59edo

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← 58edo 59edo 60edo →
Prime factorization 59 (prime)
Step size 20.339 ¢ 
Fifth 35\59 (711.864 ¢)
Semitones (A1:m2) 9:2 (183.1 ¢ : 40.68 ¢)
Dual sharp fifth 35\59 (711.864 ¢)
Dual flat fifth 34\59 (691.525 ¢)
Dual major 2nd 10\59 (203.39 ¢)
Consistency limit 7
Distinct consistency limit 7

59 equal divisions of the octave (abbreviated 59edo or 59ed2), also called 59-tone equal temperament (59tet) or 59 equal temperament (59et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of 21/59, or the 59th root of 2.

Theory

59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its 5/4 is nearly pure (stretched only 0.127 ¢), as the denominator of a convergent to log25. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit.

Using the flat fifth instead of the sharp one allows for the 12 & 35 temperament, which is a kind of bizarre cousin to garibaldi with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for flattertone temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50 & 59 temperament with a subminor third generator provides an interesting temperament.

Odd harmonics

Approximation of odd harmonics in 59edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27
Error Absolute (¢) +9.91 +0.13 +7.45 -0.52 -2.17 -6.63 +10.04 -3.26 +7.57 -2.98 +2.23 +0.25 +9.39
Relative (%) +48.7 +0.6 +36.6 -2.6 -10.6 -32.6 +49.3 -16.0 +37.2 -14.7 +11.0 +1.2 +46.2
Steps
(reduced) 		94
(35) 		137
(19) 		166
(48) 		187
(10) 		204
(27) 		218
(41) 		231
(54) 		241
(5) 		251
(15) 		259
(23) 		267
(31) 		274
(38) 		281
(45)

Subsets and supersets

59edo is the 17th prime edo, following 53edo and before 61edo. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

Intervals

Steps Cents Approximate ratios Ups and downs notation
(Dual flat fifth 34\59) 		Ups and downs notation
(Dual sharp fifth 35\59)
0 0 1/1 D D
1 20.3 ^D, E♭♭♭♭ ^D, vE♭
2 40.7 D♯, vE♭♭♭ ^^D, E♭
3 61 29/28, 30/29 ^D♯, E♭♭♭ ^3D, ^E♭
4 81.4 23/22 D𝄪, vE♭♭ ^4D, ^^E♭
5 101.7 17/16 ^D𝄪, E♭♭ v4D♯, ^3E♭
6 122 15/14 D♯𝄪, vE♭ v3D♯, ^4E♭
7 142.4 25/23 ^D♯𝄪, E♭ vvD♯, v4E
8 162.7 11/10, 34/31 D𝄪𝄪, vE vD♯, v3E
9 183.1 E D♯, vvE
10 203.4 ^E, F♭♭♭ ^D♯, vE
11 223.7 25/22, 33/29 E♯, vF♭♭ E
12 244.1 23/20 ^E♯, F♭♭ ^E, vF
13 264.4 7/6 E𝄪, vF♭ F
14 284.7 13/11, 20/17, 33/28 ^E𝄪, F♭ ^F, vG♭
15 305.1 31/26 E♯𝄪, vF ^^F, G♭
16 325.4 29/24 F ^3F, ^G♭
17 345.8 ^F, G♭♭♭♭ ^4F, ^^G♭
18 366.1 F♯, vG♭♭♭ v4F♯, ^3G♭
19 386.4 5/4 ^F♯, G♭♭♭ v3F♯, ^4G♭
20 406.8 19/15, 24/19 F𝄪, vG♭♭ vvF♯, v4G
21 427.1 32/25 ^F𝄪, G♭♭ vF♯, v3G
22 447.5 22/17 F♯𝄪, vG♭ F♯, vvG
23 467.8 17/13 ^F♯𝄪, G♭ ^F♯, vG
24 488.1 F𝄪𝄪, vG G
25 508.5 G ^G, vA♭
26 528.8 19/14, 34/25 ^G, A♭♭♭♭ ^^G, A♭
27 549.2 11/8 G♯, vA♭♭♭ ^3G, ^A♭
28 569.5 32/23 ^G♯, A♭♭♭ ^4G, ^^A♭
29 589.8 31/22 G𝄪, vA♭♭ v4G♯, ^3A♭
30 610.2 ^G𝄪, A♭♭ v3G♯, ^4A♭
31 630.5 23/16 G♯𝄪, vA♭ vvG♯, v4A
32 650.8 16/11 ^G♯𝄪, A♭ vG♯, v3A
33 671.2 25/17, 28/19 G𝄪𝄪, vA G♯, vvA
34 691.5 A ^G♯, vA
35 711.9 ^A, B♭♭♭♭ A
36 732.2 26/17, 29/19 A♯, vB♭♭♭ ^A, vB♭
37 752.5 17/11 ^A♯, B♭♭♭ ^^A, B♭
38 772.9 25/16 A𝄪, vB♭♭ ^3A, ^B♭
39 793.2 19/12, 30/19 ^A𝄪, B♭♭ ^4A, ^^B♭
40 813.6 8/5 A♯𝄪, vB♭ v4A♯, ^3B♭
41 833.9 ^A♯𝄪, B♭ v3A♯, ^4B♭
42 854.2 A𝄪𝄪, vB vvA♯, v4B
43 874.6 B vA♯, v3B
44 894.9 ^B, C♭♭♭ A♯, vvB
45 915.3 17/10, 22/13 B♯, vC♭♭ ^A♯, vB
46 935.6 12/7 ^B♯, C♭♭ B
47 955.9 33/19 B𝄪, vC♭ ^B, vC
48 976.3 ^B𝄪, C♭ C
49 996.6 B♯𝄪, vC ^C, vD♭
50 1016.9 C ^^C, D♭
51 1037.3 20/11, 31/17 ^C, D♭♭♭♭ ^3C, ^D♭
52 1057.6 C♯, vD♭♭♭ ^4C, ^^D♭
53 1078 28/15 ^C♯, D♭♭♭ v4C♯, ^3D♭
54 1098.3 32/17 C𝄪, vD♭♭ v3C♯, ^4D♭
55 1118.6 ^C𝄪, D♭♭ vvC♯, v4D
56 1139 29/15 C♯𝄪, vD♭ vC♯, v3D
57 1159.3 ^C♯𝄪, D♭ C♯, vvD
58 1179.7 C𝄪𝄪, vD ^C♯, vD
59 1200 2/1 D D

