186zpi: Difference between revisions
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It is important to note that [[124edo]] provides two possible [[3/2|fifths (3/2)]]. The closest one, from the [[val]] <124 197] (i.e. the [[patent val]]), is the [[3/2|fifth]] mapped to 73 steps of [[124edo]] with a [[relative error]] of +46.465%. The second closest, from the [[val]] <124 196] (i.e. the [[val]] 124b), is mapped to 72 steps of [[124edo]] with a [[relative error]] of -53.535%. This second [[3/2|fifth]], which appears in [[124ed8]], also corresponds to the [[3/2|fifth]] of [[31edo]]. Therefore, we choose to use the [[ups and downs notation]] of the 124b temperament, denoted as <124 196]. | It is important to note that [[124edo]] provides two possible [[3/2|fifths (3/2)]]. The closest one, from the [[val]] <124 197] (i.e. the [[patent val]]), is the [[3/2|fifth]] mapped to 73 steps of [[124edo]] with a [[relative error]] of +46.465%. The second closest, from the [[val]] <124 196] (i.e. the [[val]] 124b), is mapped to 72 steps of [[124edo]] with a [[relative error]] of -53.535%. This second [[3/2|fifth]], which appears in [[124ed8]], also corresponds to the [[3/2|fifth]] of [[31edo]]. Therefore, we choose to use the [[ups and downs notation]] of the 124b temperament, denoted as <124 196]. | ||
{| class="wikitable center-all left-1 right-2 left-3 center-4 | {| class="wikitable center-all left-1 right-2 left-3 center-4 left-5" | ||
|- | |- | ||
|colspan="3"| | | colspan="3" | JI ratios are comprised of 32-integer limit ratios,<br>and are stylized as follows to indicate their accuracy: | ||
JI ratios are comprised of 32-integer limit ratios, | |||
<br>and are stylized as follows to indicate their accuracy: | |||
* '''<u>Bold Underlined:</u>''' relative error < 8.333 % | * '''<u>Bold Underlined:</u>''' relative error < 8.333 % | ||
* '''Bold:''' relative error < 16.667 % | * '''Bold:''' relative error < 16.667 % | ||
| Line 118: | Line 116: | ||
* <small><small>Small Small:</small></small> relative error < 41.667 % | * <small><small>Small Small:</small></small> relative error < 41.667 % | ||
* <small><small><small>Small Small Small:</small></small></small> relative error < 50 % | * <small><small><small>Small Small Small:</small></small></small> relative error < 50 % | ||
|colspan="2"| | | colspan="2" | 124b at every 3-degree step<br>Pythagorean limma = 12 steps<br>whole tone = 20 steps | ||
124b at every 3-degree<br> | |||
Pythagorean limma = 12 steps<br> | |||
whole tone = 20 steps | |||
|- | |- | ||
! | ! Degree | ||
!Cents | ! Cents | ||
!Ratios | ! Ratios | ||
!Ups and Downs Notation | ! Ups and Downs Notation | ||
!Step | ! Step | ||
|- | |- | ||
| 0 | | 0 | ||
Revision as of 17:33, 10 August 2024
186 zeta peak index (abbreviated 186zpi), is the equal-step tuning system obtained from the 186st peak of the Riemann zeta function.
| Tuning | Strength | Closest EDO | Integer limit | ||||||
|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per octave | Step size (cents) | Height | Integral | Gap | EDO | Octave (cents) | Consistent | Distinct |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 1.876590 | 0.241233 | 11.567493 | 41edo | 1190.02021518380 | 2 | 2 |
Theory
Record on the Riemann zeta function with primes 2 and 3 removed
186zpi sets a height record on the Riemann zeta function with primes 2 and 3 removed. The previous record is 125zpi and the next one is 565zpi. It is important to highlight that the optimal equal tunings obtained by excluding the prime numbers 2 and 3 from the Riemann zeta function differs very slightly from the optimal equal tuning corresponding to the same peaks on the unmodified Riemann zeta function.
| Unmodified Riemann zeta function | Riemann zeta function with primes 2 and 3 removed | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Tuning | Strength | Closest EDO | Tuning | Strength | Closest EDO | |||||
| ZPI | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) | Steps per octave | Step size (cents) | Height | EDO | Octave (cents) |
| 125zpi | 30.6006474885974 | 39.2148564976330 | 1.468164 | 31edo | 1215.66055142662 | 30.5974484926723 | 39.2189564527704 | 3.769318 | 31edo | 1215.78765003588 |
| 186zpi | 41.3438354846780 | 29.0248832971658 | 1.876590 | 41edo | 1190.02021518380 | 41.3477989230936 | 29.0221010852836 | 4.469823 | 41edo | 1189.90614449663 |
| 565zpi | 98.6209462564991 | 12.1678005084130 | 2.305330 | 99edo | 1204.61225033289 | 98.6257548378926 | 12.1672072570942 | 4.883729 | 99edo | 1204.55351845233 |
Harmonic series
As a non-octave, non-tritave scale, 186zpi features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.0 | +13.7 | +9.1 | +0.1 | +3.7 | -1.9 | -0.9 | -1.7 | -9.9 | -0.8 | -6.3 | +0.3 | -11.9 | +13.8 | -10.9 |
| Relative (%) | -34.4 | +47.2 | +31.2 | +0.3 | +12.8 | -6.7 | -3.2 | -5.7 | -34.1 | -2.6 | -21.6 | +1.0 | -41.1 | +47.4 | -37.5 | |
| Step | 41 | 66 | 83 | 96 | 107 | 116 | 124 | 131 | 137 | 143 | 148 | 153 | 157 | 162 | 165 | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.2 | -11.6 | +10.9 | +9.1 | +11.7 | -10.7 | -0.6 | +12.8 | +0.2 | -9.7 | +12.0 | +7.1 | +4.4 | +3.8 | +5.1 | +8.2 |
| Relative (%) | +0.9 | -40.1 | +37.4 | +31.5 | +40.5 | -37.0 | -2.1 | +44.0 | +0.5 | -33.4 | +41.5 | +24.6 | +15.2 | +13.0 | +17.5 | +28.1 | |
| Step | 169 | 172 | 176 | 179 | 182 | 184 | 187 | 190 | 192 | 194 | 197 | 199 | 201 | 203 | 205 | 207 | |
Approximation of EDONOIs
Based on harmonics with less than 1 cent of error, 186zpi can be approximated by 96ed5, 124ed8 (or every 3 steps of 124edo), 143ed11, 153ed13, 169ed17, 187ed23, and 192ed25.
