Ed255/128

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An equal division of reduced harmonic 255 (ed255/128) is an equal-step tuning in which the octave-reduced 255th harmonic (255/128) is justly tuned and is divided in a given number of equal steps. 255/128 is very close to the octave, 2/1, but it is slightly flatter. This makes it suitable as an alternative to edos whose consonances are too sharp, such as 5edo.

5ed255/128

Harmonics

Approximation of harmonics in 5ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -7 +7 -14 +77 +0 -28 -20 +14 +71 -94 -6
relative (%) -3 +3 -6 +32 +0 -12 -9 +6 +30 -40 -3
Steps
(reduced) 		5
(0) 		8
(3) 		10
(0) 		12
(2) 		13
(3) 		14
(4) 		15
(0) 		16
(1) 		17
(2) 		17
(2) 		18
(3)


5edo, 8edt, 14ed7 for comparison:

Approximation of harmonics in 5edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0 +18 +0 +94 +18 -9 +0 +36 +94 -71 +18
relative (%) +0 +8 +0 +39 +8 -4 +0 +15 +39 -30 +8
Steps
(reduced) 		5
(0) 		8
(3) 		10
(0) 		12
(2) 		13
(3) 		14
(4) 		15
(0) 		16
(1) 		17
(2) 		17
(2) 		18
(3)
Approximation of harmonics in 8edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -11 +0 -23 +67 -11 -40 -34 +0 +55 -110 -23
relative (%) -5 +0 -9 +28 -5 -17 -14 +0 +23 -46 -9
Steps
(reduced) 		5
(5) 		8
(0) 		10
(2) 		12
(4) 		13
(5) 		14
(6) 		15
(7) 		16
(0) 		17
(1) 		17
(1) 		18
(2)
Approximation of harmonics in 14ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +3 +23 +6 +101 +26 +0 +9 +46 +104 -61 +29
relative (%) +1 +10 +3 +42 +11 +0 +4 +19 +43 -25 +12
Steps
(reduced) 		5
(5) 		8
(8) 		10
(10) 		12
(12) 		13
(13) 		14
(0) 		15
(1) 		16
(2) 		17
(3) 		17
(3) 		18
(4)

Intervals

  • 238.645
  • 477.29
  • 715.934
  • 954.579
  • 1193.224


6ed255/128

Harmonics

Approximation of harmonics in 6ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 +86.8 -13.6 -2.1 +80.0 +12.0 -20.3 -25.4 -8.9 +25.0 +73.2
relative (%) -3 +44 -7 -1 +40 +6 -10 -13 -4 +13 +37
Steps
(reduced) 		6
(0) 		10
(4) 		12
(0) 		14
(2) 		16
(4) 		17
(5) 		18
(0) 		19
(1) 		20
(2) 		21
(3) 		22
(4)


6edo for comparison:

Approximation of harmonics in 6edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +98.0 +0.0 +13.7 +98.0 +31.2 +0.0 -3.9 +13.7 +48.7 +98.0
relative (%) +0 +49 +0 +7 +49 +16 +0 -2 +7 +24 +49
Steps
(reduced) 		6
(0) 		10
(4) 		12
(0) 		14
(2) 		16
(4) 		17
(5) 		18
(0) 		19
(1) 		20
(2) 		21
(3) 		22
(4)

Intervals

  • 198.871
  • 397.741
  • 596.612
  • 795.483
  • 994.353
  • 1193.224


8ed255/128

Harmonics

Approximation of harmonics in 8ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 +37.0 -13.6 +47.6 +30.3 +61.7 -20.3 +74.1 +40.8 +25.0 +23.5
relative (%) -5 +25 -9 +32 +20 +41 -14 +50 +27 +17 +16
Steps
(reduced) 		8
(0) 		13
(5) 		16
(0) 		19
(3) 		21
(5) 		23
(7) 		24
(0) 		26
(2) 		27
(3) 		28
(4) 		29
(5)


8edo for comparison:

Approximation of harmonics in 8edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +48.0 +0.0 +63.7 +48.0 -68.8 +0.0 -53.9 +63.7 +48.7 +48.0
relative (%) +0 +32 +0 +42 +32 -46 +0 -36 +42 +32 +32
Steps
(reduced) 		8
(0) 		13
(5) 		16
(0) 		19
(3) 		21
(5) 		22
(6) 		24
(0) 		25
(1) 		27
(3) 		28
(4) 		29
(5)

