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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 flat, such as [[7edo]].
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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]].


== 7ed255/128 ==
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 22: Line 27:




== 9ed255/128 ==
== 9ed257/128 ==
=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|9|257|128|intervals=integer}}
{{Harmonics in equal|9|257|128|intervals=prime}}




[[9edo]] for comparison:
[[9edo]] for comparison:
{{Harmonics in equal|9|intervals=integer|collapsed=1}}
{{Harmonics in equal|9|intervals=prime|collapsed=1}}


=== Intervals ===
=== Intervals ===
Line 42: Line 47:




== 14ed255/128 ==
== 14ed257/128 ==
=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|14|257|128|intervals=integer}}
{{Harmonics in equal|14|257|128|intervals=prime}}




[[14edo]] for comparison:
[[14edo]] for comparison:
{{Harmonics in equal|14|intervals=integer|collapsed=1}}
{{Harmonics in equal|14|intervals=prime|collapsed=1}}


=== Intervals ===
=== Intervals ===
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== 16ed255/128 ==
== 16ed257/128 ==
=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|16|257|128|intervals=integer}}
{{Harmonics in equal|16|257|128|intervals=prime}}




[[16edo]] for comparison:
[[16edo]] for comparison:
{{Harmonics in equal|16|intervals=integer|collapsed=1}}
{{Harmonics in equal|16|intervals=prime|collapsed=1}}


=== Intervals ===
=== Intervals ===
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== 19ed255/128 ==
== 19ed257/128 ==
=== Harmonics ===
=== Harmonics ===
{{Harmonics in equal|19|257|128|intervals=integer}}
{{Harmonics in equal|19|257|128|intervals=prime}}




[[19edo]] for comparison:
[[19edo]] for comparison:
{{Harmonics in equal|19|intervals=integer|collapsed=1}}
{{Harmonics in equal|19|intervals=prime|collapsed=1}}


=== Intervals ===
=== Intervals ===
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* 1143.236
* 1143.236
* 1206.749
* 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 ==
* [[Ed255/128]]
* [[Substitute harmonic]]
* [[Substitute harmonic]]
* [[Equal-step tuning]]
* [[Equal-step tuning]]


[[Category:Edonoi]][[Category:7edo]][[Category:7-tone scales]]
[[Category:Edonoi]][[Category:7edo]][[Category:7-tone scales]]
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