Intervals and notation
There are several ways to approach notation. The simplest method involves using the notations from 41edo. However, this method does not preserve octave compression when rendered by notation software. To address this issue, consider using the ups and downs notation from 124edo at every 3-degree step (i.e., the edonoi 124ed8).
It is important to note that 124edo provides two possible fifths (3/2). The closest one, from the val <124 197] (i.e. the patent val), is the fifth mapped to 73 steps of 124edo with a relative error of +46.465%. The second closest, from the val <124 196] (i.e. the val 124b), is mapped to 72 steps of 124edo with a relative error of -53.535%. This second fifth, which appears in 124ed8, also corresponds to the fifth of 31edo. Therefore, we choose to use the ups and downs notation of the 124b temperament, denoted as <124 196].
| JI ratios are comprised of 32-integer limit ratios, and are stylized as follows to indicate their accuracy:
|
124b at every 3-degree step Pythagorean limma = 12 steps whole tone = 20 steps | |||
| Degree | Cents | Ratios | Ups and Downs Notation | Step |
|---|---|---|---|---|
| 0 | 0.000 | P1 | 0 | |
| 1 | 29.025 | ^^^1 | 3 | |
| 2 | 58.050 | 32/31, 31/30, 30/29, 29/28, 28/27, 27/26, 26/25, 25/24 | vvA1, ^^d2 | 6 |
| 3 | 87.075 | 24/23, 23/22, 22/21, 21/20, 20/19, 19/18, 18/17 | vvvm2 | 9 |
| 4 | 116.100 | 17/16, 16/15, 31/29, 15/14, 29/27, 14/13 | m2 | 12 |
| 5 | 145.124 | 27/25, 13/12, 25/23, 12/11, 23/21 | ^^^m2 | 15 |
| 6 | 174.149 | 11/10, 32/29, 21/19, 31/28, 10/9 | vvM2 | 18 |
| 7 | 203.174 | 29/26, 19/17, 28/25, 9/8, 26/23, 17/15 | ^M2 | 21 |
| 8 | 232.199 | 25/22, 8/7, 31/27, 23/20 | ^4M2 | 24 |
| 9 | 261.224 | 15/13, 22/19, 29/25, 7/6 | ^^^d3 | 27 |
| 10 | 290.249 | 27/23, 20/17, 13/11, 32/27, 19/16, 25/21, 31/26 | vvm3 | 30 |
| 11 | 319.274 | 6/5, 29/24, 23/19 | ^m3 | 33 |
| 12 | 348.299 | 17/14, 28/23, 11/9, 27/22, 16/13 | ~3 | 36 |
| 13 | 377.323 | 21/17, 26/21, 31/25, 5/4 | vM3 | 39 |
| 14 | 406.348 | 29/23, 24/19, 19/15, 14/11 | ^^M3 | 42 |
| 15 | 435.373 | 23/18, 32/25, 9/7, 31/24, 22/17 | vvvA3 | 45 |
| 16 | 464.398 | 13/10, 30/23, 17/13, 21/16, 25/19, 29/22 | v44 | 48 |
| 17 | 493.423 | 4/3 | v4 | 51 |
| 18 | 522.448 | 31/23, 27/20, 23/17, 19/14, 15/11 | ^^4 | 54 |
| 19 | 551.473 | 26/19, 11/8, 29/21, 18/13 | vvvA4 | 57 |
| 20 | 580.498 | 25/18, 32/23, 7/5, 31/22 | A4 | 60 |
| 21 | 609.523 | 24/17, 17/12, 27/19, 10/7 | vd5 | 63 |
| 22 | 638.547 | 23/16, 13/9, 29/20, 16/11 | ^^d5 | 66 |
| 23 | 667.572 | 19/13, 22/15, 25/17, 28/19, 31/21 | vvv5 | 69 |
| 24 | 696.597 | 3/2 | P5 | 72 |
| 25 | 725.622 | 32/21, 29/19, 26/17, 23/15 | ^^^5 | 75 |
| 26 | 754.647 | 20/13, 17/11, 31/20, 14/9 | vvA5, ^^d6 | 78 |
| 27 | 783.672 | 25/16, 11/7, 30/19, 19/12 | vvvm6 | 81 |
