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 2^1/44, 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.

It is on the optimal ET sequence for 7-, 11- and 13-limit nautilus temperament, for 11-limit spell temperament, and for 13-limit cantrip temperament. 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
Error	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)

Approximation of odd harmonics in 44edo (continued)
Harmonic		25	27	29	31	33	35	37	39	41	43	45
Error	Absolute (¢)	-9.0	-5.9	+6.8	+0.4	+1.3	+8.5	-5.9	+12.1	+7.3	+6.7	+9.8
Error	Relative (%)	-33.0	-21.5	+24.9	+1.5	+4.7	+31.2	-21.6	+44.2	+26.8	+24.4	+35.8
Steps (reduced)		204 (28)	209 (33)	214 (38)	218 (42)	222 (2)	226 (6)	229 (9)	233 (13)	236 (16)	239 (19)	242 (22)

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. The aforementioned 88edo, which doubles it, is a meantone tuning that corrects the 7th harmonic to near-just, although at the expense of increasing relative error of the 13th and 19th harmonics; alternatively, if it is treated as directly approximating the 9th harmonic, then it also corrects the 9th harmonic to near-just.

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).

#	Cents	Approximate ratios*	Ups and downs notation
0	0.0	1/1	Perfect 1sn	P1	D
1	27.3	65/64	Up 1sn	^1	^D
2	54.5	28/27, 32/31, 33/32, 34/33, 36/35	Minor 2nd	m2	Eb
3	81.8	19/18, 20/19, 23/22, 24/23	Upminor 2nd	^m2	^Eb
4	109.1	15/14, 16/15, 17/16, 18/17	Dupminor 2nd, downmid 2nd	^^m2, v~2	^^Eb
5	136.4	13/12, 14/13	Mid 2nd	~2	vvvE, ^^^Eb
6	163.6	11/10, 12/11, 32/29, 10/9	Dudmajor 2nd, upmid 2nd	vvM2, ^~2	vvE
7	190.9	19/17	Downmajor 2nd	vM2	vE
8	218.2	8/7, 9/8, 17/15	Major 2nd	M2	E
9	245.5	15/13, 22/19	Upmajor 2nd, downminor 3rd	^M2, vm3	^E, vF
10	272.7	7/6, 20/17	Minor 3rd	m3	F
11	300.0	13/11, 19/16	Upminor 3rd	^m3	^F
12	327.3	6/5, 17/14, 11/9	Dupminor 3rd, downmid 3rd	^^m3, v~3	^^F
13	354.5	16/13, 26/21, 39/32	Mid 3rd	~3	^^^F, vvvF#
14	381.8	5/4	Dudmajor 3rd, upmid 3rd	vvM3, ^~3	vvF#
15	409.1	19/15, 24/19	Downmajor 3rd	vM3	vF#
16	436.4	9/7, 14/11	Major 3rd	M3	F#
17	463.6	13/10, 17/13	Upmajor 3rd, down 4th	^M3, v4	^F#, vG
18	490.9	4/3	Perfect 4th	P4	G
19	518.2	19/14	Up 4th	^4	^G
20	545.5	11/8, 15/11, 26/19	Dup 4th, downmid 4th, dim 5th	^^4, v~4, d5	Ab, ^^G
21	572.7	18/13, 32/23	Mid 4th, Updim 5th	~4, ^d5	^^^G, vvvG#
22	600.0	17/12, 24/17, 7/5, 10/7	Upmid 4th, downmid 5th	^~4, v~5	vvG#, ^^Ab
23	627.3	13/9, 23/16	Downaug 4th, mid 5th	vA4, ~5	vvvA, ^^^Ab
24	654.5	16/11, 19/13, 22/15	Aug 4th, upmid 5th, dud 5th	A4, ^~5, vv5	G#, vvA
25	681.8	28/19	Down 5th	v5	vA
26	709.1	3/2	Perfect 5th	P5	A
27	736.4	20/13, 26/17	Up 5th, downminor 6th	^5, vm6	^A, vBb
28	763.6	14/9, 11/7	Minor 6th	m6	Bb
29	790.9	19/12, 30/19	Upminor 6th	^m6	^Bb
30	818.2	8/5	Dupminor 6th, downmid 6th	^^m6, v~6	^^Bb
31	845.5	13/8, 21/13	Mid 6th	~6	^^^Bb, vvvB
32	872.7	5/3, 28/17, 18/11	Dudmajor 6th, upmid 6th	vvM6, ^~6	vvB
33	900.0	22/13, 32/19	Downmajor 6th	vM6	vB
34	927.3	12/7, 17/10	Major 6th	M6	B
35	954.5	19/11, 26/15	Upmajor 6th, downminor 7th	^M6, vm7	^B, vC
36	981.8	7/4, 16/9, 30/17	Minor 7th	m7	C
37	1009.1	34/19	Upminor 7th	^m7	^C
38	1036.4	11/6, 20/11, 29/16, 9/5	Dupminor 7th, downmid 7th	^^m7, v~7	^^C
39	1063.6	13/7, 24/13	Mid 7th	~7	^^^C, vvvC#
40	1090.9	15/8, 17/9, 28/15, 32/17	Dudmajor 7th, upmid 7th	vvM7, ^~7	vvC#
41	1118.2	19/10, 36/19, 44/23, 23/12	Downmajor 7th	vM7	vC#
42	1145.5	27/14, 31/16, 33/17, 35/18, 64/33	Major 7th	M7	C#
43	1172.7	128/65	Upmajor 7th, down 8ve	^M7, v8	^C#, vD
44	1200.0	2/1	Perfect 8ve	P8	D

* As a 19-limit temperament, with additional ratios of 23, 29, and 31

Notation

Ups and downs notation

44edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

Alternative ups and downs have 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	13	14
Sharp symbol
Flat symbol

If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol
Flat symbol

Sagittal notation

This notation uses the same sagittal sequence as edos 23b, 30, and 37, and is a superset of the notations for edos 22 and 11.

Evo flavor

Revo flavor

Evo-SZ flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	3\44	81.82	22/21	Nautilus (44d)
1	7\44	190.91	9/8	Spell (44def) / cantrip (44de)
1	9\44	245.46	15/13	Immunity (44cff, 2.3.5.13)
1	13\44	354.55	11/9	Beatles / ringo (44e)
1	15\44	409.09	5/4	Hocus (44)
2	3\44	81.82	22/21	Harry (44ceff)
4	4\44	109.09	16/15	Bidia (44d, 7-limit)

* Octave-reduced form, reduced to the first half-octave

Scales

Evacuated planet^{[idiosyncratic term]} (approximated from 66 afdo): 5 13 8 12 6
Approximations of gamelan scales:
- 5-tone pelog: 4 6 15 4 15
- 7-tone pelog: 4 6 9 6 4 10 5
- 5-tone slendro: 9 9 8 9 9

Instrument layouts

Music

Bryan Deister

Improvisation in 44edo (composed and played by Bryan Deister in May 2023, transcribed by Stephen Weigel in Sept 2024)
Leaning Dream - Pizza Tower (adapted into 44edo by Bryan Deister in July 2024)
44edo improvisation (Oct 2024)
44edo improv (Oct 2025)
Buried Treasure - 44edo (2026)