55edo: Difference between revisions
→Music: ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)'' – rendered by Claudi Meneghin: Add harpsichord version (2026) |
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{{ | {{Interwiki | ||
| en = 55edo | |||
| de = 55-EDO | | de = 55-EDO | ||
| es = 55 EDO | | es = 55 EDO | ||
| ja = | | ja = | ||
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== Theory == | == Theory == | ||
55edo is a [[ | 55edo is a [[zeta valley edo]], so it does not approximate the harmonic series very well for its size. Despite this, it can be used as 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 [[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 === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 55 factors into {{ | Since 55 factors into primes as {{nowrap| 5 × 11 }}, 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 ratios | ! Approximate ratios | ||
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{{Ups and downs sharpness}} | {{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# | [[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}} | {{Sharpness-sharp4}} | ||
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| [[Twothirdtonic]] (55f) | | [[Twothirdtonic]] (55f) | ||
|- | |- | ||
|1 | | 1 | ||
|8\55 | | 8\55 | ||
|174.5 | | 174.5 | ||
|[[10/9]]~[[11/10]] | | [[10/9]]~[[11/10]] | ||
|[[Tetracot]] (55c) | | [[Tetracot]] (55c) | ||
|- | |- | ||
| 1 | | 1 | ||
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| 501.8<br>(65.5) | | 501.8<br>(65.5) | ||
| 4/3<br>(36/35) | | 4/3<br>(36/35) | ||
| [[Hendecatonic]] (55) | | [[Hendecatonic (temperament)|Hendecatonic]] (55) | ||
|} | |} | ||
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
; Subsets of | ; Subsets of Twothirdtonic[37] | ||
* Undecimal otonal-like pentatonic: 17 8 7 12 11 | * Undecimal otonal-like pentatonic: 17 8 7 12 11 | ||
; Subsets of | ; Subsets of Hendecatonic[33] | ||
* Septimal pentatonic-like: 10 13 9 13 10 | * Septimal pentatonic-like: 10 13 9 13 10 | ||
* Septimal minor blues-like: 13 10 4 5 13 10 | * Septimal minor blues-like: 13 10 4 5 13 10 | ||
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; [[James Kukula]] | ; [[James Kukula]] | ||
* | * [https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg ''55edo Melted Syntonic''] (2025) | ||
; [[Budjarn Lambeth]] | ; [[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]] | ||
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; [[Herman Miller]] | ; [[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 == | ||