45edo

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45edo

46edo →

Prime factorization	32 × 5
Step size	26.6667 ¢
Fifth	26\45 (693.333 ¢)
Semitones (A1:m2)	2:5 (53.33 ¢ : 133.3 ¢)
Consistency limit	7
Distinct consistency limit	7

45 equal divisions of the octave (abbreviated 45edo or 45ed2), also called 45-tone equal temperament (45tet) or 45 equal temperament (45et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 45 equal parts of about 26.7 ¢ each. Each step represents a frequency ratio of 2^1/45, or the 45th root of 2.

Theory

45edo effectively has two approximate major thirds, each almost equally far from just, but the flat one is slightly closer. Combined with a perfect fifth 8.6 cents flat of just, it can be used as a meantone tuning, forming a good approximation to 2/5-comma meantone (in fact falling into the flattone range). It is a flat-tending system in the 7-limit, with 3, 5, and 7 all flat, but the 11 is sharp.

Odd harmonics

Approximation of odd harmonics in 45edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-8.6	-13.0	-8.8	+9.4	+8.7	+12.8	+5.1	+1.7	-4.2	+9.2	+11.7
	Relative (%)	-32.3	-48.7	-33.1	+35.3	+32.6	+48.0	+19.0	+6.4	-15.7	+34.6	+44.0
Steps (reduced)		71 (26)	104 (14)	126 (36)	143 (8)	156 (21)	167 (32)	176 (41)	184 (4)	191 (11)	198 (18)	204 (24)

As a tuning of other temperaments

It tempers out 81/80, 525/512, 875/864, and 3125/3087 in the 7-limit, and 45/44 in the 11-limit. It provides the optimal patent val for 7- and 11-limit flattone temperament, and the 45f val is an excellent tuning for 13-limit flattone. It also provides the optimal patent val for the 7-limit rank-3 avicennmic temperament, tempering out 525/512, the 11-limit calliope temperament, tempering out 45/44 and 81/80, and the rank-4 temperament tempering out 45/44. It is also the unique equal temperament tuning whose patent val tempers out both the syntonic comma and the ennealimma.

45edo tempers out the quartisma and provides an excellent tuning for the 2.7/3.33-subgroup direct quartismic temperament, in which it approximates the 33/32 quartertone with 2 steps and 7/6 with 10 steps. A bit more broadly, it maps the 2.27.25.63.33.65.17 subgroup to great precision; this is the part of the 17-limit shared with 270edo.

Otherwise, it can be treated as a 2.5/3.7/3-subgroup system (borrowing 5/3 from 15edo and 7/3 from 9edo) and is a good tuning for gariberttet, defined by tempering out 3125/3087 in this subgroup, approximating 2/5-comma gariberttet.

Subsets and supersets

Since 45 factors into primes as 3² × 5, 45edo has subset edos 3, 5, 9, and 15. 135edo, which triples it, corrects its primes 3, 7, and 11 to near-just qualities, and 270edo offers even more.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.0	1/1	Perfect Unison	P1	D
1	26.7	49/48, 50/49	Up unison	^1	^D
2	53.3	36/35, 25/24, 64/63	Augmented Unison	A1	D#
3	80.0	21/20	Diminished 2nd	d2	Ebb
4	106.7	15/14	Downminor 2nd	vm2	vEb
5	133.3	13/12, 14/13, 16/15	Minor 2nd	m2	Eb
6	160.0	54/49	Mid 2nd	~2	vE
7	186.7	10/9, 9/8	Major 2nd	M2	E
8	213.3		Upmajor 2nd	^M2	^E
9	240.0	8/7, 15/13	Augmented 2nd	A2	E#
10	266.7	7/6	Diminished 3rd	d3	Fb
11	293.3	25/21	Downminor 3rd	vm3	vF
12	320.0	6/5	Minor 3rd	m3	F
13	346.7	49/40, 60/49	Mid 3rd	~3	^F
14	373.3	5/4, 26/21, 16/13	Major 3rd	M3	F#
15	400.0	63/50	Upmajor 3rd	^M3	^F#
16	426.7	9/7	Augmented 3rd	A3	Fx
17	453.3	13/10, 21/16	Diminished 4th	d4	Gb
18	480.0	64/49	Down 4th	v4	vG
19	506.7	4/3	Perfect 4th	P4	G
20	533.3	49/36	Up 4th or Mid 4th	^4, ~4	^G
21	560.0	18/13	Augmented 4th	A4	G#
22	586.7	7/5	Upaugmented 4th	^A4	^G#
23	613.3	10/7	Downdiminshed 5th	vd5	vAb
24	640.0	13/9	Diminished 5th	d5	Ab
25	666.7	72/49	Down 5th or Mid 5th	v5, ~5	vA
26	693.3	3/2	Perfect 5th	P5	A
27	720.0	49/32	Up 5th	^5	^A
28	746.7	20/13, 32/21	Augmented 5th	A5	A#
29	773.3	14/9	Diminished 6th	d6	Bbb
30	800.0	100/63	Downminor 6th	vm6	vBb
31	826.7	8/5, 21/13, 13/8	Minor 6th	m6	Bb
32	853.3	49/30, 80/49	Mid 6th	~6	vB
33	880.0	5/3	Major 6th	M6	B
34	906.7	42/25	Upmajor 6th	^M6	^B
35	933.3	12/7	Augmented 6th	A6	B#
36	960.0	7/4, 26/15	Diminished 7th	d7	Cb
37	986.7		Downminor 7th	vm7	vC
38	1013.3	9/5, 16/9	Minor 7th	m7	C
39	1040.0	49/27	Mid 7th	~7	^C
40	1066.7	13/7, 24/13, 15/8	Major 7th	M7	C#
41	1093.3	28/15	Upmajor 7th	^M7	^C#
42	1120.0	40/21	Augmented 7th	A7	Cx
43	1146.7	35/18, 48/25, 63/32	Diminished 8ve	d8	Db
44	1173.3	49/25, 96/49	Down 8ve	v8	vD
45	1200.0	2/1	Perfect Octave	P8	D

