58edo: Difference between revisions

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== Theory ==

58edo is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].

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. 58 = 2 × 29, and 58edo shares the same excellent fifth with [[29edo]].

As an equal temperament, 58et tempers out [[2048/2025]], [[126/125]], [[1728/1715]], [[144/143]], [[176/175]], [[896/891]], [[243/242]], [[5120/5103]], [[351/350]], [[364/363]], [[441/440]], and [[540/539]]. It [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s plus a number of [[gravity family]] extensions, and supplies the [[optimal patent val]] for the 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].

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. 58 = 2 × 29, and 58edo shares the same excellent fifth with [[29edo]].

Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.

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== Notation ==

=== Ups and downs notation ===

In 58edo, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.

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

=== Sagittal ===

The following table shows [[sagittal notation]] accidentals in one apotome for 58edo.

{| class="wikitable center-all"

! ~~Steps~~

! Step Offset

| 0

| 1

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| [[File:Sagittal sharp.png]]

|}

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

58edo can also be notated using [[ups and downs notation]]. In this case, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.

~~{{Sharpness-sharp6}}~~

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

~~{{Sharpness-sharp6-qt}}~~

== Approximation to JI ==

@@ Line 5: / Line 5: @@
 == Theory ==
 edo is a strong system in the [[11-limit|11]]-, [[13-limit|13]]- and [[17-limit]]. It is the smallest [[edo]] which is [[consistent]] through the [[17-odd-limit]], and is also the smallest distinctly consistent in the [[11-odd-limit]] (the first equal temperament to map the entire 11-odd-limit [[tonality diamond]] to distinct scale steps), and hence the first which can define a tempered version of the famous 43-note [[Harry Partch related scales|Genesis scale]] of [[Harry Partch]].
+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. 58 = 2 × 29, and 58edo shares the same excellent fifth with [[29edo]].
 As an equal temperament, 58et tempers out [[2048/2025]], [[126/125]], [[1728/1715]], [[144/143]], [[176/175]], [[896/891]], [[243/242]], [[5120/5103]], [[351/350]], [[364/363]], [[441/440]], and [[540/539]]. It [[support]]s [[hemififths]], [[myna]], [[diaschismic]], [[harry]], [[mystery]], [[buzzard]], [[thuja]] [[regular temperament|temperament]]s plus a number of [[gravity family]] extensions, and supplies the [[optimal patent val]] for the 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank-3 temperaments [[thrush]], [[bluebird]], [[aplonis]] and [[jofur]].
-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. 58 = 2 × 29, and 58edo shares the same excellent fifth with [[29edo]].
 Of all edos which map the syntonic comma ([[81/80]]) to 1 step by patent val, 58edo is the one with the step size closest to 81/80, with one step of 58edo being less than 1{{cent}} narrower than the just interval.
@@ Line 351: / Line 351: @@
 == Notation ==
+=== Ups and downs notation ===
+In 58edo, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.
+{{Sharpness-sharp6}}
+If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
+{{Sharpness-sharp6-qt}}
 === Sagittal ===
 The following table shows [[sagittal notation]] accidentals in one apotome for 58edo.
 {| class="wikitable center-all"
-! Steps
+! Step Offset
 | 0
 | 1
@@ Line 373: / Line 382: @@
 | [[File:Sagittal sharp.png]]
 |}
-=== Ups and downs notation ===
-edo can also be notated using [[ups and downs notation]]. In this case, a sharp raises by six steps, so a combination of quarter tone accidentals and arrow accidentals from [[Helmholtz–Ellis notation]] can be used to fill in the gaps.
-{{Sharpness-sharp6}}
-If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:
-{{Sharpness-sharp6-qt}}
 == Approximation to JI ==