Relative errors of small EDOs: Difference between revisions

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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.  
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]].  


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 32: Line 15:
|-
|-
|  1
|  1
|  0.00
|  41.50
|  41.50
| -32.19
| -32.19
Line 43: Line 25:
|-
|-
|  2
|  2
|  0.00
| -16.99
| -16.99
|  35.61
|  35.61
Line 54: Line 35:
|-
|-
|  3
|  3
|  0.00
|  24.51
|  24.51
|  3.42
|  3.42
Line 65: Line 45:
|-
|-
|  4
|  4
|  0.00
| -33.99
| -33.99
| -28.77
| -28.77
Line 76: Line 55:
|-
|-
|  5
|  5
|  0.00
|  7.52
|  7.52
|  39.04
|  39.04
Line 87: Line 65:
|-
|-
|  6
|  6
|  0.00
|  49.02
|  49.02
|  6.84
|  6.84
Line 98: Line 75:
|-
|-
|  7
|  7
|  0.00
|  -9.47
|  -9.47
| -25.35
| -25.35
Line 109: Line 85:
|-
|-
|  8
|  8
|  0.00
|  32.03
|  32.03
|  42.46
|  42.46
Line 120: Line 95:
|-
|-
|  9
|  9
|  0.00
| -26.47
| -26.47
|  10.26
|  10.26
Line 131: Line 105:
|-
|-
| 10
| 10
|  0.00
|  15.04
|  15.04
| -21.93
| -21.93
Line 142: Line 115:
|-
|-
| 11
| 11
|  0.00
| -43.46
| -43.46
|  45.88
|  45.88
Line 153: Line 125:
|-
|-
| 12
| 12
|  0.00
|  -1.96
|  -1.96
|  13.69
|  13.69
Line 164: Line 135:
|-
|-
| 13
| 13
|  0.00
|  39.55
|  39.55
| -18.51
| -18.51
Line 175: Line 145:
|-
|-
| 14
| 14
|  0.00
| -18.95
| -18.95
|  49.30
|  49.30
Line 186: Line 155:
|-
|-
| 15
| 15
|  0.00
|  22.56
|  22.56
|  17.11
|  17.11
Line 197: Line 165:
|-
|-
| 16
| 16
|  0.00
| -35.94
| -35.94
| -15.08
| -15.08
Line 208: Line 175:
|-
|-
| 17
| 17
|  0.00
|  5.56
|  5.56
| -47.28
| -47.28
Line 219: Line 185:
|-
|-
| 18
| 18
|  0.00
|  47.07
|  47.07
|  20.53
|  20.53
Line 230: Line 195:
|-
|-
| 19
| 19
|  0.00
| -11.43
| -11.43
| -11.66
| -11.66
Line 241: Line 205:
|-
|-
| 20
| 20
|  0.00
|  30.07
|  30.07
| -43.86
| -43.86
Line 252: Line 215:
|-
|-
| 21
| 21
|  0.00
| -28.42
| -28.42
|  23.95
|  23.95
Line 263: Line 225:
|-
|-
| 22
| 22
|  0.00
|  13.08
|  13.08
|  -8.24
|  -8.24
Line 274: Line 235:
|-
|-
| 23
| 23
|  0.00
| -45.41
| -45.41
| -40.43
| -40.43
Line 285: Line 245:
|-
|-
| 24
| 24
|  0.00
|  -3.91
|  -3.91
|  27.37
|  27.37
Line 296: Line 255:
|-
|-
| 25
| 25
|  0.00
|  37.59
|  37.59
|  -4.82
|  -4.82
Line 307: Line 265:
|-
|-
| 26
| 26
|  0.00
| -20.90
| -20.90
| -37.01
| -37.01
Line 318: Line 275:
|-
|-
| 27
| 27
|  0.00
|  20.60
|  20.60
|  30.79
|  30.79
Line 329: Line 285:
|-
|-
| 28
| 28
|  0.00
| -37.90
| -37.90
|  -1.40
|  -1.40
Line 340: Line 295:
|-
|-
| 29
| 29
|  0.00
|  3.61
|  3.61
| -33.59
| -33.59
Line 351: Line 305:
|-
|-
| 30
| 30
|  0.00
|  45.11
|  45.11
|  34.22
|  34.22
Line 362: Line 315:
|-
|-
| 31
| 31
|  0.00
| -13.38
| -13.38
|  2.02
|  2.02
Line 373: Line 325:
|-
|-
| 32
| 32
|  0.00
|  28.12
|  28.12
| -30.17
| -30.17
Line 384: Line 335:
|-
|-
| 33
| 33
|  0.00
| -30.38
| -30.38
|  37.64
|  37.64
Line 395: Line 345:
|-
|-
| 34
| 34
|  0.00
|  11.13
|  11.13
|  5.44
|  5.44
Line 406: Line 355:
|-
|-
| 35
| 35
|  0.00
| -47.37
| -47.37
| -26.75
| -26.75
Line 417: Line 365:
|-
|-
| 36
| 36