Intervals

  • 149.153
  • 298.306
  • 447.459
  • 596.612
  • 745.765
  • 894.918
  • 1044.071
  • 1193.224


11ed255/128

Harmonics

Approximation of harmonics in 11ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 +50.6 -13.6 +34.0 +43.8 -6.1 -20.3 -7.3 +27.3 -29.3 +37.0
relative (%) -6 +47 -12 +31 +40 -6 -19 -7 +25 -27 +34
Steps
(reduced) 		11
(0) 		18
(7) 		22
(0) 		26
(4) 		29
(7) 		31
(9) 		33
(0) 		35
(2) 		37
(4) 		38
(5) 		40
(7)


11edo for comparison:

Approximation of harmonics in 11edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -47.4 +0.0 +50.0 -47.4 +13.0 +0.0 +14.3 +50.0 -5.9 -47.4
relative (%) +0 -43 +0 +46 -43 +12 +0 +13 +46 -5 -43
Steps
(reduced) 		11
(0) 		17
(6) 		22
(0) 		26
(4) 		28
(6) 		31
(9) 		33
(0) 		35
(2) 		37
(4) 		38
(5) 		39
(6)

Intervals

  • 108.475
  • 216.95
  • 325.425
  • 433.9
  • 542.375
  • 650.85
  • 759.324
  • 867.799
  • 976.274
  • 1084.749
  • 1193.224


15ed255/128

See also: 5- to 10-tone scales in 47zpi


Harmonics

Approximation of harmonics in 15ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 +7.2 -13.6 -2.1 +0.4 -27.8 -20.3 +14.4 -8.9 -14.8 -6.3
relative (%) -9 +9 -17 -3 +1 -35 -26 +18 -11 -19 -8
Steps
(reduced) 		15
(0) 		24
(9) 		30
(0) 		35
(5) 		39
(9) 		42
(12) 		45
(0) 		48
(3) 		50
(5) 		52
(7) 		54
(9)


15edo for comparison:

Approximation of harmonics in 15edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 +36.1 +13.7 +8.7 +18.0
relative (%) +0 +23 +0 +17 +23 -11 +0 +45 +17 +11 +23
Steps
(reduced) 		15
(0) 		24
(9) 		30
(0) 		35
(5) 		39
(9) 		42
(12) 		45
(0) 		48
(3) 		50
(5) 		52
(7) 		54
(9)

Intervals

  • 79.548
  • 159.097
  • 238.645
  • 318.193
  • 397.741
  • 477.29
  • 556.838
  • 636.386
  • 715.934
  • 795.483
  • 875.031
  • 954.579
  • 1034.128
  • 1113.676
  • 1193.224

17ed255/128

Harmonics

Approximation of harmonics in 17ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 -6.8 -13.6 +21.3 -13.6 +0.3 -20.3 -13.7 +14.5 -10.1 -20.4
relative (%) -10 -10 -19 +30 -19 +0 -29 -19 +21 -14 -29
Steps
(reduced) 		17
(0) 		27
(10) 		34
(0) 		40
(6) 		44
(10) 		48
(14) 		51
(0) 		54
(3) 		57
(6) 		59
(8) 		61
(10)


17edo for comparison:

Approximation of harmonics in 17edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +3.9 +0.0 -33.4 +3.9 +19.4 +0.0 +7.9 -33.4 +13.4 +3.9
relative (%) +0 +6 +0 -47 +6 +27 +0 +11 -47 +19 +6
Steps
(reduced) 		17
(0) 		27
(10) 		34
(0) 		39
(5) 		44
(10) 		48
(14) 		51
(0) 		54
(3) 		56
(5) 		59
(8) 		61
(10)

Intervals

  • 70.19
  • 140.379
  • 210.569
  • 280.759
  • 350.948
  • 421.138
  • 491.328
  • 561.517
  • 631.707
  • 701.897
  • 772.086
  • 842.276
  • 912.466
  • 982.655
  • 1052.845
  • 1123.034
  • 1193.224


