25edo: Difference between revisions

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== Theory ==

25edo is a good way to tune the [[blackwood]] temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25edo is a good way to tune the [[blackwood]] temperament, which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[trismegistus]] temperament (or [[mavila]] if it is interpreted as [[3/2]]). In fact, it is a convergent to a melodically optimal "golden" tuning of trismegistus or mavila, at around 672.7 cents.

25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] ~~temperament~~.

If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

~~Since 25 is 5 x 5~~, ~~25edo is~~ the ~~smallest composite EDO that doesn't have any intervals in common with~~ [[~~12edo~~]].

Its step of 48{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having a very high [[harmonic entropy]]. This is because the harmonic entropy model is usually tuned to reflect the general perception of quarter-tones being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

=== Possible usage in Indonesian music ===

Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings.

Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings. However, many tunings of pelog are also better represented by the tuning's native 3-2-6-3-2-3-6 [[omnidiatonic]] scale.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 25 is 5 x 5, 25edo is the smallest composite EDO that doesn't have any intervals in common with [[12edo]]. Doubling 25edo to get [[50edo]] produces a good [[meantone]] tuning.

== Intervals ==

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|}

*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.

25-edo chords can be named with ups and downs, see [[Ups and ~~Downs Notation~~#Chords and Chord Progressions|Ups and ~~Downs Notation~~ - Chords and Chord Progressions]].

25-edo chords can be named with ups and downs, see [[Ups and downs notation#Chords and Chord Progressions|Ups and downs notation - Chords and Chord Progressions]].

[[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]]

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== Notation ==

=== Ups and downs notation ===

25edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

===Sagittal notation===

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

This notation uses the same sagittal sequence as [[32edo#Sagittal notation|~~32-EDO~~]], and is a superset of the notation for [[5edo#Sagittal notation|~~5-EDO~~]].

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[32edo#Sagittal notation|32edo]], and is a superset of the notation for [[5edo #Sagittal notation|5edo]].

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rect 20 80 415 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]

default [[File:25-EDO_Sagittal.svg]]

</imagemap>~~[[MisterShafXen’s 25edo~~ notation]]

</imagemap>

=== Second-best fifth (mavila) notation ===

== Regular temperament properties ==

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A subset of the Passion[13] scale. Approximated from the original Akebono I scale of [[12edo]].

; Unfair [[blackwood]][10]

; 4 1 4 1 4 1 4 1 4 1

Named "unfair" (by Igliashon Jones) because of the predominance of the larger interval. The major triads come with the large supermajor third.

; Fair [[blackwood]][10]

; 3 2 3 2 3 2 3 2 3 2

Named "fair" (by Igliashon Jones) because larger and smaller interval are more balanced. The major triads come with the nice 5/4 major third.

== Relationship to Armodue ==

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; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=nhpvxkUDBFk "Prelude in C major, No. 1" from ''The Well-Tempered Clavier I'', BWV 846] (1722) – played by [[Stephen Weigel]] on a [[lumatone]] (2024) – [[mavila]] in 25edo tuning

; {{W|Marco Uccellini}}

* [https://www.youtube.com/watch?v=2yXu0nFoIbI ''Aria Sopra La Bergamasca''] – arranged for organ and rendered by Claudi Meneghin (2025)

=== 21st century ===

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; [[Micronaive]]

* [https://youtu.be/wCqVRcU9tec ''No.27.63'']

; [[Herman Miller]]

* ''[https://soundcloud.com/morphosyntax-1/rabbit-tracks-in-the-snow Rabbit Tracks in the Snow]'' (2025)

; [[No Clue Music]]

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** ''[http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3 play]''

~~[[Category:Keyboard]]~~

[[Category:Listen]]

[[Category:Twentuning]]

