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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|90}}
{{Harmonics in equal|90}}
=== Subsets and supersets ===
Since 90 factors into primes as 2 x 3<sup>2</sup> × 5, 90 has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 }}.
As a composite edo, the smallest subsets it lacks are subsets of [[4edo|4]], [[7edo|7]] and [[8edo|8]], but 13\90 = 173.333{{cent}} offers a good approximation to 1\7 = 171.428{{c}}, and instead of 1\8 = 150{{cent}}, it has 27\80 = 146.667{{cent}}, serving a similar function.
Like [[80edo]], this may offer a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).
Some supersets of 90edo include: {{EDOs| 180, 270, 360, 450, 540, 630, 720, 810, 900... }}. Temperament mergers of these might produce [[90th-octave temperaments]].


== Interval table ==
== Interval table ==

Revision as of 01:36, 23 October 2025

← 89edo 90edo 91edo →
Prime factorization 2 × 32 × 5
Step size 13.3333 ¢ 
Fifth 53\90 (706.667 ¢)
Semitones (A1:m2) 11:5 (146.7 ¢ : 66.67 ¢)
Dual sharp fifth 53\90 (706.667 ¢)
Dual flat fifth 52\90 (693.333 ¢) (→ 26\45)
Dual major 2nd 15\90 (200 ¢) (→ 1\6)
Consistency limit 7
Distinct consistency limit 7

90 equal divisions of the octave (abbreviated 90edo or 90ed2), also called 90-tone equal temperament (90tet) or 90 equal temperament (90et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 90 equal parts of about 13.3 ¢ each. Each step represents a frequency ratio of 21/90, or the 90th root of 2.

Theory

As an equal temperament, it tempers out 2048/2025 (diaschisma) in the 5-limit, 245/243 and 3125/3087 in the 7-limit, 121/120 and 176/175 in the 11-limit, and 169/168 and 275/273 in the 13-limit. It provides the optimal patent val for the 31 & 90 temperament in the 7-, 11- and 13-limit. Notably, it is the second lowest in a series of four consecutive edos to temper out the quartisma.

Odd harmonics

Approximation of odd harmonics in 90edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +4.71 +0.35 +4.51 -3.91 -4.65 -0.53 +5.06 +1.71 -4.18 -4.11 -1.61
Relative (%) +35.3 +2.6 +33.8 -29.3 -34.9 -4.0 +38.0 +12.8 -31.3 -30.9 -12.1
Steps
(reduced) 		143
(53) 		209
(29) 		253
(73) 		285
(15) 		311
(41) 		333
(63) 		352
(82) 		368
(8) 		382
(22) 		395
(35) 		407
(47)

Subsets and supersets

Since 90 factors into primes as 2 x 32 × 5, 90 has subset edos 2, 3, 5, 6, 9, 10, 15, 18, 30, 45.

As a composite edo, the smallest subsets it lacks are subsets of 4, 7 and 8, but 13\90 = 173.333 ¢ offers a good approximation to 1\7 = 171.428 ¢, and instead of 1\8 = 150 ¢, it has 27\80 = 146.667 ¢, serving a similar function.

Like 80edo, this may offer a relatively unexplored strategy of "tempered detempering", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).

Some supersets of 90edo include: 180, 270, 360, 450, 540, 630, 720, 810, 900.... Temperament mergers of these might produce 90th-octave temperaments.