| 28 | 812.697 | 27/17, 8/5, 29/18 | m6 | 84 |
| 29 | 841.722 | 21/13, 13/8, 31/19, 18/11 | ^^^m6 | 87 |
| 30 | 870.746 | 23/14, 28/17, 5/3 | vvM6 | 90 |
| 31 | 899.771 | 32/19, 27/16, 22/13 | ^M6 | 93 |
| 32 | 928.796 | 17/10, 29/17, 12/7, 31/18 | ^4M6 | 96 |
| 33 | 957.821 | 19/11, 26/15, 7/4 | ^^^d7 | 99 |
| 34 | 986.846 | 30/17, 23/13, 16/9 | vvm7 | 102 |
| 35 | 1015.871 | 25/14, 9/5, 29/16 | ^m7 | 105 |
| 36 | 1044.896 | 20/11, 31/17, 11/6 | ~7 | 108 |
| 37 | 1073.921 | 24/13, 13/7, 28/15, 15/8 | vM7 | 111 |
| 38 | 1102.946 | 32/17, 17/9, 19/10 | ^^M7 | 114 |
| 39 | 1131.970 | 21/11, 23/12, 25/13, 27/14, 29/15, 31/16 | vvvA7 | 117 |
| 40 | 1160.995 | v41 +1oct | 120 | |
| 41 | 1190.020 | 2/1 | v1 +1oct | 123 |
| 42 | 1219.045 | ^^1 +1oct | 126 | |
| 43 | 1248.070 | 31/15, 29/14 | vvvA1 +1oct | 129 |
| 44 | 1277.095 | 27/13, 25/12, 23/11, 21/10 | v4m2 +1oct | 132 |
| 45 | 1306.120 | 19/9, 17/8, 32/15, 15/7 | vm2 +1oct | 135 |
| 46 | 1335.145 | 28/13, 13/6 | ^^m2 +1oct | 138 |
| 47 | 1364.170 | 24/11, 11/5, 31/14 | vvvM2 +1oct | 141 |
| 48 | 1393.194 | 20/9, 29/13, 9/4 | M2 +1oct | 144 |
| 49 | 1422.219 | 25/11, 16/7 | ^^^M2 +1oct | 147 |
| 50 | 1451.244 | 23/10, 30/13 | vvA2 +1oct, ^^d3 +1oct | 150 |
| 51 | 1480.269 | 7/3, 26/11 | vvvm3 +1oct | 153 |
| 52 | 1509.294 | 19/8, 31/13, 12/5 | m3 +1oct | 156 |
| 53 | 1538.319 | 29/12, 17/7, 22/9 | ^^^m3 +1oct | 159 |
| 54 | 1567.344 | 27/11, 32/13 | vvM3 +1oct | 162 |
| 55 | 1596.369 | 5/2 | ^M3 +1oct | 165 |
| 56 | 1625.393 | 28/11, 23/9, 18/7 | ^4M3 +1oct | 168 |
| 57 | 1654.418 | 31/12, 13/5 | ^^^d4 +1oct | 171 |
| 58 | 1683.443 | 21/8, 29/11 | vv4 +1oct | 174 |
| 59 | 1712.468 | 8/3, 27/10 | ^4 +1oct | 177 |
| 60 | 1741.493 | 19/7, 30/11, 11/4 | ~4 +1oct | 180 |
| 61 | 1770.518 | 25/9, 14/5 | vA4 +1oct | 183 |
| 62 | 1799.543 | 31/11, 17/6 | ^^A4 +1oct, vvd5 +1oct | 186 |
| 63 | 1828.568 | 20/7, 23/8, 26/9 | ^d5 +1oct | 189 |
| 64 | 1857.593 | 29/10, 32/11 | ~5 +1oct | 192 |
| 65 | 1886.617 | v5 +1oct | 195 | |
| 66 | 1915.642 | 3/1 | ^^5 +1oct | 198 |
| 67 | 1944.667 | 31/10 | vvvA5 +1oct | 201 |
| 68 | 1973.692 | 28/9, 25/8, 22/7 | v4m6 +1oct | 204 |
| 69 | 2002.717 | 19/6, 16/5 | vm6 +1oct | 207 |
| 70 | 2031.742 | 29/9, 13/4 | ^^m6 +1oct | 210 |
| 71 | 2060.767 | 23/7 | vvvM6 +1oct | 213 |
| 72 | 2089.792 | 10/3 | M6 +1oct | 216 |
| 73 | 2118.816 | 27/8, 17/5, 24/7 | ^^^M6 +1oct | 219 |
| 74 | 2147.841 | 31/9 | vvA6 +1oct, ^^d7 +1oct | 222 |
| 75 | 2176.866 | 7/2 | vvvm7 +1oct | 225 |
| 76 | 2205.891 | 32/9, 25/7, 18/5 | m7 +1oct | 228 |
| 77 | 2234.916 | 29/8, 11/3 | ^^^m7 +1oct | 231 |
| 78 | 2263.941 | 26/7 | vvM7 +1oct | 234 |
| 79 | 2292.966 | 15/4 | ^M7 +1oct | 237 |
| 80 | 2321.991 | 19/5, 23/6 | ^4M7 +1oct | 240 |
| 81 | 2351.016 | 27/7, 31/8 | ^^^d1 +2oct | 243 |
| 82 | 2380.040 | vv1 +2oct | 246 | |
