25edo: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

__FORCETOC__

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference: ~~

=25 tone equal temperament=

~~: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-25 18:52:38 UTC</tt>. ~~

~~: The original revision id was <tt>602808482</tt>. ~~

~~: The revision comment was: <tt></tt> ~~

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. ~~

~~<h4>Original Wikitext content:</h4>~~

~~<div style~~=~~"width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em">~~<~~pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]~~

~~=<~~span style="color: #006b2e; font-family: 'Times New Roman',Times,serif; font-size: 113%;"~~>~~25 tone equal temperament~~<~~/span~~>~~=

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~~|5/4]]) and 7 ([[~~7_4~~|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|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 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~~|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[~~128_125~~|128/125]] [[diesis]] and two [[septimal tritones]] of [[~~7_5~~|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.

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|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|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125|128/125]] [[diesis|diesis]] and two [[septimal_tritones|septimal tritones]] of [[7/5|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|50EDO]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[Mavila|mavila]] temperament.

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.

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.

=Music=

~~//[~~[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]]

''[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]

~~[[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://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3|Flat fourth blues]]// by Fabrizio Fulvio Fausto Fiale~~

[~~[media type="file" key="25edochorale~~.~~mid" width~~=~~"300" height="50"~~]~~] [~~[~~file~~:~~25edochorale.mid]] Peter Kosmorsky (10~~/14/~~10, 2~~.5.~~7 subgroup, a friend responded "The ~~25edo~~<~~/~~span> 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.")~~

[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

~~[[media type="file" key="25 edo prelude largo.mid" width="300" height="50"]] [[file:25 edo prelude largo.mid]] Peter Kosmorsky (2011, Blackwood)~~

~~=Intervals=~~

''[http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3 Flat fourth blues]'' by Fabrizio Fulvio Fausto Fiale

~~||= Degrees ||= Cents ||= Approximate~~

[[File:25edochorale.mid]] [[:File:25edochorale.mid|25edochorale.mid]] 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.")

~~Ratios* ||= Armodue~~

~~Notation ||||||=~~ [[~~xenharmonic/Ups and Downs Notation~~|~~ups and downs notation~~]] ||

[[File:25_edo_prelude_largo.mid]] [[:File:25_edo_prelude_largo.mid|25 edo prelude largo.mid]] Peter Kosmorsky (2011, Blackwood)

~~||= 0 ||= 0 ||= 1/1 ||= 1 ||= P1 ||= perfect 1sn ||= D, Eb ||~~

~~||= 1 ||= 48 ||= 33/32, 39/38, 34/33 ||= 1# ||= ^1, ^m2 ||= up 1sn, upminor 2nd ||= D^, Eb^ ||~~

~~||= 2 ||= 96 ||= 17/16, 20/19, 18/17 ||= 2b ||= ^^m2 ||= double-upminor 2nd ||= Eb^^ ||~~

~~||= 3 ||= 144 ||= 12/11, 38/35 ||= 2 ||= vvM2 ||= double-downmajor 2nd ||= Evv ||~~

~~||= 4 ||= 192 ||= 9/8,~~ 10~~/9, 19/17 ||= 2# ||= vM2 ||= downmajor 2nd ||= Ev ||~~

~~||= 5· ||= 240 ||= 8/7 ||= 3b ||= M2, m3 ||= major 2nd, minor 3rd ||= E, F ||~~

~~||= 6 ||= 288 ||= 19/16, 20/17 ||= 3 ||= ^m3 ||= upminor 3rd ||= F^ ||~~

~~||= 7 ||= 336 ||= 39/32, 17~~/14~~, 40~~/~~33 ||= 3# ||= ^^m3 ||= double-upminor 3rd ||= F^^ ||~~

~~||= 8· ||= 384 ||= 5/4 ||= 4b ||= vvM3 ||= double-downmajor 3rd ||= F#vv ||~~

~~||= 9 ||= 432 ||= 9/7, 32/25, 50/39 ||= 4 ||= vM3 ||= downmajor ||= F#v ||~~

~~||=~~ 10 ~~||= 480 ||= 33/25~~, ~~25/19 ||= 4#/5b ||= M3, P4 ||= major 3rd, perfect 4th ||= F#, G ||~~

~~||= 11· ||= 528 ||= 31/21, 34/25 ||=~~ 5 ~~||= ^4 ||= up 4th ||= G^ ||~~

~~||= 12 ||= 576 ||=~~ 7/5, ~~39/28 ||= 5# ||= ^^4,^^d5 ||= double-up 4th,~~

~~double-up dim 5th ||= G^^, Ab^^ ||~~

~~||= 13 ||= 624 ||~~= 10/7, ~~56/39 ||= 6b ||= vvA4~~,~~vv5 ||= double-down aug 4th~~,

~~double-down 5th ||= G#vv~~, ~~Avv ||~~

~~||= 14· ||= 672 ||= 42/31~~, 25~~/17 ||= 6 ||= v5 ||= down 5th ||= Av ||~~

~~||= 15 ||= 720 ||= 50/33, 38/25 ||= 6# ||= P5, m6 ||= perfect 5th, minor 6th ||= A, Bb ||~~

|~~|= 16 ||= 768 ||= 14/9,~~ 25~~/16, 39/25 ||= 7b ||= ^m6 ||= upminor 6th ||= Bb^ ||~~

~~||= 17· ||= 816 ||= 8/5 ||= 7 ||= ^^m6 ||= double-upminor 6th ||= Bb^^ ||~~

~~||= 18 ||= 864 ||= 64/39, 28/17, 33/20 ||= 7# ||= vvM6 ||= double-downmajor 6th ||= Bvv ||~~

~~||= 19 ||= 912 ||= 32/19, 17/10 ||= 8b ||= vM6 ||= downmajor 6th ||= Bv ||~~

~~||= 20· ||= 960 ||= 7/4 ||= 8 ||= M6, m7 ||= major 6th, minor 7th ||= B, C ||~~

~~||= 21 ||= 1008 ||= 16/9, 9/5, 34/19 ||= 8# ||= ^m7 ||= upminor 7th ||= C^ ||~~

~~||= 22 ||= 1056 ||= 11/6, 35/19 ||= 9b ||= ^^m7 ||= double-upminor 7th ||= C^^ ||~~

~~||= 23 ||= 1104 ||= 32/17, 17/9, 19/10 ||= 9 ||= vvM7 ||= double-downmajor 7th ||= C#vv ||~~

~~||= 24 ||= 1152 ||= 33/17, 64/33~~, ~~76/39 ||= 9#/1b ||= vM7 ||= downmajor 7th ||= C#v ||~~

~~||= 25 ||= 1200 ||= 2/1 ||= 1 ||= P8 ||= perfect 8ve ||= C#, D ||~~

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

~~[[media type~~=~~"custom" key~~=~~"25100128"]]~~

=Intervals=

~~[[file~~:~~25ed2~~-~~001.svg]]~~

{| class="wikitable"

|-

| style="text-align:center;" | Degrees

| style="text-align:center;" | Cents

| style="text-align:center;" | Approximate

Ratios*

| style="text-align:center;" | Armodue

=~~Relationship to Armodue~~=

Notation

| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]

