# 44edo

 ← 43edo 44edo 45edo →
Prime factorization 22 × 11
Step size 27.2727¢
Fifth 26\44 (709.091¢) (→13\22)
Semitones (A1:m2) 6:2 (163.6¢ : 54.55¢)
Consistency limit 5
Distinct consistency limit 5

44 equal divisions of the octave (abbreviated 44edo or 44ed2), also called 44-tone equal temperament (44tet) or 44 equal temperament (44et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 44 equal parts of about 27.3 ¢ each. Each step represents a frequency ratio of 21/44, or the 44th root of 2.

## Theory

44edo is a double of 22edo, to which it adds the ratios of 13, 19, and 23. While not the most accurate 2.3.5.7.11 tuning, 22edo is certainly a relatively compact one, and it's natural to extend it this way. The most practically useful of these additions is easily the 13th harmonic with its neutral intervals, but the 17th, 19th, and 23rd are not to be dismissed. In the 13-limit it supplies the optimal patent val for vigin temperament. The 2*44 subgroup of 44edo is 2.9.5.21.11.13.17.19.23, on which 44 tempers out the same commas as the patent val for 88edo.

### Harmonics

Approximation of odd harmonics in 44edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +7.1 -4.5 +13.0 -13.0 -5.9 +4.9 +2.6 +4.1 +2.5 -7.1 -1.0
Relative (%) +26.2 -16.5 +47.6 -47.7 -21.5 +18.1 +9.7 +15.2 +9.1 -26.2 -3.7
Steps
(reduced)
70
(26)
102
(14)
124
(36)
139
(7)
152
(20)
163
(31)
172
(40)
180
(4)
187
(11)
193
(17)
199
(23)

### Subsets and supersets

44edo has subsets 2, 4, 11, 22.

One step of 44edo is very close (only 0.0086 cents sharp) to 64/63 (the septimal comma). Ruthenium temperament realizes this proximity through a regular temperament perspective, and it is supported by a large number of edos which are a multiple of 44 - for example 1012edo, 1848edo, and 2684edo.

## Intervals

In 44edo, sharps and flats alter pitch by 6 EDOsteps. This means intervals can be notated with half sharps and half flats equal to 3 EDOsteps, in addition to ups and downs. The table below uses only sharps, flats, and ups and downs. When translating music from 22edo to 44edo, single ups and downs simply become double ups and downs (vEb in 22edo would be vvEb in 44edo).

Degrees Cents Ups and Downs Notation
0 0.000 Perfect 1sn P1 D
1 27.273 Up 1sn ^1 ^D
2 54.545 Minor 2nd m2 Eb
3 81.818 Upminor 2nd ^m2 ^Eb
4 109.091 Downmid 2nd v~2 ^^Eb
5 136.364 Mid 2nd ~2 vvvE, ^^^Eb
6 163.636 Upmid 2nd ^~2 vvE
7 190.909 Downmajor 2nd vM2 vE
8 218.182 Major 2nd M2 E
9 245.455 Upmajor 2nd, Downminor 3rd ^M2, vm3 ^E, vF
10 272.727 Minor 3rd m3 F
11 300.000 Upminor 3rd ^m3 ^F
12 327.273 Downmid 3rd v~3 ^^F
13 354.545 Mid 3rd ~3 ^^^F, vvvF#
14 381.818 Upmid 3rd ^~3 vvF#
15 409.091 Downmajor 3rd vM3 vF#
16 436.364 Major 3rd M3 F#
17 463.636 Upmajor 3rd, Down 4th ^M3, v4 ^F#, vG
18 490.909 Perfect 4th P4 G
19 518.182 Up 4th ^4 ^G
20 545.455 Dim 5th, Downmid 4th d5, v~4 Ab, ^^G
21 572.727 Mid 4th ~4 ^^^G, vvvG#
22 600.000 Upmid 4th, Downmid 5th ^~4, v~5 vvG#, ^^Ab
23 627.273 Mid 5th ~5 vvvA, ^^^Ab
24 654.545 Aug 4th, Upmid 5th A4, ^~5 G#, vvA
25 681.818 Down 5th v5 vA
26 709.091 Perfect 5th P5 A
27 736.364 Up 5th, Downminor 6th ^5, vm6 ^A, vBb
28 763.636 Minor 6th m6 Bb
29 790.909 Upminor 6th ^m6 ^Bb
30 818.182 Downmid 6th v~6 ^^Bb
31 845.455 Mid 6th ~6 ^^^Bb, vvvB
32 872.727 Upmid 6th ^~6 vvB
33 900.000 Downmajor 6th vM6 vB
34 927.273 Major 6th M6 B
35 954.545 Upmajor 6th, Downminor 7th ^M6, vm7 ^B, vC
36 981.818 Minor 7th m7 C
37 1009.091 Upminor 7th ^m7 ^C
38 1036.364 Downmid 7th v~7 ^^C
39 1063.636 Mid 7th ~7 ^^^C, vvvC#
40 1090.909 Upmid 7th ^~7 vvC#
41 1118.182 Downmajor 7th vM7 vC#
42 1145.455 Major 7th M7 C#
43 1172.727 Upmajor 7th, Down 8ve ^M7, v8 ^C#, vD
44 1200.000 Perfect 8ve P8 D