Relative errors of small EDOs: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
m 99edo was mysteriously missing
 
(15 intermediate revisions by 8 users not shown)
Line 1: Line 1:
__TOC__
The following table shows [[relative error]]s of the first 9 prime harmonics for [[edo]]s up to 99. There is no point for showing higher primes because no edo under 99 is [[consistent]] up to odd limits higher than 23. The relative errors of other intervals can be derived easily since they follow the [[Relative interval error #Computation|additive rule]].


The relative error of an interval in an [[edo]] is the error approximating [[JI]] divided by the size of a single step. The formula for closest mapping: ''error''(''n'', ''r'') = round (''n'' log<sub>2</sub>''r'') - ''n'' log<sub>2</sub>''r'', where ''n'' is the edo number and ''r'' is the frequency ratio. With closest mapping, the relative error ranges from -0.5 to +0.5. With patent val mapping, it can be farther from zero.
This article contains two lists. The first shows relative errors of the first 9 prime harmonies for edos up to 99. There is no point for showing higher primes because no edo under 99 is consistent up to them. For other intervals, the relative error follows the additive rule (see below), so they can be derived easily. Also by that rule, however, finding large errors with such ''p''/1 harmonies will not suffice that the edo does a poor approximation in the ''p''-limit overall. One must inspect every relevant interval to be sure of that. The second list comes in naturally for showing the root-mean-squared relative errors of a certain JI subgroup, and may be used as the criterion. 
== Additivity ==
There are two additivities of relative errors. 
First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1</sub>''r''<sub>2</sub> for ''n'', then error (''n'', ''r''<sub>3</sub>) = error (''n'', ''r''<sub>1</sub>) + error (''n'', ''r''<sub>2</sub>). 
If the error exceeds the range -0.5 to +0.5, it indicates that an inconsistency occurs, and there is a discrepancy in patent val mapping and closest mapping, so is the error. The patent val mapping error is unchanged, and that of closest mapping is the previous result reduced by an integer to fit it into the range. 
For example, the errors of 2/1, 3/1 and 5/1 in 19-edo are 0, -0.1143 and -0.1166, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-0.1143) - (-0.1166) = 0.0023. That shows 19-edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation. 
Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then error (''n''<sub>3</sub>, ''r'') = error (''n''<sub>1</sub>, ''r'') + error (''n''<sub>2</sub>'','' ''r''). This too needs to be reduced by an integer to fit into the range. Specially, if an edo duplicates itself, and if the mappings do not change, then the error also duplicates. 
For example, the errors of 3/1 for 26-edo and 27-edo are -0.2090 and +0.2060, repectively, and their sum -0.0030 is the error of 3/1 for 53-edo. 
== List of Relative Errors of Prime Harmonies for Small Edos ==
{| class="wikitable sortable mw-collapsible" style="text-align:right"
{| class="wikitable sortable mw-collapsible" style="text-align:right"
! rowspan="2" | Edo
! rowspan="2" | EDO
! colspan="9" | Relative Errors (Percent)
! colspan="8" | Relative Errors (Percent)
|-
|-
!  2/1
!  3/1
!  3/1
!  5/1
!  5/1
Line 34: Line 15:
|-
|-
|  1
|  1
|  0.00
|  41.50
|  41.50
| -32.19
| -32.19
Line 45: Line 25:
|-
|-
|  2
|  2
|  0.00
| -16.99
| -16.99
|  35.61
|  35.61
Line 56: Line 35:
|-
|-
|  3
|  3
|  0.00
|  24.51
|  24.51
|  3.42
|  3.42
Line 67: Line 45:
|-
|-
|  4
|  4
|  0.00
| -33.99
| -33.99
| -28.77
| -28.77
Line 78: Line 55:
|-
|-
|  5
|  5
|  0.00
|  7.52
|  7.52
|  39.04
|  39.04
Line 89: Line 65:
|-
|-
|  6
|  6
|  0.00
|  49.02
|  49.02
|  6.84
|  6.84
Line 100: Line 75:
|-
|-
|  7
|  7
|  0.00
|  -9.47
|  -9.47