|-

| style="text-align:center;" | 0

| style="text-align:center;" | 1/1

| style="text-align:center;" | 1

| style="text-align:center;" | P1

| style="text-align:center;" | perfect 1sn

| style="text-align:center;" | D, Eb

|-

| style="text-align:center;" | 1

| style="text-align:center;" | 48

| style="text-align:center;" | 33/32, 39/38, 34/33

| style="text-align:center;" | 1#

| style="text-align:center;" | ^1, ^m2

| style="text-align:center;" | up 1sn, upminor 2nd

| style="text-align:center;" | D^, Eb^

|-

| style="text-align:center;" | 2

| style="text-align:center;" | 96

| style="text-align:center;" | 17/16, 20/19, 18/17

| style="text-align:center;" | 2b

| style="text-align:center;" | ^^m2

| style="text-align:center;" | double-upminor 2nd

| style="text-align:center;" | Eb^^

|-

| style="text-align:center;" | 3

| style="text-align:center;" | 144

| style="text-align:center;" | 12/11, 38/35

| style="text-align:center;" | 2

| style="text-align:center;" | vvM2

| style="text-align:center;" | double-downmajor 2nd

| style="text-align:center;" | Evv

|-

| style="text-align:center;" | 4

| style="text-align:center;" | 192

| style="text-align:center;" | 9/8, 10/9, 19/17

| style="text-align:center;" | 2#

| style="text-align:center;" | vM2

| style="text-align:center;" | downmajor 2nd

| style="text-align:center;" | Ev

|-

| style="text-align:center;" | 5·

| style="text-align:center;" | 240

| style="text-align:center;" | 8/7

| style="text-align:center;" | 3b

| style="text-align:center;" | M2, m3

| style="text-align:center;" | major 2nd, minor 3rd

| style="text-align:center;" | E, F

|-

| style="text-align:center;" | 6

| style="text-align:center;" | 288

| style="text-align:center;" | 19/16, 20/17

| style="text-align:center;" | 3

| style="text-align:center;" | ^m3

| style="text-align:center;" | upminor 3rd

| style="text-align:center;" | F^

|-

| style="text-align:center;" | 7

| style="text-align:center;" | 336

| style="text-align:center;" | 39/32, 17/14, 40/33

| style="text-align:center;" | 3#

| style="text-align:center;" | ^^m3

| style="text-align:center;" | double-upminor 3rd

| style="text-align:center;" | F^^

|-

| style="text-align:center;" | 8·

| style="text-align:center;" | 384

| style="text-align:center;" | 5/4

| style="text-align:center;" | 4b

| style="text-align:center;" | vvM3

| style="text-align:center;" | double-downmajor 3rd

| style="text-align:center;" | F#vv

|-

| style="text-align:center;" | 9

| style="text-align:center;" | 432

| style="text-align:center;" | 9/7, 32/25, 50/39

| style="text-align:center;" | 4

| style="text-align:center;" | vM3

| style="text-align:center;" | downmajor

| style="text-align:center;" | F#v

|-

| style="text-align:center;" | 10

| style="text-align:center;" | 480

| style="text-align:center;" | 33/25, 25/19

| style="text-align:center;" | 4#/5b

| style="text-align:center;" | M3, P4

| style="text-align:center;" | major 3rd, perfect 4th

| style="text-align:center;" | F#, G

|-

| style="text-align:center;" | 11·

| style="text-align:center;" | 528

| style="text-align:center;" | 31/21, 34/25

| style="text-align:center;" | 5

| style="text-align:center;" | ^4

| style="text-align:center;" | up 4th

| style="text-align:center;" | G^

|-

| style="text-align:center;" | 12

| style="text-align:center;" | 576

| style="text-align:center;" | 7/5, 39/28

| style="text-align:center;" | 5#

| style="text-align:center;" | ^^4,^^d5

| style="text-align:center;" | double-up 4th,

~~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.~~

double-up dim 5th

| style="text-align:center;" | G^^, Ab^^

|-

| style="text-align:center;" | 13

| style="text-align:center;" | 624

| style="text-align:center;" | 10/7, 56/39

| style="text-align:center;" | 6b

| style="text-align:center;" | vvA4,vv5

| style="text-align:center;" | double-down aug 4th,

=~~Commas=~~

double-down 5th

~~25 EDO tempers out the following commas. (Note~~: ~~This assumes the val &lt~~; ~~25 40 58 70 86 93~~ |.)

| style="text-align:center;" | G#vv, Avv

||~~~ Comma~~ ||~~~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 |~~|

|-

||= ~~256/243 ||< | 8~~ -~~5 &gt~~; ||~~&gt~~; ~~90.22~~ ||= ~~Limma~~ ||= ~~Pythagorean Minor 2nd~~ ||= ||

| style="text-align:center;" | 14·

||= ~~3125/3072~~ ||~~&lt~~; | -~~10 -1 5 &gt~~; ||~~&gt~~; ~~29.61~~ ||= ~~Small Diesis~~ ||= ~~Magic Comma~~ ||= ||

| style="text-align:center;" | 672

||= ||~~&lt~~; | 38 -~~2 -15 &gt~~; ||~~&gt~~; ~~1.38~~ ||= ~~Hemithirds Comma~~ ||= ||= ||

| style="text-align:center;" | 42/31, 25/17

||= ~~16807/16384 || |~~ -~~14 0 0 5 &gt~~; ||~~&gt~~; ~~44.13 || || || |~~|

| style="text-align:center;" | 6

||= ~~49/48 ||< |~~ -~~4 -1 0 2 &gt~~; ||~~&gt~~; ~~35.70~~ ||= ~~Slendro Diesis~~ ||= ||= ||

| style="text-align:center;" | v5

||= ~~64/63~~ ||~~&lt~~; | 6 -~~2 0 -1 &gt~~; ||~~&gt~~; ~~27.26~~ ||= ~~Septimal Comma~~ ||= ~~Archytas' Comma~~ ||= ~~Leipziger Komma~~ ||

| style="text-align:center;" | down 5th

||= ~~3125/3087~~ ||~~&lt~~; | 0 -~~2 5 -3 &gt~~; ||~~&gt~~; ~~21.18~~ ||= ~~Gariboh~~ ||= ||= ||

| style="text-align:center;" | Av

||= ~~50421/50000 ||< |~~ -~~4 1 -5 5 &gt~~; ||~~&gt~~; ~~14.52~~ ||= ~~Trimyna~~ ||= ||= ||

|-

||= ~~1029/1024 ||< |~~ -~~10 1 0 3 &gt~~; ||~~&gt~~; ~~8.43~~ ||~~= Gamelisma |~~|= ||= ||

| style="text-align:center;" | 15

||= ~~3136/3125 ||&lt~~; | ~~6 0 -~~5 ~~2 >~~ |~~|&gt~~; ~~6.08~~ ||= ~~Hemimean~~ ||= ||= ||

| style="text-align:center;" | 720

||= ~~65625/65536 ||< |~~ -~~16 1 5 1 &gt~~; ||~~&gt~~; ~~2.35~~ ||= ~~Horwell~~ ||= ||= ||

| style="text-align:center;" | 50/33, 38/25

||= ~~100/99 ||&lt~~; | 2 -~~2 2 0~~ -~~1 &gt~~; ||~~> 17.40 |~~|= ~~Ptolemisma~~ ||= ||= ||

| style="text-align:center;" | 6#

||= ~~176/175 ||< | 4 0~~ -~~2 -1 1 &gt~~; |~~|>~~ 9~~.86 |~~|= ~~Valinorsma~~ ||= ||= ||

| style="text-align:center;" | P5, m6

||= ~~91/90~~ ||~~&lt~~; | -~~1 -2 -1 1 0 1 &gt~~; ||~~&gt~~; ~~19.13~~ ||= ~~Superleap~~ ||= ||= ||

| style="text-align:center;" | perfect 5th, minor 6th

||= ~~676/675~~ ||~~&lt~~; | 2 -~~3 -2 0 0 2 &gt~~; ||~~&gt~~; ~~2.56~~ ||= ~~Parizeksma~~ ||= ||= ||

| style="text-align:center;" | A, Bb

|-

| style="text-align:center;" | 16

| style="text-align:center;" | 768

| style="text-align:center;" | 14/9, 25/16, 39/25

| style="text-align:center;" | 7b

| style="text-align:center;" | ^m6

| style="text-align:center;" | upminor 6th

| style="text-align:center;" | Bb^

|-

| style="text-align:center;" | 17·

| style="text-align:center;" | 816

| style="text-align:center;" | 8/5

| style="text-align:center;" | 7

| style="text-align:center;" | ^^m6

| style="text-align:center;" | double-upminor 6th

| style="text-align:center;" | Bb^^

|-

| style="text-align:center;" | 18

| style="text-align:center;" | 864

| style="text-align:center;" | 64/39, 28/17, 33/20

| style="text-align:center;" | 7#

| style="text-align:center;" | vvM6

| style="text-align:center;" | double-downmajor 6th

| style="text-align:center;" | Bvv

|-

| style="text-align:center;" | 19

| style="text-align:center;" | 912

| style="text-align:center;" | 32/19, 17/10

| style="text-align:center;" | 8b

| style="text-align:center;" | vM6

| style="text-align:center;" | downmajor 6th

| style="text-align:center;" | Bv

|-

| style="text-align:center;" | 20·

| style="text-align:center;" | 960

| style="text-align:center;" | 7/4

| style="text-align:center;" | 8

| style="text-align:center;" | M6, m7

| style="text-align:center;" | major 6th, minor 7th

| style="text-align:center;" | B, C

|-

| style="text-align:center;" | 21

| style="text-align:center;" | 1008

| style="text-align:center;" | 16/9, 9/5, 34/19

| style="text-align:center;" | 8#

| style="text-align:center;" | ^m7

| style="text-align:center;" | upminor 7th

| style="text-align:center;" | C^

|-

| style="text-align:center;" | 22

| style="text-align:center;" | 1056

| style="text-align:center;" | 11/6, 35/19

| style="text-align:center;" | 9b

| style="text-align:center;" | ^^m7

| style="text-align:center;" | double-upminor 7th

| style="text-align:center;" | C^^

|-

| style="text-align:center;" | 23

| style="text-align:center;" | 1104

| style="text-align:center;" | 32/17, 17/9, 19/10

| style="text-align:center;" | 9

| style="text-align:center;" | vvM7

| style="text-align:center;" | double-downmajor 7th

| style="text-align:center;" | C#vv

|-

| style="text-align:center;" | 24

| style="text-align:center;" | 1152

| style="text-align:center;" | 33/17, 64/33, 76/39

| style="text-align:center;" | 9#/1b

| style="text-align:center;" | vM7

| style="text-align:center;" | downmajor 7th

| style="text-align:center;" | C#v

|-

| style="text-align:center;" | 25

| style="text-align:center;" | 1200

| style="text-align:center;" | 2/1

| style="text-align:center;" | 1

| style="text-align:center;" | P8

| style="text-align:center;" | perfect 8ve

| style="text-align:center;" | C#, D

|}

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

=~~A 25edo keyboard=~~

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

[[~~image~~:~~mm25.PNG]]</pre></div>~~

[[:File:25ed2-001.svg|25ed2-001.svg]]

~~<h4>Original HTML content:</h4>~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>25edo</title></head><body><a href="#x25 tone equal temperament">25 tone equal temperament</a> | <a href="#Music">Music</a> | <a href="#Intervals">Intervals</a> | <a href="#Relationship to Armodue">Relationship to Armodue</a> | <a href="#Commas">Commas</a> | <a href="#A 25edo keyboard">A 25edo keyboard</a>

<h1 id="toc0"><a name="x25 tone equal temperament"></a>25 tone equal temperament</h1>

~~ ~~

25EDO divides the <a class="wiki_link" href="/octave">octave</a> in 25 equal steps of exact size 48 <a class="wiki_link" href="/cent">cent</a>s each. It is a good way to tune the <a class="wiki_link" href="/Blackwood%20temperament">Blackwood temperament</a>, which takes the very sharp fifths of <a class="wiki_link" href="/5EDO">5EDO</a> as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (<a class="wiki_link" href="/5_4">5/4</a>) and 7 (<a class="wiki_link" href="/7_4">7/4</a>). 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.

