58edo: Difference between revisions

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While the [[17/1|17th harmonic]] is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.

The [[19/1|19th]] and [[23/1|23rd]] harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be [[diamond monotone]] in the 23-odd-limit, past the idiosyncratic 53e val. However, its accuracy is questionable, with primes 19 and 23 being about 13 cents sharp, so one may want to use a larger system like [[62edo]] for the 23-limit instead, which has the added benefit of being meantone.

=== Prime harmonics ===

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== Notation ==

=== ~~Ups and downs~~ notation ===

=== Stein–Zimmermann–Gould notation ===

58edo ~~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 58edo uses sharps and flats combined with quartertone accidentals and arrows:

~~Half-sharps and half-flats~~ can be ~~used~~ to ~~avoid triple arrows~~:

If double arrows are not desirable, then arrows can be attached to quartertone accidentals:

~~Alternatively, a combination of quarter tone accidentals~~ and ~~arrow accidentals from~~ [[~~Helmholtz–Ellis~~ notation]] ~~can be used~~.

=== Kite's ups and downs notation ===

58edo 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.

~~If double arrows are not desirable, then arrows~~ can be ~~attached~~ to ~~quarter-tone accidentals~~:

Half-sharps and half-flats can be used to avoid triple arrows:

=== Ivan Wyschnegradsky's notation ===

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:

=== Sagittal notation ===

==== Evo flavor ====

~~<imagemap>~~

~~File:58-EDO_Evo_Sagittal.svg~~

~~desc none~~

==== Evo-SZ flavor ====

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 662 80 [https://sagittal.org#periodic~~-~~table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 120 106 [[81/80]]~~

~~rect 120 80 230 106 [[55/54]]~~

~~rect 230 80 350 106 [[33/32]]~~

~~default [[File:58~~-~~EDO_Evo_Sagittal.svg]]~~

~~</imagemap>~~

==== Revo flavor ====

~~<imagemap>~~

~~File:58-EDO_Revo_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 120 106 [[81/80]]~~

~~rect 120 80 230 106 [[55/54]]~~

~~rect 230 80 350 106 [[33/32]]~~

~~default [[File:58-EDO_Revo_Sagittal.svg]]~~

~~</imagemap>~~

~~==== Evo-SZ flavor ====~~

~~<imagemap>~~

~~File:58-EDO_Evo-SZ_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 120 106 [[81/80]]~~

~~rect 120 80 230 106 [[55/54]]~~

~~rect 230 80 350 106 [[33/32]]~~

~~default [[File:58-EDO_Evo-SZ_Sagittal.svg]]~~

~~</imagemap>~~

=== Hemipyth notation ===

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|+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)

|-

! &#~~35;~~

! #

! Cents

! Note names<br />on D

! Note names<br>on D

|-

| 0

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| 21/16

| [[Buzzard]] / [[subfourth]]

|-

| 1

| 25\58

| 517.2

| 27/20

| [[Gravity]] / [[abergravity]] / [[gravid]]

|-

| 1

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| [[Mystery]]

|}

<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

58et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).

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; [[Cam Taylor]]

* [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)

; [[Xotla]]

* [https://www.youtube.com/watch?v=yTkPhjTEQMw "Wormhole Shmurmhole"], from [https://www.youtube.com/playlist?list=PL4HmfPDldHXueRjmTV-iJN3fsLn_O9jvT ''Just Another Microtonal Music Album''] (2025–2026) – in part, the rest being in 31edo

[[Category:Buzzard]]