|  0.00
|  -5.87
|  -5.87
|  41.06
|  41.06
Line 428: Line 375:
|-
|-
| 37
| 37
|  0.00
|  35.64
|  35.64
|  8.87
|  8.87
Line 439: Line 385:
|-
|-
| 38
| 38
|  0.00
| -22.86
| -22.86
| -23.33
| -23.33
Line 450: Line 395:
|-
|-
| 39
| 39
|  0.00
|  18.65
|  18.65
|  44.48
|  44.48
Line 461: Line 405:
|-
|-
| 40
| 40
|  0.00
| -39.85
| -39.85
|  12.29
|  12.29
Line 472: Line 415:
|-
|-
| 41
| 41
|  0.00
|  1.65
|  1.65
| -19.91
| -19.91
Line 483: Line 425:
|-
|-
| 42
| 42
|  0.00
|  43.16
|  43.16
|  47.90
|  47.90
Line 494: Line 435:
|-
|-
| 43
| 43
|  0.00
| -15.34
| -15.34
|  15.71
|  15.71
Line 505: Line 445:
|-
|-
| 44
| 44
|  0.00
|  26.16
|  26.16
| -16.48
| -16.48
Line 516: Line 455:
|-
|-
| 45
| 45
|  0.00
| -32.33
| -32.33
| -48.68
| -48.68
Line 527: Line 465:
|-
|-
| 46
| 46
|  0.00
|  9.17
|  9.17
|  19.13
|  19.13
Line 538: Line 475:
|-
|-
| 47
| 47
|  0.00
| -49.32
| -49.32
| -13.06
| -13.06
Line 549: Line 485:
|-
|-
| 48
| 48
|  0.00
|  -7.82
|  -7.82
| -45.25
| -45.25
Line 560: Line 495:
|-
|-
| 49
| 49
|  0.00
|  33.68
|  33.68
|  22.55
|  22.55
Line 571: Line 505:
|-
|-
| 50
| 50
|  0.00
| -24.81
| -24.81
|  -9.64
|  -9.64
Line 582: Line 515:
|-
|-
| 51
| 51
|  0.00
|  16.69
|  16.69
| -41.83
| -41.83
Line 593: Line 525:
|-
|-
| 52
| 52
|  0.00
| -41.81
| -41.81
|  25.97
|  25.97
Line 604: Line 535:
|-
|-
| 53
| 53
|  0.00
|  -0.30
|  -0.30
|  -6.22
|  -6.22
Line 615: Line 545:
|-
|-
| 54
| 54
|  0.00
|  41.20
|  41.20
| -38.41
| -38.41
Line 626: Line 555:
|-
|-
| 55
| 55
|  0.00
| -17.29
| -17.29
|  29.40
|  29.40
Line 637: Line 565:
|-
|-
| 56
| 56
|  0.00
|  24.21
|  24.21
|  -2.80
|  -2.80
Line 648: Line 575:
|-
|-
| 57
| 57
|  0.00
| -34.29
| -34.29
| -34.99
| -34.99
Line 659: Line 585:
|-
|-
| 58
| 58
|  0.00
|  7.22
|  7.22
|  32.82
|  32.82
Line 670: Line 595:
|-
|-
| 59
| 59
|  0.00
|  48.72
|  48.72
|  0.62
|  0.62
Line 681: Line 605:
|-
|-
| 60
| 60
|  0.00
|  -9.78
|  -9.78
| -31.57
| -31.57
Line 692: Line 615:
|-
|-
| 61
| 61
|  0.00
|  31.73
|  31.73
|  36.24
|  36.24
Line 703: Line 625:
|-
|-
| 62
| 62
|  0.00
| -26.77
| -26.77
|  4.05
|  4.05
Line 714: Line 635:
|-
|-
| 63
| 63
|  0.00
|  14.74
|  14.74
| -28.15
| -28.15
Line 725: Line 645:
|-
|-
| 64
| 64
|  0.00
| -43.76
| -43.76
|  39.66
|  39.66
Line 736: Line 655:
|-
|-
| 65
| 65
|  0.00
|  -2.26
|  -2.26
|  7.47
|  7.47
Line 747: Line 665:
|-
|-
| 66
| 66
|  0.00
|  39.25
|  39.25
| -24.73
| -24.73
Line 758: Line 675:
|-
|-
| 67
| 67
|  0.00
| -19.25
| -19.25
|  43.08
|  43.08
Line 769: Line 685:
|-
|-
| 68
| 68
|  0.00
|  22.25
|  22.25
|  10.89
|  10.89
Line 780: Line 695:
|-
|-
| 69
| 69
|  0.00
| -36.24
| -36.24
| -21.30
| -21.30
Line 791: Line 705:
|-
|-
| 70
| 70
|  0.00
|  5.26
|  5.26
|  46.50
|  46.50
Line 802: Line 715:
|-
|-
| 71
| 71
|  0.00
|  46.77
|  46.77
|  14.31
|  14.31
Line 813: Line 725:
|-
|-
| 72
| 72
|  0.00
| -11.73
| -11.73
| -17.88
| -17.88
Line 824: Line 735:
|-
|-
| 73
| 73
|  0.00
|  29.77
|  29.77
|  49.92
|  49.92
Line 835: Line 745:
|-
|-
| 74
| 74
|  0.00
| -28.72
| -28.72
|  17.73
|  17.73
Line 846: Line 755:
|-
|-
| 75
| 75
|  0.00
|  12.78
|  12.78
| -14.46
| -14.46
Line 857: Line 765:
|-
|-
| 76
| 76
|  0.00
| -45.72
| -45.72
| -46.65
| -46.65
Line 868: Line 775:
|-
|-
| 77
| 77
|  0.00
|  -4.21
|  -4.21
|  21.15
|  21.15
Line 879: Line 785:
|-
|-
| 78
| 78
|  0.00
|  37.29
|  37.29
| -11.04
| -11.04
Line 890: Line 795:
|-
|-
| 79
| 79
|  0.00
| -21.20
| -21.20
| -43.23
| -43.23
Line 901: Line 805:
|-
|-
| 80
| 80
|  0.00
|  20.30
|  20.30
|  24.58
|  24.58
Line 912: Line 815:
|-
|-
| 81
| 81
|  0.00
| -38.20
| -38.20
|  -7.62
|  -7.62
Line 923: Line 825:
|-
|-
| 82
| 82
|  0.00
|  3.31
|  3.31
| -39.81
| -39.81
Line 934: Line 835:
|-
|-
| 83
| 83
|  0.00
|  44.81
|  44.81
|  28.00
|  28.00
Line 945: Line 845:
|-
|-
| 84
| 84
|  0.00
| -13.69
| -13.69
|  -4.20
|  -4.20
Line 956: Line 855:
|-
|-
| 85
| 85
|  0.00
|  27.82
|  27.82
| -36.39
| -36.39
Line 967: Line 865:
|-
|-
| 86
| 86
|  0.00
| -30.68
| -30.68
|  31.42
|  31.42
Line 978: Line 875:
|-
|-
| 87
| 87
|  0.00
|  10.83
|  10.83
|  -0.77
|  -0.77
Line 989: Line 885:
|-
|-
| 88
| 88
|  0.00
| -47.67
| -47.67
| -32.97
| -32.97
Line 1,000: Line 895:
|-
|-
| 89
| 89
|  0.00
|  -6.17
|  -6.17
|  34.84
|  34.84
Line 1,011: Line 905:
|-
|-
| 90
| 90
|  0.00
|  35.34
|  35.34
|  2.65
|  2.65
Line 1,022: Line 915:
|-
|-
| 91
| 91
|  0.00
| -23.16
| -23.16
| -29.55
| -29.55
Line 1,033: Line 925:
|-
|-
| 92
| 92
|  0.00
|  18.34
|  18.34
|  38.26
|  38.26
Line 1,044: Line 935:
|-
|-
| 93
| 93
|  0.00
| -40.15
| -40.15
|  6.07
|  6.07
Line 1,055: Line 945:
|-
|-
| 94
| 94
|  0.00
|  1.35
|  1.35
| -26.12
| -26.12
Line 1,066: Line 955:
|-
|-
| 95
| 95
|  0.00
|  42.86
|  42.86
|  41.68
|  41.68
Line 1,077: Line 965:
|-
|-
| 96
| 96
|  0.00
| -15.64
| -15.64
|  9.49
|  9.49
Line 1,088: Line 975:
|-
|-
| 97
| 97
|  0.00
|  25.86
|  25.86
| -22.70
| -22.70
Line 1,099: Line 985:
|-
|-
| 98
| 98
|  0.00
| -32.63
| -32.63
|  45.10
|  45.10
Line 1,108: Line 993:
| -29.69
| -29.69
| -30.91
| -30.91
|}
== List of RMS Relative Errors of JI Subgroups for Small Edos ==
Notes: 
# Patent val mapping is used throughout. 
# 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. 
# The weight of intervals is determined by the [[tonality diamond]]. For example, the [[9-odd-limit]] takes account of 3/1 twice.
{| class="wikitable sortable mw-collapsible" style="text-align:right"
! rowspan="2" | Edo
! colspan="11" | Root Mean Squared Relative Errors (Percent)
|-
! 3 Odd Limit
! 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
| 99
| 6.27
|   8.87
| 7.63
| 12.91
| 6.60
|   7.19
| 8.38
| -48.37
| 31.37
| -34.35
| 33.58
| 34.12
| 33.84
| 45.52
| 35.67
| 16.74
| 38.51
| 36.92
| 35.54
|}
|}
== See also ==
* Relative errors of small ed-''p''<span/>'s:
** [[Relative errors of small EDTs]]
** [[Relative errors of small EDFs]]
** [[Relative errors of small ED5s]]
** [[Relative errors of small ED6s]]
* [[Relative errors of small equal temperaments]]
[[Category:EDO theory pages]]
[[Category:Tables]]

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

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