Relative errors of small EDOs: Difference between revisions

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Line 18: Line 18:
== List of Relative Errors of Prime Harmonies for Small Edos ==
== 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 (Permille)
! colspan="9" | Relative Errors (Percent)
|-
|-
!2/1
! 2/1
!3/1
! 3/1
!5/1
! 5/1
!7/1
! 7/1
!11/1
! 11/1
!13/1
! 13/1
!17/1
! 17/1
!19/1
! 19/1
!23/1
! 23/1
|-
|-
|1
| 1
|0.0
|   0.00
|415.0
| 41.50
| -321.9
| -32.19
|192.6
| 19.26
| -459.4
| -45.94
|299.6
| 29.96
| -87.5
| -8.75
| -247.9
| -24.79
|476.4
| 47.64
|-
|-
|2
| 2
|0.0
|   0.00
| -169.9
| -16.99
|356.1
| 35.61
|385.3
| 38.53
|81.1
|   8.11
| -400.9
| -40.09
| -174.9
| -17.49
| -495.9
| -49.59
|47.1
| -4.71
|-
|-
|3
| 3
|0.0
|   0.00
|245.1
| 24.51
|34.2
|   3.42
| -422.1
| -42.21
| -378.3
| -37.83
| -101.3
| -10.13
| -262.4
| -26.24
|256.2
| 25.62
|429.3
| 42.93
|-
|-
|4
| 4
|0.0
|   0.00
| -339.9
| -33.99
| -287.7
| -28.77
| -229.4
| -22.94
|162.3
| 16.23
|198.2
| 19.82
| -349.9
| -34.99
|8.3
|   0.83
|94.2
| -9.42
|-
|-
|5
| 5
|0.0
|   0.00
|75.2
|   7.52
|390.4
| 39.04
| -36.8
| -3.68
| -297.2
| -29.72
|497.8
| 49.78
| -437.3
| -43.73
| -239.6
| -23.96
|382.2
| 38.22
|-
|-
|6
| 6
|0.0
|   0.00
|490.2
| 49.02
|68.4
|   6.84
|155.9
| 15.59
|243.4
| 24.34
| -202.6
| -20.26
|475.2
| 47.52
| -487.6
| -48.76
| -141.4
| -14.14
|-
|-
|7
| 7
|0.0
|   0.00
| -94.7
| -9.47
| -253.5
| -25.35
|348.5
| 34.85
| -216.0
| -21.60
|96.9
|   9.69
|387.8
| 38.78
|264.5
| 26.45
|335.1
| 33.51
|-
|-
|8
| 8
|0.0
|   0.00
|320.3
| 32.03
|424.6
| 42.46
| -458.8
| -45.88
|324.5
| 32.45
|396.5
| 39.65
|300.3
| 30.03
|16.6
|   1.66
| -188.5
| -18.85
|-
|-
|9
| 9
|0.0
|   0.00
| -264.7
| -26.47
|102.6
| 10.26
| -266.2
| -26.62
| -134.9
| -13.49
| -304.0
| -30.40
|212.8
| 21.28
| -231.3
| -23.13
|287.9
| 28.79
|-
|-
|10
| 10
|0.0
|   0.00
|150.4
| 15.04
| -219.3
| -21.93
| -73.5
| -7.35
|405.7
| 40.57
| -4.4
| -0.44
|125.4
| 12.54
| -479.3
| -47.93
| -235.6
| -23.56
|-
|-
|11
| 11
|0.0
|   0.00
| -434.6
| -43.46
|458.8
| 45.88
|119.1
| 11.91
| -53.7
| -5.37
|295.2
| 29.52
|37.9
|   3.79
|272.8
| 27.28
|240.8
| 24.08
|-
|-
|12
| 12
|0.0
|   0.00
| -19.6
| -1.96
|136.9
| 13.69
|311.7
| 31.17
|486.8
| 48.68
| -405.3
| -40.53
| -49.6
| -4.96
|24.9
|   2.49
|282.7
| -28.27
|-
|-
|13
| 13
|0.0
|   0.00
|395.5
| 39.55
| -185.1
| -18.51
| -495.6
| -49.56
|27.4
|   2.74
| -105.7
| -10.57
| -137.0
| -13.70
| -223.1
| -22.31
|193.7
| 19.37
|-
|-
|14
| 14
|0.0
|   0.00
| -189.5
| -18.95
|493.0
| 49.30
| -303.0
| -30.30
| -432.0
| -43.20
|193.8
| 19.38
| -224.5
| -22.45
| -471.0
| -47.10
| -329.9
| -32.99
|-
|-
|15
| 15
|0.0
|   0.00
|225.6
| 22.56
|171.1
| 17.11
| -110.3
| -11.03
|108.5
| 10.85
|493.4
| 49.34
| -311.9
| -31.19
|281.1
| 28.11
|146.6
| 14.66
|-
|-
|16
| 16
|0.0
|   0.00
| -359.4
| -35.94
| -150.8
| -15.08
|82.3
|   8.23
| -350.9
| -35.09
| -207.0
| -20.70
| -399.4
| -39.94
|33.2
|   3.32
| -377.0
| -37.70
|-
|-
|17
| 17
|0.0
|   0.00
|55.6
|   5.56
| -472.8
| -47.28
|275.0
| 27.50
|189.7
| 18.97
|92.5
|   9.25
| -486.9
| -48.69
| -214.8
| -21.48
|99.4
|   9.94
|-
|-
|18
| 18
|0.0
|   0.00
|470.7
| 47.07
|205.3
| 20.53
|467.6
| 46.76
| -269.8
| -26.98
|392.1
| 39.21
|425.7
| 42.57
| -462.7
| -46.27
| -424.1
| -42.41
|-
|-
|19
| 19
|0.0
|   0.00
| -114.3
| -11.43
| -116.6
| -11.66
| -339.7
| -33.97
|270.8
| 27.08
| -308.4
| -30.84
|338.2
| 33.82
|289.4
| 28.94
|52.3
|   5.23
|-
|-
|20
| 20
|0.0
|   0.00
|300.7
| 30.07
| -438.6
| -43.86
| -147.1
| -14.71
| -188.6
| -18.86
| -8.8
| -0.88
|250.7
| 25.07
|41.4
|   4.14
|471.2
| -47.12
|-
|-
|21
| 21
|0.0
|   0.00
| -284.2
| -28.42
|239.5
| 23.95
|45.5
|   4.55
|351.9
| 35.19
|290.8
| 29.08
|163.3
| 16.33
| -206.5
| -20.65
|5.2
|   0.52
|-
|-
|22
| 22
|0.0
|   0.00
|130.8
| 13.08
| -82.4
| -8.24
|238.2
| 23.82
| -107.5
| -10.75
| -409.7
| -40.97
|75.8
|   7.58
| -454.4
| -45.44
|481.6
| 48.16
|-
|-
|23
| 23
|0.0
|   0.00
| -454.1
| -45.41
| -404.3
| -40.43
|430.8
| 43.08
|433.1
| 43.31
| -110.1
| -11.01
| -11.6
| -1.16
|297.7
| 29.77
| -41.9
| -4.19
|-
|-
|24
| 24
|0.0
|   0.00
| -39.1
| -3.91
|273.7
| 27.37
| -376.5
| -37.65
| -26.4
| -2.64
|189.4
| 18.94
| -99.1
| -9.91
|49.7
|   4.97
|434.5
| 43.45
|-
|-
|25
| 25
|0.0
|   0.00
|375.9
| 37.59
| -48.2
| -4.82
| -183.9
| -18.39
| -485.8
| -48.58
|489.0
| 48.90
| -186.6
| -18.66
| -198.2
| -19.82
| -89.0
| -8.90
|-
|-
|26
| 26
|0.0
|   0.00
| -209.0
| -20.90
| -370.1
| -37.01
|8.8
|   0.88
|54.8
|   5.48
| -211.4
| -21.14
| -274.0
| -27.40
|<nowiki>-446.1</nowiki>
| -44.61
|387.4
| 38.74
|-
|-
|27
| 27
|0.0
|   0.00
|206.0
| 20.60
|307.9
| 30.79
|201.4
| 20.14
| -404.7
| -40.47
|88.1
|   8.81
| -361.5
| -36.15
|306.0
| 30.60
| -136.2
| -13.62
|-
|-
|28
| 28
|0.0
|   0.00
| -379.0
| -37.90
| -14.0
| -1.40
|394.1
| 39.41
|135.9
| 13.59
|387.7
| 38.77
| -449.0
| -44.90
|58.0
|   5.80
|340.3
| 34.03
|-
|-
|29
| 29
|0.0
|   0.00
|36.1
|   3.61
| -335.9
| -33.59
| -413.3
| -41.33
| -323.5
| -32.35
| -312.8
| -31.28
|463.6
| 46.36
| -189.9
| -18.99
| -183.3
| -18.33
|-
|-
|30
| 30
|0.0
|   0.00
|451.1
| 45.11
|342.2
| 34.22
| -220.6
| -22.06
|217.1
| 21.71
| -13.2
| -1.32
|376.1
| 37.61
| -437.8
| -43.78
|293.1
| 29.31
|-
|-
|31
| 31
|0.0
|   0.00
| -133.8
| -13.38
|20.2
|   2.02
| -28.0
| -2.80
| -242.4
| -24.24
|286.4
| 28.64
|288.7
| 28.87
|314.2
| 31.42
| -230.4
| -23.04
|-
|-
|32
| 32
|0.0
|   0.00
