BudjarnLambeth/Ed257/128: Difference between revisions

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An '''equal division of reduced harmonic 257''' ('''ed257/128''') is an [[equal-step tuning]] in which the octave-reduced 257th harmonic ([[257/128]]) is [[Just intonation|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 sharp, such as [[7edo]].
{{Editable user page}}


== 7ed255/128 ==
An '''equal division of reduced harmonic 257''' ('''ed257/128''') is an [[equal-step tuning]] in which the [[4/1|4ve]]-reduced 257th harmonic ([[257/128]]) is [[Just intonation|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 [[ed255/128]]s.
 
== 7ed257/128 ==
=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|7|257|128|intervals=integer}}
{{Harmonics in equal|7|257|128|intervals=prime}}




7edo, [[16ed5]], [[22ed9]] for comparison:
7edo, [[16ed5]], [[22ed9]] for comparison:
{{Harmonics in equal|7|intervals=integer|collapsed=1}}
{{Harmonics in equal|7|intervals=prime|collapsed=1}}
{{Harmonics in equal|16|5|1|intervals=integer|collapsed=1}}
{{Harmonics in equal|16|5|1|intervals=prime|collapsed=1}}
{{Harmonics in equal|22|9|1|intervals=integer|collapsed=1}}
{{Harmonics in equal|22|9|1|intervals=prime|collapsed=1}}


=== Intervals ===
=== Intervals ===
Line 19: Line 25:
* 1034.357
* 1034.357
* 1206.749
* 1206.749
== 9ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|9|257|128|intervals=prime}}
[[9edo]] for comparison:
{{Harmonics in equal|9|intervals=prime|collapsed=1}}
=== Intervals ===
* 134.083
* 268.167
* 402.25
* 536.333
* 670.416
* 804.5
* 938.583
* 1072.666
* 1206.749
== 14ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|14|257|128|intervals=prime}}
[[14edo]] for comparison:
{{Harmonics in equal|14|intervals=prime|collapsed=1}}
=== 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 ===
{{Harmonics in equal|16|257|128|intervals=prime}}
[[16edo]] for comparison:
{{Harmonics in equal|16|intervals=prime|collapsed=1}}
=== 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 ===
{{Harmonics in equal|19|257|128|intervals=prime}}
[[19edo]] for comparison:
{{Harmonics in equal|19|intervals=prime|collapsed=1}}
=== 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 [[Meantone family#Dreamtone|dreamtone]] temperament, much better than standard 33edo. It is almost exactly the TE tuning.
=== Harmonics ===
{{Harmonics in equal|33|257|128|intervals=prime}}
[[33edo]] for comparison:
{{Harmonics in equal|33|intervals=prime|collapsed=1}}
=== 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 ===
{{Harmonics in equal|38|257|128|intervals=prime}}
[[38edo]] for comparison:
{{Harmonics in equal|38|intervals=prime|collapsed=1}}
== 42ed257/128 ==
''See [[42ed257/128]].''
== 45ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|45|257|128|intervals=prime}}
[[45edo]] for comparison:
{{Harmonics in equal|45|intervals=prime|collapsed=1}}
== 54ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|54|257|128|intervals=prime}}
[[45edo]] for comparison:
{{Harmonics in equal|54|intervals=prime|collapsed=1}}
== 59ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|59|257|128|intervals=prime}}
[[59edo]] for comparison:
{{Harmonics in equal|59|intervals=prime|collapsed=1}}
== 64ed257/128 ==
=== Harmonics ===
{{Harmonics in equal|64|257|128|intervals=prime}}
[[64edo]] for comparison:
{{Harmonics in equal|64|intervals=prime|collapsed=1}}


== Related concepts ==
== Related concepts ==
* [[Ed257/128]]
* [[Substitute harmonic]]
* [[Substitute harmonic]]
* [[Equal-step tuning]]
* [[Equal-step tuning]]
[[Category:Edonoi]][[Category:7edo]][[Category:7-tone scales]]

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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.

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

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