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 (%) +4 -3 +8 -16 +1 +46 +12 -7 -12 -8 +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)


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 -9 +0 -25 -9 +35 +0 -19 -25 -22 -9
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 (%) +11 +8 +22 +0 +19 -34 +33 +16 +11 +16 +30
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 +12 -11 +6 -48 +18 +0 -5 -1 +12
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 -18 +10 +22 -13 -12 +15 -37 +27 +4 -8
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 -26 +0 +10 -26 -27 +0 +47 +10 -13 -26
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 (%) +8 -7 +16 -33 +1 -8 +23 -13 -25 -16 +9
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 -19 +0 +49 -19 -30 +0 -38 +49 -43 -19
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 (%) +9 -22 +18 +6 -13 +33 +27 -44 +15 -4 -4
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 -36 +0 -15 -36 +8 +0 +28 -15 -35 -36
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 (%) +11 +5 +21 +13 +16 -4 +32 +11 +24 -36 +27
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 -11 +0 -12 -11 -34 +0 -23 -12 +27 -11
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 -1 +37 -20 +17 -12 -45 -2 -1 +48 +36
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 -30 +0 +38 -30 +36 +0 +39 +38 -16 -30
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 +11 +43 +26 +32 -8 -36 +22 +47 +28 -47
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 -23 +0 -23 -23 +32 +0 -46 -23 -46 -23
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 +8 -50 +10 +33 +38 -24 +15 +35 +20 -42
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 -32 +0 -49 -32 -33 +0 +35 -49 +33 -32
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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