|281.2
| 28.12
| -301.7
| -30.17
|164.6
| 16.46
|298.2
| 29.82
| -414.1
| -41.41
|201.2
| 20.12
|66.3
|   6.63
|246.0
| 24.60
|-
|-
|33
| 33
|0.0
|   0.00
| -303.8
| -30.38
|376.4
| 37.64
|357.3
| 35.73
| -161.2
| -16.12
| -114.5
| -11.45
|113.7
| 11.37
| -181.6
| -18.16
| -277.5
| -27.75
|-
|-
|34
| 34
|0.0
|   0.00
|111.3
| 11.13
|54.4
|   5.44
| -450.1
| -45.01
|379.3
| 37.93
|185.0
| 18.50
|26.3
|   2.63
| -429.5
| -42.95
|198.9
| 19.89
|-
|-
|35
| 35
|0.0
|   0.00
| -473.7
| -47.37
| -267.5
| -26.75
| -257.4
| -25.74
| -80.1
| -8.01
|484.6
| 48.46
| -61.2
| -6.12
|322.5
| 32.25
| -324.7
| -32.47
|-
|-
|36
| 36
|0.0
|   0.00
| -58.7
| -5.87
|410.6
| 41.06
| -64.8
| -6.48
|460.5
| 46.05
| -215.8
| -21.58
| -148.7
| -14.87
|74.6
|   7.46
|151.8
| 15.18
|-
|-
|37
| 37
|0.0
|   0.00
|356.4
| 35.64
|88.7
|   8.87
|127.9
| 12.79
|1.0
|   0.10
|83.7
|   8.37
| -236.1
| -23.61
| -173.3
| -17.33
| -371.8
| -37.18
|-
|-
|38
| 38
|0.0
|   0.00
| -228.6
| -22.86
| -233.3
| -23.33
|320.5
| 32.05
| -458.4
| -45.84
|383.3
| 38.33
| -323.6
| -32.36
| -421.2
| -42.12
|104.6
| 10.46
|-
|-
|39
| 39
|0.0
|   0.00
|186.5
| 18.65
|444.8
| 44.48
| -486.8
| -48.68
|82.2
|   8.22
| -317.1
| -31.71
| -411.1
| -41.11
|330.8
| 33.08
| -418.9
| -41.89
|-
|-
|40
| 40
|0.0
|   0.00
| -398.5
| -39.85
|122.9
| 12.29
| -294.2
| -29.42
| -377.3
| -37.73
| -17.6
| -1.76
| -498.5
| -49.85
|82.9
|   8.29
|57.5
|   5.75
|-
|-
|41
| 41
|0.0
|   0.00
|16.5
|   1.65
| -199.1
| -19.91
| -101.6
| -10.16
|163.3
| 16.33
|282.0
| 28.20
|414.0
| 41.40
| -165.0
| -16.50
| -466.0
| -46.60
|-
|-
|42
| 42
|0.0
|   0.00
|431.6
| 43.16
|479.0
| 47.90
|91.1
|   9.11
| -296.1
| -29.61
| -418.5
| -41.85
|326.6
| 32.66
| -413.0
| -41.30
|10.4
|   1.04
|-
|-
|43
| 43
|0.0
|   0.00
| -153.4
| -15.34
|157.1
| 15.71
|283.7
| 28.37
|244.4
| 24.44
| -118.9
| -11.89
|239.1
| 23.91
|339.1
| 33.91
|486.8
| 48.68
|-
|-
|44
| 44
|0.0
|   0.00
|261.6
| 26.16
| -164.8
| -16.48
|476.4
| 47.64
| -215.0
| -21.50
|180.7
| 18.07
|151.6
| 15.16
|91.2
|   9.12
| -36.7
| -3.67
|-
|-
|45
| 45
|0.0
|   0.00
| -323.3
| -32.33
| -486.8
| -48.68
| -331.0
| -33.10
|325.6
| 32.56
|480.2
| 48.02
|64.2
|   6.42
| -156.7
| -15.67
|439.7
| 43.97
|-
|-
|46
| 46
|0.0
|   0.00
|91.7
|   9.17
|191.3
| 19.13
| -138.3
| -13.83
| -133.9
| -13.39
| -220.2
| -22.02
| -23.3
| -2.33
| -404.7
| -40.47
| -83.8
| -8.38
|-
|-
|47
| 47
|0.0
|   0.00
| -493.2
| -49.32
| -130.6
| -13.06
|54.3
|   5.43
|406.7
| 40.67
|79.3
|   7.93
| -110.8
| -11.08
|347.4
| 34.74
|392.6
| 39.26
|-
|-
|48
| 48
|0.0
|   0.00
| -78.2
| -7.82
| -452.5
| -45.25
|247.0
| 24.70
| -52.7
| -5.27
|378.9
| 37.89
| -198.2
| -19.82
|99.5
|   9.95
| -131.0
| -13.10
|-
|-
|49
| 49
|0.0
|   0.00
|336.8
| 33.68
|225.5
| 22.55
|439.6
| 43.96
|487.9
| 48.79
| -321.5
| -32.15
| -285.7
| -28.57
| -148.4
| -14.84
|345.5
| 34.55
|-
|-
|50
| 50
|0.0
|   0.00
| -248.1
| -24.81
| -96.4
| -9.64
| -367.7
| -36.77
|28.4
|   2.84
| -22.0
| -2.20
| -373.1
| -37.31
| -396.4
| -39.64
| -178.1
| -17.81
|-
|-
|51
| 51
|0.0
|   0.00
|166.9
| 16.69
| -418.3
| -41.83
| -175.1
| -17.51
| -431.0
| -43.10
|277.6
| 27.76
| -460.6
| -46.06
|355.7
| 35.57
|298.3
| 29.83
|-
|-
|52
| 52
|0.0
|   0.00
| -418.1
| -41.81
|259.7
| 25.97
|17.5
|   1.75
|109.6
| 10.96
| -422.9
| -42.29
|451.9
| 45.19
|107.8
| 10.78
| -225.2
| -22.52
|-
|-
|53
| 53
|0.0
|   0.00
| -3.0
| -0.30
| -62.2
| -6.22
|210.2
| 21.02
| -349.9
| -34.99
| -123.3
| -12.33
|364.5
| 36.45
| -140.2
| -14.02
|251.2
| 25.12
|-
|-
|54
| 54
|0.0
|   0.00
|412.0
| 41.20
| -384.1
| -38.41
|402.8
| 40.28
|190.7
| 19.07
|176.3
| 17.63
|277.0
| 27.70
| -388.1
| -38.81
| -272.3
| -27.23
|-
|-
|55
| 55
|0.0
|   0.00
| -172.9
| -17.29
|294.0
| 29.40
| -404.5
| -40.45
| -268.7
| -26.87
|475.8
| 47.58
|189.5
| 18.95
|364.0
| 36.40
|204.1
| 20.41
|-
|-
|56
| 56
|0.0
|   0.00
|242.1
| 24.21
| -28.0
| -2.80
| -211.9
| -21.19
|271.8
| 27.18
| -224.6
| -22.46
|102.1
| 10.21
|116.1
| 11.61
| -319.5
| -31.95
|-
|-
|57
| 57
|0.0
|   0.00
| -342.9
| -34.29
| -349.9
| -34.99
| -19.2
| -1.92
| -187.6
| -18.76
|74.9
|   7.49
|14.6
|   1.46
| -131.9
| -13.19
|157.0
| 15.70
|-
|-
|58
| 58
|0.0
|   0.00
|72.2
|   7.22
|328.2
| 32.82
|173.4
| 17.34
|353.0
| 35.30
|374.5
| 37.45
| -72.8
| -7.28
| -379.8
| -37.98
| -366.6
| -36.66
|-
|-
|59
| 59
|0.0
|   0.00
|487.2
| 48.72
|6.2
|   0.62
|366.1
| 36.61
| -106.5
| -10.65
| -325.9
| -32.59
| -160.3
| -16.03
|372.3
| 37.23
|109.8
| 10.98
|-
|-
|60
| 60
|0.0
|   0.00
| -97.8
| -9.78
| -315.7
| -31.57
| -441.3
| -44.13
|434.1
| 43.41
| -26.4
| -2.64
| -247.8
| -24.78
|124.3
| 12.43
| -413.7
| -41.37
|-
|-
|61
| 61
|0.0
|   0.00
|317.3
| 31.73
|362.4
| 36.24
| -248.7
| -24.87
| -25.3
| -2.53
|273.2
| 27.32
| -335.2
| -33.52
| -123.6
| -12.36
|62.7
|   6.27
|-
|-
|62
| 62
|0.0
|   0.00
| -267.4
| -26.77
|40.5
|   4.05
| -56.0
| -5.60
| -484.8
| -48.48
| -427.3
| -42.73
| -422.7
| -42.27
| -371.5
| -37.15
| -460.8
| -46.08
|-
|-
|63
| 63
|0.0
|   0.00
|147.4
| 14.74
| -281.5
| -28.15
|136.6
| 13.66
|55.8
|   5.58
| -127.7
| -12.77
|489.8
| 48.98
|380.6
| 38.06
|15.6
|   1.56
|-
|-
|64
| 64
|0.0
|   0.00
| -437.6
| -43.76
|396.6
| 39.66
|329.3
| 32.93
| -403.6
| -40.36
|171.9
| 17.19
|402.4
| 40.24
|132.6
| 13.26
|492.0
| 49.20
|-
|-
|65
| 65
|0.0
|   0.00
| -22.6
| -2.26
|74.7
|   7.47
| -478.1
| -47.81
|136.9
| 13.69
|471.4
| 47.14
|314.9
| 31.49
| -115.3
| -11.53
| -31.5