| -25.35
| -25.35
Line 111: Line 85:
|-
|-
|  8
|  8
|  0.00
|  32.03
|  32.03
|  42.46
|  42.46
Line 122: Line 95:
|-
|-
|  9
|  9
|  0.00
| -26.47
| -26.47
|  10.26
|  10.26
Line 133: Line 105:
|-
|-
| 10
| 10
|  0.00
|  15.04
|  15.04
| -21.93
| -21.93
Line 144: Line 115:
|-
|-
| 11
| 11
|  0.00
| -43.46
| -43.46
|  45.88
|  45.88
Line 155: Line 125:
|-
|-
| 12
| 12
|  0.00
|  -1.96
|  -1.96
|  13.69
|  13.69
Line 166: Line 135:
|-
|-
| 13
| 13
|  0.00
|  39.55
|  39.55
| -18.51
| -18.51
Line 177: Line 145:
|-
|-
| 14
| 14
|  0.00
| -18.95
| -18.95
|  49.30
|  49.30
Line 188: Line 155:
|-
|-
| 15
| 15
|  0.00
|  22.56
|  22.56
|  17.11
|  17.11
Line 199: Line 165:
|-
|-
| 16
| 16
|  0.00
| -35.94
| -35.94
| -15.08
| -15.08
Line 210: Line 175:
|-
|-
| 17
| 17
|  0.00
|  5.56
|  5.56
| -47.28
| -47.28
Line 221: Line 185:
|-
|-
| 18
| 18
|  0.00
|  47.07
|  47.07
|  20.53
|  20.53
Line 232: Line 195:
|-
|-
| 19
| 19
|  0.00
| -11.43
| -11.43
| -11.66
| -11.66
Line 243: Line 205:
|-
|-
| 20
| 20
|  0.00
|  30.07
|  30.07
| -43.86
| -43.86
Line 254: Line 215:
|-
|-
| 21
| 21
|  0.00
| -28.42
| -28.42
|  23.95
|  23.95
Line 265: Line 225:
|-
|-
| 22
| 22
|  0.00
|  13.08
|  13.08
|  -8.24
|  -8.24
Line 276: Line 235:
|-
|-
| 23
| 23
|  0.00
| -45.41
| -45.41
| -40.43
| -40.43
Line 287: Line 245:
|-
|-
| 24
| 24
|  0.00
|  -3.91
|  -3.91
|  27.37
|  27.37
Line 298: Line 255:
|-
|-
| 25
| 25
|  0.00
|  37.59
|  37.59
|  -4.82
|  -4.82
Line 309: Line 265:
|-
|-
| 26
| 26
|  0.00
| -20.90
| -20.90
| -37.01
| -37.01
Line 320: Line 275:
|-
|-
| 27
| 27
|  0.00
|  20.60
|  20.60
|  30.79
|  30.79
Line 331: Line 285:
|-
|-
| 28
| 28
|  0.00
| -37.90
| -37.90
|  -1.40
|  -1.40
Line 342: Line 295:
|-
|-
| 29
| 29
|  0.00
|  3.61
|  3.61
| -33.59
| -33.59
Line 353: Line 305:
|-
|-
| 30
| 30
|  0.00
|  45.11
|  45.11
|  34.22
|  34.22
Line 364: Line 315:
|-
|-
| 31
| 31
|  0.00
| -13.38
| -13.38
|  2.02
|  2.02
Line 375: Line 325:
|-
|-
| 32
| 32
|  0.00
|  28.12
|  28.12
| -30.17
| -30.17
Line 386: Line 335:
|-
|-
| 33
| 33
|  0.00
| -30.38
| -30.38
|  37.64
|  37.64
Line 397: Line 345:
|-
|-
| 34
| 34
|  0.00
|  11.13
|  11.13
|  5.44
|  5.44
Line 408: Line 355:
|-
|-
| 35
| 35
|  0.00
| -47.37
| -47.37
| -26.75
| -26.75
Line 419: Line 365:
|-
|-
| 36
| 36
|  0.00
|  -5.87
|  -5.87
|  41.06
|  41.06
Line 430: Line 375:
|-
|-
| 37
| 37
|  0.00
|  35.64
|  35.64
|  8.87
|  8.87
Line 441: Line 385:
|-
|-
| 38
| 38
|  0.00
| -22.86
| -22.86
| -23.33
| -23.33
Line 452: Line 395:
|-
|-
| 39
| 39
|  0.00
|  18.65
|  18.65
|  44.48
|  44.48
Line 463: Line 405:
|-
|-
| 40
| 40
|  0.00
| -39.85
| -39.85
|  12.29
|  12.29
Line 474: Line 415:
|-
|-
| 41
| 41
|  0.00
|  1.65
|  1.65
| -19.91
| -19.91
Line 485: Line 425:
|-
|-
| 42
| 42
|  0.00
|  43.16
|  43.16
|  47.90
|  47.90
Line 496: Line 435:
|-
|-
| 43
| 43
|  0.00
| -15.34
| -15.34
|  15.71
|  15.71
Line 507: Line 445:
|-
|-
| 44
| 44
|  0.00
|  26.16
|  26.16
| -16.48
| -16.48
Line 518: Line 455:
|-
|-
| 45
| 45
|  0.00
| -32.33
| -32.33
| -48.68
| -48.68
Line 529: Line 465:
|-
|-
| 46
| 46
|  0.00
|  9.17
|  9.17
|  19.13
|  19.13
Line 540: Line 475:
|-
|-
| 47
| 47
|  0.00
| -49.32
| -49.32
| -13.06
| -13.06
Line 551: Line 485:
|-
|-
| 48
| 48
|  0.00
|  -7.82
|  -7.82
| -45.25
| -45.25
Line 562: Line 495:
|-
|-
| 49
| 49
|  0.00
|  33.68
|  33.68
|  22.55
|  22.55
Line 573: Line 505:
|-
|-
| 50
| 50
|  0.00
| -24.81
| -24.81
|  -9.64
|  -9.64
Line 584: Line 515:
|-
|-
| 51
| 51
|  0.00
|  16.69
|  16.69
| -41.83
| -41.83
Line 595: Line 525:
|-
|-
| 52
| 52
|  0.00
| -41.81
| -41.81
|  25.97
|  25.97
Line 606: Line 535:
|-
|-
| 53
| 53
|  0.00
|  -0.30
|  -0.30
|  -6.22
|  -6.22
Line 617: Line 545:
|-
|-
| 54
| 54
|  0.00
|  41.20
|  41.20
| -38.41
| -38.41
Line 628: Line 555:
|-
|-
| 55
| 55
|  0.00
| -17.29
| -17.29
|  29.40
|  29.40
Line 639: Line 565:
|-
|-
