>~~

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperament

|-

| 1

Line 781:

Line 832:

| 4/3 (78/77)

| [[Decic]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==

Line 788:

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See [[Lumatone mapping for 50edo]]

; Piano

A [[:Category:Piano|piano]] playing with a 50edo ensemble may wish to use the tuning [[116ed5]]. This tuning is almost exactly the same as 50edo, but with octaves [[octave stretch|stretched]] by 1 cent. Because pianos usually use stretched octaves, this tuning will sit better with the [[timbre]] of the piano, while still being close enough that it sounds perfectly in-tune with the other instruments tuned to 50edo.

== Music ==

=== Modern renderings ===

; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024, organ sound rendering)

* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) — rendered by Claudi Meneghin (2025, harpsichord sound rendering)

; {{W|Nicolaus Bruhns}}

Line 805:

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=== 21st century===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)

* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well - Deltarune (microtonal cover in 50edo)''] (2025)

; [[Francium]]

* [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|50}}
+{{ED intro}}
 == Theory ==
-As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice. In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts &ndash; 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 11/9 and 18/11.
+As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts &ndash; 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 11/9 and 18/11.
 It tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 &amp; 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.
 === Odd harmonics ===
-{{Harmonics in equal|50}}
+{{Harmonics in equal|50|columns=15}}
 === Relations ===
@@ Line 20: / Line 20: @@
 ! Ratios<ref group="note">{{sg|13-limit}}</ref>
 ! colspan="3" | [[Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and vvd2)
 |-
 | 0
@@ Line 379: / Line 380: @@
 |}
-== JI approximation ==
+== Notation ==
+=== Ups and downs notation ===
+Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
+{{sharpness-sharp3a}}
+Using [[Helmholtz–Ellis]] accidentals, 50edo can also be notated using [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]:
+{{Sharpness-sharp3}}
+Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as EDOs [[57edo#Sagittal notation|57]], [[64edo#Sagittal notation|64]], and [[71edo#Second-best fifth notation|71b]].
+==== Evo flavor ====
+<imagemap>
+File:50-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 160 106 [[1053/1024]]
+default [[File:50-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:50-EDO_Revo_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 80 160 106 [[1053/1024]]
+default [[File:50-EDO_Revo_Sagittal.svg]]
+</imagemap>
+In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
+== Approximation to JI ==
 [[File:50ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 29-limit intervals approximated in 50edo]]
-=== 15-odd-limit mappings ===
-{{Q-odd-limit intervals|50}}
+=== 15-odd-limit interval mappings ===
+{{Q-odd-limit intervals|50|15}}
 == Regular temperament properties ==
 === Temperament measures ===
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
@@ Line 422: / Line 467: @@
 | 1.57
 | 6.54
-{{comma basis end}}
+|}
 === Commas ===
-edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
+et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places. This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
 {| class="commatable wikitable center-all left-3 right-4 left-5"
 |-
-! [[Harmonic limit|Prime<br />limit]]
+! [[Harmonic limit|Prime<br>limit]]
 ! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
@@ Line 493: / Line 538: @@
 | {{monzo| 6 0 -5 2 }}
 | 6.08
-| Hemimean
+| Hemimean comma
 |-
 | 7
@@ Line 499: / Line 544: @@
 | {{monzo| 11 -10 -10 10 }}
 | 5.57
-| [[Linus]]
+| [[Linus comma]]
 |-
 | 7
@@ Line 517: / Line 562: @@
 | {{monzo| -1 0 1 2 -2 }}
 | 21.33
-| Cassacot
+| Frostma
 |-
 | 11
@@ Line 535: / Line 580: @@
 | {{monzo| 5 -1 3 0 -3 }}
 | 3.03
-| Wizardharry
+| Wizardharry comma
 |-
 | 11
@@ Line 565: / Line 610: @@
 | {{monzo| 2 3 0 -1 1 -2 }}
 | 7.30
-| Kestrel Comma
+| Kestrel comma
 |-
 | 13
@@ Line 577: / Line 622: @@
 | {{monzo| 2 -1 0 1 -2 1 }}
 | 4.76
-| Gentle comma
+| Minor minthma
 |-
 | 13
@@ Line 669: / Line 714: @@
 | Triaphonisma
 |}
-<references/>
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>ratio*
+! Temperament
 |-
 | 1
@@ Line 781: / Line 832: @@
 | 4/3<br>(78/77)
 | [[Decic]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Instruments ==
@@ Line 788: / Line 839: @@
 See [[Lumatone mapping for 50edo]]
+; Piano
+A [[:Category:Piano|piano]] playing with a 50edo ensemble may wish to use the tuning [[116ed5]]. This tuning is almost exactly the same as 50edo, but with octaves [[octave stretch|stretched]] by 1 cent. Because pianos usually use stretched octaves, this tuning will sit better with the [[timbre]] of the piano, while still being close enough that it sounds perfectly in-tune with the other instruments tuned to 50edo.
 == Music ==
 === Modern renderings ===
 ; {{W|Johann Sebastian Bach}}
+* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
 * [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) &ndash; rendered by Claudi Meneghin (2024)
-* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) &ndash; rendered by Claudi Meneghin (2024)
+* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) &ndash; rendered by Claudi Meneghin (2024, organ sound rendering)
+* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) &mdash; rendered by Claudi Meneghin (2025, harpsichord sound rendering)
 ; {{W|Nicolaus Bruhns}}
@@ Line 805: / Line 863: @@
 === 21st century===
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)
+* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well - Deltarune (microtonal cover in 50edo)''] (2025)
 ; [[Francium]]
 * [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)