| -3.15
|-
|-
|66
| 66
|0.0
|   0.00
|392.5
| 39.25
| -247.3
| -24.73
| -285.4
| -28.54
| -322.5
| -32.25
| -229.0
| -22.90
|227.5
| 22.75
| -363.2
| -36.32
|444.9
| 44.49
|-
|-
|67
| 67
|0.0
|   0.00
| -192.5
| -19.25
|430.8
| 43.08
| -92.8
| -9.28
|218.1
| 21.81
|70.5
|   7.05
|140.0
| 14.00
|388.9
| 38.89
| -78.7
| -7.87
|-
|-
|68
| 68
|0.0
|   0.00
|222.5
| 22.25
|108.9
| 10.89
|99.9
|   9.99
| -241.4
| -24.14
|370.1
| 37.01
|52.5
|   5.25
|140.9
| 14.09
|397.8
| 39.78
|-
|-
|69
| 69
|0.0
|   0.00
| -362.4
| -36.24
| -213.0
| -21.30
|292.5
| 29.25
|299.2
| 29.92
| -330.3
| -33.03
| -34.9
| -3.49
| -107.0
| -10.70
| -125.8
| -12.58
|-
|-
|70
| 70
|0.0
|   0.00
|52.6
|   5.26
|465.0
| 46.50
|485.2
| 48.52
| -160.2
| -16.02
| -30.8
| -3.08
| -122.4
| -12.24
| -354.9
| -35.49
|350.7
| 35.07
|-
|-
|71
| 71
|0.0
|   0.00
|467.7
| 46.77
|143.1
| 14.31
| -322.2
| -32.22
|380.4
| 38.04
|268.8
| 26.88
| -209.9
| -20.99
|397.1
| 39.71
| -172.9
| -17.29
|-
|-
|72
| 72
|0.0
|   0.00
| -117.3
| -11.73
| -178.8
| -17.88
| -129.6
| -12.96
| -79.1
| -7.91
| -431.7
| -43.17
| -297.3
| -29.73
|149.2
| 14.92
|303.5
| 30.35
|-
|-
|73
| 73
|0.0
|   0.00
|297.7
| 29.77
|499.2
| 49.92
|63.1
|   6.31
|461.5
| 46.15
| -132.1
| -13.21
| -384.8
| -38.48
| -98.7
| -9.87
| -220.0
| -22.00
|-
|-
|74
| 74
|0.0
|   0.00
| -287.2
| -28.72
|143.1
| 17.73
|255.7
| 25.57
|2.1
|   0.21
|167.5
| 16.75
| -472.3
| -47.23
| -346.6
| -34.66
|256.4
| 25.64
|-
|-
|75
| 75
|0.0
|   0.00
|127.8
| 12.78
| -144.6
| -14.46
|448.4
| 44.84
| -457.4
| -45.74
|467.0
| 46.70
|440.3
| 44.03
|405.4
| 40.54
| -267.1
| -26.71
|-
|-
|76
| 76
|0.0
|   0.00
| -457.2
| -45.72
| -466.5
| -46.65
| -359.0
| -35.90
|83.2
|   8.32
| -233.4
| -23.34
|352.8
| 35.28
|157.5
| 15.75
|209.3
| 20.93
|-
|-
|77
| 77
|0.0
|   0.00
| -42.1
| -4.21
|211.5
| 21.15
| -166.3
| -16.63
| -376.2
| -37.62
|66.1
|   6.61
|265.4
| 26.54
| -90.4
| -9.04
| -314.3
| -31.43
|-
|-
|78
| 78
|0.0
|   0.00
|372.9
| 37.29
| -110.4
| -11.04
|26.3
|   2.63
|164.3
| 16.43
|365.7
| 36.57
|177.9
| 17.79
| -338.3
| -33.83
|162.2
| 16.22
|-
|-
|79
| 79
|0.0
|   0.00
| -212.0
| -21.20
| -432.3
| -43.23
|219.0
| 21.90
| -295.1
| -29.51
| -334.7
| -33.47
|90.4
|   9.04
|413.7
| 41.37
| -361.4
| -36.14
|-
|-
|80
| 80
|0.0
|   0.00
|203.0
| 20.30
|245.8
| 24.58
|411.6
| 41.16
|245.5
| 24.55
| -35.2
| -3.52
|3.0
|   0.30
|165.8
| 16.58
|115.0
| 11.50
|-
|-
|81
| 81
|0.0
|   0.00
| -382.0
| -38.20
| -76.2
| -7.62
| -395.7
| -39.57
| -214.0
| -21.40
|264.4
| 26.44
| -84.5
| -8.45
| -82.1
| -8.21
| -408.5
| -40.85
|-
|-
|82
| 82
|0.0
|   0.00
|33.1
|   3.31
| -398.1
| -39.81
| -203.1
| -20.31
|326.6
| 32.66
| -436.1
| -43.61
| -172.0
| -17.20
| -330.1
| -33.01
|67.9
|   6.79
|-
|-
|83
| 83
|0.0
|   0.00
|448.1
| 44.81
|280.0
| 28.00
| -10.5
| -1.05
| -132.8
| -13.28
| -136.5
| -13.65
| -259.4
| -25.94
|422.0
| 42.20
| -455.6
| -45.56
|-
|-
|84
| 84
|0.0
|   0.00
| -136.9
| -13.69
| -42.0
| -4.20
|182.2
| 18.22
|407.7
| 40.77
|163.1
| 16.31
| -346.9
| -34.69
|174.1
| 17.41
|20.8
|   2.08
|-
|-
|85
| 85
|0.0
|   0.00
|278.2
| 27.82
| -363.9
| -36.39
|374.8
| 37.48
| -51.7
| -5.17
|462.6
| 46.26
| -434.3
| -43.43
| -73.8
| -7.38
|497.2
| 49.72
|-
|-
|86
| 86
|0.0
|   0.00
| -306.8
| -30.68
|314.2
| 31.42
| -432.5
| -43.25
|488.9
| 48.89
| -237.8
| -23.78
|478.2
| 47.82
| -321.8
| -32.18
| -26.3
| -2.63
|-
|-
|87
| 87
|0.0
|   0.00
|108.3
| 10.83
| -7.7
| -0.77
| -239.9
| -23.99
|29.4
|   2.94
|61.7
|   6.17
|390.7
| 39.07
|430.3
| 43.03
|450.1
| 45.01
|-
|-
|88
| 88
|0.0
|   0.00
| -476.7
| -47.67
| -329.7
| -32.97
| -47.2
| -4.72
| -430.0
| -43.00
|361.3
| 36.13
|303.3
| 30.33
|182.4
| 18.24
| -73.5
| -7.35
|-
|-
|89
| 89
|0.0
|   0.00
| -61.7
| -6.17
|348.4
| 34.84
|145.4
| 14.54
|110.6
| 11.06
| -339.1
| -33.91
|215.8
| 21.58
| -65.5
| -6.55
|403.0
| 40.30
|-
|-
|90
| 90
|0.0
|   0.00
|353.4
| 35.34
|26.5
|   2.65
|338.1
| 33.81
| -348.8
| -34.88
| -39.6
| -3.96
|128.3
| 12.83
| -313.5
| -31.35
| -120.6
| -12.06
|-
|-
|91
| 91
|0.0
|   0.00
| -231.6
| -23.16
| -295.5
| -29.55
| -469.3
| -46.93
|191.7
| 19.17
|260.0
| 26.00
|40.9
|   4.09
|438.6
| 43.86
|355.9
| 35.59
|-
|-
|92
| 92
|0.0
|   0.00
|183.4
| 18.34
|382.6
| 38.26
| -276.7
| -27.67
| -267.7
| -26.77
| -440.5
| -44.05
| -46.6
| -4.66
|190.7
| 19.07
| -167.7
| -16.77
|-
|-
|93
| 93
|0.0
|   0.00
| -401.5
| -40.15
|60.7
|   6.07
| -84.0
| -8.40
|272.9
| 27.29
| -140.9
| -14.09
| -134.0
| -13.40
| -57.3
| -5.73
|308.7
| 30.87
|-
|-
|94
| 94
|0.0
|   0.00
|13.5
|   1.35
| -261.2
| -26.12
|108.6
| 10.86
| -186.6
| -18.66
|158.7
| 15.87
| -221.5
| -22.15
| -305.2
| -30.52
| -214.8
| -21.48
|-
|-
|95
| 95
|0.0
|   0.00
|428.6
| 42.86
|416.8
| 41.68
|301.3
| 30.13
|354.0
| 35.40
|458.2
| 45.82
| -309.0
| -30.90
|446.9
| 44.69
|261.6
| 26.16
|-
|-
|96
| 96
|0.0
|   0.00
| -156.4
| -15.64
|94.9
|   9.49
|493.9
| 49.39
| -105.4
| -10.54
| -242.2
| -24.22
| -396.4
| -39.64
|199.0
| 19.90
| -261.9
| -26.19
|-
|-
|97
| 97
|0.0
|   0.00
|258.6
| 25.86
| -227.0
| -22.70
| -313.4
| -31.34
|435.1
| 43.51
|27.3
|   5.73
| -483.9
| -48.39
| -49.0
| -4.90
|214.5
| 21.45
|-
|-
|98
| 98
|0.0
|   0.00
| -326.3
| -32.63
|451.0
| 45.10
| -120.8
| -12.08
| -24.3
| -2.43
|356.9
| 35.69
|428.6
| 42.86
| -296.9
| -29.69
| -309.1
| -30.91
|-
|99
|0.0
|88.7
|129.1
|71.9
| -483.7
| -343.5
|341.2
|455.2
|167.4
|}
|}


Line 1,124: Line 1,113:
Notes:   
Notes:   