~~ ~~

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 <a class="wiki_link" href="/consistent">consistent</a>. It therefore makes sense to use it as a 2.5.7 <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> tuning. Looking just at 2, 5, and 7, it equates five <a class="wiki_link" href="/8_7">8/7</a>s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a <a class="wiki_link" href="/128_125">128/125</a> <a class="wiki_link" href="/diesis">diesis</a> and two <a class="wiki_link" href="/septimal%20tritones">septimal tritones</a> of <a class="wiki_link" href="/7_5">7/5</a> 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 <a class="wiki_link" href="/50EDO">50EDO</a>. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for <a class="wiki_link" href="/mavila">mavila</a> temperament.

~~ ~~

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 <a class="wiki_link" href="/k%2AN%20subgroups">2*25 subgroup</a> 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.

~~ ~~

~~<h1 id="toc1"><a name="Music"></a>Music</h1>~~

<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3" rel="nofollow">Study in Fives</a> by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow">Paul Rapoport</a>

<a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2377" rel="nofollow">Fantasy for Piano in 25 Note per Octave Tuning</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3" rel="nofollow">play</a> by Chris Vaisvil

<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3" rel="nofollow">Flat fourth blues</a> by Fabrizio Fulvio Fausto Fiale

~~ ~~

<embed type="audio/midi" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/25edochorale.mid" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed> Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded &quot;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.&quot;)

<embed type="audio/midi" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/25+edo+prelude+largo.mid" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"></embed> <div class="objectEmbed"><a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid" onclick="ws.common.trackFileLink('/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid');"><img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="25 edo prelude largo.mid" /></a><div><a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid" onclick="ws.common.trackFileLink('/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid');" class="filename" title="25 edo prelude largo.mid">25 edo prelude largo.mid</a> <ul><li><a href="/file/detail/25%20edo%20prelude%20largo.mid">Details</a></li><li><a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid">Download</a></li><li style="color: #666">6 KB</li></ul></div></div> Peter Kosmorsky (2011, Blackwood)

~~ ~~

~~<h1 id="toc2"><a name="Intervals"></a>Intervals</h1>~~

~~ ~~

=Relationship to Armodue=

~~<table class="wiki_table">~~

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.

~~<tr>~~

~~<td style="text-align: center;">Degrees ~~

~~</td>~~

~~<td style="text-align: center;">Cents ~~

~~</td>~~

~~<td style="text-align: center;">Approximate ~~

~~Ratios* ~~

~~</td>~~

~~<td style="text~~-~~align: center;">Armodue ~~

~~Notation ~~

~~</td>~~

~~<td colspan="3" style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups~~ and ~~downs notation</a> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text~~-~~align: center;">0 ~~

~~</td>~~

~~<td style="text-align: center;">0 ~~

~~</td>~~

~~<td style="text-align: center;">1/1 ~~

~~</td>~~

~~<td style="text-align: center;">1 ~~

~~</td>~~

~~<td style="text-align: center;">P1 ~~

~~</td>~~

~~<td style="text-align: center;">perfect 1sn ~~

~~</td>~~

~~<td style="text-align: center;">D~~, ~~Eb ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">1 ~~

~~</td>~~

~~<td style="text-align: center;">48 ~~

~~</td>~~

~~<td style="text-align: center;">33/32, 39/38, 34/33 ~~

~~</td>~~

~~<td style="text-align: center;">1# ~~

~~</td>~~

~~<td style="text~~-~~align: center;">^1, ^m2 ~~

~~</td>~~

~~<td style="text-align: center;">up 1sn, upminor 2nd ~~

~~</td>~~

~~<td style="text-align: center;">D^, Eb^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">2 ~~

~~</td>~~

~~<td style="text-align: center;">96 ~~

~~</td>~~

~~<td style="text-align: center;">17/16, 20/19, 18/17 ~~

~~</td>~~

~~<td style="text-align: center;">2b ~~

~~</td>~~

~~<td style="text-align: center;">^^m2 ~~

~~</td>~~

~~<td style="text-align: center;">double-upminor 2nd ~~

~~</td>~~

~~<td style="text-align: center;">Eb^^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3 ~~

~~</td>~~

~~<td style="text-align: center;">144 ~~

~~</td>~~

~~<td style="text-align: center;">12/11, 38/35 ~~

~~</td>~~

~~<td style="text-align: center;">2 ~~

~~</td>~~

~~<td style="text-align: center;">vvM2 ~~

~~</td>~~

~~<td style="text-align: center;">double-downmajor 2nd ~~

~~</td>~~

~~<td style="text-align: center;">Evv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">4 ~~

~~</td>~~

~~<td style="text-align: center;">192 ~~

~~</td>~~

~~<td style="text-align: center;">~~9~~/8, 10/9, 19/17 ~~

~~</td>~~

~~<td style="text~~-~~align: center;~~"~~>2# ~~

~~</td>~~

~~<td style=~~"~~text-align: center;">vM2 ~~

~~</td>~~

~~<td style="text-align: center;">downmajor 2nd ~~

~~</td>~~

~~<td style="text-align: center;">Ev ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text~~-~~align: center;">5· ~~

~~</td>~~

~~<td style="text-align: center;">240 ~~

~~</td>~~

~~<td style="text-align: center;">8/7 ~~

~~</td>~~

~~<td style="text-align: center;">3b ~~

~~</td>~~

~~<td style="text-align: center;">M2, m3 ~~

~~</td>~~

~~<td style="text-align: center;">major 2nd, minor 3rd ~~

~~</td>~~

~~<td style="text-align: center;">E, F ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">6 ~~

~~</td>~~

~~<td style="text-align: center;">288 ~~

~~</td>~~

~~<td style="text-align: center;">19/16, 20/17 ~~

~~</td>~~

~~<td style="text-align: center;">3 ~~

~~</td>~~

~~<td style="text-align: center;">^m3 ~~

~~</td>~~

~~<td style="text-align: center;">upminor 3rd ~~

~~</td>~~

~~<td style="text-align: center;">F^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">7 ~~

~~</td>~~

~~<td style="text-align: center;">336 ~~

~~</td>~~

~~<td style="text-align: center;">39/32, 17/14, 40/33 ~~

~~</td>~~

~~<td style="text-align: center;">~~3~~# ~~

~~<~~/~~td>~~

~~<td style="text-align: center;">^^m3 ~~

~~</td>~~

~~<td style="text-align: center;">double-upminor 3rd ~~

~~</td>~~

~~<td style="text-align: center;">F^^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">8· ~~

~~</td>~~

~~<td style="text-align: center;">384 ~~

~~</td>~~

~~<td style="text-align: center;">~~5~~/4 ~~

~~</td>~~

~~<td style="text-align: center;">4b ~~

~~</td>~~

~~<td style="text-align: center;">vvM3 ~~

~~</td>~~

~~<td style="text-align: center;">double-downmajor 3rd ~~

~~</td>~~

~~<td style="text-align: center;">F#vv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">~~9~~ ~~

~~</td>~~

~~<td style="text~~-~~align: center;">432 ~~

~~</td>~~

~~<td style="text-align: center;">9/7~~, ~~32/25, 50/39 ~~

~~</td>~~

~~<td style="text-align: center;">~~4~~ ~~

~~</td>~~

~~<td style="text~~-~~align: center;">vM3 ~~

~~</td>~~

~~<td style="text-align: center;">downmajor ~~

~~</td>~~

~~<td style="text-align: center;">F#v ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">10 ~~

~~</td>~~

~~<td style="text-align: center;">480 ~~

~~</td>~~

~~<td style="text-align: center;">33/25, 25/19 ~~

~~</td>~~

~~<td style="text-align: center;">4#/5b ~~

~~</td>~~

~~<td style="text-align: center;">M3, P4 ~~

~~</td>~~

~~<td style="text-align: center;">major 3rd, perfect 4th ~~

~~</td>~~

~~<td style="text-align: center;">F#, G ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">11· ~~

~~</td>~~

~~<td style="text-align: center;">528 ~~

~~</td>~~

~~<td style="text-align: center;">31/21, 34/~~25~~ ~~

~~</td>~~

~~<td style="text-align: center;">5 ~~

~~</td>~~

~~<td style="text-align: center;">^4 ~~

~~</td>~~

~~<td style="text-align: center;">up 4th ~~

~~</td>~~

~~<td style="text-align: center;">G^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">12 ~~

