51edo: Difference between revisions
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Using the [[patent val]], 51et [[tempering out|tempers out]] [[250/243]] in the [[5-limit]], [[225/224]] and [[2401/2400]] in the [[7-limit]], and [[55/54]] and [[100/99]] in the [[11-limit]]. It is the [[optimal patent val]] for [[sonic]], the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to [[22edo]] for [[porcupine]], with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit [[porky]] temperament, which is sonic plus 225/224. It contains an archeotonic ([[6L 1s]]) scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top. | Using the [[patent val]], 51et [[tempering out|tempers out]] [[250/243]] in the [[5-limit]], [[225/224]] and [[2401/2400]] in the [[7-limit]], and [[55/54]] and [[100/99]] in the [[11-limit]]. It is the [[optimal patent val]] for [[sonic]], the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to [[22edo]] for [[porcupine]], with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit [[porky]] temperament, which is sonic plus 225/224. It contains an archeotonic ([[6L 1s]]) scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top. | ||
Using the 51c val {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented (temperament)|augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. Alternatively, the 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]]. The 51ce val {{val| 51 81 '''119''' 143 '''177''' 189 }} supports a variant of rodan called [[aerodino]]. | Using the 51c val {{val| 51 81 '''119''' 143 }}, the [[5/4]] is mapped to 1\3 (400 cents), [[support]]ing [[augmented (temperament)|augmented]]. In the 7-limit it tempers out [[245/243]] and supports [[hemiaug]] and [[rodan]]. Alternatively, the 51cd val {{val| 51 81 '''119''' '''144''' }} takes the same [[7/4]] from 17edo, and supports [[augene]]. The 51ce val {{val| 51 81 '''119''' 143 '''177''' 189 }} supports a variant of rodan called [[Gamelismic_clan#Aerodino|aerodino]]. | ||
51edo's step is the closest direct approximation to the [[Pythagorean comma]] by | 51edo's step is the closest direct approximation to the [[Pythagorean comma]] by edosteps, though that comma itself is mapped to a different interval. | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|51}} | {{Harmonics in equal|51|intervals=odd|prec=2|columns=14}} | ||
{{Harmonics in equal|51|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 51edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
| Line 27: | Line 28: | ||
! rowspan="2" colspan="3" | [[Ups and downs notation]] | ! rowspan="2" colspan="3" | [[Ups and downs notation]] | ||
|- | |- | ||
! 2.3.7.13<br>subgroup | ! 2.3.7.11/5.13<br>subgroup | ||
! Ratios of 5 and 11<br>tending flat (51 val) | ! Ratios of 5 and 11<br>tending flat (51 val) | ||
! Ratios of 5 and 11<br>tending sharp (51ce val) | ! Ratios of 5 and 11<br>tending sharp (51ce val) | ||
| Line 96: | Line 97: | ||
| 7 | | 7 | ||
| 164.7 | | 164.7 | ||
| [[11/10]] | |||
| [[11/10]] | | ''10/9'', ''12/11'' | ||
| | | | ||
| Upmid 2nd | | Upmid 2nd | ||
| ^~2 | | ^~2 | ||
| Line 159: | Line 160: | ||
| 14 | | 14 | ||
| 329.4 | | 329.4 | ||
| [[63/52]] | | [[40/33]], [[63/52]] | ||
| ''6/5'', ''11/9'' | | ''6/5'', ''11/9'' | ||
| | | | ||
| Line 240: | Line 241: | ||
| 23 | | 23 | ||
| 541.2 | | 541.2 | ||
| [[15/11]] | |||
| [[11 | | [[11/8]], ''27/20'' | ||
| | | | ||
| Downdim 5th | | Downdim 5th | ||
| vd5 | | vd5 | ||
| Line 285: | Line 286: | ||
| 28 | | 28 | ||
| 658.8 | | 658.8 | ||
| [[22/15]] | |||
| [[ | | [[16/11]], ''40/27'' | ||
| | | | ||
| Upaug 4th | | Upaug 4th | ||
| ^A4 | | ^A4 | ||
| Line 366: | Line 367: | ||
| 37 | | 37 | ||
| 870.6 | | 870.6 | ||
| [[104/63]] | | [[33/20]], [[104/63]] | ||
| ''5/3'', ''18/11'' | | ''5/3'', ''18/11'' | ||
| | | | ||
| Line 429: | Line 430: | ||
| 44 | | 44 | ||