| 83 | 2409.065 | 4/1 | ^1 +2oct | 249 |
| 84 | 2438.090 | ^41 +2oct | 252 | |
| 85 | 2467.115 | 29/7, 25/6 | ^^^d2 +2oct | 255 |
| 86 | 2496.140 | 21/5, 17/4 | vvm2 +2oct | 258 |
| 87 | 2525.165 | 30/7, 13/3 | ^m2 +2oct | 261 |
| 88 | 2554.190 | 22/5 | ~2 +2oct | 264 |
| 89 | 2583.215 | 31/7 | vM2 +2oct | 267 |
| 90 | 2612.239 | 9/2 | ^^M2 +2oct | 270 |
| 91 | 2641.264 | 32/7, 23/5 | vvvA2 +2oct | 273 |
| 92 | 2670.289 | 14/3 | v4m3 +2oct | 276 |
| 93 | 2699.314 | 19/4 | vm3 +2oct | 279 |
| 94 | 2728.339 | 24/5, 29/6 | ^^m3 +2oct | 282 |
| 95 | 2757.364 | vvvM3 +2oct | 285 | |
| 96 | 2786.389 | 5/1 | M3 +2oct | 288 |
| 97 | 2815.414 | ^^^M3 +2oct | 291 | |
| 98 | 2844.439 | 31/6, 26/5 | vvA3 +2oct, ^^d4 +2oct | 294 |
| 99 | 2873.463 | 21/4 | vvv4 +2oct | 297 |
| 100 | 2902.488 | 16/3 | P4 +2oct | 300 |
| 101 | 2931.513 | 27/5 | ^^^4 +2oct | 303 |
| 102 | 2960.538 | 11/2 | vvA4 +2oct | 306 |
| 103 | 2989.563 | 28/5, 17/3 | ^A4 +2oct | 309 |
| 104 | 3018.588 | 23/4 | d5 +2oct | 312 |
| 105 | 3047.613 | 29/5 | ^^^d5 +2oct | 315 |
| 106 | 3076.638 | vv5 +2oct | 318 | |
| 107 | 3105.663 | 6/1 | ^5 +2oct | 321 |
| 108 | 3134.687 | ^45 +2oct | 324 | |
| 109 | 3163.712 | 31/5, 25/4 | ^^^d6 +2oct | 327 |
| 110 | 3192.737 | 19/3 | vvm6 +2oct | 330 |
| 111 | 3221.762 | 32/5 | ^m6 +2oct | 333 |
| 112 | 3250.787 | 13/2 | ~6 +2oct | 336 |
| 113 | 3279.812 | 20/3 | vM6 +2oct | 339 |
| 114 | 3308.837 | 27/4 | ^^M6 +2oct | 342 |
| 115 | 3337.862 | vvvA6 +2oct | 345 | |
| 116 | 3366.886 | 7/1 | v4m7 +2oct | 348 |
| 117 | 3395.911 | vm7 +2oct | 351 | |
| 118 | 3424.936 | 29/4 | ^^m7 +2oct | 354 |
| 119 | 3453.961 | 22/3 | vvvM7 +2oct | 357 |
| 120 | 3482.986 | 15/2 | M7 +2oct | 360 |
| 121 | 3512.011 | 23/3 | ^^^M7 +2oct | 363 |
| 122 | 3541.036 | 31/4 | vvA7 +2oct, ^^d1 +3oct | 366 |
| 123 | 3570.061 | vvv1 +3oct | 369 | |
| 124 | 3599.086 | 8/1 | P1 +3oct | 372 |
| 125 | 3628.110 | ^^^1 +3oct | 375 | |
| 126 | 3657.135 | 25/3 | vvA1 +3oct, ^^d2 +3oct | 378 |
| 127 | 3686.160 | vvvm2 +3oct | 381 | |
| 128 | 3715.185 | 17/2 | m2 +3oct | 384 |
| 129 | 3744.210 | 26/3 | ^^^m2 +3oct | 387 |
| 130 | 3773.235 | vvM2 +3oct | 390 | |
| 131 | 3802.260 | 9/1 | ^M2 +3oct | 393 |
| 132 | 3831.285 | ^4M2 +3oct | 396 | |
| 133 | 3860.309 | 28/3 | ^^^d3 +3oct | 399 |
| 134 | 3889.334 | 19/2 | vvm3 +3oct | 402 |
| 135 | 3918.359 | 29/3 | ^m3 +3oct | 405 |
| 136 | 3947.384 | ~3 +3oct | 408 | |
| 137 | 3976.409 | 10/1 | vM3 +3oct | 411 |
| 138 | 4005.434 | ^^M3 +3oct | 414 | |
| 139 | 4034.459 | 31/3 | vvvA3 +3oct | 417 |
| 140 | 4063.484 | 21/2 | v44 +3oct | 420 |
| 141 | 4092.509 | 32/3 | v4 +3oct | 423 |
| 142 | 4121.533 | ^^4 +3oct | 426 | |
| 143 | 4150.558 | 11/1 | vvvA4 +3oct | 429 |
| 144 | 4179.583 | A4 +3oct | 432 | |
| 145 | 4208.608 | vd5 +3oct | 435 | |