Interval table

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 52\90) 		Ups and downs notation
(Dual sharp fifth 53\90)
0 0 1/1 D D
1 13.3 ^D, vE♭♭♭ ^D, v4E♭
2 26.7 ^^D, E♭♭♭ ^^D, v3E♭
3 40 41/40 vD♯, ^E♭♭♭ ^3D, vvE♭
4 53.3 32/31, 33/32, 34/33 D♯, vvE♭♭ ^4D, vE♭
5 66.7 26/25 ^D♯, vE♭♭ ^5D, E♭
6 80 ^^D♯, E♭♭ v5D♯, ^E♭
7 93.3 39/37 vD𝄪, ^E♭♭ v4D♯, ^^E♭
8 106.7 17/16, 33/31 D𝄪, vvE♭ v3D♯, ^3E♭
9 120 15/14 ^D𝄪, vE♭ vvD♯, ^4E♭
10 133.3 40/37, 41/38 ^^D𝄪, E♭ vD♯, ^5E♭
11 146.7 25/23, 37/34 vD♯𝄪, ^E♭ D♯, v5E
12 160 34/31 D♯𝄪, vvE ^D♯, v4E
13 173.3 ^D♯𝄪, vE ^^D♯, v3E
14 186.7 29/26, 39/35 E ^3D♯, vvE
15 200 37/33 ^E, vF♭♭ ^4D♯, vE
16 213.3 26/23 ^^E, F♭♭ E
17 226.7 vE♯, ^F♭♭ ^E, v4F
18 240 23/20, 39/34 E♯, vvF♭ ^^E, v3F
19 253.3 22/19, 37/32 ^E♯, vF♭ ^3E, vvF
20 266.7 7/6 ^^E♯, F♭ ^4E, vF
21 280 20/17 vE𝄪, ^F♭ F
22 293.3 E𝄪, vvF ^F, v4G♭
23 306.7 31/26, 37/31 ^E𝄪, vF ^^F, v3G♭
24 320 F ^3F, vvG♭
25 333.3 17/14, 23/19, 40/33 ^F, vG♭♭♭ ^4F, vG♭
26 346.7 ^^F, G♭♭♭ ^5F, G♭
27 360 16/13 vF♯, ^G♭♭♭ v5F♯, ^G♭
28 373.3 31/25, 41/33 F♯, vvG♭♭ v4F♯, ^^G♭
29 386.7 5/4 ^F♯, vG♭♭ v3F♯, ^3G♭
30 400 29/23, 39/31 ^^F♯, G♭♭ vvF♯, ^4G♭
31 413.3 33/26 vF𝄪, ^G♭♭ vF♯, ^5G♭
32 426.7 32/25, 41/32 F𝄪, vvG♭ F♯, v5G
33 440 40/31 ^F𝄪, vG♭ ^F♯, v4G
34 453.3 13/10 ^^F𝄪, G♭ ^^F♯, v3G
35 466.7 17/13, 38/29 vF♯𝄪, ^G♭ ^3F♯, vvG
36 480 29/22, 33/25, 37/28 F♯𝄪, vvG ^4F♯, vG
37 493.3 ^F♯𝄪, vG G
38 506.7 G ^G, v4A♭
39 520 ^G, vA♭♭♭ ^^G, v3A♭
40 533.3 34/25 ^^G, A♭♭♭ ^3G, vvA♭
41 546.7 vG♯, ^A♭♭♭ ^4G, vA♭
42 560 G♯, vvA♭♭ ^5G, A♭
43 573.3 32/23, 39/28 ^G♯, vA♭♭ v5G♯, ^A♭
44 586.7 ^^G♯, A♭♭ v4G♯, ^^A♭
45 600 41/29 vG𝄪, ^A♭♭ v3G♯, ^3A♭
46 613.3 37/26 G𝄪, vvA♭ vvG♯, ^4A♭
47 626.7 23/16, 33/23 ^G𝄪, vA♭ vG♯, ^5A♭
48 640 ^^G𝄪, A♭ G♯, v5A
49 653.3 35/24 vG♯𝄪, ^A♭ ^G♯, v4A
50 666.7 25/17 G♯𝄪, vvA ^^G♯, v3A
51 680 37/25 ^G♯𝄪, vA ^3G♯, vvA
52 693.3 A ^4G♯, vA
53 706.7 ^A, vB♭♭♭ A
54 720 ^^A, B♭♭♭ ^A, v4B♭
55 733.3 26/17, 29/19 vA♯, ^B♭♭♭ ^^A, v3B♭
56 746.7 20/13, 37/24 A♯, vvB♭♭ ^3A, vvB♭
57 760 31/20 ^A♯, vB♭♭ ^4A, vB♭
58 773.3 25/16 ^^A♯, B♭♭ ^5A, B♭
59 786.7 41/26 vA𝄪, ^B♭♭ v5A♯, ^B♭
60 800 A𝄪, vvB♭ v4A♯, ^^B♭
61 813.3 8/5 ^A𝄪, vB♭ v3A♯, ^3B♭
62 826.7 ^^A𝄪, B♭ vvA♯, ^4B♭
63 840 13/8 vA♯𝄪, ^B♭ vA♯, ^5B♭
64 853.3 A♯𝄪, vvB A♯, v5B
65 866.7 28/17, 33/20, 38/23 ^A♯𝄪, vB ^A♯, v4B
66 880 B ^^A♯, v3B
67 893.3 ^B, vC♭♭ ^3A♯, vvB
68 906.7 ^^B, C♭♭ ^4A♯, vB
69 920 17/10 vB♯, ^C♭♭ B
70 933.3 12/7 B♯, vvC♭ ^B, v4C
71 946.7 19/11 ^B♯, vC♭ ^^B, v3C
72 960 40/23 ^^B♯, C♭ ^3B, vvC
73 973.3 vB𝄪, ^C♭ ^4B, vC
74 986.7 23/13 B𝄪, vvC C
75 1000 41/23 ^B𝄪, vC ^C, v4D♭
76 1013.3 C ^^C, v3D♭
77 1026.7 ^C, vD♭♭♭ ^3C, vvD♭
78 1040 31/17 ^^C, D♭♭♭ ^4C, vD♭
79 1053.3 vC♯, ^D♭♭♭ ^5C, D♭
80 1066.7 37/20 C♯, vvD♭♭ v5C♯, ^D♭
81 1080 28/15, 41/22 ^C♯, vD♭♭ v4C♯, ^^D♭
82 1093.3 32/17 ^^C♯, D♭♭ v3C♯, ^3D♭
83 1106.7 vC𝄪, ^D♭♭ vvC♯, ^4D♭
84 1120 C𝄪, vvD♭ vC♯, ^5D♭
85 1133.3 25/13 ^C𝄪, vD♭ C♯, v5D
86 1146.7 31/16, 33/17 ^^C𝄪, D♭ ^C♯, v4D
87 1160 vC♯𝄪, ^D♭ ^^C♯, v3D
88 1173.3 C♯𝄪, vvD ^3C♯, vvD
89 1186.7 ^C♯𝄪, vD ^4C♯, vD
90 1200 2/1 D D

Scales

Instruments

Music

Bryan Deister
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