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 (%) -2.8 +3.0 -5.7 +32.4 +0.2 -11.6 -8.5 +6.0 +29.6 -39.5 -2.7
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 (%) +0.0 +7.5 +0.0 +39.0 +7.5 -3.7 +0.0 +15.0 +39.0 -29.7 +7.5
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 (%) -4.7 +0.0 -9.5 +28.0 -4.7 -17.0 -14.2 +0.0 +23.3 -46.1 -9.5
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 (%) +1.3 +9.6 +2.6 +42.1 +10.9 +0.0 +3.9 +19.2 +43.4 -25.2 +12.2
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 (%) -3.4 +43.6 -6.8 -1.1 +40.2 +6.0 -10.2 -12.8 -4.5 +12.6 +36.8
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 (%) +0.0 +49.0 +0.0 +6.8 +49.0 +15.6 +0.0 -2.0 +6.8 +24.3 +49.0
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 (%) -4.5 +24.8 -9.1 +31.9 +20.3 +41.4 -13.6 +49.7 +27.4 +16.7 +15.7
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 (%) +0.0 +32.0 +0.0 +42.5 +32.0 -45.9 +0.0 -35.9 +42.5 +32.5 +32.0
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 (%) -6.2 +46.6 -12.5 +31.4 +40.4 -5.6 -18.7 -6.7 +25.1 -27.0 +34.1
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 (%) +0.0 -43.5 +0.0 +45.9 -43.5 +11.9 +0.0 +13.1 +45.9 -5.4 -43.5
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 (%) -8.5 +9.1 -17.0 -2.7 +0.5 -34.9 -25.6 +18.1 -11.2 -18.6 -8.0
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 (%) +0.0 +22.6 +0.0 +17.1 +22.6 -11.0 +0.0 +45.1 +17.1 +10.9 +22.6
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 (%) -9.7 -9.7 -19.3 +30.3 -19.4 +0.4 -29.0 -19.5 +20.7 -14.4 -29.0
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 (%) +0.0 +5.6 +0.0 -47.3 +5.6 +27.5 +0.0 +11.1 -47.3 +19.0 +5.6
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 (%) -10.2 +30.9 -20.4 -3.2 +20.6 +18.1 -30.7 -38.3 -13.4 +37.7 +10.4
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 (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1
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 (%) -15.3 -3.7 -30.7 -4.8 -19.0 -22.9 -46.0 -7.4 -20.1 +6.5 -34.4
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 (%) +0.0 +20.6 +0.0 +30.8 +20.6 +20.1 +0.0 +41.2 +30.8 -40.5 +20.6
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 (%) -22.1 -16.5 -44.3 -6.9 -38.6 -10.9 +33.6 -32.9 -29.1 +31.6 +39.3
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 (%) +0.0 +18.6 +0.0 +44.5 +18.6 -48.7 +0.0 +37.3 +44.5 +8.2 +18.6
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 (%) -27.8 -10.4 +44.3 -42.1 -38.2 -34.2 +16.5 -20.8 +30.1 -47.5 +33.9
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 (%) +0.0 +33.7 +0.0 +22.6 +33.7 +44.0 +0.0 -32.6 +22.6 +48.8 +33.7
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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