# Patent val mapping is used throughout.   
# Patent val mapping is used throughout.   
# Octave equivalence is assumed. For example, the 5-odd-limit takes account of 2/1, 3/1, 5/1, 5/3 and their octave inverses.   
# 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 9-odd-limit takes account of 3/2 twice as the [[tonality diamond]] suggests. Same goes for other non-prime limits. 
# 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"
{| class="wikitable sortable mw-collapsible" style="text-align:right"
! rowspan="2" |Edo
! rowspan="2" | Edo
! colspan="11" |Root Mean Squared Relative Errors (Permille)
! colspan="11" | Root Mean Squared Relative Errors (Percent)
|-
|-
!3 Odd Limit
! 3 Odd Limit
!5 Odd Limit
! 5 Odd Limit
!7 Odd Limit
! 7 Odd Limit
!9 Odd Limit
! 9 Odd Limit
!11 Odd Limit
! 11 Odd Limit
!13 Odd Limit
! 13 Odd Limit
!15 Odd Limit
! 15 Odd Limit
!17 Odd Limit
! 17 Odd Limit
!19 Odd Limit
! 19 Odd Limit
!21 Odd Limit
! 21 Odd Limit
!23 Odd Limit
! 23 Odd Limit
|-
|-
|1
| 1
|293.5
| 29.35
|452.5
| 42.66
|408.9
| 38.25
|585.0
| 54.88
|654.2
| 61.68
|610.7
| 57.87
|569.1
| 54.17
|544.0
| 51.99
|537.6
| 51.57
|559.1
| 53.79
|553.0
| 53.35
|-
|-
|2
| 2
|120.2
| 12.02
|328.8
| 31.00
|356.6
| 33.35
|431.1
| 40.44
|392.1
| 36.97
|431.7
| 40.91
|413.7
| 39.38
|398.1
| 38.05
|433.7
| 41.60
|428.1
| 41.18
|408.8
| 39.43
|-
|-
|3
| 3
|173.3
| 17.33
|162.6
| 15.33
|365.9
| 34.23
|455.1
| 42.69
|483.2
| 45.56
|447.9
| 42.45
|442.1
| 42.08
|435.2
| 41.59
|428.2
| 41.08
|415.8
| 39.93
|433.3
| 41.80
|-
|-
|4
| 4
|240.3
| 24.03
|224.2
| 21.13
|196.1
| 18.34
|330.7
| 31.02
|398.9
| 37.61
|429.9
| 40.74
|459.6
| 43.75
|435.4
| 41.62
|426.0
| 40.86
|431.8
| 41.54
|416.9
| 40.21
|-
|-
|5
| 5
|53.2
| 5.32
|253.7
| 23.91
|254.6
| 23.81
|228.5
| 21.44
|310.5
| 29.27
|370.4
| 35.10
|386.3
| 36.78
|455.7
| 43.55
|454.7
| 43.62
|432.3
| 41.59
|433.6
| 41.82
|-
|-
|6
| 6
|346.6
| 34.66
|325.2
| 30.66
|284.5
| 26.61
|546.1
| 51.22
|498.9
| 47.03
|535.3
| 50.72
|521.3
| 49.62
|497.3
| 47.53
|587.1
| 56.32
|585.4
| 56.32
|582.9
| 56.23
|-
|-
|7
| 7
|67.0
| 6.70
|156.9
| 14.79
|333.6
| 31.20
|318.7
| 29.90
|306.1
| 28.86
|294.8
| 27.93
|312.5
| 29.75
|365.9
| 34.97
|369.6
| 35.45
|367.3
| 35.34
|372.9
| 35.97
|-
|-
|8
| 8
|226.5
| 22.65
|271.0
| 25.55
|519.9
| 48.63
|576.8
| 54.11
|529.7
| 49.94
|497.7
| 47.16
|526.7
| 50.14
|494.6
| 47.27
|484.0
| 46.42
|491.5
| 47.29
|499.6
| 48.19
|-
|-
|9
| 9
|187.1
| 18.71
|232.1
| 21.88
|245.6
| 22.98
|335.9
| 31.51
|306.8
| 28.92
|289.6
| 27.45
|270.3
| 25.73
|316.8
| 30.28
|301.7
| 28.94
|327.3
| 31.48
|368.0
| 35.50
|-
|-
|10
| 10
|106.3
| 10.63
|227.7
| 21.47
|201.5
| 18.85
|271.0
| 25.42
|322.9
| 30.45
|310.9
| 28.61
|290.4
| 27.65
|274.4
| 26.22
|354.9
| 34.04
|338.4
| 32.56
|339.3
| 32.73
|-
|-
|11
| 11
|307.3
| 30.73
|547.1
| 51.59
|483.1
| 45.19
|695.3
| 65.23
|633.6
| 59.74
|624.4
| 59.17
|583.2
| 55.52
|549.5
| 52.52
|538.4
| 51.64
|527.3
| 50.73
|516.4
| 49.81
|-
|-
|12
| 12
|13.8
| 1.38
|104.4
| 9.84
|200.4
| 18.74
|199.3
| 18.70
|291.6
| 27.49
|393.8
| 37.31
|367.5
| 34.98
|349.8
| 33.43
|331.0
| 31.75
|330.1
| 31.76
|347.4
| 33.51
|-
|-
|13
| 13
|279.7
| 27.97
|363.2
| 34.24
|487.5
| 45.60
|677.8
| 63.59
|617.0
| 58.18
|577.4
| 54.71
|542.6
| 51.65
|519.4
| 49.64
|506.3
| 48.56
|484.3
| 46.59
|468.0
| 45.14
|-
|-
|14
| 14
|134.0
| 13.40
|431.5
| 40.68
|460.3
| 43.05
|469.9
| 44.08
|471.0
| 44.41
|466.6
| 44.22
|475.2
| 45.23
|454.4
| 43.44
|466.5
| 44.75
|473.3
| 45.54
|458.7
| 44.25
|-
|-
|15
| 15
|159.5
| 15.95
|144.1
| 13.59
|202.6
| 18.95
|291.6
| 27.35
|265.8
| 25.06
|306.3
| 29.02
|302.5
| 28.80
|375.7
| 35.91
|359.3
| 34.47
|342.4
| 32.94
|327.0
| 31.54
|-
|-
|16
| 16
|254.1
| 25.41
|221.0
| 20.84
|254.0
| 23.76
|432.9
| 40.61
|399.1
| 37.62
|368.3
| 34.90
|367.1
| 34.95
|349.4
| 33.39
|360.0
| 34.53
|342.3
| 32.93
|330.3
| 31.86
|-
|-
|17
| 17
|39.3
| 3.93
|355.6
| 33.53
|412.1
| 38.55
|378.0
| 35.46
|360.4
| 33.98
|333.8
| 31.63
|382.3
| 36.39
|419.3
| 40.08
|401.4
| 38.50
|420.3
| 40.44
|405.8
| 39.15
|-
|-
|18
| 18
|332.8
| 33.28
|289.0
| 27.25
|298.0
| 27.87
|476.3
| 44.68
|578.4
| 54.53
|534.8
| 50.68
|528.6
| 50.32
|497.3
| 47.53
|596.6
| 57.22
|630.8
| 60.69
|378.0
| 65.40
|-
|-
|19
| 19
|80.8
| 8.08
|81.7
| 7.70
|186.2
| 17.42
|174.0
| 16.32
|288.0
| 27.15
|289.1
| 27.40
|274.7
| 26.15
|338.6
| 32.36
|359.6
| 34.49
|383.2
| 36.87
|369.9
| 35.68
|-
|-
|20
| 20
|212.7
| 21.27
|455.3
| 42.93
|402.9
| 37.69
|542.0
| 50.84
|512.0
| 48.27
|471.9
| 44.72
|445.9
| 42.45
|435.0
| 41.57
|411.3
| 39.45
|394.5
| 37.96
|429.7
| 41.45
|-
|-
|21
| 21
|201.0
| 20.10
|321.1
| 30.27
|283.1
| 26.48
|425.8
| 39.94
|466.1
| 43.95
|462.0
| 43.78
|431.1
| 41.04
|411.5
| 39.33
|401.9
| 38.55
|395.0
| 38.00
|377.3
