48edo: Difference between revisions

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!#
!#
!Cents
!Cents
!Diatonic interval category
![[Ups and downs notation]]
|-
|-
|0
|0
|0.0
|0.0
|perfect unison
|{{Ups and downs note name|step=0}}
|-
|-
|1
|1
|25.0
|25.0
|superunison
|{{Ups and downs note name|step=1}}
|-
|-
|2
|2
|50.0
|50.0
|subminor second
|{{Ups and downs note name|step=2}}
|-
|-
|3
|3
|75.0
|75.0
|subminor second
|{{Ups and downs note name|step=3}}
|-
|-
|4
|4
|100.0
|100.0
|minor second
|{{Ups and downs note name|step=4}}
|-
|-
|5
|5
|125.0
|125.0
|supraminor second
|{{Ups and downs note name|step=5}}
|-
|-
|6
|6
|150.0
|150.0
|neutral second
|{{Ups and downs note name|step=6}}
|-
|-
|7
|7
|175.0
|175.0
|submajor second
|{{Ups and downs note name|step=7}}
|-
|-
|8
|8
|200.0
|200.0
|major second
|{{Ups and downs note name|step=8}}
|-
|-
|9
|9
|225.0
|225.0
|supermajor second
|{{Ups and downs note name|step=9}}
|-
|-
|10
|10
|250.0
|250.0
|ultramajor second
|{{Ups and downs note name|step=10}}
|-
|-
|11
|11
|275.0
|275.0
|subminor third
|{{Ups and downs note name|step=11}}
|-
|-
|12
|12
|300.0
|300.0
|minor third
|{{Ups and downs note name|step=12}}
|-
|-
|13
|13
|325.0
|325.0
|supraminor third
|{{Ups and downs note name|step=13}}
|-
|-
|14
|14
|350.0
|350.0
|neutral third
|{{Ups and downs note name|step=14}}
|-
|-
|15
|15
|375.0
|375.0
|submajor third
|{{Ups and downs note name|step=15}}
|-
|-
|16
|16
|400.0
|400.0
|major third
|{{Ups and downs note name|step=16}}
|-
|-
|17
|17
|425.0
|425.0
|supermajor third
|{{Ups and downs note name|step=17}}
|-
|-
|18
|18
|450.0
|450.0
|ultramajor third
|{{Ups and downs note name|step=18}}
|-
|-
|19
|19
|475.0
|475.0
|subfourth
|{{Ups and downs note name|step=19}}
|-
|-
|20
|20
|500.0
|500.0
|perfect fourth
|{{Ups and downs note name|step=20}}
|-
|-
|21
|21
|525.0
|525.0
|superfourth
|{{Ups and downs note name|step=21}}
|-
|-
|22
|22
|550.0
|550.0
|superfourth
|{{Ups and downs note name|step=22}}
|-
|-
|23
|23
|575.0
|575.0
|low tritone
|{{Ups and downs note name|step=23}}
|-
|-
|24
|24
|600.0
|600.0
|high tritone
|{{Ups and downs note name|step=24}}
|-
|-
|25
|25
|625.0
|625.0
|high tritone
|{{Ups and downs note name|step=25}}
|-
|-
|26
|26
|650.0
|650.0
|subfifth
|{{Ups and downs note name|step=26}}
|-
|-
|27
|27
|675.0
|675.0
|subfifth
|{{Ups and downs note name|step=27}}
|-
|-
|28
|28
|700.0
|700.0
|perfect fifth
|{{Ups and downs note name|step=28}}
|-
|-
|29
|29
|725.0
|725.0
|superfifth
|{{Ups and downs note name|step=29}}
|-
|-
|30
|30
|750.0
|750.0
|ultrafifth
|{{Ups and downs note name|step=30}}
|-
|-
|31
|31
|775.0
|775.0
|subminor sixth
|{{Ups and downs note name|step=31}}
|-
|-
|32
|32
|800.0
|800.0
|minor sixth
|{{Ups and downs note name|step=32}}
|-
|-
|33
|33
|825.0
|825.0
|supraminor sixth
|{{Ups and downs note name|step=33}}
|-
|-
|34
|34
|850.0
|850.0
|neutral sixth
|{{Ups and downs note name|step=34}}
|-
|-
|35
|35
|875.0
|875.0
|submajor sixth
|{{Ups and downs note name|step=35}}
|-
|-
|36
|36
|900.0
|900.0
|major sixth
|{{Ups and downs note name|step=36}}
|-
|-
|37
|37
|925.0
|925.0
|supermajor sixth
|{{Ups and downs note name|step=37}}
|-
|-
|38
|38
|950.0
|950.0
|ultramajor sixth
|{{Ups and downs note name|step=38}}
|-
|-
|39
|39
|975.0
|975.0
|subminor seventh
|{{Ups and downs note name|step=39}}
|-
|-
|40
|40
|1000.0
|1000.0
|minor seventh
|{{Ups and downs note name|step=40}}
|-
|-
|41
|41
|1025.0
|1025.0
|supraminor seventh
|{{Ups and downs note name|step=41}}
|-
|-
|42
|42
|1050.0
|1050.0
|neutral seventh
|{{Ups and downs note name|step=42}}
|-
|-
|43
|43
|1075.0
|1075.0
|submajor seventh
|{{Ups and downs note name|step=43}}
|-
|-
|44
|44
|1100.0
|1100.0
|major seventh
|{{Ups and downs note name|step=44}}
|-
|-
|45
|45
|1125.0
|1125.0
|supermajor seventh
|{{Ups and downs note name|step=45}}
|-
|-
|46
|46
|1150.0
|1150.0
|ultramajor seventh
|{{Ups and downs note name|step=46}}
|-
|-
|47
|47
|1175.0
|1175.0
|suboctave
|{{Ups and downs note name|step=47}}
|-
|-
|48
|48
|1200.0
|1200.0
|perfect octave
|{{Ups and downs note name|step=48}}
|}
|}


Revision as of 01:39, 12 June 2023

← 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

Template:EDO intro

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, 48 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.

48edo is the 10th highly melodic EDO.


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)

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

Regular temperament properties

Rank-2 temperaments

Periods

per octave

Generator

(reduced)

Cents

(reduced)

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

Music

Jon Lyle Smith

Ray Perlner

Carlo Serafini

Retrieved from "https://en.xen.wiki/index.php?title=48edo&oldid=116399"