| 146 | 4237.633 | 23/2 | ^^d5 +3oct | 438 |
| 147 | 4266.658 | vvv5 +3oct | 441 | |
| 148 | 4295.683 | 12/1 | P5 +3oct | 444 |
| 149 | 4324.708 | ^^^5 +3oct | 447 | |
| 150 | 4353.732 | vvA5 +3oct, ^^d6 +3oct | 450 | |
| 151 | 4382.757 | 25/2 | vvvm6 +3oct | 453 |
| 152 | 4411.782 | m6 +3oct | 456 | |
| 153 | 4440.807 | 13/1 | ^^^m6 +3oct | 459 |
| 154 | 4469.832 | vvM6 +3oct | 462 | |
| 155 | 4498.857 | 27/2 | ^M6 +3oct | 465 |
| 156 | 4527.882 | ^4M6 +3oct | 468 | |
| 157 | 4556.907 | 14/1 | ^^^d7 +3oct | 471 |
| 158 | 4585.932 | vvm7 +3oct | 474 | |
| 159 | 4614.956 | ^m7 +3oct | 477 | |
| 160 | 4643.981 | 29/2 | ~7 +3oct | 480 |
| 161 | 4673.006 | vM7 +3oct | 483 | |
| 162 | 4702.031 | 15/1 | ^^M7 +3oct | 486 |
| 163 | 4731.056 | 31/2 | vvvA7 +3oct | 489 |
| 164 | 4760.081 | v41 +4oct | 492 | |
| 165 | 4789.106 | 16/1 | v1 +4oct | 495 |
| 166 | 4818.131 | ^^1 +4oct | 498 | |
| 167 | 4847.156 | vvvA1 +4oct | 501 | |
| 168 | 4876.180 | v4m2 +4oct | 504 | |
| 169 | 4905.205 | 17/1 | vm2 +4oct | 507 |
| 170 | 4934.230 | ^^m2 +4oct | 510 | |
| 171 | 4963.255 | vvvM2 +4oct | 513 | |
| 172 | 4992.280 | 18/1 | M2 +4oct | 516 |
| 173 | 5021.305 | ^^^M2 +4oct | 519 | |
| 174 | 5050.330 | vvA2 +4oct, ^^d3 +4oct | 522 | |
| 175 | 5079.355 | vvvm3 +4oct | 525 | |
| 176 | 5108.379 | 19/1 | m3 +4oct | 528 |
| 177 | 5137.404 | ^^^m3 +4oct | 531 | |
| 178 | 5166.429 | vvM3 +4oct | 534 | |
| 179 | 5195.454 | 20/1 | ^M3 +4oct | 537 |
| 180 | 5224.479 | ^4M3 +4oct | 540 | |
| 181 | 5253.504 | ^^^d4 +4oct | 543 | |
| 182 | 5282.529 | 21/1 | vv4 +4oct | 546 |
| 183 | 5311.554 | ^4 +4oct | 549 | |
| 184 | 5340.579 | 22/1 | ~4 +4oct | 552 |
| 185 | 5369.603 | vA4 +4oct | 555 | |
| 186 | 5398.628 | ^^A4 +4oct, vvd5 +4oct | 558 | |
| 187 | 5427.653 | 23/1 | ^d5 +4oct | 561 |
| 188 | 5456.678 | ~5 +4oct | 564 | |
| 189 | 5485.703 | v5 +4oct | 567 | |
| 190 | 5514.728 | 24/1 | ^^5 +4oct | 570 |
| 191 | 5543.753 | vvvA5 +4oct | 573 | |
| 192 | 5572.778 | 25/1 | v4m6 +4oct | 576 |
| 193 | 5601.802 | vm6 +4oct | 579 | |
| 194 | 5630.827 | 26/1 | ^^m6 +4oct | 582 |
| 195 | 5659.852 | vvvM6 +4oct | 585 | |
| 196 | 5688.877 | M6 +4oct | 588 | |
| 197 | 5717.902 | 27/1 | ^^^M6 +4oct | 591 |
| 198 | 5746.927 | vvA6 +4oct, ^^d7 +4oct | 594 | |
| 199 | 5775.952 | 28/1 | vvvm7 +4oct | 597 |
| 200 | 5804.977 | m7 +4oct | 600 | |
| 201 | 5834.002 | 29/1 | ^^^m7 +4oct | 603 |
| 202 | 5863.026 | vvM7 +4oct | 606 | |
| 203 | 5892.051 | 30/1 | ^M7 +4oct | 609 |
| 204 | 5921.076 | ^4M7 +4oct | 612 | |
| 205 | 5950.101 | 31/1 | ^^^d1 +5oct | 615 |
| 206 | 5979.126 | vv1 +5oct | 618 | |
| 207 | 6008.151 | 32/1 | ^1 +5oct | 621 |
Approximation to JI
The following table illustrates the representation of the 32-integer limit intervals in 186zpi. Prime harmonics are in bold; inconsistent intervals are in italic.
