89edo: Difference between revisions

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89edo has a [[3/1|harmonic 3]] less than a cent flat and a [[5/1|harmonic 5]] less than five cents sharp, with a [[7/1|7]] two cents sharp and an [[11/1|11]] 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]].
89edo has a [[3/1|harmonic 3]] less than a cent flat and a [[5/1|harmonic 5]] less than five cents sharp, with a [[7/1|7]] two cents sharp and an [[11/1|11]] 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]].


89et [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.
89et [[tempering out|tempers out]] the commas 32805/32768 ([[schisma]]) in the 5-limit; [[126/125]], [[1728/1715]], and [[2401/2400]] in the 7-limit; and [[176/175]], [[243/242]], [[441/440]] and [[540/539]] in the 11-limit. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.


=== Prime harmonics ===
=== Prime harmonics ===
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| 2.3
| 2.3
| {{monzo| -141 89 }}
| {{monzo| -141 89 }}
| [{{val| 89 141 }}]
| {{mapping| 89 141 }}
| +0.262
| +0.262
| 0.262
| 0.262
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| 2.3.5
| 2.3.5
| 32805/32768, 10077696/9765625
| 32805/32768, 10077696/9765625
| [{{val| 89 141 207 }}]
| {{mapping| 89 141 207 }}
| -0.500
| -0.500
| 1.098
| 1.098
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| 2.3.5.7
| 2.3.5.7
| 126/125, 1728/1715, 32805/32768
| 126/125, 1728/1715, 32805/32768
| [{{val| 89 141 207 250 }}]
| {{mapping| 89 141 207 250 }}
| -0.550
| -0.550
| 0.955
| 0.955
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| 2.3.5.7.11
| 2.3.5.7.11
| 126/125, 176/175, 243/242, 16384/16335
| 126/125, 176/175, 243/242, 16384/16335
| [{{val| 89 141 207 250 308 }}]
| {{mapping| 89 141 207 250 308 }}
| -0.526
| -0.526
| 0.855
| 0.855
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|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator*
! Cents
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperament
! Temperament
|-
|-
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| [[Grackle]]
| [[Grackle]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Scales ==
== Scales ==
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[[Category:Myna]]
[[Category:Myna]]
[[Category:Thrush]]
[[Category:Thrush]]
[[Category:Quartismic]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 05:36, 17 June 2024

← 88edo 89edo 90edo →
Prime factorization 89 (prime)
Step size 13.4831 ¢ 
Fifth 52\89 (701.124 ¢)
Semitones (A1:m2) 8:7 (107.9 ¢ : 94.38 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

89edo has a harmonic 3 less than a cent flat and a harmonic 5 less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out quartisma.

89et tempers out the commas 32805/32768 (schisma) in the 5-limit; 126/125, 1728/1715, and 2401/2400 in the 7-limit; and 176/175, 243/242, 441/440 and 540/539 in the 11-limit. It is an especially good tuning for the myna temperament, both in the 7-limit, tempering out 126/125 and 1728/1715, and in the 11-limit, where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament thrush, tempering out 126/125 and 176/175.

Prime harmonics

Approximation of prime harmonics in 89edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.83 +4.70 +1.96 +1.49 -4.57 +2.91 -0.88 +5.43 -4.86 +1.03
Relative (%) +0.0 -6.2 +34.8 +14.5 +11.1 -33.9 +21.6 -6.6 +40.3 -36.0 +7.7
Steps
(reduced) 		89
(0) 		141
(52) 		207
(29) 		250
(72) 		308
(41) 		329
(62) 		364
(8) 		378
(22) 		403
(47) 		432
(76) 		441
(85)

Subsets and supersets

89edo is the 24th prime edo, and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.