~~</td>~~

~~<td style="text-align: center;">576 ~~

~~</td>~~

~~<td style="text-align: center;">7/5, 39/28 ~~

~~</td>~~

~~<td style="text-align: center;">5# ~~

~~</td>~~

~~<td style="text-align: center;">^^4,^^d5 ~~

~~</td>~~

~~<td style="text-align: center;">double-up 4th, ~~

~~double-up dim 5th ~~

~~</td>~~

~~<td style="text-align: center;">G^^, Ab^^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text~~-~~align: center;">13 ~~

~~</td>~~

~~<td style="text-align: center;">624 ~~

~~</td>~~

~~<td style="text-align: center;">10/7, 56/39 ~~

~~</td>~~

~~<td style="text-align: center;">6b ~~

~~</td>~~

~~<td style="text-align: center;">vvA4,vv5 ~~

~~</td>~~

~~<td style="text-align: center;">double-down aug 4th, ~~

~~double-down 5th ~~

~~</td>~~

~~<td style="text-align: center;">G#vv, Avv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">14· ~~

~~</td>~~

~~<td style="text-align: center;">~~672~~ ~~

~~</td>~~

~~<td style="text~~-~~align: center;">42/31,~~ 25~~/17 ~~

~~</td>~~

~~<td style="text~~-~~align: center;">6 ~~

~~</td>~~

~~<td style="text~~-~~align: center~~;~~">v5 ~~

~~</td>~~

~~<td style="text-align: center;">down 5th ~~

~~</td>~~

~~<td style="text-align: center;">Av ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">15 ~~

~~</td>~~

~~<td style="text-align: center;">720 ~~

~~</td>~~

~~<td style="text-align: center;">50/33, 38/~~25~~ ~~

~~</td>~~

~~<td style="text~~-~~align: center;">6# ~~

~~</td>~~

~~<td style="text-align: center;">P5, m6 ~~

~~</td>~~

~~<td style="text-align: center;">perfect 5th, minor~~ 6th~~ ~~

~~</td>~~

~~<td style="text-align: center;">A, Bb ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">~~16~~ ~~

~~</td>~~

~~<td style="text~~-~~align: center;">768 ~~

~~</td>~~

~~<td style="text-align: center;">14/9~~, ~~25/16, 39/25 ~~

~~</td>~~

~~<td style="text-align: center;">7b ~~

~~</td>~~

~~<td style="text-align: center;">^m6 ~~

~~</td>~~

~~<td style="text-align: center;">upminor 6th ~~

~~</td>~~

~~<td style="text-align: center;">Bb^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">17· ~~

~~</td>~~

~~<td style="text-align: center;">816 ~~

~~</td>~~

~~<td style="text-align: center;">8/5 ~~

~~</td>~~

~~<td style="text-align: center;">7 ~~

~~</td>~~

~~<td style="text-align: center;">^^m6 ~~

~~</td>~~

~~<td style="text-align: center;">double-upminor 6th ~~

~~</td>~~

~~<td style="text-align: center;">Bb^^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">18 ~~

~~</td>~~

~~<td style="text-align: center;">864 ~~

~~</td>~~

~~<td style="text-align: center;">64/39, 28/17, 33/20 ~~

~~</td>~~

~~<td style="text-align: center;">7# ~~

~~</td>~~

~~<td style="text-align: center;">vvM6 ~~

~~</td>~~

~~<td style="text-align: center;">double-downmajor 6th ~~

~~</td>~~

~~<td style="text-align: center;">Bvv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">19 ~~

~~</td>~~

~~<td style="text-align: center;">912 ~~

~~</td>~~

~~<td style="text-align: center;">32/19, 17/10 ~~

~~</td>~~

~~<td style="text-align: center;">8b ~~

~~</td>~~

~~<td style="text-align: center;">vM6 ~~

~~</td>~~

~~<td style="text-align: center;">downmajor 6th ~~

~~</td>~~

~~<td style="text-align: center;">Bv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">20· ~~

~~</td>~~

~~<td style="text-align: center;">960 ~~

~~</td>~~

~~<td style="text-align: center;">7/4 ~~

~~</td>~~

~~<td style="text-align: center;">8 ~~

~~</td>~~

~~<td style="text-align: center;">M6, m7 ~~

~~</td>~~

~~<td style="text-align: center;">major 6th, minor 7th ~~

~~</td>~~

~~<td style="text-align: center;">B, C ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">21 ~~

~~</td>~~

~~<td style="text-align: center;">1008 ~~

~~</td>~~

~~<td style="text-align: center;">16/9, 9/5, 34/19 ~~

~~</td>~~

~~<td style="text-align: center;">8# ~~

~~</td>~~

~~<td style="text-align: center;">^m7 ~~

~~</td>~~

~~<td style="text-align: center;">upminor 7th ~~

~~</td>~~

~~<td style="text-align: center;">C^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">22 ~~

~~</td>~~

~~<td style="text-align: center;">1056 ~~

~~</td>~~

~~<td style="text-align: center;">11/6, 35/19 ~~

~~</td>~~

~~<td style="text-align: center;">9b ~~

~~</td>~~

~~<td style="text-align: center;">^^m7 ~~

~~</td>~~

~~<td style="text-align: center;">double-upminor 7th ~~

~~</td>~~

~~<td style="text-align: center;">C^^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">23 ~~

~~</td>~~

~~<td style="text-align: center;">1104 ~~

~~</td>~~

~~<td style="text-align: center;">32/17, 17/9, 19/10 ~~

~~</td>~~

~~<td style="text-align: center;">9 ~~

~~</td>~~

~~<td style="text-align: center;">vvM7 ~~

~~</td>~~

~~<td style="text-align: center;">double-downmajor 7th ~~

~~</td>~~

~~<td style="text-align: center;">C#vv ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">24 ~~

~~</td>~~

~~<td style="text-align: center;">1152 ~~

~~</td>~~

~~<td style="text-align: center;">33/17, 64/33, 76/39 ~~

~~</td>~~

~~<td style="text-align: center;">9#/1b ~~

~~</td>~~

~~<td style="text-align: center;">vM7 ~~

~~</td>~~

~~<td style="text-align: center;">downmajor 7th ~~

~~</td>~~

~~<td style="text-align: center;">C#v ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">25 ~~

~~</td>~~

~~<td style="text-align: center;">1200 ~~

~~</td>~~

~~<td style="text-align: center;">2/1 ~~

~~</td>~~

~~<td style="text-align: center;">1 ~~

~~</td>~~

~~<td style="text-align: center;">P8 ~~

~~</td>~~

~~<td style="text-align: center;">perfect 8ve ~~

~~</td>~~

~~<td style="text-align: center;">C#, D ~~

~~</td>~~

~~</tr>~~

~~</table>~~

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

=Commas=

~~ ~~

25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)

<object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/25ed2-001.svg">alt : Your browser has no SVG support.</object>

~~ ~~

<div class="objectEmbed"><a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/25ed2-001.svg/489051424/25ed2-001.svg');"><img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="25ed2-001.svg" /></a><div><a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/25ed2-001.svg/489051424/25ed2-001.svg');" class="filename" title="25ed2-001.svg">25ed2-001.svg</a> <ul><li><a href="/file/detail/25ed2-001.svg">Details</a></li><li><a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg">Download</a></li><li style="color: #666">23 KB</li></ul></div></div>

~~ ~~

<h1 id="toc3"><a name="Relationship to Armodue"></a>Relationship to Armodue</h1>

~~ ~~

Like <a class="wiki_link" href="/16edo">16-EDO</a> and <a class="wiki_link" href="/23edo">23-EDO</a>, 25-EDO contains the 9-note &quot;Superdiatonic&quot; scale of <a class="wiki_link" href="/7L%202s">7L2s</a> (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 <a class="wiki_link" href="/Armodue%20theory">Armodue</a> notation system in much the same way that <a class="wiki_link" href="/19edo">19-EDO</a> 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.

~~ ~~

~~<h1 id="toc4"><a name="Commas"></a>Commas</h1>~~

25 EDO tempers out the following commas. (Note: This assumes the val &~~amp;~~lt; 25 40 58 70 86 93 |.)~~ ~~

{| class="wikitable"

|-

! | Comma

! | Monzo

! | Value (Cents)