Notation

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 66-EDO.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation144/14381/801053/1024
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation144/14381/801053/1024

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.

Second-best fifth notation

This notation uses the same sagittal sequence as EDOs 45 and 52.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35
Evo-SZ flavor
Sagittal notationPeriodic table of EDOs with sagittal notation36/35

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

Octave stretch or compression

59edo’s approximations of 3/1, 7/1 and 11/1 are improved by 93edt, a stretched-octave version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

211ed12 is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers compressed octaves, then 296zpi is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

What follows is a comparison of stretched- and compressed-octave 59edo tunings.

93edt
  • Octave size: 1206.62 ¢

Stretching the octave of 59edo by around 6.5 ¢ results in improved primes 3, 7 and 11 but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 8.22 ¢. The tuning 93edt does this. So does the tuning 203ed11 whose octaves are identical within 0.1 ¢.

Approximation of harmonics in 93edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.62 +0.00 -7.22 -4.96 +6.62 +5.61 -0.60 +0.00 +1.66 +0.26 -7.22
Relative (%) +32.4 +0.0 -35.3 -24.3 +32.4 +27.4 -2.9 +0.0 +8.1 +1.3 -35.3
Steps
(reduced) 		59
(59) 		93
(0) 		117
(24) 		136
(43) 		152
(59) 		165
(72) 		176
(83) 		186
(0) 		195
(9) 		203
(17) 		210
(24)
Approximation of harmonics in 93edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.63 -8.22 -4.96 +6.02 +3.32 +6.62 -5.18 +8.27 +5.61 +6.88 -8.73 -0.60
Relative (%) -12.9 -40.2 -24.3 +29.4 +16.2 +32.4 -25.3 +40.5 +27.4 +33.6 -42.7 -2.9
Steps
(reduced) 		217
(31) 		223
(37) 		229
(43) 		235
(49) 		240
(54) 		245
(59) 		249
(63) 		254
(68) 		258
(72) 		262
(76) 		265
(79) 		269
(83)
152ed6
  • Octave size: 1204.05 ¢

Stretching the octave of 59edo by around 4 ¢ results in improved primes 3 and 7, but worse primes 2, 5, 11 and 13. This approximates all harmonics up to 16 within 9.53 ¢. The tuning 152ed6 does this.

