55edo: Difference between revisions

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== Theory ==

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.

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; part of the reasoning is that 1\55 is extremely close to the size of the syntonic comma [[81/80]].<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> Indeed, by direct approximation, only [[56edo]] approximates the syntonic comma more closely until reaching much higher edo sizes; the [[patent val]] of 55edo tempers it out, but the 55c val (which supports [[tetracot]]) maps it to 1\55. 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.

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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–Zimmermann 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}}

{{Q-odd-limit intervals|54.91|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55cf val mapping}}

== Regular temperament properties ==

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| 8\55

| 174.5

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

| [[10/9]]~[[11/10]]<br>[[31/28]]

| [[Tetracot]] (55c)

| [[Tetracot]] (55c)<br>Tricesoprimal tetracot analog (55)

|-

| 1

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| 501.8

| 4/3

| [[Meantone]] (55d)

| [[Meantone]] (55d) / [[Meantone_family#Mildtone|Mildtone]] (55)

|-

| 1

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

; {W|Louis Couperin}}

; {{W|John Bull (composer)|John Bull}}

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

* [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|Gabriel Fauré}}

* [https://www.youtube.com/watch?v=XMgSJRXX6PY ''Pavane'', op. 50] (1887) — arranged for harpsichord and 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|Toru Kitajima}}

* [https://www.youtube.com/shorts/b9WZ6YIlJOk ''Unravel''] (2014) — microtonal cover in 55edo by [[Bryan Deister]] (2025)

; {{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)

* ''~~Mozart's~~ Gigue ~~KV 574~~'' – rendered by [[Claudi Meneghin]]

* ''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 9: / Line 9: @@
 == Theory ==
-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.
+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; part of the reasoning is that 1\55 is extremely close to the size of the syntonic comma [[81/80]].<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> Indeed, by direct approximation, only [[56edo]] approximates the syntonic comma more closely until reaching much higher edo sizes; the [[patent val]] of 55edo tempers it out, but the 55c val (which supports [[tetracot]]) maps it to 1\55. 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.
@@ Line 421: / 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–Zimmermann accidentals|Stein-Zimmerman accidental set]]:
-{{Sharpness-sharp4}}
 === Sagittal notation ===
@@ Line 473: / 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}}
+{{Q-odd-limit intervals|54.91|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55cf val mapping}}
 == Regular temperament properties ==
@@ Line 534: / Line 537: @@
 | 8\55
 | 174.5
-| [[10/9]]~[[11/10]]
+| [[10/9]]~[[11/10]]<br>[[31/28]]
-| [[Tetracot]] (55c)
+| [[Tetracot]] (55c)<br>Tricesoprimal tetracot analog (55)
 |-
 | 1
@@ Line 547: / Line 550: @@
 | 501.8
 | 4/3
-| [[Meantone]] (55d)
+| [[Meantone]] (55d) / [[Meantone_family#Mildtone|Mildtone]] (55)
 |-
 | 1
@@ Line 607: / Line 610: @@
 * [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
-; {W|Louis Couperin}}
+; {{W|John Bull (composer)|John Bull}}
-* [https://www.youtube.com/shorts/bcgY3k-6rao ''«La Piémontoise»''] (1658?) – rendered by Claudi Meneghin (2026)
+* [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|Gabriel Fauré}}
+* [https://www.youtube.com/watch?v=XMgSJRXX6PY ''Pavane'', op. 50] (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026)
 ; {{W|Georg Frideric Handel}}
@@ Line 615: / Line 624: @@
 ; {{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|Toru Kitajima}}
+* [https://www.youtube.com/shorts/b9WZ6YIlJOk ''Unravel''] (2014) — microtonal cover in 55edo by [[Bryan Deister]] (2025)
+; {{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)
-* ''Mozart's Gigue KV 574'' – rendered by [[Claudi Meneghin]]
+* ''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}}

55edo: Difference between revisions

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