45edo

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45edo

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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 major thirds, each almost equally far from just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5-comma meantone. It is the optimal patent val for flattone temperament, the 525/512 planar 7-limit avicennmic temperament, the 11-limit calliope temperament tempering out 45/44 and 81/80, and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp.

45edo tempers out the quartisma and provides an excellent tuning for the 2.33/32.7/6 subgroup direct quartismic temperament, in which it approximates the 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the ennealimma.

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)

Octave stretch

45edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1 and 17/1 are all improved by APS3.21farab, a stretched-octave version of 45edo. The trade-off is a slightly worse 2/1.

The tuning 126ed7 may be used for this purpose too, it improves 3/1, 5/1, 7/1, 11/1 and 13/1, at the cost of a slightly worse 2/1.

There are also some nearby Zeta peak index (ZPI) tunings which can be used for this same purpose: 207zpi, 208zpi and 209zpi. The main Zeta peak index page details all three tunings.

Intervals

Step #	ET	Just (JI)		Error (ET−JI)	Ups and downs notation
	Cents	Interval	Cents
0	0.000	1/1	0.000	0.000	Perfect Unison	P1	D
1	26.666	65/64	26.841	-0.174	Up unison	^1	^D
2	53.333	33/32	53.273	0.060	Augmented Unison	A1	D#
3	80.000	22/21	80.537	-0.537	Diminished 2nd	d2	Ebb
4	106.666	17/16	104.955	1.711	Downminor 2nd	vm2	vEb
5	133.333	27/25	133.238	0.095	Minor 2nd	m2	Eb
6	160.000	11/10	165.004	-5.004	Mid 2nd	~2	vE
7	186.666	10/9	182.404	4.262	Major 2nd	M2	E
8	213.333	9/8	203.910	9.423	Upmajor 2nd	^M2	^E
9	240.000	8/7	231.174	8.826	Augmented 2nd	A2	E#
10	266.666	7/6	266.871	-0.205	Diminished 3rd	d3	Fb
11	293.333	32/27	294.135	-0.802	Downminor 3rd	vm3	vF
12	320.000	6/5	315.641	4.359	Minor 3rd	m3	F
13	346.666	11/9	347.408	-0.741	Mid 3rd	~3	^F
14	373.333	5/4	386.314	-12.980	Major 3rd	M3	F#
15	400.000	63/50	400.108	-0.108	Upmajor 3rd	^M3	^F#
16	426.666	9/7	435.084	-8.418	Augmented 3rd	A3	Fx
17	453.333	13/10	454.294	-0.961	Diminished 4th	d4	Gb
18	480.000	21/16	470.781	9.219	Down 4th	v4	vG
19	506.666	4/3	498.045	8.622	Perfect 4th	P4	G
20	533.333	49/36	533.742	-0.409	Up 4th or Mid 4th	^4, ~4	^G
21	560.000	18/13	563.382	-3.382	Augmented 4th	A4	G#
22	586.666	7/5	582.512	4.155	Upaugmented 4th	^A4	^G#
23	613.333	10/7	617.488	-4.155	Downdiminshed 5th	vd5	vAb
24	640.000	13/9	636.618	3.382	Diminished 5th	d5	Ab
25	666.666	72/49	666.258	0.409	Down 5th or Mid 5th	v5, ~5	vA
26	693.333	3/2	701.955	-8.622	Perfect 5th	P5	A
27	720.000	32/21	729.219	-9.219	Up 5th	^5	^A
28	746.666	20/13	745.786	0.961	Augmented 5th	A5	A#
29	773.333	14/9	764.916	8.418	Diminished 6th	d6	Bbb
30	800.000	100/63	799.892	0.108	Downminor 6th	vm6	vBb
31	826.666	8/5	813.686	12.980	Minor 6th	m6	Bb
32	853.333	18/11	852.592	0.741	Mid 6th	~6	vB
33	880.000	5/3	884.359	-4.359	Major 6th	M6	B
34	906.666	27/16	905.865	0.802	Upmajor 6th	^M6	^B
35	933.333	12/7	933.129	0.205	Augmented 6th	A6	B#
36	960.000	7/4	968.826	-8.826	Diminished 7th	d7	Cb
37	986.666	16/9	996.089	-9.423	Downminor 7th	vm7	vC
38	1013.333	9/5	1017.596	-4.262	Minor 7th	m7	C
39	1040.000	20/11	1034.996	5.004	Mid 7th	~7	^C
40	1066.666	50/27	1066.762	-0.095	Major 7th	M7	C#
41	1093.333	32/17	1095.044	-1.711	Upmajor 7th	^M7	^C#
42	1120.000	21/11	1119.463	0.537	Augmented 7th	A7	Cx
43	1146.666	64/33	1146.727	-0.060	Diminished 8ve	d8	Db
44	1173.333	128/65	1173.158	0.174	Down 8ve	v8	vD
45	1200.000	2/1	1200.000	0.000	Perfect Octave	P8	D

Regular temperament properties

Commas

This is a partial list of the commas that 45edo 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

Instruments

Lumatone

See Lumatone mapping for 45edo

Music

JUMBLE

• Fishbowl (2023)
• Archipelago Arpeggio (2024)
• Fallen Angel (2024)
• Solar Guardian (2024)