Approximation of harmonics in 152ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.05 -4.05 +8.10 +9.53 +0.00 -1.57 -8.26 -8.10 -6.83 -8.58 +4.05
Relative (%) +19.8 -19.8 +39.7 +46.7 +0.0 -7.7 -40.5 -39.7 -33.5 -42.0 +19.8
Steps
(reduced) 		59
(59) 		93
(93) 		118
(118) 		137
(137) 		152
(0) 		165
(13) 		176
(24) 		186
(34) 		195
(43) 		203
(51) 		211
(59)
Approximation of harmonics in 152ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +8.33 +2.48 +5.48 -4.21 -7.13 -4.05 +4.39 -2.78 -5.62 -4.53 +0.15 +8.10
Relative (%) +40.8 +12.1 +26.8 -20.7 -34.9 -19.8 +21.5 -13.6 -27.5 -22.2 +0.7 +39.7
Steps
(reduced) 		218
(66) 		224
(72) 		230
(78) 		235
(83) 		240
(88) 		245
(93) 		250
(98) 		254
(102) 		258
(106) 		262
(110) 		266
(114) 		270
(118)
294zpi
  • Step size: 20.399 ¢, octave size: 1203.54 ¢

Stretching the octave of 59edo by around 3.5 ¢ results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.08 ¢. The tuning 294zpi does this.

Approximation of harmonics in 294zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.54 -4.85 +7.08 +8.35 -1.31 -2.99 -9.78 -9.70 -8.51 +10.08 +2.23
Relative (%) +17.4 -23.8 +34.7 +40.9 -6.4 -14.7 -47.9 -47.5 -41.7 +49.4 +11.0
Step 59 93 118 137 152 165 176 186 195 204 211
Approximation of harmonics in 294zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.45 +0.55 +3.50 -6.24 -9.20 -6.16 +2.24 -4.97 -7.84 -6.78 -2.14 +5.77
Relative (%) +31.6 +2.7 +17.2 -30.6 -45.1 -30.2 +11.0 -24.4 -38.4 -33.2 -10.5 +28.3
Step 218 224 230 235 240 245 250 254 258 262 266 270
211ed12
  • Octave size: 1202.92 ¢

Stretching the octave of 59edo by around 3 ¢ results in improved primes 3 and 7, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.82 ¢. The tuning 211ed12 does this.

Approximation of harmonics in 211ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.92 -5.83 +5.83 +6.90 -2.92 -4.74 +8.75 +8.72 +9.82 +7.92 +0.00
Relative (%) +14.3 -28.6 +28.6 +33.8 -14.3 -23.2 +42.9 +42.8 +48.1 +38.8 +0.0
Steps
(reduced) 		59
(59) 		93
(93) 		118
(118) 		137
(137) 		152
(152) 		165
(165) 		177
(177) 		187
(187) 		196
(196) 		204
(204) 		211
(0)
Approximation of harmonics in 211ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.15 -1.82 +1.07 -8.72 +8.65 -8.75 -0.41 -7.66 +9.82 -9.55 -4.96 +2.92
Relative (%) +20.3 -8.9 +5.2 -42.8 +42.4 -42.9 -2.0 -37.6 +48.2 -46.9 -24.3 +14.3
Steps
(reduced) 		218
(7) 		224
(13) 		230
(19) 		235
(24) 		241
(30) 		245
(34) 		250
(39) 		254
(43) 		259
(48) 		262
(51) 		266
(55) 		270
(59)
295zpi
  • Step size: 20.342 ¢, octave size: 1200.18 ¢

Stretching the octave of 59edo by around a fifth of a cent results in slightly improved primes 11 and 13, but slightly worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 9.97 ¢. The tuning 294zpi does this. 294zpi shares error equally between the two mappings of harmonic 3, so it is the best dual-fifth option for 59edo.

Approximation of harmonics in 295zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.18 -10.15 +0.36 +0.54 -9.97 +7.95 +0.53 +0.04 +0.72 -1.55 -9.79
Relative (%) +0.9 -49.9 +1.8 +2.7 -49.0 +39.1 +2.6 +0.2 +3.5 -7.6 -48.1
Step 59 93 118 137 152 166 177 187 196 204 211
Approximation of harmonics in 295zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.97 +8.12 -9.61 +0.71 -2.53 +0.22 +8.33 +0.90 -2.20 -1.37 +3.04 -9.62
Relative (%) -29.4 +39.9 -47.2 +3.5 -12.5 +1.1 +40.9 +4.4 -10.8 -6.7 +14.9 -47.3
Step 218 225 230 236 241 246 251 255 259 263 267 270
59edo
  • Step size: 20.339 ¢, octave size: 1200.00 ¢

Pure-octaves 59edo approximates all harmonics up to 16 within 10.04 ¢. So does the tuning 137ed5 whose octave is identical within 0.05 ¢.