| 36.40
|-
|-
|22
| 22
|92.5
| 9.25
|131.7
| 12.42
|185.3
| 17.34
|201.1
| 18.87
|219.3
| 20.68
|323.2
| 30.63
|301.7
| 28.72
|285.0
| 27.24
|340.8
| 32.68
|365.2
| 35.13
|398.3
| 38.42
|-
|-
|23
| 23
|321.1
| 32.11
|305.0
| 28.76
|539.6
| 50.48
|682.0
| 63.98
|732.8
| 69.09
|675.3
| 64.00
|720.0
| 68.54
|683.9
| 65.37
|684.4
| 65.64
|653.0
| 62.83
|625.0
| 60.29
|-
|-
|24
| 24
|27.6
| 2.76
|208.8
| 19.68
|349.0
| 32.65
|312.4
| 29.30
|283.9
| 26.76
|287.9
| 27.28
|293.4
| 27.93
|281.2
| 26.88
|266.5
| 25.56
|311.3
| 29.95
|351.0
| 33.86
|-
|-
|25
| 25
|265.8
| 26.58
|284.4
| 26.81
|313.8
| 29.35
|514.5
| 48.26
|597.1
| 56.30
|592.3
| 56.12
|560.1
| 53.32
|549.2
| 52.49
|537.4
| 51.55
|512.9
| 49.34
|495.3
| 47.78
|-
|-
|26
| 26
|147.8
| 14.78
|227.3
| 21.43
|238.4
| 22.30
|269.8
| 25.31
|282.8
| 26.66
|262.1
| 24.84
|318.2
| 30.29
|300.0
| 28.68
|300.5
| 28.82
|286.2
| 27.54
|373.4
| 36.02
|-
|-
|27
| 27
|145.7
| 14.57
|192.1
| 18.11
|168.9
| 15.80
|206.0
| 19.33
|398.9
| 37.61
|367.8
| 34.85
|394.4
| 37.55
|444.1
| 42.45
|429.0
| 41.14
|424.6
| 40.85
|422.0
| 40.71
|-
|-
|28
| 28
|268.0
| 26.80
|263.2
| 24.81
|413.4
| 38.67
|587.3
| 55.10
|557.1
| 52.53
|573.7
| 54.37
|562.7
| 53.56
|555.8
| 53.12
|531.2
| 50.95
|507.4
| 48.81
|515.1
| 49.68
|-
|-
|29
| 29
|25.5
| 2.55
|251.3
| 23.69
|300.3
| 28.09
|307.5
| 28.84
|299.9
| 28.27
|287.4
| 27.23
|273.7
| 26.06
|400.6
| 38.29
|379.8
| 36.43
|375.8
| 36.16
|359.1
| 34.64
|-
|-
|30
| 30
|319.0
| 31.90
|288.3
| 27.18
|405.2
| 37.90
|583.2
| 54.71
|531.5
| 50.11
|513.3
| 48.64
|549.9
| 52.35
|517.3
| 49.44
|592.0
| 56.78
|562.7
| 54.14
|537.8
| 51.88
|-
|-
|31
| 31
|94.6
| 9.46
|102.5
| 9.67
|89.7
| 8.39
|161.3
| 15.13
|171.8
| 16.20
|260.0
| 24.64
|244.0
| 23.22
|280.6
| 26.82
|303.9
| 29.15
|298.2
| 28.68
|299.2
| 28.86
|-
|-
|32
| 32
|198.8
| 19.88
|357.0
| 33.66
|331.2
| 30.98
|433.6
| 40.68
|403.5
| 38.04
|480.0
| 45.48
|449.9
| 42.83
|426.7
| 40.79
|403.5
| 38.70
|413.0
| 39.73
|397.8
| 38.37
|-
|-
|33
| 33
|214.8
| 21.48
|417.3
| 39.34
|424.5
| 39.71
|573.8
| 53.83
|525.9
| 49.58
|485.4
| 45.99
|457.0
| 43.50
|435.6
| 41.64
|417.2
| 40.02
|398.8
| 38.37
|393.2
| 37.93
|-
|-
|34
| 34
|78.7
| 7.87
|68.1
| 6.42
|336.1
| 31.44
|348.2
| 32.66
|384.6
| 36.26
|361.0
| 34.20
|339.2
| 32.30
|319.7
| 30.55
|376.3
| 36.09
|389.5
| 37.48
|381.2
| 36.77
|-
|-
|35
| 35
|334.9
| 33.49
|290.9
| 27.42
|254.0
| 23.76
|477.9
| 44.83
|467.2
| 44.04
|607.9
| 57.61
|621.2
| 59.14
|592.6
| 56.64
|616.7
| 59.15
|627.4
| 60.36
|599.8
| 57.86
|-
|-
|36
| 36
|41.5
| 4.15
|313.1
| 29.52
|298.2
| 27.89
|289.4
| 27.14
|354.9
| 33.46
|367.3
| 34.81
|371.0
| 35.32
|366.5
| 35.03
|346.5
| 33.23
|339.3
| 32.64
|326.5
| 31.49
|-
|-
|37
| 37
|252.0
| 25.20
|227.2
| 21.42
|198.8
| 18.60
|386.5
| 36.26
|379.1
| 35.75
|355.9
| 33.72
|353.2
| 33.63
|395.5
| 37.80
|404.4
| 38.79
|411.0
| 39.54
|450.0
| 43.41
|-
|-
|38
| 38
|161.6
| 16.16
|163.3
| 15.40
|341.8
| 31.97
|397.0
| 37.24
|407.3
| 38.40
|475.5
| 45.06
|477.8
| 45.49
|456.5
| 43.64
|446.6
| 42.83
|440.7
| 42.40
|434.5
| 41.91
|-
|-
|39
| 39
|131.8
| 13.18
|273.6
| 25.79
|515.2
| 48.19
|502.3
| 47.12
|456.4
| 43.03
|473.5
| 44.87
|527.9
| 50.25
|552.6
| 52.82
|536.4
| 51.45
|535.0
| 51.48
|544.9
| 52.56
|-
|-
|40
| 40
|281.8
| 28.18
|333.8
| 31.47
|320.1
| 29.94
|487.1
| 45.70
|446.2
| 42.07
|435.1
| 41.23
|405.3
| 38.58
|397.1
| 37.96
|408.6
| 39.19
|432.0
| 41.56
|435.3
| 41.99
|-
|-
|41
| 41
|11.7
| 1.17
|146.9
| 13.85
|131.0
| 12.25
|132.6
| 12.44
|169.6
| 15.99
|220.0
| 20.85
|229.4
| 21.84
|288.5
| 27.57
|288.4
| 27.66
|278.1
| 26.76
|329.7
| 31.80
|-
|-
|42
| 42
|305.2
| 30.52
|323.3
| 30.48
|314.6
| 29.43
|463.9
| 43.52
|563.4
| 53.12
|628.8
| 59.59
|690.9
| 65.77
|649.5
| 62.08
|683.4
| 65.55
|664.0
| 63.88
|640.1
| 61.75
|-
|-
|43
| 43
|108.5
| 10.85
|190.1
| 17.93
|248.5
| 23.24
|318.2
| 29.85
|320.6
| 30.23
|306.4
| 29.04
|285.4
| 27.17
|287.7
| 27.50
|302.7
| 29.04
|288.9
| 27.80
|322.6
| 31.12
|-
|-
|44
| 44
|185.0
| 18.50
|263.4
| 24.83
|370.7
| 34.67
|402.2
| 37.73
|438.7
| 41.36
|404.4
| 38.33
|377.6
| 35.94
|354.5
| 33.89
|336.0
| 32.23
|407.3
| 39.18
|400.0
| 38.59
|-
|-
|45
| 45
|228.6
| 22.86
|303.4
| 28.60
|267.8
| 25.05
|323.3
| 30.33
|481.7
| 45.42
|582.9
| 55.24
|634.8
| 60.43
|610.9
| 58.39
|577.8
| 55.42
|583.4
| 56.13
|619.8
| 59.79
|-
|-
|46
| 46
|64.9
| 6.49
|117.2
| 11.05
|183.5
| 17.16
|186.2
| 17.47
|202.6
| 19.10
|228.5
| 21.65
|255.2
| 24.29
|241.0
| 23.04
|297.0
| 28.49
|282.6
| 27.19
|271.1
| 26.15
|-
|-
|47
| 47
|348.8
| 34.88
|313.0
| 29.51
|322.7
| 30.18
|584.7
| 54.85
|665.7
| 62.76
|629.3
| 59.63
|630.0
| 59.97
|593.4
| 56.72
|611.1
| 58.62
|594.0
| 57.15
|611.6
| 59.00
|-
|-
|48
| 48
|55.3
| 5.53
|296.2
| 27.93
|379.3
| 35.48
|342.2
| 32.10
