BudjarnLambeth/Ed257/128

From Xenharmonic Wiki
Jump to navigation Jump to search
This user page is editable by any wiki editor.

As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages.

However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits).


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.

Ed257/128s really only make sense for that purpose with 65 or fewer tones per pseudo-octave. With more tones than that, the relative error on 2/1 becomes unacceptably high and it makes more sense to switch to a different tuning like a zpi or ed513/256.

Ed257/128s are the complementary opposite of ed255/128s.

7ed257/128

Harmonics

Approximation of prime harmonics in 7ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 -5.6 -28.0 +79.0 -13.9 +41.7 -78.0 +74.3 -84.1 +31.8 -83.7
Relative (%) +3.9 -3.3 -16.3 +45.8 -8.1 +24.2 -45.2 +43.1 -48.8 +18.4 -48.5
Steps
(reduced) 		7
(0) 		11
(4) 		16
(2) 		20
(6) 		24
(3) 		26
(5) 		28
(0) 		30
(2) 		31
(3) 		34
(6) 		34
(6)


7edo, 16ed5, 22ed9 for comparison:

Approximation of prime harmonics in 7edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -16.2 -43.5 +59.7 -37.0 +16.6 +66.5 +45.3 +57.4 -1.0 +55.0
Relative (%) +0.0 -9.5 -25.3 +34.9 -21.6 +9.7 +38.8 +26.5 +33.5 -0.6 +32.1
Steps
(reduced) 		7
(0) 		11
(4) 		16
(2) 		20
(6) 		24
(3) 		26
(5) 		29
(1) 		30
(2) 		32
(4) 		34
(6) 		35
(0)
Approximation of prime harmonics in 16ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +19.0 +13.6 +0.0 -60.1 +28.2 -86.9 -28.9 -47.3 -29.8 -82.8 -24.1
Relative (%) +10.9 +7.8 +0.0 -34.5 +16.2 -49.9 -16.6 -27.2 -17.1 -47.5 -13.8
Steps
(reduced) 		7
(7) 		11
(11) 		16
(0) 		19
(3) 		24
(8) 		25
(9) 		28
(12) 		29
(13) 		31
(15) 		33
(1) 		34
(2)
Approximation of prime harmonics in 22ed9
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +10.3 +0.0 -19.8 -83.6 -1.6 +55.0 -63.6 -83.3 -68.2 +49.2 -66.3
Relative (%) +6.0 +0.0 -11.5 -48.4 -0.9 +31.8 -36.8 -48.2 -39.5 +28.5 -38.3
Steps
(reduced) 		7
(7) 		11
(11) 		16
(16) 		19
(19) 		24
(2) 		26
(4) 		28
(6) 		29
(7) 		31
(9) 		34
(12) 		34
(12)

Intervals

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


9ed257/128

Harmonics

Approximation of prime harmonics in 9ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 -24.8 +29.4 -16.7 +5.3 -15.8 +56.1 -2.3 -64.9 -64.0 -45.4
Relative (%) +5.0 -18.5 +22.0 -12.5 +3.9 -11.8 +41.9 -1.8 -48.4 -47.7 -33.8
Steps
(reduced) 		9
(0) 		14
(5) 		21
(3) 		25
(7) 		31
(4) 		33
(6) 		37
(1) 		38
(2) 		40
(4) 		43
(7) 		44
(8)


9edo for comparison:

Approximation of prime harmonics in 9edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -35.3 +13.7 -35.5 -18.0 -40.5 +28.4 -30.8 +38.4 +37.1 +55.0
Relative (%) +0.0 -26.5 +10.3 -26.6 -13.5 -30.4 +21.3 -23.1 +28.8 +27.8 +41.2
Steps
(reduced) 		9
(0) 		14
(5) 		21
(3) 		25
(7) 		31
(4) 		33
(6) 		37
(1) 		38
(2) 		41
(5) 		44
(8) 		45
(0)

Intervals

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


14ed257/128

Harmonics

Approximation of prime harmonics in 14ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 -5.6 -28.0 -7.2 -13.9 +41.7 +8.2 -11.9 +2.1 +31.8 +2.5
Relative (%) +7.8 -6.5 -32.5 -8.3 -16.1 +48.4 +9.6 -13.8 +2.4 +36.9 +2.9
Steps
(reduced) 		14
(0) 		22
(8) 		32
(4) 		39
(11) 		48
(6) 		52
(10) 		57
(1) 		59
(3) 		63
(7) 		68
(12) 		69
(13)


14edo for comparison:

