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{{interwiki
| de = 25-EDO
| en = 25edo
| es = 25 EDO
| ja =
}}
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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 divides the [[octave]] in 25 equal steps of exact size 48 [[cent]]s each. It 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 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 [[Just intonation subgroups|subgroup]] tuning. Looking just at 2, 5, and 7, it equates five [[8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125]] [[diesis]] and two [[septimal tritones]] 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.  


If 5/4 and 7/4 aren't 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.
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. 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 ==
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 38: Line 56:
| 17/16, 20/19, 18/17
| 17/16, 20/19, 18/17
| 2b
| 2b
| v~2
| ^^m2
| downmid 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
|-
|-
Line 46: Line 64:
| 12/11, 38/35
| 12/11, 38/35
| 2
| 2
| ^~2
| vvM2
| upmid 2nd
| dudmajor 2nd
| vvE
| vvE
|-
|-
Line 78: Line 96:
| 39/32, 17/14, 40/33
| 39/32, 17/14, 40/33
| 3#
| 3#
| v~3
| ^^m3
| downmid 3rd
| dupminor 3rd
| ^^F
| ^^F
|-
|-
Line 86: Line 104:
| 5/4
| 5/4
| 4b
| 4b
| ^~3
| vvM3
| upmid 3rd
| dudmajor 3rd
| vvF#
| vvF#
|-
|-
Line 118: Line 136:
| 7/5, 39/28
| 7/5, 39/28
| 5#
| 5#
| v~4,v~5
| ^^4,^^b5
| downmid 4th, <br> downmid 5th
| dup 4th, dupdim 5th
| ^^G, ^^Ab
| ^^G, ^^Ab
|-
|-
Line 126: Line 144:
| 10/7, 56/39
| 10/7, 56/39
| 6b
| 6b
| ^~4,^~5
| vvA4,vv5
| upmid 4th, <br> upmid 5th
| dudaug 4th, dud 5th
| vvG#, vvA
| vvG#, vvA
|-
|-
Line 158: Line 176:
| 8/5
| 8/5
| 7
| 7
| v~6
| ^^m6
| downmid 6th
| dupminor 6th
| ^^Bb
| ^^Bb
|-
|-
Line 166: Line 184:
| 64/39, 28/17, 33/20
| 64/39, 28/17, 33/20
| 7#
| 7#
| ^~6
| vvM6
| upmid 6th
| dudmajor 6th
| vvB
| vvB
|-
|-
Line 198: Line 216:
| 11/6, 35/19
| 11/6, 35/19
| 9b
| 9b
| v~7
| ^^m7
| downmid 7th
| dupminor 7th
| ^^C
| ^^C
|-
|-
Line 206: Line 224:
| 32/17, 17/9, 19/10
| 32/17, 17/9, 19/10
| 9
| 9
| ^~7
| vvM7
| upmid 7th
| dudmajor 7th
| vvC#
| vvC#
|-
|-
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|}
|}
*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.
*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.]]
[[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]]
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[[:File:25ed2-001.svg|25ed2-001.svg]]
[[:File:25ed2-001.svg|25ed2-001.svg]]


== Relationship to Armodue ==
== 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).
{{Sharpness-sharp5a}}
 
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]]:
 
{{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.


Like [[16edo|16-EDO]] and [[23edo|23-EDO]], 25-EDO contains the 9-note "Superdiatonic" scale of [[7L_2s|7L2s]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9-EDO or 666.67 cents, to 4\7-EDO or 685.71 cents). The 25-EDO generator for this scale is the 672-cent interval. This allows 25-EDO to be used with the [[Armodue_theory|Armodue]] notation system in much the same way that [[19edo|19-EDO]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25-EDO Armodue 6th is flatter than that of 16-EDO (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.
=== 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]].


== Commas ==
<imagemap>
File:25-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 415 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 415 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:25-EDO_Sagittal.svg]]
</imagemap>


25 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)
=== Second-best fifth (mavila) notation ===
{{Mavila}}
 
== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable"
|-
! Generator
! Periods per octave
! "Sharp 3/2" temperaments
! "Flat 3/2" temperaments (25b val)
! MOS scales
|-
| 1\25
| 1
|
|
|
|-
| 2\25
| 1
| [[Passion]]
|
| [[1L 11s]], [[12L 1s]]
|-
| 3\25
| 1
| [[Bleu]]
|
| [[1L 7s]], [[8L 1s]], [[8L 9s]]
|-
| 4\25
| 1
|
| [[Luna]] / [[didacus]]
| [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]]
|-
| 6\25
| 1
|
| [[Gariberttet]]
| [[4L 1s]], [[4L 5s]], [[4L 9s]], [[4L 13s]], [[4L 17s]]
|-
| 7\25
| 1
|
| [[Sixix]]
| [[4L 3s]], [[7L 4s]], [[7L 11s]]
|-
| 8\25
| 1
| [[Magic]]
|
| [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]], [[3L 16s]], [[3L 19s]]
|-
| 9\25
| 1
| [[Hamity]]
|
| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]]
|-
| 11\25
| 1
| [[Mabila]] / [[trismegistus]]
| [[Armodue]] / [[Pelogic]] (25bd)
| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[9L 7s]]
|-
| 12\25
| 1
| [[Tritonic]]
| [[Triton]]
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]], [[2L 21s]]
|-
| 1\25
| 5
| [[Blackwood]] favouring 9/7
|
| [[5L 5s]], [[5L 10s]], [[5L 15s]]
|-
| 2\25
| 5
| [[Blackwood]] favouring 5/4
|
| [[5L 5s]], [[10L 5s]]
|-
| 1\25
| 25
|
|
|
|-
|}
 
=== Commas ===
25et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 25 40 58 70 86 93 }}.)


{| class="wikitable center-all left-3 right-4 left-6"
{| class="wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime <br> limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Monzo]]
Line 255: Line 384:
| 90.22
| 90.22
| Sawa
| Sawa
| Limma, Pythagorean Minor 2nd
| Blackwood comma, Pythagorean limma
|-
|-
| 5
| 5
Line 262: Line 391:
| 29.61
| 29.61
| Laquinyo
| Laquinyo
| Small Diesis, Magic Comma
| Magic comma
|-
|-
| 5
| 5
Line 276: Line 405:
| 44.13
| 44.13
| Laquinzo
| Laquinzo
| Cloudy
| Cloudy comma
|-
|-
| 7
| 7
Line 283: Line 412:
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro Diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 290: Line 419:
| 27.26
| 27.26
| Ru
| Ru
| Septimal Comma, Archytas' Comma, Leipziger Komma
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|-
| 7
| 7
Line 297: Line 426:
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
Line 304: Line 433:
| 14.52
| 14.52
| Quinzogu
| Quinzogu
| Trimyna
| Trimyna comma
|-
|-
| 7
| 7
Line 325: Line 454:
| 2.35
| 2.35
| Lazoquinyo
| Lazoquinyo
| Horwell
| Horwell comma
|-
|-
| 11
| 11
Line 346: Line 475:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 353: Line 482:
| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma, parizeksma
|}
|}
<references/>
<references/>


== A 25edo keyboard ==
== Scales ==


; [[Antipental blues]]
; 6 4 2 3 5 5
Approximated from a [[hexatonic]] subset of the [[dwarf17marv]] scale. Contains lots of [[consonance]]s from the 2.3.7.11 [[subgroup]] while excluding the familiar [[harmonic]] 5.
; [[Armodue]]/[[pelogic]][5]
; 3 3 8 3 8
A [[pentatonic]] and [[MOS]] scale somewhat resembling [[pelog]].
; Armodue/pelogic[9]
; 3 3 2 3 3 3 2 2 3 3
A [[:Category:9-tone scales|9-tone]] MOS scale somewhat resembling pelog.
; [[Equipentatonic]]
; 5 5 5 5 5
Somewhat resembles [[slendro]]. Is the [[blacksmith]]/[[blackwood]][5] MOS. Is the same as [[5edo]].
; [[Mabila]]/[[trismegistus]] justified pentatonic
; 3 3 9 2 8
A pentatonic subset of the mabila/trismegistus[16] MOS scale, it is those temperaments' pentatonic MOS, but with their complex 3/2 substituted in.
; Mabila/trismegistus justified nonatonic
; 3 3 2 3 4 2 2 2 3 3
A 9-tone subset of the mabila/trismegistus[16] MOS scale, it is those temperaments' 9-tone MOS, but with their complex 3/2 substituted in.
; [[Magic]][13]
; 1 1 5 1 1 1 5 1 1 1 5 1 1
A [[:Category:13-tone scales|13-tone]] MOS scale. A useful starting point for a [[scalesmith]] to find [[MODMOS]]es, or to find 4- to 9-tone subsets.
; Amulet{{idiosyncratic}}
; 2 1 2 2 1 2 3 2 2 1 2 3 2
A MODMOS of Magic[13]. It is the same as Magic[13], but with 4 tones shifted over by one [[chroma]] (the difference between MOS step sizes, in this case 4\25). This gives its intervals a more even spread, which makes it very useable as a chromatic-like scale. Can also be used to take 4- to 9-tone subsets.
; Fennec{{idiosyncratic}}
; 3 4 3 5 2 5 2
A subset of the amulet scale. Approximated from the original fennec scale of [[14edo]].
; [[Passion]][13]
; 2 2 2 2 2 2 1 2 2 2 2 2 2
A 13-tone MOS scale with a lot of consonances available. Can be used as a chromatic-like scale. Can also be used to take 4- to 9-tone subsets.
; Akebono I
; 4 2 9 4 6
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 ==
Like [[16edo]] and [[23edo]], 25edo contains the 9-note superdiatonic scale of [[7L 2s]] (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9 or 666.67 cents, to 4\7 or 685.71 cents). The 25edo generator for this scale is the 672-cent interval. This allows 25edo to be used with the [[Armodue theory|Armodue]] notation system in much the same way that [[19edo]] is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25edo Armodue 6th is flatter than that of 16edo (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.
== Keyboard layout ==
'''Piano keyboard'''
[[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]]
[[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]]
'''Lumatone'''
See [[Lumatone mapping for 25edo]]