! | Name 1

! | Name 2

! | Name 3

|-

| style="text-align:center;" | 256/243

| | | 8 -5 >

| style="text-align:right;" | 90.22

| style="text-align:center;" | Limma

| style="text-align:center;" | Pythagorean Minor 2nd

| style="text-align:center;" |

|-

| style="text-align:center;" | 3125/3072

| | | -10 -1 5 >

| style="text-align:right;" | 29.61

| style="text-align:center;" | Small Diesis

| style="text-align:center;" | Magic Comma

| style="text-align:center;" |

|-

| style="text-align:center;" |

| | | 38 -2 -15 >

| style="text-align:right;" | 1.38

| style="text-align:center;" | Hemithirds Comma

| style="text-align:center;" |

|-

| style="text-align:center;" | 16807/16384

| | | -14 0 0 5 >

| style="text-align:right;" | 44.13

| |

|-

| style="text-align:center;" | 49/48

| | | -4 -1 0 2 >

| style="text-align:right;" | 35.70

| style="text-align:center;" | Slendro Diesis

| style="text-align:center;" |

|-

| style="text-align:center;" | 64/63

| | | 6 -2 0 -1 >

| style="text-align:right;" | 27.26

| style="text-align:center;" | Septimal Comma

| style="text-align:center;" | Archytas' Comma

| style="text-align:center;" | Leipziger Komma

|-

| style="text-align:center;" | 3125/3087

| | | 0 -2 5 -3 >

| style="text-align:right;" | 21.18

| style="text-align:center;" | Gariboh

| style="text-align:center;" |

|-

| style="text-align:center;" | 50421/50000

| | | -4 1 -5 5 >

| style="text-align:right;" | 14.52

| style="text-align:center;" | Trimyna

| style="text-align:center;" |

|-

| style="text-align:center;" | 1029/1024

| | | -10 1 0 3 >

| style="text-align:right;" | 8.43

| style="text-align:center;" | Gamelisma

| style="text-align:center;" |

|-

| style="text-align:center;" | 3136/3125

| | | 6 0 -5 2 >

| style="text-align:right;" | 6.08

| style="text-align:center;" | Hemimean

| style="text-align:center;" |

|-

| style="text-align:center;" | 65625/65536

| | | -16 1 5 1 >

| style="text-align:right;" | 2.35

| style="text-align:center;" | Horwell

| style="text-align:center;" |

|-

| style="text-align:center;" | 100/99

| | | 2 -2 2 0 -1 >

| style="text-align:right;" | 17.40

| style="text-align:center;" | Ptolemisma

| style="text-align:center;" |

|-

| style="text-align:center;" | 176/175

| | | 4 0 -2 -1 1 >

| style="text-align:right;" | 9.86

| style="text-align:center;" | Valinorsma

| style="text-align:center;" |

|-

| style="text-align:center;" | 91/90

| | | -1 -2 -1 1 0 1 >

| style="text-align:right;" | 19.13

| style="text-align:center;" | Superleap

| style="text-align:center;" |

|-

| style="text-align:center;" | 676/675

| | | 2 -3 -2 0 0 2 >

| style="text-align:right;" | 2.56

| style="text-align:center;" | Parizeksma

| style="text-align:center;" |

|}

~~<table class~~=~~"wiki_table">~~

=A 25edo keyboard=

~~<tr>~~

~~<th>Comma ~~

~~</th>~~

~~<th>Monzo ~~

~~</th>~~

~~<th>Value (Cents) ~~

~~</th>~~

~~<th>Name 1 ~~

~~</th>~~

~~<th>Name 2 ~~

~~</th>~~

~~<th>Name 3 ~~

~~</th>~~

~~</tr>~~

~~<tr>~~

~~<td style~~=~~"text-align: center;">256/243 ~~

~~</td>~~

~~<td style="text-align: left;">| 8 -5 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">90.22 ~~

~~</td>~~

~~<td style="text-align: center;">Limma ~~

~~</td>~~

~~<td style="text-align: center;">Pythagorean Minor 2nd ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3125/3072 ~~

~~</td>~~

~~<td style="text-align: left;">| -10 -1 5 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">29.61 ~~

~~</td>~~

~~<td style="text-align: center;">Small Diesis ~~

~~</td>~~

~~<td style="text-align: center;">Magic Comma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: left;">| 38 -2 -15 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">1.38 ~~

~~</td>~~

~~<td style="text-align: center;">Hemithirds Comma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">16807/16384 ~~

~~</td>~~

~~<td>| -14 0 0 5 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">44.13 ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">49/48 ~~

~~</td>~~

~~<td style="text-align: left;">| -4 -1 0 2 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">35.70 ~~

~~</td>~~

~~<td style="text-align: center;">Slendro Diesis ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">64/63 ~~

~~</td>~~

~~<td style="text-align: left;">| 6 -2 0 -1 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">27.26 ~~

~~</td>~~

~~<td style="text-align: center;">Septimal Comma ~~

~~</td>~~

~~<td style="text-align: center;">Archytas' Comma ~~

~~</td>~~

~~<td style="text-align: center;">Leipziger Komma ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3125/3087 ~~

~~</td>~~

~~<td style="text-align: left;">| 0 -2 5 -3 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">21.18 ~~

~~</td>~~

~~<td style="text-align: center;">Gariboh ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">50421/50000 ~~

~~</td>~~

~~<td style="text-align: left;">| -4 1 -5 5 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">14.52 ~~

~~</td>~~

~~<td style="text-align: center;">Trimyna ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">1029/1024 ~~

~~</td>~~

~~<td style="text-align: left;">| -10 1 0 3 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">8.43 ~~

~~</td>~~

~~<td style="text-align: center;">Gamelisma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3136/3125 ~~

~~</td>~~

~~<td style="text-align: left;">| 6 0 -5 2 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">6.08 ~~

~~</td>~~

~~<td style="text-align: center;">Hemimean ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">65625/65536 ~~

~~</td>~~

~~<td style="text-align: left;">| -16 1 5 1 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">2.35 ~~

~~</td>~~

~~<td style="text-align: center;">Horwell ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">100/99 ~~

~~</td>~~

~~<td style="text-align: left;">| 2 -2 2 0 -1 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">17.40 ~~

~~</td>~~

~~<td style="text-align: center;">Ptolemisma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">176/175 ~~

~~</td>~~

~~<td style="text-align: left;">| 4 0 -2 -1 1 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">9.86 ~~

~~</td>~~

~~<td style="text-align: center;">Valinorsma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">91/90 ~~

~~</td>~~

~~<td style="text-align: left;">| -1 -2 -1 1 0 1 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">19.13 ~~

~~</td>~~

~~<td style="text-align: center;">Superleap ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">676/675 ~~

~~</td>~~

~~<td style="text-align: left;">| 2 -3 -2 0 0 2 &gt; ~~

~~</td>~~

~~<td style="text-align: right;">2.56 ~~

~~</td>~~

~~<td style="text-align: center;">Parizeksma ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~<td style="text-align: center;"> ~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~ ~~

[[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]] [[Category:25edo]]

~~<h1 id="toc5"><a name="A 25edo keyboard"></a>A 25edo keyboard</h1>~~

[[Category:edo]]

~~ ~~

[[Category:keyboard]]

~~<img src="/file/view/~~mm25.PNG~~/179204243/~~mm25.PNG~~" alt="mm25.PNG" title="mm25.PNG" /></body></html></pre></div>~~

[[Category:listen]]

[[Category:subgroup]]

[[Category:todo:unify_precision]]

