@@ Line 1: / Line 1: @@
-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 flat, 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 in equal|7|257|128|intervals=integer}}
+{{Harmonics in equal|7|257|128|intervals=prime}}
 edo, [[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 ===
@@ Line 19: / Line 25: @@
 * 1034.357
 * 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 ==
-* [[Ed255/128]]
 * [[Substitute harmonic]]
 * [[Equal-step tuning]]
+[[Category:Edonoi]][[Category:7edo]][[Category:7-tone scales]]

BudjarnLambeth/Ed257/128: Difference between revisions