|311.4
| 29.36
|374.5
| 35.49
|430.7
| 41.00
|408.1
| 39.01
|395.2
| 37.90
|389.4
| 37.47
|373.2
| 36.00
|-
|-
|49
| 49
|238.2
| 23.82
|210.2
| 19.82
|246.8
| 23.09
|337.0
| 31.61
|317.8
| 29.96
|461.1
| 43.69
|453.8
| 43.20
|505.8
| 48.35
|507.5
| 48.67
|541.0
| 52.04
|518.0
| 49.97
|-
|-
|50
| 50
|175.5
| 17.55
|153.2
| 14.44
|212.8
| 19.91
|269.9
| 25.32
|287.9
| 27.15
|280.5
| 26.58
|274.6
| 26.14
|271.1
| 25.91
|268.8
| 25.78
|302.7
| 29.12
|291.3
| 28.10
|-
|-
|51
| 51
|118.0
| 11.80
|369.2
| 34.81
|327.8
| 30.66
|395.0
| 37.05
|426.5
| 40.21
|436.6
| 41.37
|420.2
| 40.00
|435.9
| 41.66
|460.0
| 44.12
|437.8
| 42.12
|442.0
| 42.63
|-
|-
|52
| 52
|295.6
| 29.56
|418.8
| 39.49
|368.5
| 34.47
|584.2
| 54.80
|557.3
| 52.54
|533.5
| 50.56
|496.9
| 47.31
|551.7
| 52.73
|530.5
| 50.88
|521.1
| 50.14
|499.6
| 48.19
|-
|-
|53
| 53
|2.1
| 0.21
|42.9
| 4.05
|156.4
| 14.63
|141.8
| 13.30
|247.3
| 23.32
|232.4
| 22.02
|216.6
| 20.62
|280.2
| 26.78
|271.6
| 26.05
|275.1
| 26.47
|281.3
| 27.14
|-
|-
|54
| 54
|291.3
| 29.13
|487.6
| 45.97
|497.5
| 46.54
|619.3
| 58.09
|563.5
| 53.13
|520.2
| 49.29
|494.6
| 47.08
|465.6
| 44.50
|514.5
| 49.35
|563.2
| 54.18
|572.7
| 55.25
|-
|-
|55
| 55
|122.3
| 12.23
|289.1
| 27.26
|385.3
| 36.05
|381.6
| 35.80
|355.7
| 33.53
|462.7
| 43.84
|439.9
| 41.88
|426.4
| 40.75
|436.1
| 41.83
|486.5
| 46.81
|474.1
| 45.74
|-
|-
|56
| 56
|171.2
| 17.12
|181.9
| 17.15
|244.1
| 22.83
|363.9
| 34.14
|344.7
| 32.50
|366.6
| 34.74
|348.0
| 33.13
|326.8
| 31.24
|309.1
| 29.65
|295.2
| 28.40
|327.2
| 31.56
|-
|-
|57
| 57
|242.4
| 24.24
|245.0
| 23.10
|254.8
| 23.84
|381.3
| 35.77
|350.1
| 33.01
|367.9
| 34.87
|415.1
| 39.51
|412.4
| 39.42
|393.0
| 37.70
|377.6
| 36.33
|395.5
| 38.15
|-
|-
|58
| 58
|51.0
| 5.10
|211.2
| 19.91
|186.2
| 17.42
|166.1
| 15.58
|190.9
| 18.00
|203.5
| 19.28
|212.0
| 20.19
|244.1
| 23.33
|344.0
| 33.00
|330.1
| 31.76
|378.1
| 36.48
|-
|-
|59
| 59
|344.5
| 34.45
|342.3
| 32.27
|326.6
| 30.56
|542.8
| 50.92
|559.5
| 52.75
|607.0
| 57.52
|583.2
| 55.52
|578.0
| 55.24
|552.1
| 52.96
|590.9
| 56.85
|567.9
| 54.79
|-
|-
|60
| 60
|69.1
| 6.91
|197.9
| 18.66
|263.3
| 24.63
|235.1
| 22.05
|418.6
| 39.47
|387.7
| 36.74
|394.0
| 37.51
|373.9
| 35.74
|373.0
| 35.78
|396.7
| 38.17
|392.9
| 37.90
|-
|-
|61
| 61
|224.4
| 22.44
|241.9
| 22.80
|376.0
| 35.17
|461.9
| 43.33
|440.2
| 41.50
|408.8
| 38.74
|450.9
| 42.93
|503.1
| 48.09
|494.7
| 47.45
|471.7
| 45.38
|451.6
| 43.57
|-
|-
|62
| 62
|189.3
| 18.93
|205.1
| 19.34
|179.5
| 16.79
|322.6
| 30.27
|343.6
| 32.40
|335.0
| 31.75
|312.1
| 29.71
|304.5
| 29.11
|291.7
| 27.98
|278.1
| 26.75
|275.5
| 26.58
|-
|-
|63
| 63
|104.2
| 10.42
|266.9
| 25.16
|261.4
| 24.46
|292.9
| 27.48
|266.1
| 25.09
|263.8
| 24.99
|258.8
| 24.64
|329.5
| 31.49
|341.4
| 32.75
|334.1
| 32.14
|321.5
| 31.01
|-
|-
|64
| 64
|309.4
| 30.94
|511.0
| 48.18
|499.4
| 46.71
|724.3
| 67.95
|677.1
| 63.84
|648.0
| 61.41
|604.6
| 57.55
|615.2
| 58.80
|587.2
| 56.33
|559.1
| 53.79
|575.7
| 55.53
|-
|-
|65
| 65
|16.0
| 1.60
|62.3
| 5.88
|328.9
| 30.76
|295.6
| 27.73
|298.0
| 28.10
|388.8
| 36.84
|362.3
| 34.49
|366.2
| 35.00
|354.4
| 33.99
|406.8
| 39.13
|388.5
| 37.48
|-
|-
|66
| 66
|277.5
| 27.75
|395.1
| 37.25
|408.3
| 38.19
|614.1
| 57.61
|612.2
| 57.72
|582.9
| 55.24
|546.6
| 52.04
|521.5
| 49.84
|526.0
| 50.46
|501.7
| 48.27
|509.2
| 49.12
|-
|-
|67
| 67
|136.1
| 13.61
|390.9
| 36.85
|359.3
| 33.61
|409.0
| 38.37
|400.1
| 37.73
|370.6
| 35.12
|363.3
| 34.59
|344.6
| 32.93
|359.0
| 34.44
|374.9
| 36.06
|361.6
| 34.88
|-
|-
|68
| 68
|157.4
| 15.74
|136.3
| 12.85
|119.2
| 11.15
|229.5
| 21.53
|312.6
| 29.47
|319.3
| 30.25
|311.3
| 29.63
|297.1
| 28.40
|280.9
| 26.95
|276.5
| 26.60
|279.9
| 27.00
|-
|-
|69
| 69
|256.3
| 25.63
|223.1
| 21.03
|372.1
| 34.80
|515.5
| 48.35
|545.5
| 51.43
|514.6
| 48.76
|523.0
| 49.79
|497.2
| 47.53
|471.3
| 45.20
|450.2
| 43.31
|430.2
| 41.50
|-
|-
|70
| 70
|37.2
| 3.72
|311.9
| 29.41
|340.6
| 31.86
|316.2
| 29.66
|357.8
| 33.73
|344.0
| 32.60
|372.4
| 35.45
|376.8
| 36.02
|419.5
| 40.23
|440.3
| 42.36
|429.4
| 41.42
|-
|-
|71
| 71
|330.7
| 33.07
|293.5
| 27.67
|429.1
| 40.14
|646.1
| 60.61
|591.7
| 55.78
|545.0
| 51.65
|534.8
| 50.91
|557.4
| 53.28
|530.8
| 50.91
|507.2
| 48.80
|514.5
| 49.64
|-
|-
|72
| 72
|82.9
| 8.29
|111.3
| 10.49
|99.2
| 9.28
|117.5
| 11.02
|110.7
| 10.44
|190.7
| 18.07
|188.5
| 17.94
|184.7
| 17.66
|232.0
| 22.25
|223.5
| 21.50
|286.8
| 27.66
|-
|-
|73
| 73
|210.5
| 21.05
|307.6
| 29.00
|299.5
| 28.01
|352.1
| 33.03
|333.7
| 31.46
|387.4
| 36.71
|448.4
| 42.68
|534.3
| 51.07
|527.4
| 50.59
|505.4
| 48.62
|515.0
| 49.68
|-
|-
|74
| 74
|203.1
| 20.31
|287.1
| 27.07
|315.3
| 29.50
|463.5
| 43.48
|423.8
| 39.96
|409.8
| 38.83
|383.2
| 36.48
|410.8
| 39.27
|404.2
| 38.77