[[Category:twentuning]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+__FORCETOC__
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+=<span style="color: #006b2e; font-family: 'Times New Roman',Times,serif; font-size: 113%;">25 tone equal temperament</span>=
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-25 18:52:38 UTC</tt>.<br>
-: The original revision id was <tt>602808482</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
-=&lt;span style="color: #006b2e; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;25 tone equal temperament&lt;/span&gt;=
-EDO 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|5/4]]) and 7 ([[7_4|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 divides the [[Octave|octave]] in 25 equal steps of exact size 48 [[cent|cent]]s each. It is a good way to tune the [[Blackwood_temperament|Blackwood temperament]], which takes the very sharp fifths of [[5edo|5EDO]] as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 ([[5/4|5/4]]) and 7 ([[7/4|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 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|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128_125|128/125]] [[diesis]] and two [[septimal tritones]] of [[7_5|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.
+EDO 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|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|8/7]]s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a [[128/125|128/125]] [[diesis|diesis]] and two [[septimal_tritones|septimal tritones]] of [[7/5|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|50EDO]]. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for [[Mavila|mavila]] temperament.
-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.
+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.
 =Music=
-//[[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]]
+''[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]
-[[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://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3|Flat fourth blues]]// by Fabrizio Fulvio Fausto Fiale
-[[media type="file" key="25edochorale.mid" width="300" height="50"]] [[file:25edochorale.mid]] Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The &lt;span class="il"&gt;25edo&lt;/span&gt; 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.")
+[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
-[[media type="file" key="25 edo prelude largo.mid" width="300" height="50"]] [[file:25 edo prelude largo.mid]] Peter Kosmorsky (2011, Blackwood)
-=Intervals=
+''[http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3 Flat fourth blues]'' by Fabrizio Fulvio Fausto Fiale
-||= Degrees ||= Cents ||= Approximate
+[[File:25edochorale.mid]] [[:File:25edochorale.mid|25edochorale.mid]] Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The <span style="">25edo</span> 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.")
-Ratios* ||= Armodue
-Notation ||||||= [[xenharmonic/Ups and Downs Notation|ups and downs notation]] ||
+[[File:25_edo_prelude_largo.mid]] [[:File:25_edo_prelude_largo.mid|25 edo prelude largo.mid]] Peter Kosmorsky (2011, Blackwood)
-||= 0 ||= 0 ||= 1/1 ||= 1 ||= P1 ||= perfect 1sn ||= D, Eb ||
-||= 1 ||= 48 ||= 33/32, 39/38, 34/33 ||= 1# ||= ^1, ^m2 ||= up 1sn, upminor 2nd ||= D^, Eb^ ||
-||= 2 ||= 96 ||= 17/16, 20/19, 18/17 ||= 2b ||= ^^m2 ||= double-upminor 2nd ||= Eb^^ ||
-||= 3 ||= 144 ||= 12/11, 38/35 ||= 2 ||= vvM2 ||= double-downmajor 2nd ||= Evv ||
-||= 4 ||= 192 ||= 9/8, 10/9, 19/17 ||= 2# ||= vM2 ||= downmajor 2nd ||= Ev ||
-||= 5· ||= 240 ||= 8/7 ||= 3b ||= M2, m3 ||= major 2nd, minor 3rd ||= E, F ||
-||= 6 ||= 288 ||= 19/16, 20/17 ||= 3 ||= ^m3 ||= upminor 3rd ||= F^ ||
-||= 7 ||= 336 ||= 39/32, 17/14, 40/33 ||= 3# ||= ^^m3 ||= double-upminor 3rd ||= F^^ ||
-||= 8· ||= 384 ||= 5/4 ||= 4b ||= vvM3 ||= double-downmajor 3rd ||= F#vv ||
-||= 9 ||= 432 ||= 9/7, 32/25, 50/39 ||= 4 ||= vM3 ||= downmajor ||= F#v ||
-||= 10 ||= 480 ||= 33/25, 25/19 ||= 4#/5b ||= M3, P4 ||= major 3rd, perfect 4th ||= F#, G ||
-||= 11· ||= 528 ||= 31/21, 34/25 ||= 5 ||= ^4 ||= up 4th ||= G^ ||
-||= 12 ||= 576 ||= 7/5, 39/28 ||= 5# ||= ^^4,^^d5 ||= double-up 4th,
-double-up dim 5th ||= G^^, Ab^^ ||
-||= 13 ||= 624 ||= 10/7, 56/39 ||= 6b ||= vvA4,vv5 ||= double-down aug 4th,
-double-down 5th ||= G#vv, Avv ||
-||= 14· ||= 672 ||= 42/31, 25/17 ||= 6 ||= v5 ||= down 5th ||= Av ||
-||= 15 ||= 720 ||= 50/33, 38/25 ||= 6# ||= P5, m6 ||= perfect 5th, minor 6th ||= A, Bb ||
-||= 16 ||= 768 ||= 14/9, 25/16, 39/25 ||= 7b ||= ^m6 ||= upminor 6th ||= Bb^ ||
-||= 17· ||= 816 ||= 8/5 ||= 7 ||= ^^m6 ||= double-upminor 6th ||= Bb^^ ||
-||= 18 ||= 864 ||= 64/39, 28/17, 33/20 ||= 7# ||= vvM6 ||= double-downmajor 6th ||= Bvv ||
-||= 19 ||= 912 ||= 32/19, 17/10 ||= 8b ||= vM6 ||= downmajor 6th ||= Bv ||
-||= 20· ||= 960 ||= 7/4 ||= 8 ||= M6, m7 ||= major 6th, minor 7th ||= B, C ||
-||= 21 ||= 1008 ||= 16/9, 9/5, 34/19 ||= 8# ||= ^m7 ||= upminor 7th ||= C^ ||
-||= 22 ||= 1056 ||= 11/6, 35/19 ||= 9b ||= ^^m7 ||= double-upminor 7th ||= C^^ ||
-||= 23 ||= 1104 ||= 32/17, 17/9, 19/10 ||= 9 ||= vvM7 ||= double-downmajor 7th ||= C#vv ||
-||= 24 ||= 1152 ||= 33/17, 64/33, 76/39 ||= 9#/1b ||= vM7 ||= downmajor 7th ||= C#v ||
-||= 25 ||= 1200 ||= 2/1 ||= 1 ||= P8 ||= perfect 8ve ||= C#, D ||
-*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.
-[[media type="custom" key="25100128"]]
+=Intervals=
-[[file:25ed2-001.svg]]
+{| class="wikitable"
+|-
+| style="text-align:center;" | Degrees
+| style="text-align:center;" | Cents
+| style="text-align:center;" | Approximate
+Ratios*
+| style="text-align:center;" | Armodue
-=Relationship to Armodue=
+Notation
+| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]]
+|-
+| style="text-align:center;" | 0
+| style="text-align:center;" | 0
+| style="text-align:center;" | 1/1
+| style="text-align:center;" | 1
+| style="text-align:center;" | P1
+| style="text-align:center;" | perfect 1sn
+| style="text-align:center;" | D, Eb
+|-
+| style="text-align:center;" | 1
+| style="text-align:center;" | 48
+| style="text-align:center;" | 33/32, 39/38, 34/33
+| style="text-align:center;" | 1#
+| style="text-align:center;" | ^1, ^m2
+| style="text-align:center;" | up 1sn, upminor 2nd
+| style="text-align:center;" | D^, Eb^
+|-
+| style="text-align:center;" | 2
+| style="text-align:center;" | 96
+| style="text-align:center;" | 17/16, 20/19, 18/17
+| style="text-align:center;" | 2b
+| style="text-align:center;" | ^^m2
+| style="text-align:center;" | double-upminor 2nd
+| style="text-align:center;" | Eb^^
+|-
+| style="text-align:center;" | 3
+| style="text-align:center;" | 144
+| style="text-align:center;" | 12/11, 38/35
+| style="text-align:center;" | 2
+| style="text-align:center;" | vvM2
+| style="text-align:center;" | double-downmajor 2nd
+| style="text-align:center;" | Evv
+|-
+| style="text-align:center;" | 4
+| style="text-align:center;" | 192
+| style="text-align:center;" | 9/8, 10/9, 19/17
+| style="text-align:center;" | 2#
+| style="text-align:center;" | vM2
+| style="text-align:center;" | downmajor 2nd
+| style="text-align:center;" | Ev
+|-
+| style="text-align:center;" | 5·
+| style="text-align:center;" | 240
+| style="text-align:center;" | 8/7
+| style="text-align:center;" | 3b
+| style="text-align:center;" | M2, m3
+| style="text-align:center;" | major 2nd, minor 3rd
+| style="text-align:center;" | E, F
+|-
+| style="text-align:center;" | 6
+| style="text-align:center;" | 288
+| style="text-align:center;" | 19/16, 20/17
+| style="text-align:center;" | 3
+| style="text-align:center;" | ^m3
+| style="text-align:center;" | upminor 3rd
+| style="text-align:center;" | F^
+|-
+| style="text-align:center;" | 7
+| style="text-align:center;" | 336
+| style="text-align:center;" | 39/32, 17/14, 40/33
+| style="text-align:center;" | 3#
+| style="text-align:center;" | ^^m3
+| style="text-align:center;" | double-upminor 3rd
+| style="text-align:center;" | F^^
+|-
+| style="text-align:center;" | 8·
+| style="text-align:center;" | 384
+| style="text-align:center;" | 5/4
+| style="text-align:center;" | 4b
+| style="text-align:center;" | vvM3
+| style="text-align:center;" | double-downmajor 3rd
+| style="text-align:center;" | F#vv
+|-
+| style="text-align:center;" | 9
+| style="text-align:center;" | 432
+| style="text-align:center;" | 9/7, 32/25, 50/39
+| style="text-align:center;" | 4
+| style="text-align:center;" | vM3
+| style="text-align:center;" | downmajor
+| style="text-align:center;" | F#v
+|-
+| style="text-align:center;" | 10
+| style="text-align:center;" | 480
+| style="text-align:center;" | 33/25, 25/19
+| style="text-align:center;" | 4#/5b
+| style="text-align:center;" | M3, P4
+| style="text-align:center;" | major 3rd, perfect 4th
+| style="text-align:center;" | F#, G
+|-
+| style="text-align:center;" | 11·
+| style="text-align:center;" | 528
+| style="text-align:center;" | 31/21, 34/25
+| style="text-align:center;" | 5
+| style="text-align:center;" | ^4
+| style="text-align:center;" | up 4th
+| style="text-align:center;" | G^
+|-
+| style="text-align:center;" | 12
+| style="text-align:center;" | 576
+| style="text-align:center;" | 7/5, 39/28
+| style="text-align:center;" | 5#
+| style="text-align:center;" | ^^4,^^d5
+| style="text-align:center;" | double-up 4th,
-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.
+double-up dim 5th
+| style="text-align:center;" | G^^, Ab^^
+|-
+| style="text-align:center;" | 13
+| style="text-align:center;" | 624
+| style="text-align:center;" | 10/7, 56/39
+| style="text-align:center;" | 6b
+| style="text-align:center;" | vvA4,vv5
+| style="text-align:center;" | double-down aug 4th,
-=Commas=
+double-down 5th
-EDO tempers out the following commas. (Note: This assumes the val &lt; 25 40 58 70 86 93 |.)
+| style="text-align:center;" | G#vv, Avv
-||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
+|-
-||= 256/243 ||&lt; | 8 -5 &gt; ||&gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=   ||
+| style="text-align:center;" | 14·
-||= 3125/3072 ||&lt; | -10 -1 5 &gt; ||&gt; 29.61 ||= Small Diesis ||= Magic Comma ||=   ||
+| style="text-align:center;" | 672
-||=   ||&lt; | 38 -2 -15 &gt; ||&gt; 1.38 ||= Hemithirds Comma ||=   ||=   ||
+| style="text-align:center;" | 42/31, 25/17
-||= 16807/16384 || | -14 0 0 5 &gt; ||&gt; 44.13 ||   ||   ||   ||
+| style="text-align:center;" | 6
-||= 49/48 ||&lt; | -4 -1 0 2 &gt; ||&gt; 35.70 ||= Slendro Diesis ||=   ||=   ||
+| style="text-align:center;" | v5
-||= 64/63 ||&lt; | 6 -2 0 -1 &gt; ||&gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
+| style="text-align:center;" | down 5th
-||= 3125/3087 ||&lt; | 0 -2 5 -3 &gt; ||&gt; 21.18 ||= Gariboh ||=   ||=   ||
+| style="text-align:center;" | Av
-||= 50421/50000 ||&lt; | -4 1 -5 5 &gt; ||&gt; 14.52 ||= Trimyna ||=   ||=   ||
+|-
-||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=   ||=   ||
+| style="text-align:center;" | 15
-||= 3136/3125 ||&lt; | 6 0 -5 2 &gt; ||&gt; 6.08 ||= Hemimean ||=   ||=   ||
+| style="text-align:center;" | 720
-||= 65625/65536 ||&lt; | -16 1 5 1 &gt; ||&gt; 2.35 ||= Horwell ||=   ||=   ||
+| style="text-align:center;" | 50/33, 38/25
-||= 100/99 ||&lt; | 2 -2 2 0 -1 &gt; ||&gt; 17.40 ||= Ptolemisma ||=   ||=   ||
+| style="text-align:center;" | 6#
-||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=   ||=   ||
+| style="text-align:center;" | P5, m6
-||= 91/90 ||&lt; | -1 -2 -1 1 0 1 &gt; ||&gt; 19.13 ||= Superleap ||=   ||=   ||
+| style="text-align:center;" | perfect 5th, minor 6th
-||= 676/675 ||&lt; | 2 -3 -2 0 0 2 &gt; ||&gt; 2.56 ||= Parizeksma ||=   ||=   ||
+| style="text-align:center;" | A, Bb
+|-
+| style="text-align:center;" | 16
+| style="text-align:center;" | 768
+| style="text-align:center;" | 14/9, 25/16, 39/25
+| style="text-align:center;" | 7b
+| style="text-align:center;" | ^m6
+| style="text-align:center;" | upminor 6th
+| style="text-align:center;" | Bb^
+|-
+| style="text-align:center;" | 17·
+| style="text-align:center;" | 816
+| style="text-align:center;" | 8/5
+| style="text-align:center;" | 7
+| style="text-align:center;" | ^^m6
+| style="text-align:center;" | double-upminor 6th
+| style="text-align:center;" | Bb^^
+|-
+| style="text-align:center;" | 18
+| style="text-align:center;" | 864
+| style="text-align:center;" | 64/39, 28/17, 33/20
+| style="text-align:center;" | 7#
+| style="text-align:center;" | vvM6
+| style="text-align:center;" | double-downmajor 6th
+| style="text-align:center;" | Bvv
+|-
+| style="text-align:center;" | 19
+| style="text-align:center;" | 912
+| style="text-align:center;" | 32/19, 17/10
+| style="text-align:center;" | 8b
+| style="text-align:center;" | vM6
+| style="text-align:center;" | downmajor 6th
+| style="text-align:center;" | Bv
+|-
+| style="text-align:center;" | 20·
+| style="text-align:center;" | 960
+| style="text-align:center;" | 7/4
+| style="text-align:center;" | 8
+| style="text-align:center;" | M6, m7
+| style="text-align:center;" | major 6th, minor 7th
+| style="text-align:center;" | B, C
+|-
+| style="text-align:center;" | 21
+| style="text-align:center;" | 1008