Approximation of harmonics in 59edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 +9.91 +0.00 +0.13 +9.91 +7.45 +0.00 -0.52 +0.13 -2.17 +9.91
Relative (%) +0.0 +48.7 +0.0 +0.6 +48.7 +36.6 +0.0 -2.6 +0.6 -10.6 +48.7
Steps
(reduced) 		59
(0) 		94
(35) 		118
(0) 		137
(19) 		153
(35) 		166
(48) 		177
(0) 		187
(10) 		196
(19) 		204
(27) 		212
(35)
Approximation of harmonics in 59edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -6.63 +7.45 +10.04 +0.00 -3.26 -0.52 +7.57 +0.13 -2.98 -2.17 +2.23 +9.91
Relative (%) -32.6 +36.6 +49.3 +0.0 -16.0 -2.6 +37.2 +0.6 -14.7 -10.6 +11.0 +48.7
Steps
(reduced) 		218
(41) 		225
(48) 		231
(54) 		236
(0) 		241
(5) 		246
(10) 		251
(15) 		255
(19) 		259
(23) 		263
(27) 		267
(31) 		271
(35)
59et, 13-limit WE tuning
  • Step size: 20.320 ¢, octave size: 1198.88 ¢

Compressing the octave of 59edo by around 1 ¢ results in slightly improved primes 3, 7 and 13, but slightly worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.95 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 59et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.12 +8.12 -2.24 -2.47 +7.00 +4.29 -3.36 -4.07 -3.59 -6.04 +5.88
Relative (%) -5.5 +40.0 -11.0 -12.2 +34.5 +21.1 -16.5 -20.0 -17.7 -29.7 +29.0
Step 59 94 118 137 153 166 177 187 196 204 212
Approximation of harmonics in 59et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.55 +3.17 +5.65 -4.48 -7.84 -5.19 +2.81 -4.71 -7.90 -7.16 -2.83 +4.76
Relative (%) +47.0 +15.6 +27.8 -22.0 -38.6 -25.5 +13.8 -23.2 -38.9 -35.2 -13.9 +23.4
Step 219 225 231 236 241 246 251 255 259 263 267 271
59et, 7-limit WE tuning
  • Step size: 20.301 ¢, octave size: 1197.76 ¢

Compressing the octave of 59edo by around 2 ¢ results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.91 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this.

Approximation of harmonics in ETNAME, 59et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.24 +6.34 -4.48 -5.08 +4.10 +1.14 -6.72 -7.62 -7.32 -9.91 +1.86
Relative (%) -11.0 +31.2 -22.1 -25.0 +20.2 +5.6 -33.1 -37.5 -36.0 -48.8 +9.1
Step 59 94 118 137 153 166 177 187 196 204 212
Approximation of harmonics in 59et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.39 -1.10 +1.26 -8.96 +7.89 -9.86 -1.96 -9.56 +7.48 +8.15 -7.91 -0.38
Relative (%) +26.6 -5.4 +6.2 -44.2 +38.8 -48.6 -9.7 -47.1 +36.8 +40.1 -39.0 -1.9
Step 219 225 231 236 242 246 251 255 260 264 267 271
166ed7
  • Octave size: 1197.35 ¢

Compressing the octave of 59edo by around 2.5 ¢ results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.71 ¢. The tuning 166ed7 does this.