| 56
| 56
|  0.00
|  24.21
|  24.21
|  -2.80
|  -2.80
Line 650: Line 575:
|-
|-
| 57
| 57
|  0.00
| -34.29
| -34.29
| -34.99
| -34.99
Line 661: Line 585:
|-
|-
| 58
| 58
|  0.00
|  7.22
|  7.22
|  32.82
|  32.82
Line 672: Line 595:
|-
|-
| 59
| 59
|  0.00
|  48.72
|  48.72
|  0.62
|  0.62
Line 683: Line 605:
|-
|-
| 60
| 60
|  0.00
|  -9.78
|  -9.78
| -31.57
| -31.57
Line 694: Line 615:
|-
|-
| 61
| 61
|  0.00
|  31.73
|  31.73
|  36.24
|  36.24
Line 705: Line 625:
|-
|-
| 62
| 62
|  0.00
| -26.77
| -26.77
|  4.05
|  4.05
Line 716: Line 635:
|-
|-
| 63
| 63
|  0.00
|  14.74
|  14.74
| -28.15
| -28.15
Line 727: Line 645:
|-
|-
| 64
| 64
|  0.00
| -43.76
| -43.76
|  39.66
|  39.66
Line 738: Line 655:
|-
|-
| 65
| 65
|  0.00
|  -2.26
|  -2.26
|  7.47
|  7.47
Line 749: Line 665:
|-
|-
| 66
| 66
|  0.00
|  39.25
|  39.25
| -24.73
| -24.73
Line 760: Line 675:
|-
|-
| 67
| 67
|  0.00
| -19.25
| -19.25
|  43.08
|  43.08
Line 771: Line 685:
|-
|-
| 68
| 68
|  0.00
|  22.25
|  22.25
|  10.89
|  10.89
Line 782: Line 695:
|-
|-
| 69
| 69
|  0.00
| -36.24
| -36.24
| -21.30
| -21.30
Line 793: Line 705:
|-
|-
| 70
| 70
|  0.00
|  5.26
|  5.26
|  46.50
|  46.50
Line 804: Line 715:
|-
|-
| 71
| 71
|  0.00
|  46.77
|  46.77
|  14.31
|  14.31
Line 815: Line 725:
|-
|-
| 72
| 72
|  0.00
| -11.73
| -11.73
| -17.88
| -17.88
Line 826: Line 735:
|-
|-
| 73
| 73
|  0.00
|  29.77
|  29.77
|  49.92
|  49.92
Line 837: Line 745:
|-
|-
| 74
| 74
|  0.00
| -28.72
| -28.72
|  17.73
|  17.73
Line 848: Line 755:
|-
|-
| 75
| 75
|  0.00
|  12.78
|  12.78
| -14.46
| -14.46
Line 859: Line 765:
|-
|-
| 76
| 76
|  0.00
| -45.72
| -45.72
| -46.65
| -46.65
Line 870: Line 775:
|-
|-
| 77
| 77
|  0.00
|  -4.21
|  -4.21
|  21.15
|  21.15
Line 881: Line 785:
|-
|-
| 78
| 78
|  0.00
|  37.29
|  37.29
| -11.04
| -11.04
Line 892: Line 795:
|-
|-
| 79
| 79
|  0.00
| -21.20
| -21.20
| -43.23
| -43.23
Line 903: Line 805:
|-
|-
| 80
| 80
|  0.00
|  20.30
|  20.30
|  24.58
|  24.58
Line 914: Line 815:
|-
|-
| 81
| 81
|  0.00
| -38.20
| -38.20
|  -7.62
|  -7.62
Line 925: Line 825:
|-
|-
| 82
| 82
|  0.00
|  3.31
|  3.31
| -39.81
| -39.81
Line 936: Line 835:
|-
|-
| 83
| 83
|  0.00
|  44.81
|  44.81
|  28.00
|  28.00
Line 947: Line 845:
|-
|-
| 84
| 84
|  0.00
| -13.69
| -13.69
|  -4.20
|  -4.20
Line 958: Line 855:
|-
|-
| 85
| 85
|  0.00
|  27.82
|  27.82
| -36.39
| -36.39
Line 969: Line 865:
|-
|-
| 86
| 86
|  0.00
| -30.68
| -30.68
|  31.42
|  31.42
Line 980: Line 875:
|-
|-
| 87
| 87
|  0.00
|  10.83
|  10.83
|  -0.77
|  -0.77
Line 991: Line 885:
|-
|-
| 88
| 88
|  0.00
| -47.67
| -47.67
| -32.97
| -32.97
Line 1,002: Line 895:
|-
|-
| 89
| 89
|  0.00
|  -6.17
|  -6.17
|  34.84
|  34.84
Line 1,013: Line 905:
|-
|-
| 90
| 90
|  0.00
|  35.34
|  35.34
|  2.65
|  2.65
Line 1,024: Line 915:
|-
|-
| 91
| 91
|  0.00
| -23.16
| -23.16
| -29.55
| -29.55
Line 1,035: Line 925:
|-
|-
| 92
| 92
|  0.00
|  18.34
|  18.34
|  38.26
|  38.26
Line 1,046: Line 935:
|-
|-
| 93
| 93
|  0.00
| -40.15
| -40.15
|  6.07
|  6.07
Line 1,057: Line 945:
|-
|-
| 94
| 94
|  0.00
|  1.35
|  1.35
| -26.12
| -26.12
Line 1,068: Line 955:
|-
|-
| 95
| 95
|  0.00
|  42.86
|  42.86
|  41.68
|  41.68
Line 1,079: Line 965:
|-
|-
| 96
| 96
|  0.00
| -15.64
| -15.64
|  9.49
|  9.49
Line 1,090: Line 975:
|-
|-
| 97
| 97
|  0.00
|  25.86
|  25.86
| -22.70
| -22.70
Line 1,101: Line 985:
|-
|-
| 98
| 98
|  0.00
| -32.63
| -32.63
|  45.10
|  45.10
Line 1,112: Line 995:
|-
|-
| 99
| 99
|  0.00
|  8.87
|  8.87
|  12.91
|  12.91
Line 1,123: Line 1,005:
|}
|}


== List of RMS Relative Errors of JI Subgroups for Small Edos ==
== See also ==
Notes:
* Relative errors of small ed-''p''<span/>'s:  
# Patent val mapping is used throughout. 
** [[Relative errors of small EDTs]]
# Octave equivalence is assumed. For example, the [[5-odd-limit]] takes account of 1/1, 3/1, 5/1, 5/3 and their octave inverses. 
