55edo: Difference between revisions
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{{interwiki | {{interwiki | ||
| de = | | de = 55-EDO | ||
| en = 55edo | | en = 55edo | ||
| es = 55 EDO | | es = 55 EDO | ||
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}} | }} | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== 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. | 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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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets. | Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets. | ||
== Intervals == | == Intervals == | ||
{| class="wikitable center-1 right-2 left-3" | {| class="wikitable center-1 right-2 left-3" | ||
|- | |- | ||
! [[Degree|#]] | ! [[Degree|#]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
! Approximate | ! Approximate ratios | ||
! colspan="3" | [[Ups and downs notation | ! colspan="3" | [[Ups and downs notation]] | ||
|- | |- | ||
| 0 | | 0 | ||
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| 26 | | 26 | ||
| 567.3 | | 567.3 | ||
| 18/13 | | [[7/5]], [[18/13]] | ||
| vA4 | | vA4 | ||
| downaug 4th | | downaug 4th | ||
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| 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 | ||
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| D | | D | ||
|} | |} | ||
<nowiki>* | <nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic'' | ||
== Notation == | |||
=== Ups and downs 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. | |||
{{Sharpness-sharp4a}} | |||
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | |||
{{Sharpness-sharp4}} | |||
=== Sagittal notation === | |||
==== Evo flavor ==== | |||
<imagemap> | |||
File:55-EDO_Evo_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 160 106 [[896/891]] | |||
rect 160 80 280 106 [[33/32]] | |||
default [[File:55-EDO_Evo_Sagittal.svg]] | |||
</imagemap> | |||
==== Revo flavor ==== | |||
<imagemap> | |||
File:55-EDO_Revo_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 160 106 [[896/891]] | |||
rect 160 80 280 106 [[33/32]] | |||
default [[File:55-EDO_Revo_Sagittal.svg]] | |||
</imagemap> | |||
==== Evo-SZ flavor ==== | |||
<imagemap> | |||
File:55-EDO_Evo-SZ_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 160 106 [[896/891]] | |||
rect 160 80 280 106 [[33/32]] | |||
default [[File:55-EDO_Evo-SZ_Sagittal.svg]] | |||
</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 == | ||
[[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]] | [[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]] | ||
=== Selected just intervals by error === | === Selected just intervals by error === | ||
{{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}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{mapping| 55 87 }} | | {{mapping| 55 87 }} | ||
| +1.31 | | +1.31 | ||
| 1. | | 1.19 | ||
| 7.21 | | 7.21 | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| 81/80, | | 81/80, {{monzo| 31 1 -14 }} | ||
| {{mapping| 55 87 128 }} | | {{mapping| 55 87 128 }} | ||
| −0.13 | | −0.13 | ||
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| 9.63 | | 9.63 | ||
|} | |} | ||
=== Uniform maps === | |||
{{Uniform map|edo=55}} | |||
=== Commas === | === Commas === | ||
{{Todo|cleanup|inline=true}} | |||
'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }} | '''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }} | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| 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* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! | ! Temperament | ||
|- | |- | ||
| 1 | | 1 | ||
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| 14/13 | | 14/13 | ||
| [[Twothirdtonic]] (55f) | | [[Twothirdtonic]] (55f) | ||
|- | |||
|1 | |||
|8\55 | |||
|174.5 | |||
|[[10/9]]~[[11/10]] | |||
|[[Tetracot]] (55c) | |||
|- | |- | ||
| 1 | | 1 | ||
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| [[Hendecatonic]] (55) | | [[Hendecatonic]] (55) | ||
|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | |||
; Subsets of twothirdtonic[37] | |||
* Undecimal otonal-like pentatonic: 17 8 7 12 11 | |||
; Subsets of hendecatonic[33] | |||
* Septimal pentatonic-like: 10 13 9 13 10 | |||
* Septimal minor blues-like: 13 10 4 5 13 10 | |||
* Septimal heptatonic blues-like: 13 10 4 5 8 5 10 | |||
; Others | |||
* Sakura-like scale containing [[phi]]: 9 6 18 5 17 | |||
* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7 | |||
== Instruments == | == Instruments == | ||
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; {{W|Johann Sebastian Bach}} | ; {{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) – arranged for two organs, rendered by Claudi Meneghin (2021) | ||
* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024) | |||
* [https://www.youtube.com/watch?v=OkRVNo19guo "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025) | |||
* [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}} | ; {{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"''] – 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"''] – rendered by Claudi Meneghin (2024) | ||
; {{W|Georg Frideric Handel}} | |||
* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025) | |||
; {{W|Scott Joplin}} | ; {{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) – arranged for harpsichord and rendered by [[Claudi Meneghin]] (2024) | ||
; {{W|Wolfgang Amadeus Mozart}} | ; {{W|Wolfgang Amadeus Mozart}} | ||
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* [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) – 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) – 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) | |||
; {{W|Keiichi Okabe}} | ; {{W|Keiichi Okabe}} | ||
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=== 21st century === | === 21st century === | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025) | |||
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025) | |||
; [[James Kukula]] | |||
* ''[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=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025) | |||
; [[Claudi Meneghin]] | ; [[Claudi Meneghin]] | ||
* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020) | * [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020) | ||
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* [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023) | * [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023) | ||
* [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021) | * [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021) | ||
* [https://www.youtube.com/watch?v=jOiub14Cskw ''Double Fugue on "Old McDonald" + "Shave & a Haircut"''] (2024) | |||
; [[Herman Miller]] | |||
* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021) | |||
* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025) | |||
== External links == | == External links == | ||
* [http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo | * ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]] | ||
and its close relative, 1/6-comma meantone] (containing another listening example) on [[Tonalsoft Encyclopedia]] | |||
== References == | |||
<references /> | |||
[[Category:Meantone]] | [[Category:Meantone]] | ||
[[Category:Historical]] | [[Category:Historical]] | ||
[[Category:Listen]] | [[Category:Listen]] |