|386.0
| 37.14
|397.5
| 38.34
|-
|-
|75
| 75
|90.4
| 9.04
|166.9
| 15.74
|331.0
| 30.96
|308.4
| 28.93
|434.8
| 40.99
|458.8
| 43.48
|431.0
| 41.03
|437.1
| 41.78
|431.5
| 41.39
|447.6
| 43.06
|466.0
| 44.95
|-
|-
|76
| 76
|323.3
| 32.33
|326.6
| 30.79
|287.1
| 26.85
|440.1
| 41.29
|495.1
| 46.68
|460.3
| 43.62
|516.9
| 49.21
|600.9
| 57.43
|606.7
| 58.19
|620.2
| 59.67
|630.6
| 60.83
|-
|-
|77
| 77
|29.8
| 2.98
|166.5
| 15.70
|205.9
| 19.26
|190.6
| 17.88
|265.5
| 25.03
|255.6
| 24.22
|262.5
| 24.99
|281.6
| 26.92
|269.6
| 25.86
|270.3
| 26.01
|283.8
| 27.38
|-
|-
|78
| 78
|263.7
| 26.37
|310.2
| 29.24
|273.7
| 25.60
|473.9
| 44.46
|431.1
| 40.65
|406.5
| 38.52
|378.9
| 36.07
|356.6
| 34.09
|418.6
| 40.15
|409.9
| 39.44
|391.5
| 37.77
|-
|-
|79
| 79
|149.9
| 14.99
|264.8
| 24.96
|366.1
| 34.25
|378.6
| 35.51
|350.9
| 33.09
|331.9
| 31.45
|375.2
| 35.72
|389.2
| 37.20
|464.2
| 44.52
|446.9
| 43.00
|435.6
| 42.02
|-
|-
|80
| 80
|143.5
| 14.35
|160.8
| 15.16
|221.6
| 20.73
|228.8
| 21.46
|207.8
| 19.59
|243.2
| 23.04
|254.8
| 24.26
|263.6
| 25.19
|250.1
| 23.99
|293.0
| 28.19
|284.7
| 27.47
|-
|-
|81
| 81
|270.1
| 27.01
|247.6
| 23.34
|268.4
| 25.10
|409.2
| 38.39
|376.7
| 35.52
|457.1
| 43.31
|440.9
| 41.97
|421.4
| 40.27
|404.0
| 38.75
|450.8
| 43.37
|434.1
| 41.87
|-
|-
|82
| 82
|23.4
| 2.34
|293.9
| 27.71
|262.0
| 24.51
|265.3
| 24.88
|339.2
| 31.98
|377.8
| 35.80
|377.3
| 35.92
|355.1
| 33.94
|347.4
| 33.32
|330.4
| 31.78
|327.6
| 31.60
|-
|-
|83
| 83
|346.9
| 31.69
|277.2
| 26.14
|293.3
| 27.44
|505.3
| 47.40
|524.9
| 49.48
|523.3
| 49.59
|553.8
| 52.72
|573.2
| 54.79
|550.6
| 52.81
|531.2
| 51.10
|580.9
| 56.03
|-
|-
|84
| 84
|96.8
| 9.68
|85.9
| 8.10
|175.1
| 16.38
|227.6
| 21.35
|330.7
| 31.18
|313.3
| 29.69
|311.4
| 29.65
|337.4
| 32.25
|330.9
| 31.74
|315.3
| 30.33
|301.2
| 29.05
|-
|-
|85
| 85
|196.7
| 19.67
|394.4
| 37.18
|433.9
| 40.58
|484.7
| 45.47
|457.2
| 43.11
|455.1
| 43.13
|443.4
| 42.21
|496.2
| 47.43
|474.1
| 45.48
|514.4
| 49.49
|514.5
| 49.63
|-
|-
|86
| 86
|216.9
| 21.69
|380.3
| 35.85
|437.3
| 40.91
|495.7
| 46.50
|595.3
| 56.12
|553.6
| 52.46
|518.9
| 49.39
|555.9
| 53.13
|540.8
| 51.87
|588.3
| 56.60
|563.1
| 54.32
|-
|-
|87
| 87
|76.6
| 7.66
|79.4
| 7.49
|191.9
| 17.95
|228.6
| 21.45
|207.8
| 19.59
|192.8
| 18.27
|183.4
| 17.46
|239.2
| 22.87
|275.9
| 26.46
|281.2
| 27.06
|306.2
| 29.54
|-
|-
|88
| 88
|337.1
| 33.71
|299.0
| 28.19
|298.6
| 27.93
|519.6
| 48.75
|473.5
| 44.65
|580.7
| 55.03
|602.0
| 57.31
|638.5
| 61.03
|635.1
| 60.92
|617.3
| 59.39
|593.6
| 57.26
|-
|-
|89
| 89
|43.6
| 4.36
|270.8
| 25.53
|238.6
| 22.32
|254.5
| 23.87
|232.7
| 21.94
|302.6
| 28.68
|312.6
| 29.76
|304.0
| 29.05
|293.1
| 28.11
|278.9
| 26.83
|301.4
| 29.08
|-
|-
|90
| 90
|249.9
| 24.99
|241.1
| 22.73
|251.9
| 23.56
|389.3
| 36.52
|500.6
| 47.19
|475.0
| 45.01
|456.7
| 43.48
|429.5
| 41.05
|459.2
| 44.05
|498.2
| 47.93
|490.6
| 47.32
|-
|-
|91
| 91
|163.8
| 16.38
|190.4
| 17.95
|254.1
| 23.77
|260.8
| 24.47
|359.5
| 33.89
|412.6
| 39.10
|427.8
| 40.72
|416.0
| 39.76
|475.6
| 45.62
|517.4
| 49.78
|536.4
| 51.74
|-
|-
|92
| 92
|129.7
| 12.97
|234.4
| 22.10
|367.0
| 34.33
|372.4
| 34.93
|405.1
| 38.20
|457.0
| 43.31
|510.4
| 48.59
|482.0
| 46.07
|459.8
| 44.10
|442.2
| 42.54
|431.5
| 41.62
|-
|-
|93
| 93
|283.9
| 28.39
|307.6
| 29.00
|269.2
| 25.18
|484.0
| 45.40
|526.5
| 49.64
|485.1
| 45.97
|460.7
| 43.85
|433.1
| 41.39
|412.7
| 39.58
|415.3
| 39.96
|445.5
| 42.98
|-
|-
|94
| 94
|9.6
| 0.96
|189.7
| 17.88
|207.6
| 19.42
|188.8
| 17.71
|194.5
| 18.34
|209.7
| 19.87
|225.0
| 21.42
|226.1
| 21.61
|238.2
| 22.84
|243.7
| 23.45
|239.8
| 23.13
|-
|-
|95
| 95
|303.0
| 30.30
|299.0
| 28.19
|261.3
| 24.44
|415.5
| 38.98
|378.3
| 35.67
|350.1
| 33.18
|392.0
| 37.32
|512.4
| 48.98
|485.6
| 46.57
|484.7
| 46.63
|466.8
| 45.03
|-
|-
|96
| 96
|110.6
| 11.06
|155.4
| 14.65
|363.1
| 33.96
|411.4
| 38.60
|380.6
| 35.89
|373.2
| 35.36
|347.7
| 33.10
|367.0
| 35.08
|367.7
| 35.27
|385.7
| 37.11
|381.8
| 36.83
|-
|-
|97
| 97
|182.9
| 18.29
|297.6
| 28.06
|335.4
| 31.37
|463.8
| 43.50
|473.8
| 44.67
|437.4
| 41.44
|408.8
| 38.92
|468.8
| 44.81
|444.6
| 42.64
|423.9
| 40.78
|412.7
| 39.81
|-
|-
|98
| 98
|230.7
| 23.07
|478.1
| 45.07
|430.6
| 40.28
|550.3
| 51.63
|503.3
| 47.45
|524.4
| 49.69
|495.3
| 47.15
|510.5
| 48.80
|503.4
| 48.28
|513.5
| 49.40
|501.8
| 48.41
|-
|-
|99
| 99
|62.7
| 6.27
|80.9
| 7.63
|70.6
| 6.60
|89.3
| 8.38
|332.7
| 31.37
|354.4
| 33.58
|355.5
| 33.84
|373.2
| 35.67
|401.5
| 38.51
|383.7
| 36.92
|368.4
| 35.54
|}
|}

Revision as of 12:57, 17 May 2020

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 log2r) - n log2r, 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 r3 = r1r2 for n, then error (n, r3) = error (n, r1) + error (n, r2).