** [[Relative errors of small EDFs]]
# The weight of intervals is determined by the [[tonality diamond]]. For example, the [[9-odd-limit]] takes account of 3/1 twice.
** [[Relative errors of small ED5s]]
{| class="wikitable sortable mw-collapsible" style="text-align:right"
** [[Relative errors of small ED6s]]
! rowspan="2" | Edo
* [[Relative errors of small equal temperaments]]
! colspan="11" | Root Mean Squared Relative Errors (Percent)
 
|-
[[Category:EDO theory pages]]
! 3 Odd Limit
[[Category:Tables]]
! 5 Odd Limit
! 7 Odd Limit
! 9 Odd Limit
! 11 Odd Limit
! 13 Odd Limit
! 15 Odd Limit
! 17 Odd Limit
! 19 Odd Limit
! 21 Odd Limit
! 23 Odd Limit
|-
| 1
| 29.35
| 42.66
| 38.25
| 54.88
| 61.68
| 57.87
| 54.17
| 51.99
| 51.57
| 53.79
| 53.35
|-
| 2
| 12.02
| 31.00
| 33.35
| 40.44
| 36.97
| 40.91
| 39.38
| 38.05
| 41.60
| 41.18
| 39.43
|-
| 3
| 17.33
| 15.33
| 34.23
| 42.69
| 45.56
| 42.45
| 42.08
| 41.59
| 41.08
| 39.93
| 41.80
|-
| 4
| 24.03
| 21.13
| 18.34
| 31.02
| 37.61
| 40.74
| 43.75
| 41.62
| 40.86
| 41.54
| 40.21
|-
| 5
| 5.32
| 23.91
| 23.81
| 21.44
| 29.27
| 35.10
| 36.78
| 43.55
| 43.62
| 41.59
| 41.82
|-
| 6
| 34.66
| 30.66
| 26.61
| 51.22
| 47.03
| 50.72
| 49.62
| 47.53
| 56.32
| 56.32
| 56.23
|-
| 7
| 6.70
| 14.79
| 31.20
| 29.90
| 28.86
| 27.93
| 29.75
| 34.97
| 35.45
| 35.34
| 35.97
|-
| 8
| 22.65
| 25.55
| 48.63
| 54.11
| 49.94
| 47.16
| 50.14
| 47.27
| 46.42
| 47.29
| 48.19
|-
| 9
| 18.71
| 21.88
| 22.98
| 31.51
| 28.92
| 27.45
| 25.73
| 30.28
| 28.94
| 31.48
| 35.50
|-
| 10
| 10.63
| 21.47
| 18.85
| 25.42
| 30.45
| 28.61
| 27.65
| 26.22
| 34.04
| 32.56
| 32.73
|-
| 11
| 30.73
| 51.59
| 45.19
| 65.23
| 59.74
| 59.17
| 55.52
| 52.52
| 51.64
| 50.73
| 49.81
|-
| 12
| 1.38
| 9.84
| 18.74
| 18.70
| 27.49
| 37.31
| 34.98
| 33.43
| 31.75
| 31.76
| 33.51
|-
| 13
| 27.97
| 34.24
| 45.60
| 63.59
| 58.18
| 54.71
| 51.65
| 49.64
| 48.56
| 46.59
| 45.14
|-
| 14
| 13.40
| 40.68
| 43.05
| 44.08
| 44.41
| 44.22
| 45.23
| 43.44
| 44.75
| 45.54
| 44.25
|-
| 15
| 15.95
| 13.59
| 18.95
| 27.35
| 25.06
| 29.02
| 28.80
| 35.91
| 34.47
| 32.94
| 31.54
|-
| 16
| 25.41
| 20.84
| 23.76
| 40.61
| 37.62
| 34.90
| 34.95
| 33.39
| 34.53
| 32.93
| 31.86
|-
| 17
| 3.93
| 33.53
| 38.55
| 35.46
| 33.98
| 31.63
| 36.39
| 40.08
| 38.50
| 40.44
| 39.15
|-
| 18
| 33.28
| 27.25
| 27.87
| 44.68
| 54.53
| 50.68
| 50.32
| 47.53
| 57.22
| 60.69
| 65.40
|-
| 19
| 8.08
| 7.70
| 17.42
| 16.32
| 27.15
| 27.40
| 26.15
| 32.36
| 34.49
| 36.87
| 35.68
|-
| 20
| 21.27
| 42.93
| 37.69
| 50.84
| 48.27
| 44.72
| 42.45
| 41.57
| 39.45
| 37.96
| 41.45
|-
| 21
| 20.10
| 30.27
| 26.48
| 39.94
| 43.95
| 43.78
| 41.04
| 39.33
| 38.55
| 38.00
| 36.40
|-
| 22
| 9.25
| 12.42
| 17.34
| 18.87
| 20.68
| 30.63
| 28.72
| 27.24
| 32.68
| 35.13
| 38.42
|-
| 23
| 32.11
| 28.76
| 50.48
| 63.98
| 69.09
| 64.00
| 68.54
| 65.37
| 65.64
| 62.83
| 60.29
|-
| 24
| 2.76
| 19.68
| 32.65
| 29.30
| 26.76
| 27.28
| 27.93
| 26.88
| 25.56
| 29.95
| 33.86
|-
| 25
| 26.58
| 26.81
| 29.35
| 48.26
| 56.30
| 56.12
| 53.32
| 52.49
| 51.55
| 49.34
| 47.78
|-
| 26
| 14.78
| 21.43
| 22.30
| 25.31
| 26.66
| 24.84
| 30.29
| 28.68
| 28.82
| 27.54
| 36.02
|-
| 27
| 14.57
| 18.11
| 15.80
| 19.33
| 37.61
| 34.85
| 37.55
