48edo
| ← 47edo | 48edo | 49edo → |
48 equal divisions of the octave (abbreviated 48edo or 48ed2), also called 48-tone equal temperament (48tet) or 48 equal temperament (48et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 48 equal parts of exactly 25 ¢ each. Each step represents a frequency ratio of 21/48, or the 48th root of 2.
Theory
Since 48 is a multiple of 12, it has attracted a small amount of interest. However, its best major third, of 375 cents, is over 11 cents flat. An alternative third is the familiar 400 cent major third. Using this third, 48 tunes to the same values as 12 in the 5-limit, but tempers out 2401/2400 in the 7-limit, making it a tuning for squares temperament. In the 11-limit we can add 99/98 and 121/120 to the list, and in the 13-limit, 66/65. While 31edo can also do 13-limit squares, 48 might be preferred for some purposes.
Using its best major third, 48edo tempers out 20000/19683, but 27edo and 34edo do a much better job for this temperament, known as tetracot, since (for example) 48edo's 7L 6s has a rather awkward 6:1 step ratio, while 27edo and 34edo have 3:1 and 4:1 step ratios for the same scale. However in the 7-limit it can be used for doublewide temperament, the half-octave period temperament with minor third generator tempering out 50/49 and 875/864, for which it is the optimal patent val. In the 11-limit, we may add 99/98, leading to 11-limit doublewide for which 48 again gives the optimal patent val. It is also the optimal patent val for the rank three temperament jubilee, which tempers out 50/49 and 99/98.
If 48 is treated as a no-fives system, it still tempers out 99/98 and 243/242 in the 11-limit, leading to a no-fives version of squares for which it does well as a tuning. In the 13 no-fives limit, we can add 144/143 to the list of commas, and we get the no-fives version of 13-limit squares, for which 48 actually defines the optimal patent val. No-fives squares should probably be considered by anyone interested in 48edo; the generator is 17\48, a 425 ¢ interval serving as both 9/7 and 14/11.
Something close to 48edo is what you get if you cross 16edo with pure fifths, for instance, on a 16-tone guitar. The presence of 12/11 in 16edo allows a string offset of 11/8 to also work for producing perfect fifths. Since 16edo supports mavila temperament, and 48edo has two viable major and minor thirds, 48edo could be used for an unexplored system of "adaptive mavila".
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.0 | -11.3 | +6.2 | -3.9 | -1.3 | +9.5 | +11.7 | -5.0 | +2.5 | +4.2 | -3.3 | +2.4 | -5.9 | -4.6 |
| Relative (%) | -7.8 | -45.3 | +24.7 | -15.6 | -5.3 | +37.9 | +46.9 | -19.8 | +9.9 | +16.9 | -13.1 | +9.5 | -23.5 | -18.3 | |
| Steps (reduced) |
76 (28) |
111 (15) |
135 (39) |
152 (8) |
166 (22) |
178 (34) |
188 (44) |
196 (4) |
204 (12) |
211 (19) |
217 (25) |
223 (31) |
228 (36) |
233 (41) | |
| Harmonic | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | 47 | 49 | 51 | 53 | 55 | 57 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.0 | -3.3 | -5.1 | -1.3 | +7.5 | -4.1 | -11.5 | +9.8 | +9.5 | +12.3 | -6.9 | +1.5 | +12.4 | +0.5 |
| Relative (%) | +19.9 | -13.1 | -20.6 | -5.4 | +30.1 | -16.2 | -46.1 | +39.1 | +38.0 | +49.4 | -27.6 | +6.0 | +49.5 | +2.1 | |
| Steps (reduced) |
238 (46) |
242 (2) |
246 (6) |
250 (10) |
254 (14) |
257 (17) |
260 (20) |
264 (24) |
267 (27) |
270 (30) |
272 (32) |
275 (35) |
278 (38) |
280 (40) | |
As a tuning standard
A step of 48edo is known as a doamu (second MIDI-resolution unit, 2mu, 22 = 4 equal divisions of the 12edo semitone). The internal data structure of the 2mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and three other bits which are not used.
Subsets and supersets
48edo is the 10th highly composite edo. Since 48 factors into 24 × 3, 48edo has subset edos 2, 3, 4, 6, 8, 12, 16, and 24.