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 n3 = n1 + n2 for r, then error (n3, r) = error (n1, r) + error (n2, 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

Edo Relative Errors (Percent)
2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1
1 0.00 41.50 -32.19 19.26 -45.94 29.96 -8.75 -24.79 47.64
2 0.00 -16.99 35.61 38.53 8.11 -40.09 -17.49 -49.59 -4.71
3 0.00 24.51 3.42 -42.21 -37.83 -10.13 -26.24 25.62 42.93
4 0.00 -33.99 -28.77 -22.94 16.23 19.82 -34.99 0.83 -9.42
5 0.00 7.52 39.04 -3.68 -29.72 49.78 -43.73 -23.96 38.22
6 0.00 49.02 6.84 15.59 24.34 -20.26 47.52 -48.76 -14.14
7 0.00 -9.47 -25.35 34.85 -21.60 9.69 38.78 26.45 33.51
8 0.00 32.03 42.46 -45.88 32.45 39.65 30.03 1.66 -18.85
9 0.00 -26.47 10.26 -26.62 -13.49 -30.40 21.28 -23.13 28.79
10 0.00 15.04 -21.93 -7.35 40.57 -0.44 12.54 -47.93 -23.56
11 0.00 -43.46 45.88 11.91 -5.37 29.52 3.79 27.28 24.08
12 0.00 -1.96 13.69 31.17 48.68 -40.53 -4.96 2.49 -28.27
13 0.00 39.55 -18.51 -49.56 2.74 -10.57 -13.70 -22.31 19.37
14 0.00 -18.95 49.30 -30.30 -43.20 19.38 -22.45 -47.10 -32.99
15 0.00 22.56 17.11 -11.03 10.85 49.34 -31.19 28.11 14.66
16 0.00 -35.94 -15.08 8.23 -35.09 -20.70 -39.94 3.32 -37.70
17 0.00 5.56 -47.28 27.50 18.97 9.25 -48.69 -21.48 9.94
18 0.00 47.07 20.53 46.76 -26.98 39.21 42.57 -46.27 -42.41
19 0.00 -11.43 -11.66 -33.97 27.08 -30.84 33.82 28.94 5.23
20 0.00 30.07 -43.86 -14.71 -18.86 -0.88 25.07 4.14 -47.12
21 0.00 -28.42 23.95 4.55 35.19 29.08 16.33 -20.65 0.52
22 0.00 13.08 -8.24 23.82 -10.75 -40.97 7.58 -45.44 48.16
23 0.00 -45.41 -40.43 43.08 43.31 -11.01 -1.16 29.77 -4.19
24 0.00 -3.91 27.37 -37.65 -2.64 18.94 -9.91 4.97 43.45
25 0.00 37.59 -4.82 -18.39 -48.58 48.90 -18.66 -19.82 -8.90
26 0.00 -20.90 -37.01 0.88 5.48 -21.14 -27.40 -44.61 38.74
27 0.00 20.60 30.79 20.14 -40.47 8.81 -36.15 30.60 -13.62
28 0.00 -37.90 -1.40 39.41 13.59 38.77 -44.90 5.80 34.03
29 0.00 3.61 -33.59 -41.33 -32.35 -31.28 46.36 -18.99 -18.33
30 0.00 45.11 34.22 -22.06 21.71 -1.32 37.61 -43.78 29.31
31 0.00 -13.38 2.02 -2.80 -24.24 28.64 28.87 31.42 -23.04
32 0.00 28.12 -30.17 16.46 29.82 -41.41 20.12 6.63 24.60
33 0.00 -30.38 37.64 35.73 -16.12 -11.45 11.37 -18.16 -27.75
34 0.00 11.13 5.44 -45.01 37.93 18.50 2.63 -42.95 19.89
35 0.00 -47.37 -26.75 -25.74 -8.01 48.46 -6.12 32.25 -32.47
36 0.00 -5.87 41.06 -6.48 46.05 -21.58 -14.87 7.46 15.18
37 0.00 35.64 8.87 12.79 0.10 8.37 -23.61 -17.33 -37.18
38 0.00 -22.86 -23.33 32.05 -45.84 38.33 -32.36 -42.12 10.46
39 0.00 18.65 44.48 -48.68 8.22 -31.71 -41.11 33.08 -41.89
40 0.00 -39.85 12.29 -29.42 -37.73 -1.76 -49.85 8.29 5.75
41 0.00 1.65 -19.91 -10.16 16.33 28.20 41.40 -16.50 -46.60
42 0.00 43.16 47.90 9.11 -29.61 -41.85 32.66 -41.30 1.04
43 0.00 -15.34 15.71 28.37 24.44 -11.89 23.91 33.91 48.68
44 0.00 26.16 -16.48 47.64 -21.50 18.07 15.16 9.12 -3.67
45 0.00 -32.33 -48.68 -33.10 32.56 48.02 6.42 -15.67 43.97
46 0.00 9.17 19.13 -13.83 -13.39 -22.02 -2.33 -40.47 -8.38
47 0.00 -49.32 -13.06 5.43 40.67 7.93 -11.08 34.74 39.26
48 0.00 -7.82 -45.25 24.70 -5.27 37.89 -19.82 9.95 -13.10
49 0.00 33.68 22.55 43.96 48.79 -32.15 -28.57 -14.84 34.55
50 0.00 -24.81 -9.64 -36.77 2.84 -2.20 -37.31 -39.64 -17.81
51 0.00 16.69 -41.83 -17.51 -43.10 27.76 -46.06 35.57 29.83
52 0.00 -41.81 25.97 1.75 10.96 -42.29 45.19 10.78 -22.52
53 0.00 -0.30 -6.22 21.02 -34.99 -12.33 36.45 -14.02 25.12
54 0.00 41.20 -38.41 40.28 19.07 17.63 27.70 -38.81 -27.23
55 0.00 -17.29 29.40 -40.45 -26.87 47.58 18.95 36.40 20.41
56 0.00 24.21 -2.80 -21.19 27.18 -22.46 10.21 11.61 -31.95
57 0.00 -34.29 -34.99 -1.92 -18.76 7.49 1.46 -13.19 15.70
58 0.00 7.22 32.82 17.34 35.30 37.45 -7.28 -37.98 -36.66
59 0.00 48.72 0.62 36.61 -10.65 -32.59 -16.03 37.23 10.98
60 0.00 -9.78 -31.57 -44.13 43.41 -2.64 -24.78 12.43 -41.37
61 0.00 31.73 36.24 -24.87 -2.53 27.32 -33.52 -12.36 6.27
62 0.00 -26.77 4.05 -5.60 -48.48 -42.73 -42.27 -37.15 -46.08
63 0.00 14.74 -28.15 13.66 5.58 -12.77 48.98 38.06 1.56
64 0.00 -43.76 39.66 32.93 -40.36 17.19 40.24 13.26 49.20
65 0.00 -2.26 7.47 -47.81 13.69 47.14 31.49 -11.53 -3.15
66 0.00 39.25 -24.73 -28.54 -32.25 -22.90 22.75 -36.32 44.49
67 0.00 -19.25 43.08 -9.28 21.81 7.05 14.00 38.89 -7.87
68 0.00 22.25 10.89 9.99 -24.14 37.01 5.25 14.09 39.78
69 0.00 -36.24 -21.30 29.25 29.92 -33.03 -3.49 -10.70 -12.58
70 0.00 5.26 46.50 48.52 -16.02 -3.08 -12.24 -35.49 35.07
71 0.00 46.77 14.31 -32.22 38.04 26.88 -20.99 39.71 -17.29
72 0.00 -11.73 -17.88 -12.96 -7.91 -43.17 -29.73 14.92 30.35
73 0.00 29.77 49.92 6.31 46.15 -13.21 -38.48 -9.87 -22.00
74 0.00 -28.72 17.73 25.57 0.21 16.75 -47.23 -34.66 25.64
75 0.00 12.78 -14.46 44.84 -45.74 46.70 44.03 40.54 -26.71
76 0.00 -45.72 -46.65 -35.90 8.32 -23.34 35.28 15.75 20.93
77 0.00 -4.21 21.15 -16.63 -37.62 6.61 26.54 -9.04 -31.43
78 0.00 37.29 -11.04 2.63 16.43 36.57 17.79 -33.83 16.22
79 0.00 -21.20 -43.23 21.90 -29.51 -33.47 9.04 41.37 -36.14
80 0.00 20.30 24.58 41.16 24.55 -3.52 0.30 16.58 11.50
81 0.00 -38.20 -7.62 -39.57 -21.40 26.44 -8.45 -8.21 -40.85
82 0.00 3.31 -39.81 -20.31 32.66 -43.61 -17.20 -33.01 6.79
83 0.00 44.81 28.00 -1.05 -13.28 -13.65 -25.94 42.20 -45.56
84 0.00 -13.69 -4.20 18.22 40.77 16.31 -34.69 17.41 2.08
85 0.00 27.82 -36.39 37.48 -5.17 46.26 -43.43 -7.38 49.72
86 0.00 -30.68 31.42 -43.25 48.89 -23.78 47.82 -32.18 -2.63
87 0.00 10.83 -0.77 -23.99 2.94 6.17 39.07 43.03 45.01
88 0.00 -47.67 -32.97 -4.72 -43.00 36.13 30.33 18.24 -7.35
89 0.00 -6.17 34.84 14.54 11.06 -33.91 21.58 -6.55 40.30
90 0.00 35.34 2.65 33.81 -34.88 -3.96 12.83 -31.35 -12.06
91 0.00 -23.16 -29.55 -46.93 19.17 26.00 4.09 43.86 35.59
92 0.00 18.34 38.26 -27.67 -26.77 -44.05 -4.66 19.07 -16.77
93 0.00 -40.15 6.07 -8.40 27.29 -14.09 -13.40 -5.73 30.87
94 0.00 1.35 -26.12 10.86 -18.66 15.87 -22.15 -30.52 -21.48
95 0.00 42.86 41.68 30.13 35.40 45.82 -30.90 44.69 26.16
96 0.00 -15.64 9.49 49.39 -10.54 -24.22 -39.64 19.90 -26.19
97 0.00 25.86 -22.70 -31.34 43.51 5.73 -48.39 -4.90 21.45
98 0.00 -32.63 45.10 -12.08 -2.43 35.69 42.86 -29.69 -30.91

List of RMS Relative Errors of JI Subgroups for Small Edos

Notes:

  1. Patent val mapping is used throughout.
  2. 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.
  3. The weight of intervals is determined by the tonality diamond. For example, the 9-odd-limit takes account of 3/1 twice.
Edo 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 6.27 7.63 6.60 8.38 31.37 33.58 33.84 35.67 38.51 36.92 35.54
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