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)

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

Todo: complete table

Consistency check in column of approximated JI ratios in the following table; recommend dual ratios for the 7th harmonic which has a lot of relative error; might need some additional ratios.

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	Approximate JI ratios*	Ups and downs notation
0	0.000	1/1	Perfect 1sn	P1	D
1	27.273	64/63	Up 1sn	^1	^D
2	54.545	33/32, 32/31	Minor 2nd	m2	Eb
3	81.818	23/22	Upminor 2nd	^m2	^Eb
4	109.091	16/15	Dupminor 2nd, Downmid 2nd	^^m2, v~2	^^Eb
5	136.364	13/12	Mid 2nd	~2	vvvE, ^^^Eb
6	163.636	11/10, 32/29	Dudmajor 2nd, Upmid 2nd	vvM2, ^~2	vvE
7	190.909	28/25	Downmajor 2nd	vM2	vE
8	218.182	9/8	Major 2nd	M2	E
9	245.455	8/7	Upmajor 2nd, Downminor 3rd	^M2, vm3	^E, vF
10	272.727	7/6, 32/27	Minor 3rd	m3	F
11	300.000	19/16	Upminor 3rd	^m3	^F
12	327.273	11/9, 29/24	Dupminor 3rd, Downmid 3rd	^^m3, v~3	^^F
13	354.545	16/13	Mid 3rd	~3	^^^F, vvvF#
14	381.818	5/4	Dudmajor 3rd, Upmid 3rd	vvM3, ^~3	vvF#
15	409.091	80/63	Downmajor 3rd	vM3	vF#
16	436.364	81/64, 9/7	Major 3rd	M3	F#
17	463.636	21/16	Upmajor 3rd, Down 4th	^M3, v4	^F#, vG
18	490.909	4/3	Perfect 4th	P4	G
19	518.182	19/14	Up 4th	^4	^G
20	545.455	11/8	Dup 4th, Downmid 4th, Dim 5th	^^4, v~4, d5	Ab, ^^G
21	572.727	7/5	Mid 4th, Updim 5th	~4, ^d5	^^^G, vvvG#
22	600.000	99/70, 140/99	Upmid 4th, Downmid 5th	^~4, v~5	vvG#, ^^Ab
23	627.273	10/7	Downaug 4th, Mid 5th	vA4, ~5	vvvA, ^^^Ab
24	654.545	16/11	Aug 4th, Upmid 5th, Dud 5th	A4, ^~5, vv5	G#, vvA
25	681.818	28/19	Down 5th	v5	vA
26	709.091	3/2	Perfect 5th	P5	A
27	736.364	32/21	Up 5th, Downminor 6th	^5, vm6	^A, vBb
28	763.636	14/9	Minor 6th	m6	Bb
29	790.909	128/81	Upminor 6th	^m6	^Bb
30	818.182	8/5	Dupminor 6th, Downmid 6th	^^m6, v~6	^^Bb
31	845.455	13/8	Mid 6th	~6	^^^Bb, vvvB
32	872.727	5/3, 48/29	Dudmajor 6th, Upmid 6th	vvM6, ^~6	vvB
33	900.000	27/16	Downmajor 6th	vM6	vB
34	927.273	12/7	Major 6th	M6	B
35	954.545	7/4?	Upmajor 6th, Downminor 7th	^M6, vm7	^B, vC
36	981.818	7/4, 16/9	Minor 7th	m7	C
37	1009.091	9/5	Upminor 7th	^m7	^C
38	1036.364	20/11, 29/16	Dupminor 7th, Downmid 7th	^^m7, v~7	^^C
39	1063.636	24/13	Mid 7th	~7	^^^C, vvvC#
40	1090.909	15/8	Dudmajor 7th, Upmid 7th	vvM7, ^~7	vvC#
41	1118.182	40/21	Downmajor 7th	vM7	vC#
42	1145.455	27/14, 31/16	Major 7th	M7	C#
43	1172.727	63/32	Upmajor 7th, Down 8ve	^M7, v8	^C#, vD
44	1200.000	2/1	Perfect 8ve	P8	D

* In 2.3.5.7.11.13.17.19.23.29.31 subgroup

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)