Interval table

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 13.5 ^D, ^^E♭♭
2 27 ^^D, ^3E♭♭
3 40.4 41/40, 42/41 ^3D, v4E♭
4 53.9 31/30, 32/31, 33/32, 34/33 ^4D, v3E♭
5 67.4 27/26 v3D♯, vvE♭
6 80.9 22/21 vvD♯, vE♭
7 94.4 19/18, 37/35 vD♯, E♭
8 107.9 33/31 D♯, ^E♭
9 121.3 15/14, 29/27 ^D♯, ^^E♭
10 134.8 40/37 ^^D♯, ^3E♭
11 148.3 12/11, 37/34 ^3D♯, v4E
12 161.8 34/31 ^4D♯, v3E
13 175.3 21/19, 31/28, 41/37 v3D𝄪, vvE
14 188.8 29/26 vvD𝄪, vE
15 202.2 9/8 E
16 215.7 17/15 ^E, ^^F♭
17 229.2 8/7 ^^E, ^3F♭
18 242.7 23/20, 38/33 ^3E, v4F
19 256.2 22/19, 36/31 ^4E, v3F
20 269.7 7/6 v3E♯, vvF
21 283.1 20/17, 33/28 vvE♯, vF
22 296.6 19/16, 32/27 F
23 310.1 ^F, ^^G♭♭
24 323.6 41/34 ^^F, ^3G♭♭
25 337.1 17/14 ^3F, v4G♭
26 350.6 38/31 ^4F, v3G♭
27 364 21/17, 37/30 v3F♯, vvG♭
28 377.5 41/33 vvF♯, vG♭
29 391 vF♯, G♭
30 404.5 24/19 F♯, ^G♭
31 418 14/11 ^F♯, ^^G♭
32 431.5 41/32 ^^F♯, ^3G♭
33 444.9 22/17, 31/24 ^3F♯, v4G
34 458.4 30/23 ^4F♯, v3G
35 471.9 21/16 v3F𝄪, vvG
36 485.4 41/31 vvF𝄪, vG
37 498.9 4/3 G
38 512.4 39/29 ^G, ^^A♭♭
39 525.8 23/17, 42/31 ^^G, ^3A♭♭
40 539.3 15/11, 41/30 ^3G, v4A♭
41 552.8 11/8 ^4G, v3A♭
42 566.3 v3G♯, vvA♭
43 579.8 7/5 vvG♯, vA♭
44 593.3 31/22, 38/27 vG♯, A♭
45 606.7 27/19 G♯, ^A♭
46 620.2 10/7 ^G♯, ^^A♭
47 633.7 ^^G♯, ^3A♭
48 647.2 16/11 ^3G♯, v4A
49 660.7 22/15, 41/28 ^4G♯, v3A
50 674.2 31/21, 34/23 v3G𝄪, vvA
51 687.6 vvG𝄪, vA
52 701.1 3/2 A
53 714.6 ^A, ^^B♭♭
54 728.1 32/21, 35/23 ^^A, ^3B♭♭
55 741.6 23/15 ^3A, v4B♭
56 755.1 17/11 ^4A, v3B♭
57 768.5 v3A♯, vvB♭
58 782 11/7 vvA♯, vB♭
59 795.5 19/12 vA♯, B♭
60 809 A♯, ^B♭
61 822.5 37/23 ^A♯, ^^B♭
62 836 34/21 ^^A♯, ^3B♭
63 849.4 31/19 ^3A♯, v4B
64 862.9 28/17 ^4A♯, v3B
65 876.4 v3A𝄪, vvB
66 889.9 vvA𝄪, vB
67 903.4 27/16, 32/19 B
68 916.9 17/10 ^B, ^^C♭
69 930.3 12/7 ^^B, ^3C♭
70 943.8 19/11, 31/18 ^3B, v4C
71 957.3 33/19, 40/23 ^4B, v3C
72 970.8 7/4 v3B♯, vvC
73 984.3 30/17 vvB♯, vC
74 997.8 16/9 C
75 1011.2 ^C, ^^D♭♭
76 1024.7 38/21 ^^C, ^3D♭♭
77 1038.2 31/17 ^3C, v4D♭
78 1051.7 11/6 ^4C, v3D♭
79 1065.2 37/20 v3C♯, vvD♭
80 1078.7 28/15, 41/22 vvC♯, vD♭
81 1092.1 vC♯, D♭
82 1105.6 36/19 C♯, ^D♭
83 1119.1 21/11 ^C♯, ^^D♭
84 1132.6 ^^C♯, ^3D♭
85 1146.1 31/16, 33/17 ^3C♯, v4D
86 1159.6 41/21 ^4C♯, v3D
87 1173 v3C𝄪, vvD
88 1186.5 vvC𝄪, vD
89 1200 2/1 D

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-141 89⟩ [⟨89 141]] +0.262 0.262 1.95
2.3.5 32805/32768, 10077696/9765625 [⟨89 141 207]] -0.500 1.098 8.15
2.3.5.7 126/125, 1728/1715, 32805/32768 [⟨89 141 207 250]] -0.550 0.955 7.08
2.3.5.7.11 126/125, 176/175, 243/242, 16384/16335 [⟨89 141 207 250 308]] -0.526 0.855 6.35

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperament
1 13\89 175.28 72/65 Sesquiquartififths / Sesquart
1 21\89 283.15 13/11 Neominor
1 23\89 310.11 6/5 Myna
1 29\89 391.01 5/4 Amigo
1 37\89 498.87 4/3 Grackle

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

Scales

Music

Francium
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