# 48edo

 ← 47edo 48edo 49edo →
Prime factorization 24 × 3
Step size 25¢
Fifth 28\48 (700¢) (→7\12)
Semitones (A1:m2) 4:4 (100¢ : 100¢)
Consistency limit 5
Distinct consistency limit 5
Special properties

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, the equal temperament tempers out 20000/19683, but 34edo does a much better job for this temperament, known as tetracot. However in the 7-limit it can be used for doublewide temperament, the 1/2 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 cent 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.

### Odd harmonics

Approximation of odd harmonics in 48edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -2.0 -11.3 +6.2 -3.9 -1.3 +9.5 +11.7 -5.0 +2.5 +4.2 -3.3
Relative (%) -7.8 -45.3 +24.7 -15.6 -5.3 +37.9 +46.9 -19.8 +9.9 +16.9 -13.1
Steps
(reduced)
76
(28)
111
(15)
135
(39)
152
(8)
166
(22)
178
(34)
188
(44)
196
(4)
204
(12)
211
(19)
217
(25)

### 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.

## Notation

48edo can be notated using ups and downs notation. This can be accomplished by combining Helmholtz-Ellis accidentals with semisharps and semiflats:

 Semitones Sharp Symbol Flat Symbol 0 1⁄4 1⁄2 3⁄4 1 1+1⁄4 1+1⁄2 1+3⁄4 2 2+1⁄4

## Intervals

# Cents Ups and Downs Notation
0 0.0 D
1 25.0 ^D, v3E♭
2 50.0 ^^D, vvE♭
3 75.0 ^3D, vE♭
4 100.0 D♯, E♭
5 125.0 ^D♯, v3E
6 150.0 ^^D♯, vvE
7 175.0 ^3D♯, vE
8 200.0 E
9 225.0 ^E, v3F
10 250.0 ^^E, vvF
11 275.0 ^3E, vF
12 300.0 F
13 325.0 ^F, v3G♭
14 350.0 ^^F, vvG♭
15 375.0 ^3F, vG♭
16 400.0 F♯, G♭
17 425.0 ^F♯, v3G
18 450.0 ^^F♯, vvG
19 475.0 ^3F♯, vG
20 500.0 G
21 525.0 ^G, v3A♭
22 550.0 ^^G, vvA♭
23 575.0 ^3G, vA♭
24 600.0 G♯, A♭
25 625.0 ^G♯, v3A
26 650.0 ^^G♯, vvA
27 675.0 ^3G♯, vA
28 700.0 A
29 725.0 ^A, v3B♭
30 750.0 ^^A, vvB♭
31 775.0 ^3A, vB♭
32 800.0 A♯, B♭
33 825.0 ^A♯, v3B
34 850.0 ^^A♯, vvB
35 875.0 ^3A♯, vB
36 900.0 B
37 925.0 ^B, v3C
38 950.0 ^^B, vvC
39 975.0 ^3B, vC
40 1000.0 C
41 1025.0 ^C, v3D♭
42 1050.0 ^^C, vvD♭
43 1075.0 ^3C, vD♭
44 1100.0 C♯, D♭
45 1125.0 ^C♯, v3D
46 1150.0 ^^C♯, vvD
47 1175.0 ^3C♯, vD
48 1200.0 D

## Regular temperament properties

### Rank-2 temperaments

Periods
8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 5\48 125.00 16/15 Negri
1 7\48 175.00 10/9 Tetracot
1 13\48 325.00 6/5, 77/64 Orgone
1 17\48 425.00 9/7 Squares
1 19\48 475.00 21/16 Buzzard
2 13\48 325.00 6/5 Doublewide

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

E8 Heterotic
norokusi
Ray Perlner
Carlo Serafini
Jon Lyle Smith