Ed257/128

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An equal division of reduced harmonic 257 (ed257/128) is an equal-step tuning in which the 4ve-reduced 257th harmonic (257/128) is justly tuned and is divided in a given number of equal steps. 257/128 is very close to the octave, 2/1, but it is slightly sharper. This makes it suitable as an alternative to edos whose consonances are too flat, such as 7edo.

7ed257/128

Harmonics

Approximation of harmonics in 7ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -5.6 +13.5 -28.0 +1.1 +79.0 +20.2 -11.3 -21.3 -13.9 +7.9
Relative (%) +3.9 -3.3 +7.8 -16.3 +0.6 +45.8 +11.7 -6.5 -12.3 -8.1 +4.6
Steps
(reduced) 		7
(0) 		11
(4) 		14
(0) 		16
(2) 		18
(4) 		20
(6) 		21
(0) 		22
(1) 		23
(2) 		24
(3) 		25
(4)


7edo, 16ed5, 22ed9 for comparison:

Approximation of harmonics in 7edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 -43.5 -16.2 +59.7 +0.0 -32.5 -43.5 -37.0 -16.2
Relative (%) +0.0 -9.5 +0.0 -25.3 -9.5 +34.9 +0.0 -18.9 -25.3 -21.6 -9.5
Steps
(reduced) 		7
(0) 		11
(4) 		14
(0) 		16
(2) 		18
(4) 		20
(6) 		21
(0) 		22
(1) 		23
(2) 		24
(3) 		25
(4)
Approximation of harmonics in 16ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +19.0 +13.6 +38.0 +0.0 +32.6 -60.1 +57.0 +27.3 +19.0 +28.2 +51.7
Relative (%) +10.9 +7.8 +21.8 +0.0 +18.7 -34.5 +32.8 +15.7 +10.9 +16.2 +29.7
Steps
(reduced) 		7
(7) 		11
(11) 		14
(14) 		16
(0) 		18
(2) 		19
(3) 		21
(5) 		22
(6) 		23
(7) 		24
(8) 		25
(9)
Approximation of harmonics in 22ed9
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(11) 		14
(14) 		16
(16) 		18
(18) 		19
(19) 		21
(21) 		22
(0) 		23
(1) 		24
(2) 		25
(3)

Intervals

  • 172.393
  • 344.786
  • 517.178
  • 689.571
  • 861.964
  • 1034.357
  • 1206.749


9ed257/128

Harmonics

Approximation of harmonics in 9ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -24.8 +13.5 +29.4 -18.0 -16.7 +20.2 -49.6 +36.2 +5.3 -11.3
Relative (%) +5.0 -18.5 +10.1 +22.0 -13.5 -12.5 +15.1 -37.0 +27.0 +3.9 -8.4
Steps
(reduced) 		9
(0) 		14
(5) 		18
(0) 		21
(3) 		23
(5) 		25
(7) 		27
(0) 		28
(1) 		30
(3) 		31
(4) 		32
(5)


9edo for comparison:

Approximation of harmonics in 9edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -35.3 +0.0 +13.7 -35.3 -35.5 +0.0 +62.8 +13.7 -18.0 -35.3
Relative (%) +0.0 -26.5 +0.0 +10.3 -26.5 -26.6 +0.0 +47.1 +10.3 -13.5 -26.5
Steps
(reduced) 		9
(0) 		14
(5) 		18
(0) 		21
(3) 		23
(5) 		25
(7) 		27
(0) 		29
(2) 		30
(3) 		31
(4) 		32
(5)

Intervals

  • 134.083
  • 268.167
  • 402.25
  • 536.333
  • 670.416
  • 804.5
  • 938.583
  • 1072.666
  • 1206.749


14ed257/128

Harmonics

Approximation of harmonics in 14ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -5.6 +13.5 -28.0 +1.1 -7.2 +20.2 -11.3 -21.3 -13.9 +7.9
Relative (%) +7.8 -6.5 +15.7 -32.5 +1.3 -8.3 +23.5 -13.1 -24.7 -16.1 +9.1
Steps
(reduced) 		14
(0) 		22
(8) 		28
(0) 		32
(4) 		36
(8) 		39
(11) 		42
(0) 		44
(2) 		46
(4) 		48
(6) 		50
(8)