| 42.45
| 41.14
| 40.85
| 40.71
|-
| 28
| 26.80
| 24.81
| 38.67
| 55.10
| 52.53
| 54.37
| 53.56
| 53.12
| 50.95
| 48.81
| 49.68
|-
| 29
| 2.55
| 23.69
| 28.09
| 28.84
| 28.27
| 27.23
| 26.06
| 38.29
| 36.43
| 36.16
| 34.64
|-
| 30
| 31.90
| 27.18
| 37.90
| 54.71
| 50.11
| 48.64
| 52.35
| 49.44
| 56.78
| 54.14
| 51.88
|-
| 31
| 9.46
| 9.67
| 8.39
| 15.13
| 16.20
| 24.64
| 23.22
| 26.82
| 29.15
| 28.68
| 28.86
|-
| 32
| 19.88
| 33.66
| 30.98
| 40.68
| 38.04
| 45.48
| 42.83
| 40.79
| 38.70
| 39.73
| 38.37
|-
| 33
| 21.48
| 39.34
| 39.71
| 53.83
| 49.58
| 45.99
| 43.50
| 41.64
| 40.02
| 38.37
| 37.93
|-
| 34
| 7.87
| 6.42
| 31.44
| 32.66
| 36.26
| 34.20
| 32.30
| 30.55
| 36.09
| 37.48
| 36.77
|-
| 35
| 33.49
| 27.42
| 23.76
| 44.83
| 44.04
| 57.61
| 59.14
| 56.64
| 59.15
| 60.36
| 57.86
|-
| 36
| 4.15
| 29.52
| 27.89
| 27.14
| 33.46
| 34.81
| 35.32
| 35.03
| 33.23
| 32.64
| 31.49
|-
| 37
| 25.20
| 21.42
| 18.60
| 36.26
| 35.75
| 33.72
| 33.63
| 37.80
| 38.79
| 39.54
| 43.41
|-
| 38
| 16.16
| 15.40
| 31.97
| 37.24
| 38.40
| 45.06
| 45.49
| 43.64
| 42.83
| 42.40
| 41.91
|-
| 39
| 13.18
| 25.79
| 48.19
| 47.12
| 43.03
| 44.87
| 50.25
| 52.82
| 51.45
| 51.48
| 52.56
|-
| 40
| 28.18
| 31.47
| 29.94
| 45.70
| 42.07
| 41.23
| 38.58
| 37.96
| 39.19
| 41.56
| 41.99
|-
| 41
| 1.17
| 13.85
| 12.25
| 12.44
| 15.99
| 20.85
| 21.84
| 27.57
| 27.66
| 26.76
| 31.80
|-
| 42
| 30.52
| 30.48
| 29.43
| 43.52
| 53.12
| 59.59
| 65.77
| 62.08
| 65.55
| 63.88
| 61.75
|-
| 43
| 10.85
| 17.93
| 23.24
| 29.85
| 30.23
| 29.04
| 27.17
| 27.50
| 29.04
| 27.80
| 31.12
|-
| 44
| 18.50
| 24.83
| 34.67
| 37.73
| 41.36
| 38.33
| 35.94
| 33.89
| 32.23
| 39.18
| 38.59
|-
| 45
| 22.86
| 28.60
| 25.05
| 30.33
| 45.42
| 55.24
| 60.43
| 58.39
| 55.42
| 56.13
| 59.79
|-
| 46
| 6.49
| 11.05
| 17.16
| 17.47
| 19.10
| 21.65
| 24.29
| 23.04
| 28.49
| 27.19
| 26.15
|-
| 47
| 34.88
| 29.51
| 30.18
| 54.85
| 62.76
| 59.63
| 59.97
| 56.72
| 58.62
| 57.15
| 59.00
|-
| 48
| 5.53
| 27.93
| 35.48
| 32.10
| 29.36
| 35.49
| 41.00
| 39.01
| 37.90
| 37.47
| 36.00
|-
| 49
| 23.82
| 19.82
| 23.09
| 31.61
| 29.96
| 43.69
| 43.20
| 48.35
| 48.67
| 52.04
| 49.97
|-
| 50
| 17.55
| 14.44
| 19.91
| 25.32
| 27.15
| 26.58
| 26.14
| 25.91
| 25.78
| 29.12
| 28.10
|-
| 51
| 11.80
| 34.81
| 30.66
| 37.05
| 40.21
| 41.37
| 40.00
| 41.66
| 44.12
| 42.12
| 42.63
|-
| 52
| 29.56
| 39.49
| 34.47
| 54.80
| 52.54
| 50.56
| 47.31
| 52.73
| 50.88
| 50.14
| 48.19
|-
| 53
| 0.21
| 4.05
| 14.63
| 13.30
| 23.32
| 22.02
| 20.62
| 26.78
| 26.05
| 26.47
| 27.14
|-
| 54
| 29.13
| 45.97
| 46.54
| 58.09
| 53.13
| 49.29
| 47.08
| 44.50
| 49.35
| 54.18
| 55.25
|-
| 55
| 12.23
| 27.26
| 36.05
| 35.80
| 33.53
| 43.84
| 41.88
| 40.75
| 41.83
| 46.81
| 45.74
|-
| 56
| 17.12
| 17.15
| 22.83
| 34.14
| 32.50
| 34.74
| 33.13
| 31.24
| 29.65
| 28.40
| 31.56
|-
| 57
| 24.24
| 23.10
| 23.84
| 35.77
| 33.01
| 34.87
| 39.51
| 39.42
| 37.70
| 36.33
| 38.15
|-
| 58
| 5.10
| 19.91
| 17.42
| 15.58
| 18.00
| 19.28
| 20.19
| 23.33
| 33.00
| 31.76
| 36.48
|-
| 59
| 34.45
| 32.27
| 30.56
| 50.92
| 52.75
| 57.52
| 55.52
| 55.24
| 52.96
| 56.85
| 54.79
|-
| 60
| 6.91
| 18.66
| 24.63
| 22.05
| 39.47
| 36.74
| 37.51
| 35.74
| 35.78
| 38.17
| 37.90
|-
| 61
