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.

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 550:

| 501.8

| 4/3

| [[Meantone]] (55d)

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

|-

| 1

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; {{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 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 537: / 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 550: / Line 550: @@
 | 501.8
 | 4/3
-| [[Meantone]] (55d)
+| [[Meantone]] (55d) / [[Meantone_family#Mildtone|Mildtone]] (55)
 |-
 | 1
@@ Line 615: / Line 615: @@
 ; {{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}}