18ed255/128

Harmonics

Approximation of harmonics in 18ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 +20.5 -13.6 -2.1 +13.7 +12.0 -20.3 -25.4 -8.9 +25.0 +6.9
relative (%) -10 +31 -20 -3 +21 +18 -31 -38 -13 +38 +10
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		63
(9) 		65
(11)


18edo for comparison:

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4
relative (%) +0 +47 +0 +21 +47 +47 +0 -6 +21 -27 +47
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11)

Intervals

  • 66.29
  • 132.58
  • 198.871
  • 265.161
  • 331.451
  • 397.741
  • 464.032
  • 530.322
  • 596.612
  • 662.902
  • 729.193
  • 795.483
  • 861.773
  • 928.063
  • 994.353
  • 1060.644
  • 1126.934
  • 1193.224


27ed255/128

Harmonics

Approximation of harmonics in 27ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 -1.6 -13.6 -2.1 -8.4 -10.1 -20.3 -3.3 -8.9 +2.9 -15.2
relative (%) -15 -4 -31 -5 -19 -23 -46 -7 -20 +6 -34
Steps
(reduced) 		27
(0) 		43
(16) 		54
(0) 		63
(9) 		70
(16) 		76
(22) 		81
(0) 		86
(5) 		90
(9) 		94
(13) 		97
(16)


27edo for comparison:

Approximation of harmonics in 27edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +9.2 +0.0 +13.7 +9.2 +9.0 +0.0 +18.3 +13.7 -18.0 +9.2
relative (%) +0 +21 +0 +31 +21 +20 +0 +41 +31 -40 +21
Steps
(reduced) 		27
(0) 		43
(16) 		54
(0) 		63
(9) 		70
(16) 		76
(22) 		81
(0) 		86
(5) 		90
(9) 		93
(12) 		97
(16)

Intervals

  • 44.193
  • 88.387
  • 132.58
  • 176.774
  • 220.967
  • 265.161
  • 309.354
  • 353.548
  • 397.741
  • 441.935
  • 486.128
  • 530.322
  • 574.515
  • 618.709
  • 662.902
  • 707.096
  • 751.289
  • 795.483
  • 839.676
  • 883.87
  • 928.063
  • 972.257
  • 1016.45
  • 1060.644
  • 1104.837
  • 1149.031
  • 1193.224


39ed255/128

Harmonics

Approximation of harmonics in 39ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 -5.0 -13.6 -2.1 -11.8 -3.3 +10.3 -10.1 -8.9 +9.7 +12.0
relative (%) -22 -16 -44 -7 -39 -11 +34 -33 -29 +32 +39
Steps
(reduced) 		39
(0) 		62
(23) 		78
(0) 		91
(13) 		101
(23) 		110
(32) 		118
(1) 		124
(7) 		130
(13) 		136
(19) 		141
(24)


19edo for comparison:

Approximation of harmonics in 39edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +5.7 +0.0 +13.7 +5.7 -15.0 +0.0 +11.5 +13.7 +2.5 +5.7
relative (%) +0 +19 +0 +44 +19 -49 +0 +37 +44 +8 +19
Steps
(reduced) 		39
(0) 		62
(23) 		78
(0) 		91
(13) 		101
(23) 		109
(31) 		117
(0) 		124
(7) 		130
(13) 		135
(18) 		140
(23)


49ed255/128

Harmonics

Approximation of harmonics in 49ed255/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) -6.8 -2.5 +10.8 -10.2 -9.3 -8.3 +4.0 -5.1 +7.3 -11.6 +8.3
relative (%) -28 -10 +44 -42 -38 -34 +17 -21 +30 -47 +34
Steps
(reduced) 		49
(0) 		78
(29) 		99
(1) 		114
(16) 		127
(29) 		138
(40) 		148
(1) 		156
(9) 		164
(17) 		170
(23) 		177
(30)


19edo for comparison:

Approximation of harmonics in 49edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 +8.2 +0.0 +5.5 +8.2 +10.8 +0.0 -8.0 +5.5 +11.9 +8.2
relative (%) +0 +34 +0 +23 +34 +44 +0 -33 +23 +49 +34
Steps
(reduced) 		49
(0) 		78
(29) 		98
(0) 		114
(16) 		127
(29) 		138
(40) 		147
(0) 		155
(8) 		163
(16) 		170
(23) 		176
(29)


Related concepts

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