| 22.44
| 22.80
| 35.17
| 43.33
| 41.50
| 38.74
| 42.93
| 48.09
| 47.45
| 45.38
| 43.57
|-
| 62
| 18.93
| 19.34
| 16.79
| 30.27
| 32.40
| 31.75
| 29.71
| 29.11
| 27.98
| 26.75
| 26.58
|-
| 63
| 10.42
| 25.16
| 24.46
| 27.48
| 25.09
| 24.99
| 24.64
| 31.49
| 32.75
| 32.14
| 31.01
|-
| 64
| 30.94
| 48.18
| 46.71
| 67.95
| 63.84
| 61.41
| 57.55
| 58.80
| 56.33
| 53.79
| 55.53
|-
| 65
| 1.60
| 5.88
| 30.76
| 27.73
| 28.10
| 36.84
| 34.49
| 35.00
| 33.99
| 39.13
| 37.48
|-
| 66
| 27.75
| 37.25
| 38.19
| 57.61
| 57.72
| 55.24
| 52.04
| 49.84
| 50.46
| 48.27
| 49.12
|-
| 67
| 13.61
| 36.85
| 33.61
| 38.37
| 37.73
| 35.12
| 34.59
| 32.93
| 34.44
| 36.06
| 34.88
|-
| 68
| 15.74
| 12.85
| 11.15
| 21.53
| 29.47
| 30.25
| 29.63
| 28.40
| 26.95
| 26.60
| 27.00
|-
| 69
| 25.63
| 21.03
| 34.80
| 48.35
| 51.43
| 48.76
| 49.79
| 47.53
| 45.20
| 43.31
| 41.50
|-
| 70
| 3.72
| 29.41
| 31.86
| 29.66
| 33.73
| 32.60
| 35.45
| 36.02
| 40.23
| 42.36
| 41.42
|-
| 71
| 33.07
| 27.67
| 40.14
| 60.61
| 55.78
| 51.65
| 50.91
| 53.28
| 50.91
| 48.80
| 49.64
|-
| 72
| 8.29
| 10.49
| 9.28
| 11.02
| 10.44
| 18.07
| 17.94
| 17.66
| 22.25
| 21.50
| 27.66
|-
| 73
| 21.05
| 29.00
| 28.01
| 33.03
| 31.46
| 36.71
| 42.68
| 51.07
| 50.59
| 48.62
| 49.68
|-
| 74
| 20.31
| 27.07
| 29.50
| 43.48
| 39.96
| 38.83
| 36.48
| 39.27
| 38.77
| 37.14
| 38.34
|-
| 75
| 9.04
| 15.74
| 30.96
| 28.93
| 40.99
| 43.48
| 41.03
| 41.78
| 41.39
| 43.06
| 44.95
|-
| 76
| 32.33
| 30.79
| 26.85
| 41.29
| 46.68
| 43.62
| 49.21
| 57.43
| 58.19
| 59.67
| 60.83
|-
| 77
| 2.98
| 15.70
| 19.26
| 17.88
| 25.03
| 24.22
| 24.99
| 26.92
| 25.86
| 26.01
| 27.38
|-
| 78
| 26.37
| 29.24
| 25.60
| 44.46
| 40.65
| 38.52
| 36.07
| 34.09
| 40.15
| 39.44
| 37.77
|-
| 79
| 14.99
| 24.96
| 34.25
| 35.51
| 33.09
| 31.45
| 35.72
| 37.20
| 44.52
| 43.00
| 42.02
|-
| 80
| 14.35
| 15.16
| 20.73
| 21.46
| 19.59
| 23.04
| 24.26
| 25.19
| 23.99
| 28.19
| 27.47
|-
| 81
| 27.01
| 23.34
| 25.10
| 38.39
| 35.52
| 43.31
| 41.97
| 40.27
| 38.75
| 43.37
| 41.87
|-
| 82
| 2.34
| 27.71
| 24.51
| 24.88
| 31.98
| 35.80
| 35.92
| 33.94
| 33.32
| 31.78
| 31.60
|-
| 83
| 31.69
| 26.14
| 27.44
| 47.40
| 49.48
| 49.59
| 52.72
| 54.79
| 52.81
| 51.10
| 56.03
|-
| 84
| 9.68
| 8.10
| 16.38
| 21.35
| 31.18
| 29.69
| 29.65
| 32.25
| 31.74
| 30.33
| 29.05
|-
| 85
| 19.67
| 37.18
| 40.58
| 45.47
| 43.11
| 43.13
| 42.21
| 47.43
| 45.48
| 49.49
| 49.63
|-
| 86
| 21.69
| 35.85
| 40.91
| 46.50
| 56.12
| 52.46
| 49.39
| 53.13
| 51.87
| 56.60
| 54.32
|-
| 87
| 7.66
| 7.49
| 17.95
| 21.45
| 19.59
| 18.27
| 17.46
| 22.87
| 26.46
| 27.06
| 29.54
|-
| 88
| 33.71
| 28.19
| 27.93
| 48.75
| 44.65
| 55.03
| 57.31
| 61.03
| 60.92
| 59.39
| 57.26
|-
| 89
| 4.36
| 25.53
| 22.32
| 23.87
| 21.94
| 28.68
| 29.76
| 29.05
| 28.11
| 26.83
| 29.08
|-
| 90
| 24.99
| 22.73
| 23.56
| 36.52
| 47.19
| 45.01
| 43.48
| 41.05
| 44.05
| 47.93
| 47.32
|-
| 91
| 16.38
| 17.95
| 23.77
| 24.47
| 33.89
| 39.10
| 40.72
| 39.76
| 45.62
| 49.78
| 51.74
|-
| 92
| 12.97
| 22.10
| 34.33
| 34.93
| 38.20
| 43.31
| 48.59
| 46.07
| 44.10
| 42.54
| 41.62
|-
| 93
| 28.39
| 29.00
| 25.18
| 45.40
| 49.64
| 45.97
| 43.85
| 41.39
| 39.58
| 39.96
| 42.98
|-
| 94
| 0.96
| 17.88
| 19.42
| 17.71
| 18.34
| 19.87