Intervals
| # | Cents | Ups and downs notation |
|---|---|---|
| 0 | 0.0 | D |
| 1 | 25.0 | ^D, ^E♭♭ |
| 2 | 50.0 | ^^D, vvE♭ |
| 3 | 75.0 | vD♯, vE♭ |
| 4 | 100.0 | D♯, E♭ |
| 5 | 125.0 | ^D♯, ^E♭ |
| 6 | 150.0 | ^^D♯, vvE |
| 7 | 175.0 | vD𝄪, vE |
| 8 | 200.0 | E |
| 9 | 225.0 | ^E, ^F♭ |
| 10 | 250.0 | ^^E, vvF |
| 11 | 275.0 | vE♯, vF |
| 12 | 300.0 | F |
| 13 | 325.0 | ^F, ^G♭♭ |
| 14 | 350.0 | ^^F, vvG♭ |
| 15 | 375.0 | vF♯, vG♭ |
| 16 | 400.0 | F♯, G♭ |
| 17 | 425.0 | ^F♯, ^G♭ |
| 18 | 450.0 | ^^F♯, vvG |
| 19 | 475.0 | vF𝄪, vG |
| 20 | 500.0 | G |
| 21 | 525.0 | ^G, ^A♭♭ |
| 22 | 550.0 | ^^G, vvA♭ |
| 23 | 575.0 | vG♯, vA♭ |
| 24 | 600.0 | G♯, A♭ |
| 25 | 625.0 | ^G♯, ^A♭ |
| 26 | 650.0 | ^^G♯, vvA |
| 27 | 675.0 | vG𝄪, vA |
| 28 | 700.0 | A |
| 29 | 725.0 | ^A, ^B♭♭ |
| 30 | 750.0 | ^^A, vvB♭ |
| 31 | 775.0 | vA♯, vB♭ |
| 32 | 800.0 | A♯, B♭ |
| 33 | 825.0 | ^A♯, ^B♭ |
| 34 | 850.0 | ^^A♯, vvB |
| 35 | 875.0 | vA𝄪, vB |
| 36 | 900.0 | B |
| 37 | 925.0 | ^B, ^C♭ |
| 38 | 950.0 | ^^B, vvC |
| 39 | 975.0 | vB♯, vC |
| 40 | 1000.0 | C |
| 41 | 1025.0 | ^C, ^D♭♭ |
| 42 | 1050.0 | ^^C, vvD♭ |
| 43 | 1075.0 | vC♯, vD♭ |
| 44 | 1100.0 | C♯, D♭ |
| 45 | 1125.0 | ^C♯, ^D♭ |
| 46 | 1150.0 | ^^C♯, vvD |
| 47 | 1175.0 | vC𝄪, vD |
| 48 | 1200.0 | D |
Notation
Stein–Zimmermann–Gould notation
Stein–Zimmermann–Gould notation uses sharps and flats combined with quartertone accidentals and arrows:
| Semitones | 0 | 1⁄4 | 1⁄2 | 3⁄4 | 1 | 1+1⁄4 | 1+1⁄2 | 1+3⁄4 | 2 | 2+1⁄4 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | | | | | | | | | | |
| Flat symbol | | | | | | | | | |
Kite's ups and downs notation
48edo can also be notated with Kite's ups and downs, spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
| Semitones | 0 | 1⁄4 | 1⁄2 | 3⁄4 | 1 | 1 1⁄4 | 1 1⁄2 | 1 3⁄4 | 2 | 2 1⁄4 |
|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
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| Flat symbol | |
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Sagittal notation
This notation is a superset of the notations for edos 24, 12, 8, and 6.
Evo flavor
Revo flavor
Evo-SZ flavor
Approximation to JI
Interval mappings
The following tables show how 15-odd-limit intervals are represented in 48edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/6, 12/11 | 0.637 | 2.5 |
| 11/8, 16/11 | 1.318 | 5.3 |
| 3/2, 4/3 | 1.955 | 7.8 |
| 15/13, 26/15 | 2.259 | 9.0 |
| 11/9, 18/11 | 2.592 | 10.4 |
| 13/7, 14/13 | 3.298 | 13.2 |
| 9/8, 16/9 | 3.910 | 15.6 |
| 13/10, 20/13 | 4.214 | 16.9 |
| 15/14, 28/15 | 5.557 | 22.2 |
| 7/4, 8/7 | 6.174 | 24.7 |
| 9/5, 10/9 | 7.404 | 29.6 |
| 11/7, 14/11 | 7.492 | 30.0 |
| 7/5, 10/7 | 7.512 | 30.0 |
| 7/6, 12/7 | 8.129 | 32.5 |
| 5/3, 6/5 | 9.359 | 37.4 |
| 13/8, 16/13 | 9.472 | 37.9 |
| 11/10, 20/11 | 9.996 | 40.0 |
| 9/7, 14/9 | 10.084 | 40.3 |
| 13/11, 22/13 | 10.790 | 43.2 |
| 5/4, 8/5 | 11.314 | 45.3 |
| 13/12, 24/13 | 11.427 | 45.7 |
| 13/9, 18/13 | 11.618 | 46.5 |
| 15/8, 16/15 | 11.731 | 46.9 |
| 15/11, 22/15 | 11.951 | 47.8 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/6, 12/11 | 0.637 | 2.5 |
| 11/8, 16/11 | 1.318 | 5.3 |