+| style="text-align:center;" | 16/9, 9/5, 34/19
+| style="text-align:center;" | 8#
+| style="text-align:center;" | ^m7
+| style="text-align:center;" | upminor 7th
+| style="text-align:center;" | C^
+|-
+| style="text-align:center;" | 22
+| style="text-align:center;" | 1056
+| style="text-align:center;" | 11/6, 35/19
+| style="text-align:center;" | 9b
+| style="text-align:center;" | ^^m7
+| style="text-align:center;" | double-upminor 7th
+| style="text-align:center;" | C^^
+|-
+| style="text-align:center;" | 23
+| style="text-align:center;" | 1104
+| style="text-align:center;" | 32/17, 17/9, 19/10
+| style="text-align:center;" | 9
+| style="text-align:center;" | vvM7
+| style="text-align:center;" | double-downmajor 7th
+| style="text-align:center;" | C#vv
+|-
+| style="text-align:center;" | 24
+| style="text-align:center;" | 1152
+| style="text-align:center;" | 33/17, 64/33, 76/39
+| style="text-align:center;" | 9#/1b
+| style="text-align:center;" | vM7
+| style="text-align:center;" | downmajor 7th
+| style="text-align:center;" | C#v
+|-
+| style="text-align:center;" | 25
+| style="text-align:center;" | 1200
+| style="text-align:center;" | 2/1
+| style="text-align:center;" | 1
+| style="text-align:center;" | P8
+| style="text-align:center;" | perfect 8ve
+| style="text-align:center;" | C#, D
+|}
+*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.
-=A 25edo keyboard=
+[[File:25ed2-001.svg|alt=alt : Your browser has no SVG support.]]
-[[image:mm25.PNG]]</pre></div>
+[[:File:25ed2-001.svg|25ed2-001.svg]]
-<h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;25edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:15:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;a href="#x25 tone equal temperament"&gt;25 tone equal temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Relationship to Armodue"&gt;Relationship to Armodue&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#A 25edo keyboard"&gt;A 25edo keyboard&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x25 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;&lt;span style="color: #006b2e; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&gt;25 tone equal temperament&lt;/span&gt;&lt;/h1&gt;
- &lt;br /&gt;
-EDO divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; in 25 equal steps of exact size 48 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is a good way to tune the &lt;a class="wiki_link" href="/Blackwood%20temperament"&gt;Blackwood temperament&lt;/a&gt;, which takes the very sharp fifths of &lt;a class="wiki_link" href="/5EDO"&gt;5EDO&lt;/a&gt; as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;) and 7 (&lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&amp;amp;25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.&lt;br /&gt;
-&lt;br /&gt;
-EDO 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 &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt;. It therefore makes sense to use it as a 2.5.7 &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; tuning. Looking just at 2, 5, and 7, it equates five &lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt; &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; and two &lt;a class="wiki_link" href="/septimal%20tritones"&gt;septimal tritones&lt;/a&gt; of &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt; 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 &lt;a class="wiki_link" href="/50EDO"&gt;50EDO&lt;/a&gt;. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for &lt;a class="wiki_link" href="/mavila"&gt;mavila&lt;/a&gt; temperament.&lt;br /&gt;
-&lt;br /&gt;
-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 &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;2*25 subgroup&lt;/a&gt; 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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Music&lt;/h1&gt;
- &lt;em&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Rapoport/StudyInFives.mp3" rel="nofollow"&gt;Study in Fives&lt;/a&gt;&lt;/em&gt; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow"&gt;Paul Rapoport&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2377" rel="nofollow"&gt;Fantasy for Piano in 25 Note per Octave Tuning&lt;/a&gt; &lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/25edo/fantasy_for_piano_in_25_edo.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/em&gt; by Chris Vaisvil&lt;br /&gt;
-&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Fiale/flat%20fourth%20blues.mp3" rel="nofollow"&gt;Flat fourth blues&lt;/a&gt;&lt;/em&gt; by Fabrizio Fulvio Fausto Fiale&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/25edochorale.mid?h=50&amp;amp;w=300&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;25edochorale.mid&amp;amp;quot; width=&amp;amp;quot;300&amp;amp;quot; height=&amp;amp;quot;50&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;50&amp;quot; width=&amp;quot;300&amp;quot;/&amp;gt; --&gt;&lt;embed type="audio/midi" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/25edochorale.mid" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt; &lt;!-- ws:start:WikiTextFileRule:680:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/25edochorale.mid?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@25edochorale.mid&amp;quot; title=&amp;quot;File: 25edochorale.mid&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/25edochorale.mid/314425794/25edochorale.mid" onclick="ws.common.trackFileLink('/file/view/25edochorale.mid/314425794/25edochorale.mid');"&gt;&lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="25edochorale.mid" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/25edochorale.mid/314425794/25edochorale.mid" onclick="ws.common.trackFileLink('/file/view/25edochorale.mid/314425794/25edochorale.mid');" class="filename" title="25edochorale.mid"&gt;25edochorale.mid&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/25edochorale.mid"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/25edochorale.mid/314425794/25edochorale.mid"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;4 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:680 --&gt; Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded &amp;quot;The &lt;span class="il"&gt;25edo&lt;/span&gt; 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.&amp;quot;)&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMediaRule:1:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/25%20edo%20prelude%20largo.mid?h=50&amp;amp;w=300&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;25 edo prelude largo.mid&amp;amp;quot; width=&amp;amp;quot;300&amp;amp;quot; height=&amp;amp;quot;50&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;50&amp;quot; width=&amp;quot;300&amp;quot;/&amp;gt; --&gt;&lt;embed type="audio/midi" style="cursor:hand; cursor:pointer;" src="http://xenharmonic.wikispaces.com/file/view/25+edo+prelude+largo.mid" width="300" height="50" autoplay="false" target="myself" controller="true" loop="false" scale="aspect" bgcolor="#FFFFFF" pluginspage="http://www.apple.com/quicktime/download/"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:1 --&gt; &lt;!-- ws:start:WikiTextFileRule:681:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/25%20edo%20prelude%20largo.mid?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@25 edo prelude largo.mid&amp;quot; title=&amp;quot;File: 25 edo prelude largo.mid&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid" onclick="ws.common.trackFileLink('/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid');"&gt;&lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="25 edo prelude largo.mid" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid" onclick="ws.common.trackFileLink('/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid');" class="filename" title="25 edo prelude largo.mid"&gt;25 edo prelude largo.mid&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/25%20edo%20prelude%20largo.mid"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/25%20edo%20prelude%20largo.mid/314425914/25%20edo%20prelude%20largo.mid"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;6 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:681 --&gt; Peter Kosmorsky (2011, Blackwood)&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Intervals&lt;/h1&gt;
- &lt;br /&gt;
+=Relationship to Armodue=
-&lt;table class="wiki_table"&gt;
+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.
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;Degrees&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Cents&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
-Ratios*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Armodue&lt;br /&gt;
-Notation&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td colspan="3" style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;ups and downs notation&lt;/a&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;perfect 1sn&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D, Eb&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;48&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;33/32, 39/38, 34/33&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^1, ^m2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 1sn, upminor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D^, Eb^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;96&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;17/16, 20/19, 18/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^^m2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-upminor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Eb^^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;144&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;12/11, 38/35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vvM2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-downmajor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Evv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;192&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9/8, 10/9, 19/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vM2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;downmajor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Ev&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;240&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;M2, m3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;major 2nd, minor 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;E, F&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;288&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;19/16, 20/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^m3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;upminor 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;336&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;39/32, 17/14, 40/33&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;3#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^^m3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-upminor 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F^^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;8·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;384&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;4b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vvM3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-downmajor 3rd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F#vv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;432&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9/7, 32/25, 50/39&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vM3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;downmajor&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F#v&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;480&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;33/25, 25/19&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;4#/5b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;M3, P4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;major 3rd, perfect 4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F#, G&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;11·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;528&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;31/21, 34/25&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;up 4th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;576&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/5, 39/28&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^^4,^^d5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-up 4th,&lt;br /&gt;
-double-up dim 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G^^, Ab^^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;624&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10/7, 56/39&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;6b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vvA4,vv5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-down aug 4th,&lt;br /&gt;