| 21.42
| 21.61
| 22.84
| 23.45
| 23.13
|-
| 95
| 30.30
| 28.19
| 24.44
| 38.98
| 35.67
| 33.18
| 37.32
| 48.98
| 46.57
| 46.63
| 45.03
|-
| 96
| 11.06
| 14.65
| 33.96
| 38.60
| 35.89
| 35.36
| 33.10
| 35.08
| 35.27
| 37.11
| 36.83
|-
| 97
| 18.29
| 28.06
| 31.37
| 43.50
| 44.67
| 41.44
| 38.92
| 44.81
| 42.64
| 40.78
| 39.81
|-
| 98
| 23.07
| 45.07
| 40.28
| 51.63
| 47.45
| 49.69
| 47.15
| 48.80
| 48.28
| 49.40
| 48.41
|-
| 99
| 6.27
| 7.63
| 6.60
| 8.38
| 31.37
| 33.58
| 33.84
| 35.67
| 38.51
| 36.92
| 35.54
|}
[[Category:Edo]]
[[Category:List]]
[[Category:Table]]

Latest revision as of 09:46, 19 May 2024

The following table shows relative errors of the first 9 prime harmonics for edos up to 99. There is no point for showing higher primes because no edo under 99 is consistent up to odd limits higher than 23. The relative errors of other intervals can be derived easily since they follow the additive rule.

EDO Relative Errors (Percent)
3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1
1 41.50 -32.19 19.26 -45.94 29.96 -8.75 -24.79 47.64
2 -16.99 35.61 38.53 8.11 -40.09 -17.49 -49.59 -4.71
3 24.51 3.42 -42.21 -37.83 -10.13 -26.24 25.62 42.93
4 -33.99 -28.77 -22.94 16.23 19.82 -34.99 0.83 -9.42
5 7.52 39.04 -3.68 -29.72 49.78 -43.73 -23.96 38.22
6 49.02 6.84 15.59 24.34 -20.26 47.52 -48.76 -14.14
7 -9.47 -25.35 34.85 -21.60 9.69 38.78 26.45 33.51
8 32.03 42.46 -45.88 32.45 39.65 30.03 1.66 -18.85
9 -26.47 10.26 -26.62 -13.49 -30.40 21.28 -23.13 28.79
10 15.04 -21.93 -7.35 40.57 -0.44 12.54 -47.93 -23.56
11 -43.46 45.88 11.91 -5.37 29.52 3.79 27.28 24.08
12 -1.96 13.69 31.17 48.68 -40.53 -4.96 2.49 -28.27
13 39.55 -18.51 -49.56 2.74 -10.57 -13.70 -22.31 19.37
14 -18.95 49.30 -30.30 -43.20 19.38 -22.45 -47.10 -32.99
15 22.56 17.11 -11.03 10.85 49.34 -31.19 28.11 14.66
16 -35.94 -15.08 8.23 -35.09 -20.70 -39.94 3.32 -37.70
17 5.56 -47.28 27.50 18.97 9.25 -48.69 -21.48 9.94
18 47.07 20.53 46.76 -26.98 39.21 42.57 -46.27 -42.41
19 -11.43 -11.66 -33.97 27.08 -30.84 33.82 28.94 5.23
20 30.07 -43.86 -14.71 -18.86 -0.88 25.07 4.14 -47.12
21 -28.42 23.95 4.55 35.19 29.08 16.33 -20.65 0.52
22 13.08 -8.24 23.82 -10.75 -40.97 7.58 -45.44 48.16
23 -45.41 -40.43 43.08 43.31 -11.01 -1.16 29.77 -4.19
24 -3.91 27.37 -37.65 -2.64 18.94 -9.91 4.97 43.45
25 37.59 -4.82 -18.39 -48.58 48.90 -18.66 -19.82 -8.90
26 -20.90 -37.01 0.88 5.48 -21.14 -27.40 -44.61 38.74
27 20.60 30.79 20.14 -40.47 8.81 -36.15 30.60 -13.62
28 -37.90 -1.40 39.41 13.59 38.77 -44.90 5.80 34.03
29 3.61 -33.59 -41.33 -32.35 -31.28 46.36 -18.99 -18.33
30 45.11 34.22 -22.06 21.71 -1.32 37.61 -43.78 29.31
31 -13.38 2.02 -2.80 -24.24 28.64 28.87 31.42 -23.04
32 28.12 -30.17 16.46 29.82 -41.41 20.12 6.63 24.60
33 -30.38 37.64 35.73 -16.12 -11.45 11.37 -18.16 -27.75
34 11.13 5.44 -45.01 37.93 18.50 2.63 -42.95 19.89
35 -47.37 -26.75 -25.74 -8.01 48.46 -6.12 32.25 -32.47
36 -5.87 41.06 -6.48 46.05 -21.58 -14.87 7.46 15.18
37 35.64 8.87 12.79 0.10 8.37 -23.61 -17.33 -37.18
38 -22.86 -23.33 32.05 -45.84 38.33 -32.36 -42.12 10.46
39 18.65 44.48 -48.68 8.22 -31.71 -41.11 33.08 -41.89
40 -39.85 12.29 -29.42 -37.73 -1.76 -49.85 8.29 5.75
41 1.65 -19.91 -10.16 16.33 28.20 41.40 -16.50 -46.60