14edo for comparison:

Approximation of harmonics in 14edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 +42.3 -16.2 -26.0 +0.0 -32.5 +42.3 -37.0 -16.2
Relative (%) +0.0 -18.9 +0.0 +49.3 -18.9 -30.3 +0.0 -37.9 +49.3 -43.2 -18.9
Steps
(reduced) 		14
(0) 		22
(8) 		28
(0) 		33
(5) 		36
(8) 		39
(11) 		42
(0) 		44
(2) 		47
(5) 		48
(6) 		50
(8)

Intervals

  • 86.196
  • 172.393
  • 258.589
  • 344.786
  • 430.982
  • 517.178
  • 603.375
  • 689.571
  • 775.768
  • 861.964
  • 948.16
  • 1034.357
  • 1120.553
  • 1206.749


16ed257/128

Harmonics

Approximation of harmonics in 16ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -16.4 +13.5 +4.3 -9.7 +25.2 +20.2 -32.8 +11.0 -3.1 -2.9
Relative (%) +8.9 -21.8 +17.9 +5.7 -12.8 +33.4 +26.8 -43.5 +14.6 -4.1 -3.9
Steps
(reduced) 		16
(0) 		25
(9) 		32
(0) 		37
(5) 		41
(9) 		45
(13) 		48
(0) 		50
(2) 		53
(5) 		55
(7) 		57
(9)


16edo for comparison:

Approximation of harmonics in 16edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -27.0 +0.0 -11.3 -27.0 +6.2 +0.0 +21.1 -11.3 -26.3 -27.0
Relative (%) +0.0 -35.9 +0.0 -15.1 -35.9 +8.2 +0.0 +28.1 -15.1 -35.1 -35.9
Steps
(reduced) 		16
(0) 		25
(9) 		32
(0) 		37
(5) 		41
(9) 		45
(13) 		48
(0) 		51
(3) 		53
(5) 		55
(7) 		57
(9)

Intervals

  • 75.422
  • 150.844
  • 226.266
  • 301.687
  • 377.109
  • 452.531
  • 527.953
  • 603.375
  • 678.797
  • 754.218
  • 829.64
  • 905.062
  • 980.484
  • 1055.906
  • 1131.328
  • 1206.749


19ed257/128

Harmonics

Approximation of harmonics in 19ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 +3.4 +13.5 +8.3 +10.2 -2.6 +20.2 +6.9 +15.0 -23.0 +16.9
Relative (%) +10.6 +5.4 +21.3 +13.0 +16.0 -4.1 +31.9 +10.8 +23.6 -36.2 +26.7
Steps
(reduced) 		19
(0) 		30
(11) 		38
(0) 		44
(6) 		49
(11) 		53
(15) 		57
(0) 		60
(3) 		63
(6) 		65
(8) 		68
(11)


19edo for comparison:

Approximation of harmonics in 19edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 -21.5 +0.0 -14.4 -7.4 +17.1 -7.2
Relative (%) +0.0 -11.4 +0.0 -11.7 -11.4 -34.0 +0.0 -22.9 -11.7 +27.1 -11.4
Steps
(reduced) 		19
(0) 		30
(11) 		38
(0) 		44
(6) 		49
(11) 		53
(15) 		57
(0) 		60
(3) 		63
(6) 		66
(9) 		68
(11)

Intervals

  • 63.513
  • 127.026
  • 190.539
  • 254.053
  • 317.566
  • 381.079
  • 444.592
  • 508.105
  • 571.618
  • 635.131
  • 698.644
  • 762.158
  • 825.671
  • 889.184
  • 952.697
  • 1016.21
  • 1079.723
  • 1143.236
  • 1206.749


