55edo: Difference between revisions
→Rank-2 temperaments: Add Tetracot (55c) to go with Lumatone mapping for 55edo as Tetracot (6L 1s) |
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== Theory == | == 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 [[ | 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 === | === Odd harmonics === | ||
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| 26 | | 26 | ||
| 567.3 | | 567.3 | ||
| 18/13 | | [[7/5]], [[18/13]] | ||
| vA4 | | vA4 | ||
| downaug 4th | | downaug 4th | ||
| Line 216: | Line 216: | ||
| 27 | | 27 | ||
| 589.1 | | 589.1 | ||
| | | 24/17 | ||
| A4, vd5 | | A4, vd5 | ||
| aug 4th, downdim 5th | | aug 4th, downdim 5th | ||
| Line 223: | Line 223: | ||
| 28 | | 28 | ||
| 610.9 | | 610.9 | ||
| | | 17/12 | ||
| ^A4, d5 | | ^A4, d5 | ||
| upaug 4th, dim 5th | | upaug 4th, dim 5th | ||
| Line 230: | Line 230: | ||
| 29 | | 29 | ||
| 632.7 | | 632.7 | ||
| 13/9 | | [[10/7]], [[13/9]] | ||
| ^d5 | | ^d5 | ||
| updim 5th | | updim 5th | ||
| Line 458: | Line 458: | ||
default [[File:55-EDO_Evo-SZ_Sagittal.svg]] | default [[File:55-EDO_Evo-SZ_Sagittal.svg]] | ||
</imagemap> | </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 == | == Approximation to JI == | ||
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{{Q-odd-limit intervals|55}} | {{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|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025) | * [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025) | ||
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025) | |||
; [[James Kukula]] | ; [[James Kukula]] | ||