@@ Line 7: / Line 7: @@
 While the [[17/1|17th harmonic]] is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. Since {{nowrap|58 {{=}} 2 × 29}}, 58edo shares the same excellent perfect fifth with [[29edo]]. It is the last edo to have exactly one [[5L 2s|diatonic]] perfect fifth and no [[5edo]] or [[7edo]] fifths.
+The [[19/1|19th]] and [[23/1|23rd]] harmonics are very flat, so it makes sense to use their second-best approximations in the 58hi val. This val is, in fact, one of the first to be [[diamond monotone]] in the 23-odd-limit, past the idiosyncratic 53e val. However, its accuracy is questionable, with primes 19 and 23 being about 13 cents sharp, so one may want to use a larger system like [[62edo]] for the 23-limit instead, which has the added benefit of being meantone.
 === Prime harmonics ===
@@ Line 325: / Line 327: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo 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 58edo uses sharps and flats combined with quartertone accidentals and arrows:
-{{Sharpness-sharp6a}}
+{{Sharpness-sharp6-szg}}
-Half-sharps and half-flats can be used to avoid triple arrows:
+If double arrows are not desirable, then arrows can be attached to quartertone accidentals:
-{{Sharpness-sharp6b}}
+{{Sharpness-sharp6-qt-szg}}
-Alternatively, a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used.
+=== Kite's ups and downs notation ===
-{{Sharpness-sharp6}}
+edo 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.
+{{Ups and downs sharpness}}
-If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
+Half-sharps and half-flats can be used to avoid triple arrows:
-{{Sharpness-sharp6-qt}}
+{{Ups and downs sharpness|58|true}}
 === Ivan Wyschnegradsky's notation ===
 Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from [[72edo]] can also be used:
 {{Sharpness-sharp6-iw}}
 === Sagittal notation ===
 ==== Evo flavor ====
-<imagemap>
+{{Sagittal chart|Evo}}
-File:58-EDO_Evo_Sagittal.svg
-desc none
+==== Evo-SZ flavor ====
-rect 80 0 300 50 [[Sagittal_notation]]
+{{Sagittal chart|Evo-SZ}}
-rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Evo_Sagittal.svg]]
-</imagemap>
 ==== Revo flavor ====
-<imagemap>
+{{Sagittal chart}}
-File:58-EDO_Revo_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 662 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Revo_Sagittal.svg]]
-</imagemap>
-==== Evo-SZ flavor ====
-<imagemap>
-File:58-EDO_Evo-SZ_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 120 106 [[81/80]]
-rect 120 80 230 106 [[55/54]]
-rect 230 80 350 106 [[33/32]]
-default [[File:58-EDO_Evo-SZ_Sagittal.svg]]
-</imagemap>
 === Hemipyth notation ===
@@ Line 384: / Line 359: @@
 |+ style="font-size: 105%; white-space: nowrap;" | Hemipyth notation for 58edo (SW3-style)
 |-
-! &#35;
+! #
 ! Cents
-! Note names<br />on D
+! Note names<br>on D
 |-
 | 0
@@ Line 601: / Line 576: @@
 | 21/16
 | [[Buzzard]] / [[subfourth]]
+|-
+| 1
+| 25\58
+| 517.2
+| 27/20
+| [[Gravity]] / [[abergravity]] / [[gravid]]
 |-
 | 1
@@ Line 656: / Line 637: @@
 | [[Mystery]]
 |}
-<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
 et can also be detempered to [[semihemi]] ({{nowrap| 58 & 140 }}), [[supers]] ({{nowrap| 58 & 152 }}), [[condor]] ({{nowrap| 58 & 159 }}), and [[eagle]] ({{nowrap| 58 & 212 }}).
@@ Line 690: / Line 671: @@
 ; [[Cam Taylor]]
 * [https://www.youtube.com/watch?v=Keclakcqie8 ''58EDO, Mystery temperament and 2 rings of Pythagorean on the Lumatone''] (2021)
+; [[Xotla]]
+* [https://www.youtube.com/watch?v=yTkPhjTEQMw "Wormhole Shmurmhole"], from [https://www.youtube.com/playlist?list=PL4HmfPDldHXueRjmTV-iJN3fsLn_O9jvT ''Just Another Microtonal Music Album''] (2025–2026) – in part, the rest being in 31edo
 [[Category:Buzzard]]