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|25}}
+{{ED intro}}
 == Theory ==
+edo is a good way to tune the [[blackwood]] temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
-edo is a good way to tune the [[blackwood]] temperament, which takes the very sharp fifths of [[5edo]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4]]) and 7 ([[7/4]]). It also tunes [[sixix]] temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
+edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[trismegistus]] temperament (or [[mavila]] if it is interpreted as [[3/2]]). In fact, it is a convergent to a melodically optimal "golden" tuning of trismegistus or mavila, at around 672.7 cents.
-edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not [[consistent]]. It therefore makes sense to use it as a 2.5.7 [[subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]'s with the octave, and so tempers out (8/7)<sup>5</sup> / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal]] [[tritone]]s of [[7/5]] with the octave, and hence tempers out [[3136/3125]]. If we want to temper out both of these and also have decent fifths, the obvious solution is [[50edo]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[mavila]] temperament.
 If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the [[k*N subgroups|2*25 subgroup]] 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.
-Since 25 is 5 x 5, 25edo is the smallest composite EDO that doesn't have any intervals in common with [[12edo]].
+Its step of 48{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having a very high [[harmonic entropy]]. This is because the harmonic entropy model is usually tuned to reflect the general perception of quarter-tones being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
 === Possible usage in Indonesian music ===
-Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings.
+Since 25edo contains [[5edo]] as a subset, and it features an [[antidiatonic]] scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both [[Slendro]] (~5edo) and [[Pelog]] (~antidiatonic scale) tunings. However, many tunings of pelog are also better represented by the tuning's native 3-2-6-3-2-3-6 [[omnidiatonic]] scale.
 === Odd harmonics ===
 {{Harmonics in equal|25}}
+=== Subsets and supersets ===
+Since 25 is 5 x 5, 25edo is the smallest composite EDO that doesn't have any intervals in common with [[12edo]]. Doubling 25edo to get [[50edo]] produces a good [[meantone]] tuning.
 == Intervals ==
@@ Line 242: / Line 245: @@
 |}
 *based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.
--edo chords can be named with ups and downs, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
+-edo chords can be named with ups and downs, see [[Ups and downs notation#Chords and Chord Progressions|Ups and downs notation - Chords and Chord Progressions]].
 [[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]]
@@ Line 249: / Line 252: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
+{{Sharpness-sharp5a}}
-===Sagittal notation===
+Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
-This notation uses the same sagittal sequence as [[32edo#Sagittal notation|32-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
+{{Sharpness-sharp5}}
+If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as [[32edo#Sagittal notation|32edo]], and is a superset of the notation for [[5edo #Sagittal notation|5edo]].
 <imagemap>
@@ Line 260: / Line 272: @@
 rect 20 80 415 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
 default [[File:25-EDO_Sagittal.svg]]
-</imagemap>[[MisterShafXen’s 25edo notation]]
+</imagemap>
+=== Second-best fifth (mavila) notation ===
+{{Mavila}}
 == Regular temperament properties ==
@@ Line 548: / Line 563: @@
 A subset of the Passion[13] scale. Approximated from the original Akebono I scale of [[12edo]].
+; Unfair [[blackwood]][10]
+; 4 1 4 1 4 1 4 1 4 1
+Named "unfair" (by Igliashon Jones) because of the predominance of the larger interval. The major triads come with the large supermajor third.
+; Fair [[blackwood]][10]
+; 3 2 3 2 3 2 3 2 3 2
+Named "fair" (by Igliashon Jones) because larger and smaller interval are more balanced. The major triads come with the nice 5/4 major third.
 == Relationship to Armodue ==
@@ Line 565: / Line 594: @@
 ; {{W|Johann Sebastian Bach}}
 * [https://www.youtube.com/watch?v=nhpvxkUDBFk "Prelude in C major, No. 1" from ''The Well-Tempered Clavier I'', BWV 846] (1722) – played by [[Stephen Weigel]] on a [[lumatone]] (2024) – [[mavila]] in 25edo tuning
+; {{W|Marco Uccellini}}
+* [https://www.youtube.com/watch?v=2yXu0nFoIbI ''Aria Sopra La Bergamasca''] – arranged for organ and rendered by Claudi Meneghin (2025)
 === 21st century ===
@@ Line 594: / Line 626: @@
 ; [[Micronaive]]
 * [https://youtu.be/wCqVRcU9tec ''No.27.63'']
+; [[Herman Miller]]
+* ''[https://soundcloud.com/morphosyntax-1/rabbit-tracks-in-the-snow Rabbit Tracks in the Snow]'' (2025)
 ; [[No Clue Music]]
@@ Line 611: / Line 646: @@
 ** ''[http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3 play]''
-[[Category:Keyboard]]
 [[Category:Listen]]
 [[Category:Twentuning]]