33ed257/128

Harmonics

Approximation of harmonics in 33ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -0.4 +13.5 -7.1 +6.3 -4.6 -16.3 -0.8 -0.4 +17.5 +13.1
Relative (%) +18.5 -1.1 +36.9 -19.5 +17.3 -12.5 -44.6 -2.2 -1.0 +47.7 +35.8
Steps
(reduced) 		33
(0) 		52
(19) 		66
(0) 		76
(10) 		85
(19) 		92
(26) 		98
(32) 		104
(5) 		109
(10) 		114
(15) 		118
(19)


33edo for comparison:

Approximation of harmonics in 33edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -11.0 +0.0 +13.7 -11.0 +13.0 +0.0 +14.3 +13.7 -5.9 -11.0
Relative (%) +0.0 -30.4 +0.0 +37.6 -30.4 +35.7 +0.0 +39.2 +37.6 -16.1 -30.4
Steps
(reduced) 		33
(0) 		52
(19) 		66
(0) 		77
(11) 		85
(19) 		93
(27) 		99
(0) 		105
(6) 		110
(11) 		114
(15) 		118
(19)

Intervals

  • 36.568
  • 73.136
  • 109.704
  • 146.273
  • 182.841
  • 219.409
  • 255.977
  • 292.545
  • 329.113
  • 365.682
  • 402.25
  • 438.818
  • 475.386
  • 511.954
  • 548.522
  • 585.09
  • 621.659
  • 658.227
  • 694.795
  • 731.363
  • 767.931
  • 804.499
  • 841.067
  • 877.636
  • 914.204
  • 950.772
  • 987.34
  • 1023.908
  • 1060.476
  • 1097.045
  • 1133.613
  • 1170.181
  • 1206.749

Regular temperament properties

This is an excellent tuning for dreamtone temperament, much better than standard 33edo. It is almost exactly the TE tuning.


38ed257/128

Harmonics

Approximation of harmonics in 38ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 +3.4 +13.5 +8.3 +10.2 -2.6 -11.5 +6.9 +15.0 +8.8 -14.8
Relative (%) +21.3 +10.8 +42.5 +26.0 +32.1 -8.3 -36.2 +21.7 +47.3 +27.7 -46.7
Steps
(reduced) 		38
(0) 		60
(22) 		76
(0) 		88
(12) 		98
(22) 		106
(30) 		113
(37) 		120
(6) 		126
(12) 		131
(17) 		135
(21)


38edo for comparison:

Approximation of harmonics in 38edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 +10.1 +0.0 -14.4 -7.4 -14.5 -7.2
Relative (%) +0.0 -22.9 +0.0 -23.3 -22.9 +32.1 +0.0 -45.7 -23.3 -45.8 -22.9
Steps
(reduced) 		38
(0) 		60
(22) 		76
(0) 		88
(12) 		98
(22) 		107
(31) 		114
(0) 		120
(6) 		126
(12) 		131
(17) 		136
(22)


45ed257/128

Harmonics

Approximation of harmonics in 45ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 +2.0 -13.3 +2.6 +8.8 +10.1 -6.6 +4.1 +9.4 +5.3 -11.3
Relative (%) +25.2 +7.6 -49.7 +9.8 +32.7 +37.6 -24.5 +15.1 +34.9 +19.6 -42.1
Steps
(reduced) 		45
(0) 		71
(26) 		89
(44) 		104
(14) 		116
(26) 		126
(36) 		134
(44) 		142
(7) 		149
(14) 		155
(20) 		160
(25)


45edo for comparison:

Approximation of harmonics in 45edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -8.6 +0.0 -13.0 -8.6 -8.8 +0.0 +9.4 -13.0 +8.7 -8.6
Relative (%) +0.0 -32.3 +0.0 -48.7 -32.3 -33.1 +0.0 +35.3 -48.7 +32.6 -32.3
Steps
(reduced) 		45
(0) 		71
(26) 		90
(0) 		104
(14) 		116
(26) 		126
(36) 		135
(0) 		143
(8) 		149
(14) 		156
(21) 		161
(26)


Related concepts

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