| 3/2, 4/3 | 1.955 | 7.8 |
| 11/9, 18/11 | 2.592 | 10.4 |
| 13/7, 14/13 | 3.298 | 13.2 |
| 9/8, 16/9 | 3.910 | 15.6 |
| 7/4, 8/7 | 6.174 | 24.7 |
| 9/5, 10/9 | 7.404 | 29.6 |
| 11/7, 14/11 | 7.492 | 30.0 |
| 7/6, 12/7 | 8.129 | 32.5 |
| 5/3, 6/5 | 9.359 | 37.4 |
| 13/8, 16/13 | 9.472 | 37.9 |
| 11/10, 20/11 | 9.996 | 40.0 |
| 9/7, 14/9 | 10.084 | 40.3 |
| 13/11, 22/13 | 10.790 | 43.2 |
| 5/4, 8/5 | 11.314 | 45.3 |
| 13/12, 24/13 | 11.427 | 45.7 |
| 15/11, 22/15 | 11.951 | 47.8 |
| 15/8, 16/15 | 13.269 | 53.1 |
| 13/9, 18/13 | 13.382 | 53.5 |
| 7/5, 10/7 | 17.488 | 70.0 |
| 15/14, 28/15 | 19.443 | 77.8 |
| 13/10, 20/13 | 20.786 | 83.1 |
| 15/13, 26/15 | 22.741 | 91.0 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 11/6, 12/11 | 0.637 | 2.5 |
| 11/8, 16/11 | 1.318 | 5.3 |
| 3/2, 4/3 | 1.955 | 7.8 |
| 15/13, 26/15 | 2.259 | 9.0 |
| 11/9, 18/11 | 2.592 | 10.4 |
| 13/7, 14/13 | 3.298 | 13.2 |
| 9/8, 16/9 | 3.910 | 15.6 |
| 13/10, 20/13 | 4.214 | 16.9 |
| 15/14, 28/15 | 5.557 | 22.2 |
| 7/4, 8/7 | 6.174 | 24.7 |
| 11/7, 14/11 | 7.492 | 30.0 |
| 7/5, 10/7 | 7.512 | 30.0 |
| 7/6, 12/7 | 8.129 | 32.5 |
| 13/8, 16/13 | 9.472 | 37.9 |
| 9/7, 14/9 | 10.084 | 40.3 |
| 13/11, 22/13 | 10.790 | 43.2 |
| 13/12, 24/13 | 11.427 | 45.7 |
| 15/8, 16/15 | 11.731 | 46.9 |
| 15/11, 22/15 | 13.049 | 52.2 |
| 13/9, 18/13 | 13.382 | 53.5 |
| 5/4, 8/5 | 13.686 | 54.7 |
| 11/10, 20/11 | 15.004 | 60.0 |
| 5/3, 6/5 | 15.641 | 62.6 |
| 9/5, 10/9 | 17.596 | 70.4 |
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 5\48 | 125.0 | 16/15 | Negri |
| 1 | 7\48 | 175.0 | 10/9 | Tetracot |
| 1 | 13\48 | 325.0 | 77/64 | Orgone |
| 1 | 17\48 | 425.0 | 9/7 | Squares |
| 1 | 19\48 | 475.0 | 21/16 | Buzzard |
| 2 | 13\48 | 325.0 | 6/5 | Doublewide |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Octave stretch or compression
Most of 48edo's simple primes have low error, but its 5 is substantially flat, so 48edo can benefit from slight octave stretching. Some stretched-octave 48edo tunings (least to most stretched) include 266zpi, 172ed12, 124ed6, 76edt or 28edf.
Instruments
Music
48 equal divisions of the octave (48edo proper)
- 48edo layout (2025)
- Elements - Fire from Elements (2019–2020)
- from Works for Strings Vol.1 (2020)
- Elysium Planitia for strings (2022)
- Octatonic Groove (2020) – jubilismic[8] in 48edo tuning
- Ricercar, Premier Moment (2011)
- Ricercar, Deuxième Moment (2011)
- Perdu dans le Landmannalaugar (2012)
- Boundary Breaker (2025)
- Neutral Steel (2009) – blog | play
- Two At Once (2009) – blog | play
- Tim's Flutes (2009) – blog | play
- Two At Once 2 (2018) – blog | play
- The Dolomites (2018) – blog | YouTube
- October...walking alone (2025)
Well-tempered derivatives of 48edo
- passing the edge of the universe, my thoughts were of you (for microtonal orchestra) (2026 — video description specifies quarter-tones, but in comments, intervals of ±20 ¢ and ±33 ¢ are listed, and while these can be fudged together for consideration as a 48-well-tempered tuning system, an even higher number of divisions of the octave may be needed)
See also
- Equal multiplications of MIDI-resolution units