25edo
Theory
25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the blackwood temperament, which takes the very sharp fifths of 5edo as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (5/4) and 7 (7/4). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not consistent. It therefore makes sense to use it as a 2.5.7 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is 50edo. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.
If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the 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.
Intervals
| Degrees | Cents | Approximate Ratios* |
Armodue Notation |
Ups and Downs notation | ||
| 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 | v~2 | downmid 2nd | ^^Eb |
| 3 | 144 | 12/11, 38/35 | 2 | ^~2 | upmid 2nd | vvE |
| 4 | 192 | 9/8, 10/9, 19/17 | 2# | vM2 | downmajor 2nd | vE |
| 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# | v~3 | downmid 3rd | ^^F |
| 8· | 384 | 5/4 | 4b | ^~3 | upmid 3rd | vvF# |
| 9 | 432 | 9/7, 32/25, 50/39 | 4 | vM3 | downmajor | vF# |
| 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# | v~4,v~5 | downmid 4th, downmid 5th |
^^G, ^^Ab |
| 13 | 624 | 10/7, 56/39 | 6b | ^~4,^~5 | upmid 4th, upmid 5th |
vvG#, vvA |
| 14· | 672 | 42/31, 25/17 | 6 | v5 | down 5th | vA |
| 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 | v~6 | downmid 6th | ^^Bb |
| 18 | 864 | 64/39, 28/17, 33/20 | 7# | ^~6 | upmid 6th | vvB |
| 19 | 912 | 32/19, 17/10 | 8b | vM6 | downmajor 6th | vB |
| 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 | v~7 | downmid 7th | ^^C |
| 23 | 1104 | 32/17, 17/9, 19/10 | 9 | ^~7 | upmid 7th | vvC# |
| 24 | 1152 | 33/17, 64/33, 76/39 | 9#/1b | vM7 | downmajor 7th | vC# |
| 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.
25-edo chords can be named with ups and downs, see Ups and Downs Notation - Chords and Chord Progressions.
Relationship to Armodue
Like 16-EDO and 23-EDO, 25-EDO contains the 9-note "Superdiatonic" scale of 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 notation system in much the same way that 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.
Commas
25 EDO tempers out the following commas. (Note: This assumes the val ⟨25 40 58 70 86 93].)
| Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 3 | 256/243 | [8 -5⟩ | 90.22 | Sawa | Limma, Pythagorean Minor 2nd |
| 5 | 3125/3072 | [-10 -1 5⟩ | 29.61 | Laquinyo | Small Diesis, Magic Comma |
| 5 | (24 digits) | [38 -2 -15⟩ | 1.38 | Sasa-quintrigu | Hemithirds comma |
| 7 | 16807/16384 | [-14 0 0 5⟩ | 44.13 | Laquinzo | Cloudy |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Slendro Diesis |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal Comma, Archytas' Comma, Leipziger Komma |
| 7 | 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyo | Gariboh |
| 7 | 50421/50000 | [-4 1 -5 5⟩ | 14.52 | Quinzogu | Trimyna |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.43 | Latrizo | Gamelisma |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquingu | Hemimean comma |
| 7 | 65625/65536 | [-16 1 5 1⟩ | 2.35 | Lazoquinyo | Horwell |
| 11 | 100/99 | [2 -2 2 0 -1⟩ | 17.40 | Luyoyo | Ptolemisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma |
| 13 | 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.56 | Bithogu | Parizeksma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
A 25edo keyboard
Music
- Study in Fives by Paul Rapoport
- Fantasy for Piano in 25 Note per Octave Tuning play by Chris Vaisvil
- Flat fourth blues by Fabrizio Fulvio Fausto Fiale
- File:25edochorale.mid by Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")
- File:25_edo_prelude_largo.mid by Peter Kosmorsky (2011, Blackwood)