| Ratio | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 17/13 | 0.030 | 0.102 |
| 5/1 | 0.075 | 0.259 |
| 25/17 | 0.100 | 0.344 |
| 25/13 | 0.129 | 0.446 |
| 23/11 | 0.138 | 0.477 |
| 25/1 | 0.150 | 0.517 |
| 11/8 | 0.155 |
0.533 |
| 17/5 | 0.175 | 0.602 |
| 13/5 | 0.204 | 0.704 |
| 17/1 | 0.250 | 0.861 |
| 13/1 | 0.279 | 0.963 |
| 0.289 | 0.996 | |
| 23/8 | 0.293 | 1.011 |
| 23/1 | 0.621 | 2.140 |
| 31/29 | 0.641 | 2.209 |
| 30/29 | 0.642 | 2.211 |
| 23/5 | 0.696 | 2.399 |
| 29/6 | 0.717 | 2.470 |
| 0.736 | 2.535 | |
| 11/1 | 0.760 | 2.617 |
| 25/23 | 0.771 | 2.657 |
| 11/5 | 0.835 | 2.876 |
| 23/17 | 0.871 | 3.001 |
| 21/19 | 0.881 | 3.037 |
| 11/9 | 0.891 | 3.069 |
| 23/13 | 0.901 | 3.103 |
| 25/11 | 0.910 | 3.135 |
| 0.914 | 3.151 | |
| 0.990 | 3.409 | |
| 17/11 | 1.009 | 3.478 |
| 1.025 | 3.531 | |
| 23/9 | 1.029 | 3.546 |
| 13/11 | 1.039 | 3.580 |
| 25/8 | 1.065 | 3.668 |
| 17/8 | 1.164 | 4.012 |
| 27/19 | 1.171 |
4.033 |
| 11/7 | 1.180 | 4.065 |
| 13/8 | 1.194 | 4.114 |
| 31/30 | 1.283 | 4.420 |
| 23/7 | 1.318 | 4.542 |
| 31/6 | 1.358 | 4.679 |
| 1.650 | 5.686 | |
| 1.725 | 5.944 | |
| 20/19 | 1.726 | 5.947 |
| 25/9 | 1.800 | 6.203 |
| 19/4 | 1.801 | 6.205 |
| 17/9 | 1.900 | 6.547 |
| 24/19 | 1.906 | 6.568 |
| 13/9 | 1.930 | 6.649 |
| 7/1 | 1.939 | 6.682 |
| 2.015 | 6.941 | |
| 31/28 | 2.060 |
7.099 |
| 25/7 | 2.090 | 7.199 |
| 17/7 | 2.189 | 7.543 |
| 13/7 | 2.219 | 7.645 |
| 21/20 | 2.607 | 8.984 |
| 21/4 | 2.683 | 9.242 |
| 29/28 | 2.702 | 9.308 |
| 32/19 | 2.716 |
9.356 |
| 19/3 | 2.821 | 9.719 |
| 19/15 | 2.896 | 9.977 |
| 27/20 | 2.897 | 9.980 |
| 27/4 | 2.972 | 10.238 |
| 32/31 | 3.085 | 10.630 |
| 15/14 | 3.343 | 11.519 |
| 14/3 | 3.418 | 11.777 |
| 13/6 | 3.428 | 11.811 |
| 17/6 | 3.458 | 11.913 |
| 30/13 | 3.503 | 12.069 |
| 30/17 | 3.533 | 12.171 |
| 25/6 | 3.557 | 12.256 |
| 32/21 | 3.597 |
12.393 |
| 6/5 | 3.632 | 12.515 |
| 6/1 | 3.708 | 12.774 |
| 32/29 | 3.726 | 12.839 |
| 28/19 | 3.741 | 12.887 |
| 30/1 | 3.783 | 13.032 |
| 32/27 | 3.886 | 13.389 |
| 31/4 | 4.000 | 13.781 |
| 31/20 | 4.075 | 14.039 |
| 29/13 | 4.145 | 14.280 |
| 29/17 | 4.174 | 14.382 |
| 29/25 | 4.274 | 14.726 |
| 23/6 | 4.329 | 14.914 |
| 12/7 | 4.333 | 14.928 |
| 29/5 | 4.349 | 14.985 |
| 16/15 | 4.368 | 15.050 |
| 30/23 | 4.404 | 15.172 |
| 29/1 | 4.424 | 15.243 |
| 16/3 | 4.443 | 15.309 |
| 11/6 | 4.467 | 15.391 |
| 22/15 | 4.523 | 15.583 |
| 30/11 | 4.542 | 15.649 |
| 20/3 | 4.547 | 15.666 |
| 22/3 | 4.598 | 15.842 |
| 4.622 | 15.924 | |
| 29/4 | 4.641 | 15.990 |
| 15/4 | 4.697 | 16.183 |
| 29/20 | 4.716 | 16.248 |
| 31/13 | 4.786 | 16.489 |
| 31/17 | 4.816 | 16.591 |
| 28/27 | 4.911 | 16.920 |
| 31/25 | 4.915 | 16.935 |
| 31/5 | 4.990 | 17.194 |
| 29/23 | 5.046 | 17.383 |
| 31/1 | 5.066 | 17.452 |
| 27/14 | 5.069 | 17.463 |
| 29/11 | 5.184 | 17.860 |
| 15/2 | 5.283 | 18.201 |
| 29/8 | 5.339 | 18.394 |
