55edo: Difference between revisions

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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]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. 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]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. 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.

=== Odd harmonics ===

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

|-

! [[Degree|#]]

! [[Cent]]s

! Approximate Ratios

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

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

|-

| 0

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| D

|}

<nowiki>*~~</nowiki>~~ 55f val (tending flat), inconsistent intervals labeled in ''italic''

<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''

== Approximation to JI ==

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

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

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

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

! colspan="2" | Tuning Error

|-

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=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated Ratio*

! Associated Ratio*

! Temperaments

|-

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|-

| 5

| 17\55 (5\55)

| 17\55 (5\55)

| 370.9 (109.1)

| 370.9 (109.1)

| 99/80 (16/15)

| 99/80 (16/15)

| [[Quintosec]]

|-

| 11

| 23\55 (3\55)

| 23\55 (3\55)

| 501.8 (65.5)

| 501.8 (65.5)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Hendecatonic]] (55)

|}

<nowiki />* [[Normal lists|Octave-reduced form]]

== Instruments ==

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=== Modern renderings ===

; {{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=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (~~1742–1749~~) – rendered by Claudi Meneghin (2024)

* [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)

* [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=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|Scott Joplin}}

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

; {{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=XgRksdk6zyQ ''Fugue in G minor'', KV 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])

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

; {{W|Keiichi Okabe}}

* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)

=== 21st century ===

@@ Line 9: / Line 9: @@
 == 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]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. 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]], in which he was followed by {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}}. 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.
 === Odd harmonics ===
@@ Line 20: / Line 20: @@
 {| class="wikitable center-1 right-2 left-3"
 |-
-! [[Degree|#]]
+! [[Degree|&#35;]]
 ! [[Cent]]s
 ! Approximate Ratios
-! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 |-
 | 0
@@ Line 417: / Line 417: @@
 | D
 |}
-<nowiki>*</nowiki> 55f val (tending flat), inconsistent intervals labeled in ''italic''
+<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''
 == Approximation to JI ==
@@ Line 430: / Line 430: @@
 ! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve Stretch (¢)
 ! colspan="2" | Tuning Error
 |-
@@ Line 462: / Line 462: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />Ratio*
 ! Temperaments
 |-
@@ Line 500: / Line 501: @@
 |-
 | 5
-| 17\55<br>(5\55)
+| 17\55<br />(5\55)
-| 370.9<br>(109.1)
+| 370.9<br />(109.1)
-| 99/80<br>(16/15)
+| 99/80<br />(16/15)
 | [[Quintosec]]
 |-
 | 11
-| 23\55<br>(3\55)
+| 23\55<br />(3\55)
-| 501.8<br>(65.5)
+| 501.8<br />(65.5)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Hendecatonic]] (55)
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]]
 == Instruments ==
@@ Line 518: / Line 520: @@
 === Modern renderings ===
 ; {{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=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) &ndash; arranged for two organs, rendered by Claudi Meneghin (2021)
-* [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=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; 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)
+* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742&ndash;1749) &ndash; 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"''] &ndash; rendered by Claudi Meneghin (2023)
-* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
+* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] &ndash; rendered by Claudi Meneghin (2024)
 ; {{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) &ndash; arranged for harpsichord and rendered by Claudi Meneghin (2024)
 ; {{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, KV 331] (1778) &ndash; 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'', KV 401] (1782) &ndash; 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'', KV 540] (1788) &ndash; 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, KV 545] (1788) &ndash; rendered by Francium (2023)
 ; {{W|Keiichi Okabe}}
-* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)
+* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) &ndash; rendered by MortisTheneRd (2024)
 === 21st century ===