Approximation of harmonics in 166ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.65 +5.69 -5.29 -6.02 +3.05 +0.00 -7.94 -8.91 -8.66 +8.98 +0.40
Relative (%) -13.0 +28.1 -26.1 -29.7 +15.0 +0.0 -39.1 -43.9 -42.7 +44.2 +2.0
Steps
(reduced) 		59
(59) 		94
(94) 		118
(118) 		137
(137) 		153
(153) 		166
(0) 		177
(11) 		187
(21) 		196
(30) 		205
(39) 		212
(46)
Approximation of harmonics in 166ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.89 -2.65 -0.32 +9.71 +6.22 +8.74 -3.69 +8.98 +5.69 +6.33 -9.74 -2.25
Relative (%) +19.2 -13.0 -1.6 +47.8 +30.7 +43.1 -18.2 +44.3 +28.1 +31.2 -48.0 -11.1
Steps
(reduced) 		219
(53) 		225
(59) 		231
(65) 		237
(71) 		242
(76) 		247
(81) 		251
(85) 		256
(90) 		260
(94) 		264
(98) 		267
(101) 		271
(105)
212ed12
  • Octave size: 1197.24 ¢

Compressing the octave of 59edo by around 3 ¢ results in improved primes 3, 7 and 13, but worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 9.26 ¢. The tuning 212ed12 does this.

Approximation of harmonics in 212ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.76 +5.52 -5.52 -6.28 +2.76 -0.31 -8.27 -9.26 -9.03 +8.59 +0.00
Relative (%) -13.6 +27.2 -27.2 -30.9 +13.6 -1.5 -40.8 -45.6 -44.5 +42.3 +0.0
Steps
(reduced) 		59
(59) 		94
(94) 		118
(118) 		137
(137) 		153
(153) 		166
(166) 		177
(177) 		187
(187) 		196
(196) 		205
(205) 		212
(0)
Approximation of harmonics in 212ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.47 -3.07 -0.76 +9.26 +5.77 +8.27 -4.16 +8.50 +5.20 +5.83 +10.05 -2.76
Relative (%) +17.1 -15.1 -3.8 +45.6 +28.4 +40.8 -20.5 +41.9 +25.6 +28.7 +49.5 -13.6
Steps
(reduced) 		219
(7) 		225
(13) 		231
(19) 		237
(25) 		242
(30) 		247
(35) 		251
(39) 		256
(44) 		260
(48) 		264
(52) 		268
(56) 		271
(59)
296zpi
  • Step size: 20.282 ¢, octave size: 1196.64 ¢

Compressing the octave of 59edo by around 3.5 ¢ results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 10.09 ¢. The tuning 296zpi does this.

Approximation of harmonics in 296zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.36 +4.55 -6.72 -7.68 +1.19 -2.01 -10.09 +9.11 +9.24 +6.49 -2.17
Relative (%) -16.6 +22.4 -33.2 -37.9 +5.9 -9.9 -49.7 +44.9 +45.6 +32.0 -10.7
Step 59 94 118 137 153 166 177 188 197 205 212
Approximation of harmonics in 296zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.23 -5.38 -3.13 +6.83 +3.29 +5.74 -6.73 +5.88 +2.54 +3.13 +7.30 -5.53
Relative (%) +6.1 -26.5 -15.4 +33.7 +16.2 +28.3 -33.2 +29.0 +12.5 +15.4 +36.0 -27.3
Step 219 225 231 237 242 247 251 256 260 264 268 271
153ed6
  • Octave size: 1196.18 ¢

Compressing the octave of 59edo by around 4 ¢ results in greatly improved primes 3, 7 and 13, but far worse primes 2, 5 and 11. This approximates all harmonics up to 16 within 8.81 ¢. The tuning 153ed6 does this.

Approximation of harmonics in 153ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.82 +3.82 -7.64 -8.75 +0.00 -3.31 +8.81 +7.64 +7.71 +4.90 -3.82
Relative (%) -18.8 +18.8 -37.7 -43.1 +0.0 -16.3 +43.5 +37.7 +38.0 +24.2 -18.8
Steps
(reduced) 		59
(59) 		94
(94) 		118
(118) 		137
(137) 		153
(0) 		166
(13) 		178
(25) 		188
(35) 		197
(44) 		205
(52) 		212
(59)
Approximation of harmonics in 153ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.47 -7.13 -4.92 +4.99 +1.40 +3.82 -8.68 +3.89 +0.52 +1.08 +5.22 -7.64
Relative (%) -2.3 -35.2 -24.3 +24.6 +6.9 +18.8 -42.8 +19.2 +2.5 +5.3 +25.7 -37.7
Steps
(reduced) 		219
(66) 		225
(72) 		231
(78) 		237
(84) 		242
(89) 		247
(94) 		251
(98) 		256
(103) 		260
(107) 		264
(111) 		268
(115) 		271
(118)

Instruments

Lumatone

See Lumatone mapping for 59edo.

Music

Bryan Deister
Francium
Ray Perlner
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