* As a 2.3.5.7.13-subgroup temperament, using the 45f val

Notation

Ups and downs notation

Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

Step offset	0	1	2	3	4	5
Sharp symbol
Flat symbol

Quarter-tone notation

Since a sharp raises by two steps, quarter-tone accidentals can also be used.

Step offset	−4	−3	−2	−1	0	+1	+2	+3	+4
Symbol

Sagittal notation

This notation uses the same sagittal sequence as EDOs 52 and 59b.

Evo flavor

Revo flavor

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

In the following diagrams, 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

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-71 45⟩	[⟨45 71]]	+2.72	2.73	10.2
2.3.5	81/80, [-27 1 11⟩	[⟨45 71 104]]	+3.68	2.61	9.75
2.3.5.7	81/80, 525/512, 2401/2400	[⟨45 71 104 126]]	+3.55	2.27	8.49
2.3.5.7.13	65/64, 81/80, 105/104, 2401/2400	[⟨45 71 104 126 166]] (45f)	+3.59	2.03	7.60

Commas

This is a partial list of the commas that 45et tempers out with its patent val, ⟨45 71 104 126 143 156 167].

Prime limit	Ratio	Monzo	Cents	Color name	Name(s)
5	81/80	[-4 4 -1⟩	21.51	Gu	Syntonic comma, Didymus' comma, meantone comma
5	(26 digits)	[1 -27 18⟩	0.86	Satritribiyo	Ennealimma
7	16807/16384	[-14 0 0 5⟩	44.13	Laquinzo	Cloudy comma
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma, Avicenna's enharmonic diesis
7	875/864	[-5 -3 3 1⟩	21.90	Zotrigu	Keema
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh comma
7	(16 digits)	[-11 -9 0 9⟩	1.84	Tritrizo	Septimal ennealimma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
11	45/44	[-2 2 1 0 -1⟩	38.91	Luyo	Undecimal 1/5-tone
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	(18 digits)	[24 -6 0 1 -5⟩	0.51	Saquinlu-azo	Quartisma

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	1\45	26.7	49/48	Sfourth
1	2\45	53.3	36/35	Chromo
1	7\45	186.7	10/9	Mintone
1	11\45	293.3	25/21	Quasitemp
1	14\45	373.3	5/4	Submerged
1	16\45	426.7	9/7	Squares
1	23\45	453.3	13/10	Maja
1	19\45	506.7	4/3	Flattone
3	19\45 (4\45)	506.7 (106.7)	4/3 (15/14)	Lithium
5	19\45 (1\45)	506.7 (26.7)	4/3 (49/48)	Cloudtone
9	12\45 (2\45)	320.0 (53.3)	6/5 (36/35)	Ennealimmal
15	19\45 (1\45)	506.7 (26.7)	4/3 (126/125)	Pentadecal

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

Octave stretch and compression

45edo's approximations of 3/1, 5/1, 7/1, 11/1 and 13/1 and 17/1 are all improved by a stretched-octave version of 45edo, such as 161ed12 or 116ed6. The trade-off is a slightly worse 2/1.

The tuning 183ed17 may also be used, it improves 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 (with different mappings for many) but at the cost of a noticeably worse 2/1 than the others.

Scales

• Cloudtone[10] - recommended by Maeve Gutierrez: 8 1 8 1 8 1 8 1 8 1
• 13-tone 5&9edo scale: 5 4 1 5 3 2 5 2 3 5 1 4 5
• 12-tone 5&9edo scale^{[idiosyncratic term]}: 5 4 1 5 3 2 5 2 3 5 5 5

Instruments

Lumatone

See Lumatone mapping for 45edo

Music

Bryan Deister

• (short clip) Fantasy in 45edo (2025)

JUMBLE

• Fishbowl (2023)
• Archipelago Arpeggio (2024)
• Fallen Angel (2024)
• Solar Guardian (2024)
• Qúchze úzeq Qávka (2025)
• Sodium Light (45edo Microtonal Chillwave) (2026)