42 43.16 47.90 9.11 -29.61 -41.85 32.66 -41.30 1.04
43 -15.34 15.71 28.37 24.44 -11.89 23.91 33.91 48.68
44 26.16 -16.48 47.64 -21.50 18.07 15.16 9.12 -3.67
45 -32.33 -48.68 -33.10 32.56 48.02 6.42 -15.67 43.97
46 9.17 19.13 -13.83 -13.39 -22.02 -2.33 -40.47 -8.38
47 -49.32 -13.06 5.43 40.67 7.93 -11.08 34.74 39.26
48 -7.82 -45.25 24.70 -5.27 37.89 -19.82 9.95 -13.10
49 33.68 22.55 43.96 48.79 -32.15 -28.57 -14.84 34.55
50 -24.81 -9.64 -36.77 2.84 -2.20 -37.31 -39.64 -17.81
51 16.69 -41.83 -17.51 -43.10 27.76 -46.06 35.57 29.83
52 -41.81 25.97 1.75 10.96 -42.29 45.19 10.78 -22.52
53 -0.30 -6.22 21.02 -34.99 -12.33 36.45 -14.02 25.12
54 41.20 -38.41 40.28 19.07 17.63 27.70 -38.81 -27.23
55 -17.29 29.40 -40.45 -26.87 47.58 18.95 36.40 20.41
56 24.21 -2.80 -21.19 27.18 -22.46 10.21 11.61 -31.95
57 -34.29 -34.99 -1.92 -18.76 7.49 1.46 -13.19 15.70
58 7.22 32.82 17.34 35.30 37.45 -7.28 -37.98 -36.66
59 48.72 0.62 36.61 -10.65 -32.59 -16.03 37.23 10.98
60 -9.78 -31.57 -44.13 43.41 -2.64 -24.78 12.43 -41.37
61 31.73 36.24 -24.87 -2.53 27.32 -33.52 -12.36 6.27
62 -26.77 4.05 -5.60 -48.48 -42.73 -42.27 -37.15 -46.08
63 14.74 -28.15 13.66 5.58 -12.77 48.98 38.06 1.56
64 -43.76 39.66 32.93 -40.36 17.19 40.24 13.26 49.20
65 -2.26 7.47 -47.81 13.69 47.14 31.49 -11.53 -3.15
66 39.25 -24.73 -28.54 -32.25 -22.90 22.75 -36.32 44.49
67 -19.25 43.08 -9.28 21.81 7.05 14.00 38.89 -7.87
68 22.25 10.89 9.99 -24.14 37.01 5.25 14.09 39.78
69 -36.24 -21.30 29.25 29.92 -33.03 -3.49 -10.70 -12.58
70 5.26 46.50 48.52 -16.02 -3.08 -12.24 -35.49 35.07
71 46.77 14.31 -32.22 38.04 26.88 -20.99 39.71 -17.29
72 -11.73 -17.88 -12.96 -7.91 -43.17 -29.73 14.92 30.35
73 29.77 49.92 6.31 46.15 -13.21 -38.48 -9.87 -22.00
74 -28.72 17.73 25.57 0.21 16.75 -47.23 -34.66 25.64
75 12.78 -14.46 44.84 -45.74 46.70 44.03 40.54 -26.71
76 -45.72 -46.65 -35.90 8.32 -23.34 35.28 15.75 20.93
77 -4.21 21.15 -16.63 -37.62 6.61 26.54 -9.04 -31.43
78 37.29 -11.04 2.63 16.43 36.57 17.79 -33.83 16.22
79 -21.20 -43.23 21.90 -29.51 -33.47 9.04 41.37 -36.14
80 20.30 24.58 41.16 24.55 -3.52 0.30 16.58 11.50
81 -38.20 -7.62 -39.57 -21.40 26.44 -8.45 -8.21 -40.85
82 3.31 -39.81 -20.31 32.66 -43.61 -17.20 -33.01 6.79
83 44.81 28.00 -1.05 -13.28 -13.65 -25.94 42.20 -45.56
84 -13.69 -4.20 18.22 40.77 16.31 -34.69 17.41 2.08
85 27.82 -36.39 37.48 -5.17 46.26 -43.43 -7.38 49.72
86 -30.68 31.42 -43.25 48.89 -23.78 47.82 -32.18 -2.63
87 10.83 -0.77 -23.99 2.94 6.17 39.07 43.03 45.01
88 -47.67 -32.97 -4.72 -43.00 36.13 30.33 18.24 -7.35
89 -6.17 34.84 14.54 11.06 -33.91 21.58 -6.55 40.30
90 35.34 2.65 33.81 -34.88 -3.96 12.83 -31.35 -12.06
91 -23.16 -29.55 -46.93 19.17 26.00 4.09 43.86 35.59
92 18.34 38.26 -27.67 -26.77 -44.05 -4.66 19.07 -16.77
93 -40.15 6.07 -8.40 27.29 -14.09 -13.40 -5.73 30.87
94 1.35 -26.12 10.86 -18.66 15.87 -22.15 -30.52 -21.48
95 42.86 41.68 30.13 35.40 45.82 -30.90 44.69 26.16
96 -15.64 9.49 49.39 -10.54 -24.22 -39.64 19.90 -26.19
97 25.86 -22.70 -31.34 43.51 5.73 -48.39 -4.90 21.45
98 -32.63 45.10 -12.08 -2.43 35.69 42.86 -29.69 -30.91
99 8.87 12.91 7.19 -48.37 -34.35 34.12 45.52 16.74

See also

Retrieved from "https://en.xen.wiki/index.php?title=Relative_errors_of_small_EDOs&oldid=143596"