55edo: Difference between revisions

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{{~~interwiki~~

{{Interwiki

| en = 55edo

| de = 55-EDO

~~| en = 55edo~~

| es = 55 EDO

| ja =

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== Theory ==

55edo ~~is a [[The Riemann zeta function and tuning#Valley edos|zeta valley edo]], so it does not approximate the harmonic series very well for its size. Despite this, it can be used as 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.

55edo supports [[meantone]], and is close to [[1/6-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 [[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 is a [[zeta valley edo]], suggesting it does not approximate the harmonic series very well for its size.

=== Odd harmonics ===

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=== Subsets and supersets ===

Since 55 factors into {{~~factorization~~|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets.

Since 55 factors into primes as {{nowrap| 5 × 11 }}, 55edo contains [[5edo]] and [[11edo]] as its subsets.

== Intervals ==

{| class="wikitable center-1 right-2 left-3"

|-

! [[Degree|&#~~35;~~]]

! [[Degree|#]]

! [[Cent]]s

! Approximate ratios

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== Notation ==

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould 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.

[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:

=== Kite's ups and downs notation ===

55edo can also be notated with [[Kite's ups and downs notation|Kite's 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, flats, half- and sesquisharps, and half- and sesquiflats with arrows, borrowed from extended [[Helmholtz–Ellis notation]] and [[24edo#Stein-Zimmerman Accidentals|Stein-Zimmerman accidental set]]:

~~{{Sharpness-sharp4}}~~

=== Sagittal notation ===

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{{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}

{{Q-odd-limit intervals|54.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55f val mapping}}

== Regular temperament properties ==

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| [[Twothirdtonic]] (55f)

|-

|1

| 1

|8\55

| 8\55

|174.5

| 174.5

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

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

|[[Tetracot]] (55c)

| [[Tetracot]] (55c)

|-

| 1

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| 501.8<br>(65.5)

| 4/3<br>(36/35)

| [[Hendecatonic]] (55)

| [[Hendecatonic (temperament)|Hendecatonic]] (55)

|}

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

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==

; Subsets of ~~twothirdtonic~~[37]

; Subsets of Twothirdtonic[37]

* Undecimal otonal-like pentatonic: 17 8 7 12 11

; Subsets of ~~hendecatonic~~[33]

; Subsets of Hendecatonic[33]

* Septimal pentatonic-like: 10 13 9 13 10

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

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* [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|John Bull (composer)|John Bull}}

* [https://www.youtube.com/watch?v=I2d-540RuoM ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)

; {{W|Louis Couperin}}

* [https://www.youtube.com/shorts/bcgY3k-6rao ''La Piémontoise''] (1658?) – rendered by Claudi Meneghin (2026)

; {{W|Georg Frideric Handel}}

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; {{W|Scott Joplin}}

* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] ([https://www.youtube.com/watch?v=GbhpuoIJgxk 2024 version]; [https://www.youtube.com/shorts/3Y9y9I6q1as 2026 version])

; {{W|Mladen Milićević}}

* [https://www.youtube.com/shorts/1BhIK0s2-vs "The Room" from ''The Room''] (2003) – covered by [[Bryan Deister]] (2026)

; {{W|Wolfgang Amadeus Mozart}}

* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)

* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, K. 331] (1778) – rendered by Francium (2023)

* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)

* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', K. 401] (1782) – rendered by Francium (2023)

* [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])

* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', K. 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=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, K. 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~~)

* ''Kleine Gigue in G'', K. 574 (1789) – rendered by [[Claudi Meneghin]]

** [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 Fortepiano rendition] (2025)

** [https://www.youtube.com/shorts/7jt-jzziDBg Harpsichord rendition] (2026)

** [https://www.youtube.com/shorts/9J1DkS5UGzE Organ rendition] (2026)

; {{W|Keiichi Okabe}}

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; [[James Kukula]]

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

* [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=9c5MtrZFNhA ''Improvisation One in 55edo''] (2025)

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

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

; [[Claudi Meneghin]]

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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/road-trip-to-nowhere ''Road Trip to Nowhere''] (2021)

