55edo: Difference between revisions

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{{interwiki

| de =

| de = 55-EDO

| en = 55edo

| es = 55 EDO

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}}

== Theory ==

55edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[~~Meantone family~~|~~mohajira~~ and ~~liese~~]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.

55edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone_family#Mohajira|Mohajira]] and [[Meantone_family#Liese|Liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.

=== Odd harmonics ===

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| 26

| 567.3

| 18/13

| [[7/5]], [[18/13]]

| vA4

| downaug 4th

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| 27

| 589.1

| ~~7/5,~~ 24/17

| 24/17

| A4, vd5

| aug 4th, downdim 5th

Line 223:

| 28

| 610.9

| ~~10/7,~~ 17/12

| 17/12

| ^A4, d5

| upaug 4th, dim 5th

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| 29

| 632.7

| 13/9

| [[10/7]], [[13/9]]

| ^d5

| updim 5th

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== Notation ==

=== Ups and downs notation ===

55edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

=== Sagittal notation ===

==== Evo flavor ====

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default [[File:55-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

=== 31-tone subset ===

The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.

[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]

== Approximation to JI ==

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| 9.63

|}

=== Uniform maps ===

=== Commas ===

'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }}

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| 14/13

| [[Twothirdtonic]] (55f)

|-

|1

|8\55

|174.5

|[[10/9]]~[[11/10]]

|[[Tetracot]] (55c)

|-

| 1

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* Septimal minor blues-like: 13 10 4 5 13 10

* Septimal heptatonic blues-like: 13 10 4 5 8 5 10

; Others

* Sakura-like scale containing [[phi]]: 9 6 18 5 17

* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7

== Instruments ==

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; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)

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

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

* [https://www.youtube.com/watch?v=OkRVNo19guo "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)

* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

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

; {{W|Nicolaus Bruhns}}

* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)

* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)

* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|Georg Frideric Handel}}

* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025)

; {{W|Scott Joplin}}

* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] (2024)

; {{W|Wolfgang Amadeus Mozart}}

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* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])

* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)

* [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)''] – rendered by [[Claudi Meneghin]] (2025)

; {{W|Keiichi Okabe}}

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=== 21st century ===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)

* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)

; [[James Kukula]]

* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)

; [[Budjarn Lambeth]]

* ''[https://www.youtube.com/watch?v=9c5MtrZFNhA Improvisation One in 55edo]'' (2025)

* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)

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; [[Herman Miller]]

* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)

* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)

== External links ==

* [http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo

* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]

and its close relative, 1/6-comma meantone] (containing another listening example) on [[Tonalsoft Encyclopedia]]

== References ==

@@ Line 1: / Line 1: @@
 {{interwiki
-| de =
+| de = 55-EDO
 | en = 55edo
 | es = 55 EDO
@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|55}}
+{{ED intro}}
 == Theory ==
-edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone family|mohajira and liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.
+edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone_family#Mohajira|Mohajira]] and [[Meantone_family#Liese|Liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.
 === Odd harmonics ===
@@ Line 209: / Line 209: @@
 | 26
 | 567.3
-| 18/13
+| [[7/5]], [[18/13]]
 | vA4
 | downaug 4th
@@ Line 216: / Line 216: @@
 | 27
 | 589.1
-| 7/5, 24/17
+| 24/17
 | A4, vd5
 | aug 4th, downdim 5th
@@ Line 223: / Line 223: @@
 | 28
 | 610.9
-| 10/7, 17/12
+| 17/12
 | ^A4, d5
 | upaug 4th, dim 5th
@@ Line 230: / Line 230: @@
 | 29
 | 632.7
-| 13/9
+| [[10/7]], [[13/9]]
 | ^d5
 | updim 5th
@@ Line 420: / Line 420: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+{{Sharpness-sharp4a}}
+[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+{{Sharpness-sharp4}}
 === Sagittal notation ===
 ==== Evo flavor ====
@@ Line 453: / Line 458: @@
 default [[File:55-EDO_Evo-SZ_Sagittal.svg]]
 </imagemap>
+=== 31-tone subset ===
+The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.
+[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]
 == Approximation to JI ==
@@ Line 487: / Line 497: @@
 | 9.63
 |}
+=== Uniform maps ===
+{{Uniform map|edo=55}}
 === Commas ===
+{{Todo|cleanup|inline=true}}
 '''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }}
@@ Line 512: / Line 527: @@
 | 14/13
 | [[Twothirdtonic]] (55f)
+|-
+|1
+|8\55
+|174.5
+|[[10/9]]~[[11/10]]
+|[[Tetracot]] (55c)
 |-
 | 1
@@ Line 559: / Line 580: @@
 * Septimal minor blues-like: 13 10 4 5 13 10
 * Septimal heptatonic blues-like: 13 10 4 5 8 5 10
+; Others
+* Sakura-like scale containing [[phi]]: 9 6 18 5 17
+* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7
 == Instruments ==
@@ Line 567: / Line 592: @@
 ; {{W|Johann Sebastian Bach}}
 * [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
-* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
+* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)
+* [https://www.youtube.com/watch?v=OkRVNo19guo "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)
 * [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
 * [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
 ; {{W|Nicolaus Bruhns}}
-* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
+* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)
 * [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
+; {{W|Georg Frideric Handel}}
+* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025)
 ; {{W|Scott Joplin}}
-* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
+* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] (2024)
 ; {{W|Wolfgang Amadeus Mozart}}
@@ Line 583: / Line 612: @@
 * [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
 * [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)
+* [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)''] – rendered by [[Claudi Meneghin]] (2025)
 ; {{W|Keiichi Okabe}}
@@ Line 588: / Line 618: @@
 === 21st century ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)
+* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)
+; [[James Kukula]]
+* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)
+; [[Budjarn Lambeth]]
+* ''[https://www.youtube.com/watch?v=9c5MtrZFNhA Improvisation One in 55edo]'' (2025)
+* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)
 ; [[Claudi Meneghin]]
 * [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)
@@ Line 597: / Line 638: @@
 ; [[Herman Miller]]
 * ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)
+* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)
 == External links ==
-* [http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo
+* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]
-and its close relative, 1/6-comma meantone] (containing another listening example) on [[Tonalsoft Encyclopedia]]
 == References ==