-double-down 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G#vv, Avv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;14·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;672&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;42/31, 25/17&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;v5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;down 5th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Av&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;720&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;50/33, 38/25&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;6#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;P5, m6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;perfect 5th, minor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;A, Bb&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;768&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;14/9, 25/16, 39/25&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^m6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;upminor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Bb^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;17·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;816&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^^m6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-upminor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Bb^^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;864&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;64/39, 28/17, 33/20&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vvM6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-downmajor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Bvv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;912&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;32/19, 17/10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vM6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;downmajor 6th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Bv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;20·&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;960&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;M6, m7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;major 6th, minor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;B, C&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;21&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1008&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;16/9, 9/5, 34/19&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;8#&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^m7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;upminor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1056&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;11/6, 35/19&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;^^m7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-upminor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C^^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;23&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1104&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;32/17, 17/9, 19/10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vvM7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;double-downmajor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C#vv&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;24&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1152&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;33/17, 64/33, 76/39&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;9#/1b&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;vM7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;downmajor 7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C#v&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;25&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1200&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;perfect 8ve&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;C#, D&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.&lt;br /&gt;
+=Commas=
-&lt;br /&gt;
+EDO tempers out the following commas. (Note: This assumes the val &lt; 25 40 58 70 86 93 |.)
-&lt;!-- ws:start:WikiTextMediaRule:2:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/25100128?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;25100128&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/25ed2-001.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextFileRule:682:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/25ed2-001.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@25ed2-001.svg&amp;quot; title=&amp;quot;File: 25ed2-001.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/25ed2-001.svg/489051424/25ed2-001.svg');"&gt;&lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="25ed2-001.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/25ed2-001.svg/489051424/25ed2-001.svg');" class="filename" title="25ed2-001.svg"&gt;25ed2-001.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/25ed2-001.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/25ed2-001.svg/489051424/25ed2-001.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;23 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:682 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Relationship to Armodue"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Relationship to Armodue&lt;/h1&gt;
- &lt;br /&gt;
-Like &lt;a class="wiki_link" href="/16edo"&gt;16-EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/23edo"&gt;23-EDO&lt;/a&gt;, 25-EDO contains the 9-note &amp;quot;Superdiatonic&amp;quot; scale of &lt;a class="wiki_link" href="/7L%202s"&gt;7L2s&lt;/a&gt; (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 &lt;a class="wiki_link" href="/Armodue%20theory"&gt;Armodue&lt;/a&gt; notation system in much the same way that &lt;a class="wiki_link" href="/19edo"&gt;19-EDO&lt;/a&gt; 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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Commas&lt;/h1&gt;
-EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 25 40 58 70 86 93 |.)&lt;br /&gt;
+{| class="wikitable"
+|-
+! | Comma
+! | Monzo
+! | Value (Cents)
+! | Name 1
+! | Name 2
+! | Name 3
+|-
+| style="text-align:center;" | 256/243
+| | | 8 -5 &gt;
+| style="text-align:right;" | 90.22
+| style="text-align:center;" | Limma
+| style="text-align:center;" | Pythagorean Minor 2nd
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 3125/3072
+| | | -10 -1 5 &gt;
+| style="text-align:right;" | 29.61
+| style="text-align:center;" | Small Diesis
+| style="text-align:center;" | Magic Comma
+| style="text-align:center;" |
+|-
+| style="text-align:center;" |
+| | | 38 -2 -15 &gt;
+| style="text-align:right;" | 1.38
+| style="text-align:center;" | Hemithirds Comma
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 16807/16384
+| | | -14 0 0 5 &gt;
+| style="text-align:right;" | 44.13
+| |
+| |
+| |
+|-
+| style="text-align:center;" | 49/48
+| | | -4 -1 0 2 &gt;
+| style="text-align:right;" | 35.70
+| style="text-align:center;" | Slendro Diesis
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 64/63
+| | | 6 -2 0 -1 &gt;
+| style="text-align:right;" | 27.26
+| style="text-align:center;" | Septimal Comma
+| style="text-align:center;" | Archytas' Comma
+| style="text-align:center;" | Leipziger Komma
+|-
+| style="text-align:center;" | 3125/3087
+| | | 0 -2 5 -3 &gt;
+| style="text-align:right;" | 21.18
+| style="text-align:center;" | Gariboh
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 50421/50000
+| | | -4 1 -5 5 &gt;
+| style="text-align:right;" | 14.52
+| style="text-align:center;" | Trimyna
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 1029/1024
+| | | -10 1 0 3 &gt;
+| style="text-align:right;" | 8.43
+| style="text-align:center;" | Gamelisma
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 3136/3125
+| | | 6 0 -5 2 &gt;
+| style="text-align:right;" | 6.08
+| style="text-align:center;" | Hemimean
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 65625/65536
+| | | -16 1 5 1 &gt;
+| style="text-align:right;" | 2.35
+| style="text-align:center;" | Horwell
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 100/99
+| | | 2 -2 2 0 -1 &gt;
+| style="text-align:right;" | 17.40
+| style="text-align:center;" | Ptolemisma
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 176/175
+| | | 4 0 -2 -1 1 &gt;
+| style="text-align:right;" | 9.86
+| style="text-align:center;" | Valinorsma
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 91/90
+| | | -1 -2 -1 1 0 1 &gt;
+| style="text-align:right;" | 19.13
+| style="text-align:center;" | Superleap
+| style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" | 676/675
+| | | 2 -3 -2 0 0 2 &gt;
+| style="text-align:right;" | 2.56
+| style="text-align:center;" | Parizeksma
+| style="text-align:center;" |
+| style="text-align:center;" |
+|}
-&lt;table class="wiki_table"&gt;
+=A 25edo keyboard=
-    &lt;tr&gt;
-        &lt;th&gt;Comma&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Monzo&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Value (Cents)&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name 1&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name 2&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;Name 3&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 8 -5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;90.22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Limma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Pythagorean Minor 2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3125/3072&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -10 -1 5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;29.61&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Small Diesis&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Magic Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 38 -2 -15 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;1.38&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Hemithirds Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;16807/16384&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;| -14 0 0 5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;44.13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -4 -1 0 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;35.70&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Slendro Diesis&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 6 -2 0 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;27.26&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Septimal Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Archytas' Comma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Leipziger Komma&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3125/3087&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 0 -2 5 -3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;21.18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Gariboh&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;50421/50000&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -4 1 -5 5 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;14.52&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Trimyna&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3136/3125&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 6 0 -5 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;6.08&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Hemimean&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;65625/65536&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -16 1 5 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;2.35&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Horwell&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;100/99&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 2 -2 2 0 -1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;17.40&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Ptolemisma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;91/90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| -1 -2 -1 1 0 1 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;19.13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Superleap&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;676/675&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: left;"&gt;| 2 -3 -2 0 0 2 &amp;gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: right;"&gt;2.56&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Parizeksma&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
+[[File:mm25.PNG|alt=mm25.PNG|mm25.PNG]]      [[Category:25edo]]
-&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="A 25edo keyboard"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;A 25edo keyboard&lt;/h1&gt;
+[[Category:edo]]
- &lt;br /&gt;
+[[Category:keyboard]]
-&lt;!-- ws:start:WikiTextLocalImageRule:679:&amp;lt;img src=&amp;quot;/file/view/mm25.PNG/179204243/mm25.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/mm25.PNG/179204243/mm25.PNG" alt="mm25.PNG" title="mm25.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:679 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+[[Category:listen]]
+[[Category:subgroup]]
+[[Category:todo:unify_precision]]
+[[Category:twentuning]]