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

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

== External links ==

@@ Line 1: / Line 1: @@
-{{interwiki
+{{Interwiki
+| en = 55edo
 | de = 55-EDO
-| en = 55edo
 | es = 55 EDO
 | ja =
@@ Line 9: / Line 9: @@
 == Theory ==
-edo is a [[The Riemann zeta function and tuning#Valley edos|zeta valley edo]], so it does not approximate the harmonic series very well for its size. Despite this, it can be used as 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.
+edo supports [[meantone]], and is close to [[1/6-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 [[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 is a [[zeta valley edo]], suggesting it does not approximate the harmonic series very well for its size.
 === Odd harmonics ===
@@ Line 15: / Line 16: @@
 === Subsets and supersets ===
-Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets.
+Since 55 factors into primes as {{nowrap| 5 × 11 }}, 55edo contains [[5edo]] and [[11edo]] as its subsets.
 == Intervals ==
 {| class="wikitable center-1 right-2 left-3"
 |-
-! [[Degree|&#35;]]
+! [[Degree|#]]
 ! [[Cent]]s
 ! Approximate ratios
@@ Line 420: / Line 421: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould 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.
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
+{{Sharpness-sharp4-szg}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's 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.
 {{Ups and downs sharpness}}
-[[Alternative symbols for ups and downs notation]] uses sharps, flats, half- and sesquisharps, and half- and sesquiflats with arrows, borrowed from extended [[Helmholtz–Ellis notation]] and [[24edo#Stein-Zimmerman Accidentals|Stein-Zimmerman accidental set]]:
-{{Sharpness-sharp4}}
 === Sagittal notation ===
@@ Line 472: / Line 474: @@
 {{Q-odd-limit intervals|55}}
 {{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}
+{{Q-odd-limit intervals|54.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55f val mapping}}
 == Regular temperament properties ==
@@ Line 530: / Line 533: @@
 | [[Twothirdtonic]] (55f)
 |-
-|1
+| 1
-|8\55
+| 8\55
-|174.5
+| 174.5
-|[[10/9]]~[[11/10]]
+| [[10/9]]~[[11/10]]
-|[[Tetracot]] (55c)
+| [[Tetracot]] (55c)
 |-
 | 1
@@ Line 570: / Line 573: @@
 | 501.8<br>(65.5)
 | 4/3<br>(36/35)
-| [[Hendecatonic]] (55)
+| [[Hendecatonic (temperament)|Hendecatonic]] (55)
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==
-; Subsets of twothirdtonic[37]
+; Subsets of Twothirdtonic[37]
 * Undecimal otonal-like pentatonic: 17 8 7 12 11
-; Subsets of hendecatonic[33]
+; Subsets of Hendecatonic[33]
 * Septimal pentatonic-like: 10 13 9 13 10
 * Septimal minor blues-like: 13 10 4 5 13 10
@@ Line 605: / Line 608: @@
 * [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|John Bull (composer)|John Bull}}
+* [https://www.youtube.com/watch?v=I2d-540RuoM ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)
+; {{W|Louis Couperin}}
+* [https://www.youtube.com/shorts/bcgY3k-6rao ''La Piémontoise''] (1658?) – rendered by Claudi Meneghin (2026)
 ; {{W|Georg Frideric Handel}}
@@ Line 611: / Line 620: @@
 ; {{W|Scott Joplin}}
 * ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] ([https://www.youtube.com/watch?v=GbhpuoIJgxk 2024 version]; [https://www.youtube.com/shorts/3Y9y9I6q1as 2026 version])
+; {{W|Mladen Milićević}}
+* [https://www.youtube.com/shorts/1BhIK0s2-vs "The Room" from ''The Room''] (2003) – covered by [[Bryan Deister]] (2026)
 ; {{W|Wolfgang Amadeus Mozart}}
-* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)
+* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, K. 331] (1778) – rendered by Francium (2023)
-* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)
+* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', K. 401] (1782) – rendered by Francium (2023)
-* [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])
+* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', K. 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=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, K. 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)
+* ''Kleine Gigue in G'', K. 574 (1789) – rendered by [[Claudi Meneghin]]
+** [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 Fortepiano rendition] (2025)
+** [https://www.youtube.com/shorts/7jt-jzziDBg Harpsichord rendition] (2026)
+** [https://www.youtube.com/shorts/9J1DkS5UGzE Organ rendition] (2026)
 ; {{W|Keiichi Okabe}}
@@ Line 629: / Line 644: @@
 ; [[James Kukula]]
-* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)
+* [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=9c5MtrZFNhA ''Improvisation One in 55edo''] (2025)
-* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)
+* [https://www.youtube.com/watch?v=ggFGUn1Ya2A ''Improvisation Two in 55edo''] (2025)
 ; [[Claudi Meneghin]]
@@ Line 643: / Line 658: @@
 ; [[Herman Miller]]
-* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)
+* [https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere ''Road Trip to Nowhere''] (2021)
-* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)
+* [https://soundcloud.com/morphosyntax-1/migration ''Migration''] (2025)
 == External links ==