| 5.358 | 18.459 | |
| 10/3 | 5.433 | 18.718 |
| 12/11 | 5.513 | 18.993 |
| 32/3 | 5.536 | 19.075 |
| 26/15 | 5.562 | 19.164 |
| 32/15 | 5.612 | 19.334 |
| 26/3 | 5.637 |
19.422 |
| 7/6 | 5.647 | 19.456 |
| 23/12 | 5.651 | 19.470 |
| 31/23 | 5.687 | 19.592 |
| 30/7 | 5.722 | 19.714 |
| 31/19 | 5.801 | 19.986 |
| 31/11 | 5.825 | 20.069 |
| 31/8 | 5.980 | 20.603 |
| 29/9 | 6.075 | 20.929 |
| 27/16 | 6.094 | 20.994 |
| 19/14 | 6.239 | 21.496 |
| 27/22 | 6.248 | 21.528 |
| 12/1 | 6.272 | 21.610 |
| 12/5 | 6.347 | 21.869 |
| 29/7 | 6.364 | 21.925 |
| 21/16 | 6.383 | 21.991 |
| 25/12 | 6.422 | 22.127 |
| 29/19 | 6.442 | 22.195 |
| 17/12 | 6.522 | 22.471 |
| 19/18 | 6.528 | 22.492 |
| 22/21 | 6.538 | 22.524 |
| 13/12 | 6.552 | 22.573 |
| 28/3 | 6.561 | 22.606 |
| 28/15 | 6.637 | 22.865 |
| 31/21 | 6.682 | 23.023 |
| 31/9 | 6.716 | 23.138 |
| 28/13 | 6.846 | 23.588 |
| 28/17 | 6.876 | 23.690 |
| 31/27 | 6.972 | 24.019 |
| 28/25 | 6.976 | 24.034 |
| 31/7 | 7.005 | 24.134 |
| 27/2 | 7.008 | 24.145 |
| 28/5 | 7.051 | 24.292 |
| 27/10 | 7.083 | 24.404 |
| 30/19 | 7.084 | 24.406 |
| 28/1 | 7.126 | 24.551 |
| 19/6 | 7.159 | 24.665 |
| 19/16 | 7.264 | 25.027 |
| 27/26 | 7.288 | 25.108 |
| 21/2 | 7.297 | 25.141 |
| 29/21 | 7.324 | 25.232 |
| 21/10 | 7.372 | 25.400 |
| 22/19 | 7.419 | 25.561 |
| 26/21 | 7.577 | 26.104 |
| 29/27 | 7.613 | 26.228 |
| 31/24 | 7.707 | 26.554 |
| 28/23 | 7.747 | 26.691 |
| 26/7 | 7.761 | 26.739 |
| 32/13 | 7.871 | 27.119 |
| 28/11 | 7.886 | 27.168 |
| 32/17 | 7.901 | 27.221 |
| 10/7 | 7.965 | 27.443 |
| 32/25 | 8.001 |
27.565 |
| 7/2 | 8.040 | 27.702 |
| 26/9 | 8.050 | 27.735 |
| 32/5 | 8.076 | 27.824 |
| 32/1 | 8.151 | 28.082 |
| 19/2 | 8.179 | 28.178 |
| 19/10 | 8.254 | 28.437 |
| 10/9 | 8.254 | 28.439 |
| 8.329 | 28.698 | |
| 29/24 | 8.348 | 28.763 |
| 26/19 | 8.458 | 29.141 |
| 31/3 | 8.622 | 29.705 |
| 31/15 | 8.697 | 29.964 |
| 32/23 | 8.772 | 30.222 |
| 28/9 | 8.776 | 30.237 |
| 13/4 | 8.786 | 30.270 |
| 22/7 | 8.800 | 30.319 |
| 17/4 | 8.815 | 30.372 |
| 20/13 | 8.861 | 30.529 |
| 20/17 | 8.891 | 30.631 |
| 32/11 | 8.910 | 30.699 |
| 25/4 | 8.915 | 30.716 |
| 26/11 | 8.941 | 30.803 |
| 16/7 | 8.955 | 30.852 |
| 8.990 | 30.974 | |
| 9.065 | 31.233 | |
| 26/23 | 9.079 | 31.281 |
| 22/9 | 9.089 | 31.315 |
| 20/1 | 9.140 | 31.492 |
| 11/10 | 9.145 | 31.508 |
| 11/2 | 9.220 | 31.766 |
| 16/9 | 9.244 | 31.848 |
| 29/3 | 9.263 | 31.914 |
| 23/10 | 9.284 | 31.985 |
| 29/15 | 9.338 | 32.173 |
| 23/2 | 9.359 | 32.243 |
| 23/4 | 9.686 | 33.373 |
| 18/7 | 9.691 | 33.387 |
| 26/1 | 9.700 | 33.421 |
| 23/20 | 9.762 | 33.632 |
| 26/5 | 9.775 | 33.679 |
| 32/9 | 9.801 | 33.768 |
| 11/4 | 9.825 | 33.850 |
| 26/25 | 9.850 | 33.938 |
| 20/11 | 9.900 | 34.109 |
| 10/1 | 9.905 | 34.125 |
| 26/17 | 9.950 | 34.282 |
| 2/1 | 9.980 | 34.384 |
| 5/2 | 10.055 | 34.642 |
| 32/7 |
10.090 |
34.764 |
| 23/22 | 10.118 | 34.861 |
