25edo
| ← 24edo | 25edo | 26edo → |
25 equal divisions of the octave (abbreviated 25edo or 25ed2), also called 25-tone equal temperament (25tet) or 25 equal temperament (25et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 25 equal parts of exactly 48 ¢ each. Each step represents a frequency ratio of 21/25, or the 25th root of 2.
Theory
25edo is a good way to tune the blackwood temperament, which closes each circle of fifths at five fifths, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (5/4) and 7 (7/4). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.
25edo has fifths 18 cents sharp, but its major thirds of 5/4 are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not consistent. It therefore makes sense to use it as a 2.5.7 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7's with the octave, and so tempers out (8/7)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. In fact, it is a convergent to a melodically optimal "golden" tuning of mavila, at around 672.7 cents.
If 5/4 and 7/4 are not good enough, it also does 17/16 and 19/16, just like 12edo. In fact, on the 2*25 subgroup 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.
Its step of 48 ¢, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having a very high harmonic entropy. This is because the harmonic entropy model is usually tuned to reflect the general perception of quarter-tones being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
Possible usage in Indonesian music
Since 25edo contains 5edo as a subset, and it features an antidiatonic scale generated by the 672 cent fifth, it can theoretically be used to represent Indonesian music in both Slendro (~5edo) and Pelog (~antidiatonic scale) tunings. However, many tunings of pelog are also better represented by the tuning's native 3-2-6-3-2-3-6 omnidiatonic scale.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.0 | -2.3 | -8.8 | -11.9 | -23.3 | +23.5 | +15.7 | -9.0 | -9.5 | +9.2 | -4.3 |
| Relative (%) | +37.6 | -4.8 | -18.4 | -24.8 | -48.6 | +48.9 | +32.8 | -18.7 | -19.8 | +19.2 | -8.9 | |
| Steps (reduced) |
40 (15) |
58 (8) |
70 (20) |
79 (4) |
86 (11) |
93 (18) |
98 (23) |
102 (2) |
106 (6) |
110 (10) |
113 (13) | |
Subsets and supersets
Since 25 is 5 x 5, 25edo is the smallest composite EDO that doesn't have any intervals in common with 12edo. Doubling 25edo to get 50edo produces a good meantone tuning.
Intervals
| 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 | ^^m2 | dupminor 2nd | ^^Eb |
| 3 | 144 | 12/11, 38/35 | 2 | vvM2 | dudmajor 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# | ^^m3 | dupminor 3rd | ^^F |
| 8· | 384 | 5/4 | 4b | vvM3 | dudmajor 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# | ^^4,^^b5 | dup 4th, dupdim 5th | ^^G, ^^Ab |
| 13 | 624 | 10/7, 56/39 | 6b | vvA4,vv5 | dudaug 4th, dud 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 | ^^m6 | dupminor 6th | ^^Bb |
| 18 | 864 | 64/39, 28/17, 33/20 | 7# | vvM6 | dudmajor 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 | ^^m7 | dupminor 7th | ^^C |
| 23 | 1104 | 32/17, 17/9, 19/10 | 9 | vvM7 | dudmajor 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.
Notation
Ups and downs notation
25edo can be notated with ups and downs, spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using triple arrows.
Sagittal notation
This notation uses the same sagittal sequence as 32edo, and is a superset of the notation for 5edo.
Second-best fifth (mavila) notation
This tuning can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.
The first and most common defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. M2 + M2 isn't M3, and D + M2 isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules. Note that the notes that form chords are different from in diatonic: for example, a major chord, P1 – M3 – P5, is approximately 4:5:6 as would be expected, but is notated C-E#-G on C.
Alternatively, one can essentially pretend the native antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. The primary purpose of doing this is to allow music notated in 12edo or another diatonic system to be directly translated "on the fly" (or to allow support for this tuning in tools that only allow chain-of-fifths notation), and it carries over the way interval arithmetic works from diatonic notation, at the cost of notating the sizes of intervals and the shapes of chords incorrectly: that is, a major chord, P1-M3-P5, is notated C-E-G on C, but is no longer ~4:5:6 (since the third is closer to a minor third).
For the sake of clarity, the first notation is commonly called "melodic notation", and the second is called "harmonic notation", but this is a bit of a misnomer as both preserve different features of the notation of harmony.