| 1035.3 | | 1035.3 | ||
| [[20/11]] | | [[20/11]] | ||
| | | | ||
| ''9/5'', ''11/6'' | |||
| Downmid 7th | | Downmid 7th | ||
| v~7 | | v~7 | ||
| Line 502: | Line 503: | ||
== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
51edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. | [[Stein–Zimmermann–Gould notation]] for 51edo uses sharps and flats combined with quartertone accidentals and arrows: | ||
{{Sharpness-sharp6-szg}} | |||
If double arrows are not desirable, then arrows can be attached to quartertone accidentals: | |||
{{Sharpness-sharp6-qt-szg}} | |||
=== Kite's ups and downs notation === | |||
51edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. | |||
{{Sharpness-sharp6a}} | {{Sharpness-sharp6a}} | ||
Half-sharps and half-flats can be used to avoid triple arrows: | Half-sharps and half-flats can be used to avoid triple arrows: | ||
{{Sharpness-sharp6b}} | {{Sharpness-sharp6b}} | ||
=== Ivan Wyschnegradsky's notation === | === Ivan Wyschnegradsky's notation === | ||
Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used: | Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used: | ||
{{Sharpness-sharp6-iw}} | {{Sharpness-sharp6-iw}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
In the following diagrams, 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. | In the following diagrams, 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. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
| Line 634: | Line 633: | ||
| 117.6 | | 117.6 | ||
| 15/14 | | 15/14 | ||
| [[Miracle]] (51e, out of tune | | [[Miracle]] (51e, out of tune) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 646: | Line 645: | ||
| 235.3 | | 235.3 | ||
| 8/7 | | 8/7 | ||
| [[Rodan]] ( | | [[Rodan]] (51cf, out of tune) / aerodino (51ce) | ||
|- | |||
| 1 | |||
| 19\51 | |||
| 447.1 | |||
| 13/10 | |||
| [[Supersensi]] (51cde) | |||
|- | |||
| 1 | |||
| 22\51 | |||
| 517.6 | |||
| 27/20 | |||
| [[Gravity]] (51ce) / [[abergravity]] (51ce) | |||
|- | |- | ||
| 1 | | 1 | ||
| | | 23\51 | ||
| 541.2 | | 541.2 | ||
| 15/11 | | 15/11 | ||
| [[Necromanteion]] (51ce) | | [[Necromanteion]] (51ce)<br>[[Oracle]] (51)<br>[[Cypress]] (51cde…) | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 664: | Line 675: | ||
| rowspan="2" | 494.1<br>(94.1) | | rowspan="2" | 494.1<br>(94.1) | ||
| 4/3<br>(16/15) | | 4/3<br>(16/15) | ||
| [[Augmented (temperament)|Augmented]] ( | | [[Augmented (temperament)|Augmented]] (7-limit, 51cd) | ||
|- | |- | ||
| style="text-align: center;" | 4/3<br>(21/20) | | style="text-align: center;" | 4/3<br>(21/20) | ||
| style="text-align: left;" | [[Fog]] (51) | | style="text-align: left;" | [[Fog]] (51) | ||
|} | |} | ||
<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|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 == | ||
| Line 682: | Line 693: | ||
== Music == | == Music == | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/watch?v=sySLQUXnQ70 ''Preludio Sentimentale (microtonal improvisation in 28edo)''] (2023) | |||
* [https://www.youtube.com/watch?v=sCE0MjUyRUk ''28edo blues''] (2023) | |||
* [https://www.youtube.com/shorts/sTPJtuHUwkg ''51edo improv''] (2025-02-03) | * [https://www.youtube.com/shorts/sTPJtuHUwkg ''51edo improv''] (2025-02-03) | ||
* [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025-05-02) | * [https://www.youtube.com/shorts/5pM8OC0fV98 ''51edo improv''] (2025-05-02) | ||
* [https://www.youtube.com/shorts/Fymg9vYO6iQ ''Northernlight - Deltarune (microtonal cover in 51edo)''] (2025) | * [https://www.youtube.com/shorts/Fymg9vYO6iQ ''Northernlight - Deltarune (microtonal cover in 51edo)''] (2025) | ||
* [https://www.youtube.com/shorts/SJW-JTHyeIA ''51edo prelude''] (2026) | |||
* [https://www.youtube.com/watch?v=k3NOBYbiqpo ''51edo improv''] (2026-04-22) | |||
; [[Frédéric Gagné]] | ; [[Frédéric Gagné]] | ||