| 25/2 | 10.130 | 34.901 |
| 16/11 |
10.135 |
34.917 |
| 17/10 | 10.155 | 34.986 |
| 13/10 | 10.184 | 35.088 |
| 17/2 | 10.230 | 35.244 |
| 13/2 | 10.259 | 35.346 |
| 14/9 |
10.269 |
35.380 |
| 23/16 |
10.273 |
35.394 |
| 19/13 | 10.587 | 36.475 |
| 19/17 | 10.617 | 36.577 |
| 29/12 | 10.697 | 36.853 |
| 9/4 |
10.716 |
36.919 |
| 25/19 | 10.716 | 36.921 |
| 22/1 | 10.739 | 37.001 |
| 20/9 |
10.791 |
37.177 |
| 19/5 | 10.791 | 37.180 |
| 22/5 | 10.814 | 37.259 |
| 19/1 | 10.866 | 37.438 |
| 18/11 |
10.870 |
37.452 |
| 25/22 | 10.890 | 37.518 |
| 16/1 |
10.894 |
37.534 |
| 16/5 |
10.969 |
37.793 |
| 22/17 | 10.989 | 37.862 |
| 7/4 |
11.005 |
37.915 |
| 23/18 |
11.009 |
37.929 |
| 22/13 | 11.019 | 37.964 |
| 25/16 |
11.044 |
38.052 |
| 20/7 |
11.080 |
38.174 |
| 17/16 |
11.144 |
38.395 |
| 14/11 | 11.160 | 38.448 |
| 16/13 |
11.174 |
38.497 |
| 23/14 | 11.298 | 38.925 |
| 31/12 | 11.338 | 39.062 |
| 21/13 | 11.468 | 39.512 |
| 23/19 | 11.488 | 39.579 |
| 21/17 | 11.498 | 39.614 |
| 25/21 | 11.598 | 39.958 |
| 19/11 | 11.626 | 40.056 |
| 18/1 |
11.630 |
40.069 |
| 21/5 | 11.673 | 40.216 |
| 18/5 |
11.705 |
40.328 |
| 21/1 | 11.748 | 40.475 |
| 27/13 |
11.758 |
40.508 |
| 25/18 |
11.780 |
40.587 |
| 19/8 |
11.781 |
40.589 |
| 27/17 |
11.787 |
40.610 |
| 18/17 |
11.880 |
40.930 |
| 19/12 |
11.886 |
40.952 |
| 27/25 |
11.887 |
40.954 |
| 18/13 |
11.910 |
41.032 |
| 14/1 | 11.919 | 41.066 |
| 27/5 |
11.962 |
41.213 |
| 14/5 | 11.994 | 41.324 |
| 27/1 |
12.037 |
41.471 |
| 31/14 |
12.040 |
41.482 |
| 25/14 | 12.069 | 41.583 |
| 17/14 | 12.169 | 41.926 |
| 14/13 | 12.199 | 42.028 |
| 31/18 | 12.329 | 42.478 |
| 23/21 | 12.369 | 42.615 |
| 24/13 |
12.493 |
43.044 |
| 21/11 | 12.507 | 43.092 |
| 19/9 |
12.517 |
43.124 |
| 24/17 |
12.523 |
43.146 |
| 25/24 |
12.623 |
43.489 |
| 27/23 |
12.658 |
43.611 |
| 21/8 |
12.662 |
43.626 |
| 29/14 |
12.681 |
43.691 |
| 24/5 |
12.698 |
43.748 |
| 24/1 |
12.773 |
44.006 |
| 27/11 |
12.797 |
44.089 |
| 19/7 | 12.806 | 44.120 |
| 27/8 |
12.951 |
44.622 |
| 29/18 | 12.970 | 44.687 |
| 31/16 |
13.065 |
45.014 |
| 31/22 |
13.220 |
45.547 |
| 15/7 |
13.323 |
45.902 |
| 24/23 |
13.394 |
46.147 |
| 7/3 |
13.398 |
46.161 |
| 13/3 | 13.408 | 46.194 |
| 17/3 | 13.437 | 46.296 |
| 15/13 | 13.483 | 46.453 |
| 17/15 | 13.513 | 46.555 |
| 24/11 |
13.532 |
46.624 |
| 25/3 | 13.537 | 46.640 |
| 5/3 | 13.612 | 46.898 |
| 3/1 | 13.687 | 47.157 |
| 29/16 |
13.706 |
47.223 |
| 15/1 | 13.762 | 47.416 |
| 29/22 |
13.861 |
47.756 |
| 27/7 |
13.976 |
48.153 |
| 31/2 |
13.980 |
48.164 |
| 31/10 |
14.055 |
48.423 |
| 29/26 | 14.125 | 48.664 |
| 31/26 |
14.259 |
49.127 |
| 23/3 | 14.308 | 49.297 |
| 24/7 | 14.313 | 49.312 |
| 29/10 | 14.329 | 49.368 |
| 15/8 |
14.348 |
49.434 |
| 23/15 | 14.384 | 49.556 |
| 29/2 | 14.404 | 49.627 |
| 8/3 |
14.423 |
49.692 |
| 11/3 | 14.447 | 49.774 |
| 15/11 |
14.503 |
49.967 |