57edo: Difference between revisions
m →Intervals: Move inconsistent ratios to last in their table cells as recommended by edit partially lost due to edit clash |
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Using the full prime-limit [[patent val]], the equal temperament tempers out [[81/80]], [[1029/1024]], and [[3125/3072]] in the 7-limit; and [[99/98]], [[385/384]], [[441/440]], and [[625/616]] in the [[11-limit]]. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 & 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]]. | Using the full prime-limit [[patent val]], the equal temperament tempers out [[81/80]], [[1029/1024]], and [[3125/3072]] in the 7-limit; and [[99/98]], [[385/384]], [[441/440]], and [[625/616]] in the [[11-limit]]. A good generator to exploit the 2.5/3.7.11.13.17.19 aspect of 57 is the approximate [[11/8]], which is 26\57. This gives the [[19-limit]] 46 & 57 temperament [[heinz]]. It can also be used to tune [[mothra]] as well as [[trismegistus]]. | ||
57edo is interesting in that it is a [[dual-fifth]] system where both fifths are somewhat plausible. The 57b val uses the alternative fifth of 715.789 cents. While not very accurate in the 5 and 7-limits (being even worse than 19edo), it does support some interesting temperaments like [[sixix]] and a near-perfect 1/2-comma [[archy]] (such that the diatonic minor seventh is almost exactly [[7/4]]). | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|57}} | {{Harmonics in equal|57}} | ||
{{Harmonics in equal|57|intervals=odd|columns=11|prec=2|start=12|collapsed=true|title=Approximation of odd harmonics in 57edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 174: | Line 177: | ||
| 25 | | 25 | ||
| 526.32 | | 526.32 | ||
|[[19/14]], [[23/17]] | |[[19/14]], [[42/31]], [[23/17]] | ||
| {{UDnote|step=25}} | | {{UDnote|step=25}} | ||
| {{UDnote|fifth=34|step=25}} | | {{UDnote|fifth=34|step=25}} | ||
| Line 234: | Line 237: | ||
| 35 | | 35 | ||
| 736.84 | | 736.84 | ||
|[[26/17]], [[32/21]] [[29/19]] | |[[26/17]], [[32/21]], [[29/19]] | ||
| {{UDnote|step=35}} | | {{UDnote|step=35}} | ||
| {{UDnote|fifth=34|step=35}} | | {{UDnote|fifth=34|step=35}} | ||
| Line 370: | Line 373: | ||
| {{UDnote|fifth=34|step=57}} | | {{UDnote|fifth=34|step=57}} | ||
|} | |} | ||
<nowiki>*</nowiki> As a 2.5 | <nowiki>*</nowiki> As a 2.3.5.7.11.13.17.19.23.29.31-subgroup temperament, in ''italics'' if inconsistent | ||
== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
57edo can be notated using [[Stein–Zimmermann–Gould notation]]: | |||
{{Sharpness-sharp3-szg}} | |||
Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows. | |||
=== Kite's ups and downs notation === | |||
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud. | ||
{{Ups and downs sharpness}} | {{Ups and downs sharpness}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | This notation uses the same sagittal sequence as edos [[50edo #Sagittal notation|50]], [[64edo #Sagittal notation|64]], and [[71edo #Sagittal notation|71b]], and is a superset of the notation for [[19edo #Sagittal notation|19edo]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
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</imagemap> | </imagemap> | ||
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this | In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo. | ||
== Scales == | == Scales == | ||
| Line 420: | Line 425: | ||
== Music == | == Music == | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025) | * [https://www.youtube.com/watch?v=4NZvSgS3ryY ''57edo improv''] (2025-05-12) | ||
* ''Prelude in 57edo'' (2025) ( | * ''Prelude in 57edo'' (2025) (Uses different temperaments in different sections) | ||
** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)''] | ** [https://www.youtube.com/shorts/FylRB5MGBz4 ''Part 1 (short clip)''] (Lumatone view using [[5L 2s|diatonic]] sharp fifth + [[mothra]] flat fifth mapping) | ||
** [https://www.youtube.com/shorts/yVv0s8t--pg ''Part 2 (short clip)''] | ** [https://www.youtube.com/shorts/yVv0s8t--pg ''Part 2 (short clip)''] (Lumatone view using [[lithium]] mapping) | ||
** [https://www.youtube.com/watch?v=RH2QkHqVb2c ''Whole composition''] (but no view of Lumatone) | ** [https://www.youtube.com/watch?v=RH2QkHqVb2c ''Whole composition''] (but no view of Lumatone) | ||
* [https://www.youtube.com/shorts/MTMRlRxRbjQ ''57edo groove''] (2025) | * [https://www.youtube.com/shorts/MTMRlRxRbjQ ''57edo groove''] (2025) | ||
* [https://www.youtube.com/shorts/TrgdgDduDIk ''57edo improv''] (2026-09-27) | |||