== Music ==
== Music ==
=== Modern renderings ===
; {{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 ===
; [[Alisich]]
* [https://www.youtube.com/watch?v=tMtFIfpju7c ''StartingnStoppinLeftnRight''] (2024)
; [[Fabrizio Fulvio Fausto Fiale]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3 ''Flat fourth blues'']
; [[JUMBLE]]
* [https://www.youtube.com/watch?v=rdzP4y8fXF4 ''Hammock Hut''] (2024)
* [https://www.youtube.com/watch?v=rrmyuYMKvnM ''ANDROMEDA''] (2024)
* [https://www.youtube.com/watch?v=3FctBwJlcCw ''Patchouli''] (2024)
; [[Peter Kosmorsky]]
* [[:File:25edochorale.mid]] (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")
* [[:File:25_edo_prelude_largo.mid]] (2011, Blackwood)
; [[Budjarn Lambeth]]
* ''[https://www.youtube.com/watch?v=6VC5FWWMwR4 Improvisation in 25edo (Akebono I scale)]'' (2025)
* ''[https://www.youtube.com/watch?v=ZVsQAZdbvMI Improvisation in 25edo (Antipental Blues scale, glass unison timbre)]'' (2025)
* ''[https://www.youtube.com/watch?v=iy5Zwc8vipw Improvisation in 25edo (Antipental Blues scale, platinum inharmonic timbre)]'' (2025)
* ''[https://www.youtube.com/watch?v=B6HGVZzUze4 Improvisation in 25edo (Armodue5 scale)]'' (2025)
* ''[https://www.youtube.com/watch?v=EK7O0SxI9dI Improvisation in 25edo (Fennec scale)]'' (2025)
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=oMJjfbdUddU ''Happy Birthday Canon · 6-in-1 Canon in 25edo]''
; [[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]]
* [https://www.youtube.com/watch?v=JQY989Is9vg ''New File''] (2024)
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=kkPavppWUCg ''Edolian - Sepia''] (2020)
; [[Paul Rapoport]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3 ''Study in Fives'']
; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=oHDXCB_x7yo ''Embark''] (2022)


* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3 Study in Fives] by [http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Paul Rapoport]
; [[Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=2377 Fantasy for Piano in 25 Note per Octave Tuning] ''[http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3 play]'' by [[Chris Vaisvil]]
* [http://chrisvaisvil.com/?p=2377 ''Fantasy for Piano in 25 Note per Octave Tuning'']  
* [http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3 Flat fourth blues] by [[Fabrizio Fulvio Fausto Fiale]]
** ''[http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3 play]''
* [[:File:25edochorale.mid]] by [[Peter Kosmorsky]] (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")
* [[:File:25_edo_prelude_largo.mid]] by Peter Kosmorsky (2011, Blackwood)


[[Category:25edo]]
[[Category:Equal divisions of the octave]]
[[Category:Keyboard]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Subgroup]]
[[Category:Twentuning]]
[[Category:Twentuning]]
Retrieved from "https://en.xen.wiki/w/25edo"