| P1-M3-P5 ~ 4-5-6 | P1-M3-P5 = C-E-G on C | |
|---|---|---|
| Diatonic notation | No | Yes |
| Antidiatonic notation | Yes | No |
Regular temperament properties
Rank-2 temperaments
| Generator | Periods per octave | "Sharp 3/2" temperaments | "Flat 3/2" temperaments (25b val) | MOS scales |
|---|---|---|---|---|
| 1\25 | 1 | |||
| 2\25 | 1 | Passion | 1L 11s, 12L 1s | |
| 3\25 | 1 | Bleu | 1L 7s, 8L 1s, 8L 9s | |
| 4\25 | 1 | Luna / didacus | 1L 5s, 6L 1s, 6L 7s, 6L 13s | |
| 6\25 | 1 | Gariberttet | 4L 1s, 4L 5s, 4L 9s, 4L 13s, 4L 17s | |
| 7\25 | 1 | Sixix | 4L 3s, 7L 4s, 7L 11s | |
| 8\25 | 1 | Magic | 3L 4s, 3L 7s, 3L 10s, 3L 13s, 3L 16s, 3L 19s | |
| 9\25 | 1 | Hamity | 3L 2s, 3L 5s, 3L 8s, 11L 3s | |
| 11\25 | 1 | Mabila / trismegistus | Armodue / Pelogic (25bd) | 2L 3s, 2L 5s, 7L 2s, 9L 7s |
| 12\25 | 1 | Tritonic | Triton | 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, 2L 13s, 2L 15s, 2L 17s, 2L 19s, 2L 21s |
| 1\25 | 5 | Blackwood favouring 9/7 | 5L 5s, 5L 10s, 5L 15s | |
| 2\25 | 5 | Blackwood favouring 5/4 | 5L 5s, 10L 5s | |
| 1\25 | 25 |
Commas
25et 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 | Blackwood comma, Pythagorean limma |
| 5 | 3125/3072 | [-10 -1 5⟩ | 29.61 | Laquinyo | 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 comma |
| 7 | 49/48 | [-4 -1 0 2⟩ | 35.70 | Zozo | Semaphoresma, 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 comma |
| 7 | 50421/50000 | [-4 1 -5 5⟩ | 14.52 | Quinzogu | Trimyna comma |
| 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 comma |
| 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 comma, biome comma |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.56 | Bithogu | Island comma, parizeksma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Scales
- 6 4 2 3 5 5
Approximated from a hexatonic subset of the dwarf17marv scale. Contains lots of consonances from the 2.3.7.11 subgroup while excluding the familiar harmonic 5.
- 3 3 8 3 8
A pentatonic and MOS scale somewhat resembling pelog.
- Armodue/pelogic[9]
- 3 3 2 3 3 3 2 2 3 3
A 9-tone MOS scale somewhat resembling pelog.
- 5 5 5 5 5
Somewhat resembles slendro. Is the blacksmith/blackwood[5] MOS. Is the same as 5edo.
- Mabila/trismegistus justified pentatonic
- 3 3 9 2 8
A pentatonic subset of the mabila/trismegistus[16] MOS scale, it is those temperaments' pentatonic MOS, but with their complex 3/2 substituted in.
- Mabila/trismegistus justified nonatonic
- 3 3 2 3 4 2 2 2 3 3
A 9-tone subset of the mabila/trismegistus[16] MOS scale, it is those temperaments' 9-tone MOS, but with their complex 3/2 substituted in.
- Magic[13]
- 1 1 5 1 1 1 5 1 1 1 5 1 1
A 13-tone MOS scale. A useful starting point for a scalesmith to find MODMOSes, or to find 4- to 9-tone subsets.
- Amulet[idiosyncratic term]
- 2 1 2 2 1 2 3 2 2 1 2 3 2
A MODMOS of Magic[13]. It is the same as Magic[13], but with 4 tones shifted over by one chroma (the difference between MOS step sizes, in this case 4\25). This gives its intervals a more even spread, which makes it very useable as a chromatic-like scale. Can also be used to take 4- to 9-tone subsets.
- Fennec[idiosyncratic term]
- 3 4 3 5 2 5 2
A subset of the amulet scale. Approximated from the original fennec scale of 14edo.
- Passion[13]
- 2 2 2 2 2 2 1 2 2 2 2 2 2
A 13-tone MOS scale with a lot of consonances available. Can be used as a chromatic-like scale. Can also be used to take 4- to 9-tone subsets.
- Akebono I
- 4 2 9 4 6
A subset of the Passion[13] scale. Approximated from the original Akebono I scale of 12edo.
- Unfair blackwood[10]
- 4 1 4 1 4 1 4 1 4 1
Named "unfair" (by Igliashon Jones) because of the predominance of the larger interval. The major triads come with the large supermajor third.
- Fair blackwood[10]
- 3 2 3 2 3 2 3 2 3 2
Named "fair" (by Igliashon Jones) because larger and smaller interval are more balanced. The major triads come with the nice 5/4 major third.
Relationship to Armodue
Like 16edo and 23edo, 25edo contains the 9-note superdiatonic scale of 7L 2s (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9 or 666.67 cents, to 4\7 or 685.71 cents). The 25edo generator for this scale is the 672-cent interval. This allows 25edo to be used with the Armodue notation system in much the same way that 19edo is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25edo Armodue 6th is flatter than that of 16edo (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.
Keyboard layout
Piano keyboard
Lumatone
See Lumatone mapping for 25edo
Music
Modern renderings
- "Prelude in C major, No. 1" from The Well-Tempered Clavier I, BWV 846 (1722) – played by Stephen Weigel on a lumatone (2024) – mavila in 25edo tuning
- Aria Sopra La Bergamasca – arranged for organ and rendered by Claudi Meneghin (2025)
21st century
- StartingnStoppinLeftnRight (2024)
- Hammock Hut (2024)
- ANDROMEDA (2024)
- Patchouli (2024)
- File:25edochorale.mid (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")
- File:25_edo_prelude_largo.mid (2011, Blackwood)
- Improvisation in 25edo (Akebono I scale) (2025)
- Improvisation in 25edo (Antipental Blues scale, glass unison timbre) (2025)
- Improvisation in 25edo (Antipental Blues scale, platinum inharmonic timbre) (2025)
- Improvisation in 25edo (Armodue5 scale) (2025)
- Improvisation in 25edo (Fennec scale) (2025)
- Rabbit Tracks in the Snow (2025)
- New File (2024)
- Edolian - Sepia (2020)
- Embark (2022)