Approximation of prime harmonics in 14edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -16.2 +42.3 -26.0 -37.0 +16.6 -19.2 -40.4 -28.3 -1.0 -30.7
Relative (%) +0.0 -18.9 +49.3 -30.3 -43.2 +19.4 -22.4 -47.1 -33.0 -1.2 -35.9
Steps
(reduced) 		14
(0) 		22
(8) 		33
(5) 		39
(11) 		48
(6) 		52
(10) 		57
(1) 		59
(3) 		63
(7) 		68
(12) 		69
(13)

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 prime harmonics in 16ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 -16.4 +4.3 +25.2 -3.1 +9.4 -2.5 +31.2 +2.1 -22.1 +13.3
Relative (%) +8.9 -21.8 +5.7 +33.4 -4.1 +12.4 -3.4 +41.3 +2.8 -29.3 +17.6
Steps
(reduced) 		16
(0) 		25
(9) 		37
(5) 		45
(13) 		55
(7) 		59
(11) 		65
(1) 		68
(4) 		72
(8) 		77
(13) 		79
(15)


16edo for comparison:

Approximation of prime harmonics in 16edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -27.0 -11.3 +6.2 -26.3 -15.5 -30.0 +2.5 -28.3 +20.4 -20.0
Relative (%) +0.0 -35.9 -15.1 +8.2 -35.1 -20.7 -39.9 +3.3 -37.7 +27.2 -26.7
Steps
(reduced) 		16
(0) 		25
(9) 		37
(5) 		45
(13) 		55
(7) 		59
(11) 		65
(1) 		68
(4) 		72
(8) 		78
(14) 		79
(15)

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 prime harmonics in 19ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 +3.4 +8.3 -2.6 -23.0 +5.4 -14.4 -16.5 -29.7 +13.6 +25.2
Relative (%) +10.6 +5.4 +13.0 -4.1 -36.2 +8.5 -22.7 -25.9 -46.7 +21.5 +39.7
Steps
(reduced) 		19
(0) 		30
(11) 		44
(6) 		53
(15) 		65
(8) 		70
(13) 		77
(1) 		80
(4) 		85
(9) 		92
(16) 		94
(18)


19edo for comparison:

Approximation of prime harmonics in 19edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 -21.5 +17.1 -19.5 +21.4 +18.3 +3.3 -19.1 -8.2
Relative (%) +0.0 -11.4 -11.7 -34.0 +27.1 -30.8 +33.8 +28.9 +5.2 -30.2 -13.0
Steps
(reduced) 		19
(0) 		30
(11) 		44
(6) 		53
(15) 		66
(9) 		70
(13) 		78
(2) 		81
(5) 		86
(10) 		92
(16) 		94
(18)

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

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

Harmonics

Approximation of prime harmonics in 33ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 -0.4 -7.1 -4.6 +17.5 -15.8 -4.8 -14.5 -16.2 -15.2 +15.6
Relative (%) +18.5 -1.1 -19.5 -12.5 +47.7 -43.2 -13.2 -39.8 -44.3 -41.7 +42.6
Steps
(reduced) 		33
(0) 		52
(19) 		76
(10) 		92
(26) 		114
(15) 		121
(22) 		134
(2) 		139
(7) 		148
(16) 		159
(27) 		163
(31)


33edo for comparison:

Approximation of prime harmonics in 33edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -11.0 +13.7 +13.0 -5.9 -4.2 +4.1 -6.6 -10.1 -11.4 -17.8
Relative (%) +0.0 -30.4 +37.6 +35.7 -16.1 -11.5 +11.4 -18.2 -27.8 -31.3 -48.8
Steps
(reduced) 		33
(0) 		52
(19) 		77
(11) 		93
(27) 		114
(15) 		122
(23) 		135
(3) 		140
(8) 		149
(17) 		160
(28) 		163
(31)

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


38ed257/128

Harmonics

Approximation of prime harmonics in 38ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 +3.4 +8.3 -2.6 +8.8 +5.4 -14.4 +15.3 +2.1 +13.6 -6.6
Relative (%) +21.3 +10.8 +26.0 -8.3 +27.7 +17.0 -45.5 +48.2 +6.6 +42.9 -20.7
Steps
(reduced) 		38
(0) 		60
(22) 		88
(12) 		106
(30) 		131
(17) 		140
(26) 		154
(2) 		161
(9) 		171
(19) 		184
(32) 		187
(35)


38edo for comparison:

Approximation of prime harmonics in 38edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 +10.1 -14.5 +12.1 -10.2 -13.3 +3.3 +12.5 -8.2
Relative (%) +0.0 -22.9 -23.3 +32.1 -45.8 +38.3 -32.4 -42.1 +10.5 +39.7 -25.9
Steps
(reduced) 		38
(0) 		60
(22) 		88
(12) 		107
(31) 		131
(17) 		141
(27) 		155
(3) 		161
(9) 		172
(20) 		185
(33) 		188
(36)


42ed257/128

See 42ed257/128.


45ed257/128

Harmonics

Approximation of prime harmonics in 45ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 +2.0 +2.6 +10.1 +5.3 +11.0 +2.5 -2.3 -11.3 -10.4 +8.3
Relative (%) +25.2 +7.6 +9.8 +37.6 +19.6 +41.2 +9.3 -8.8 -42.2 -38.6 +30.8
Steps
(reduced) 		45
(0) 		71
(26) 		104
(14) 		126
(36) 		155
(20) 		166
(31) 		183
(3) 		190
(10) 		202
(22) 		217
(37) 		222
(42)


45edo for comparison:

Approximation of prime harmonics in 45edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -8.6 -13.0 -8.8 +8.7 +12.8 +1.7 -4.2 +11.7 +10.4 +1.6
Relative (%) +0.0 -32.3 -48.7 -33.1 +32.6 +48.0 +6.4 -15.7 +44.0 +39.1 +6.1
Steps
(reduced) 		45
(0) 		71
(26) 		104
(14) 		126
(36) 		156
(21) 		167
(32) 		184
(4) 		191
(11) 		204
(24) 		219
(39) 		223
(43)


54ed257/128

Harmonics

Approximation of prime harmonics in 54ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.75 -2.44 +7.09 +5.60 +5.26 +6.57 -10.92 -2.35 +2.10 +3.05 -0.68
Relative (%) +30.2 -10.9 +31.7 +25.1 +23.6 +29.4 -48.8 -10.5 +9.4 +13.6 -3.0
Steps
(reduced) 		54
(0) 		85
(31) 		125
(17) 		151
(43) 		186
(24) 		199
(37) 		219
(3) 		228
(12) 		243
(27) 		261
(45) 		266
(50)


45edo for comparison:

Approximation of prime harmonics in 54edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +9.16 -8.54 +8.95 +4.24 +3.92 +6.16 -8.62 -6.05 -7.35 +10.52
Relative (%) +0.0 +41.2 -38.4 +40.3 +19.1 +17.6 +27.7 -38.8 -27.2 -33.1 +47.3
Steps
(reduced) 		54
(0) 		86
(32) 		125
(17) 		152
(44) 		187
(25) 		200
(38) 		221
(5) 		229
(13) 		244
(28) 		262
(46) 		268
(52)


59ed257/128

Harmonics

Approximation of prime harmonics in 59ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.75 +0.21 -4.65 +5.98 +0.72 -2.14 +3.86 -4.62 -8.13 -0.36 +6.90
Relative (%) +33.0 +1.0 -22.8 +29.2 +3.5 -10.5 +18.9 -22.6 -39.7 -1.8 +33.7
Steps
(reduced) 		59
(0) 		93
(34) 		136
(18) 		165
(47) 		203
(26) 		217
(40) 		240
(4) 		249
(13) 		265
(29) 		285
(49) 		291
(55)


59edo for comparison:

Approximation of prime harmonics in 59edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +9.91 +0.13 +7.45 -2.17 -6.63 -3.26 +7.57 +2.23 +7.71 -6.05
Relative (%) +0.0 +48.7 +0.6 +36.6 -10.6 -32.6 -16.0 +37.2 +11.0 +37.9 -29.8
Steps
(reduced) 		59
(0) 		94
(35) 		137
(19) 		166
(48) 		204
(27) 		218
(41) 		241
(5) 		251
(15) 		267
(31) 		287
(51) 		292
(56)


64ed257/128

Harmonics

Approximation of prime harmonics in 64ed257/128
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.75 +2.45 +4.29 +6.30 -3.12 +9.36 -2.54 -6.54 +2.10 -3.24 -5.57
Relative (%) +35.8 +13.0 +22.8 +33.4 -16.5 +49.6 -13.4 -34.7 +11.1 -17.2 -29.5
Steps
(reduced) 		64
(0) 		101
(37) 		148
(20) 		179
(51) 		220
(28) 		236
(44) 		260
(4) 		270
(14) 		288
(32) 		309
(53) 		315
(59)


64edo for comparison:

Approximation of prime harmonics in 64edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -8.21 +7.44 +6.17 -7.57 +3.22 +7.54 +2.49 +9.23 +1.67 -1.29
Relative (%) +0.0 -43.8 +39.7 +32.9 -40.4 +17.2 +40.2 +13.3 +49.2 +8.9 -6.9
Steps
(reduced) 		64
(0) 		101
(37) 		149
(21) 		180
(52) 		221
(29) 		237
(45) 		262
(6) 		272
(16) 		290
(34) 		311
(55) 		317
(61)


Related concepts

Retrieved from "https://en.xen.wiki/index.php?title=BudjarnLambeth/Ed257/128&oldid=196428"