60edf: Difference between revisions

(11 intermediate revisions by 2 users not shown)

Line 3:

== Theory ==

~~{{todo~~|~~complete section|inline=1}}~~

60edf can be thought of as a very [[octave stretch]]ed version of [[103edo]], or a very compressed version of [[102edo]], but it actually inherits few properties from either.

It makes available [[dual-n|dual]] versions of [[prime]]s 2 and 3 from both systems. Yet its mappings of primes 5, 7, 11, 13 and up are actually all different from either of those edos. For example mapping prime 5 to the 238th step (not 237 as in 102edo, nor 239 as in 103edo).

60edf is very similar to [[205ed4]].

=== Harmonics ===

60edf's approximations of primes are strange. Because of its small step size, it's difficult not to hear primes 2, 3, or even 13, even though they have a lot of [[relative error]].

{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of ~~harmonics~~ in 60edf (continued)}}

60edf is much more accurate on higher primes than on smaller primes. It approximates all primes from 17 through 31 with less than 29% relative error, but has over 43% rel. err. on 2, 3 and 13.

So perhaps a reasonable - if clunky - way to interpret 60edf, is as a [[dual-n|dual]]-2, dual-3, dual-13 [[31-limit]] tuning. Extending it to the [[37-limit]] could also be an option.

{{Harmonics in equal|60|3|2|intervals=prime|columns=13|title=Approximation of primes in 60edf}}

{{Harmonics in equal|60|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of integers in 60edf }}

{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of integers in 60edf (continued)}}

=== Subsets and supersets ===

As a highly composite edf, 60edf includes as subsets: [[2edf]], [[3edf]], [[5edf]], [[6edf]], [[9edf]], [[10edf]], [[12edf]], [[15edf]], [[20edf]] and [[30edf]].

This makes it potentially a good [[polymicrotonal]] system for using multiple edfs (or their stretched/compressed relative edos/ed4s) simultaneously.

The relative edos/ed4s of its subsets are: [[7ed4]], [[5edo]], [[9edo]]/[[17ed4]], [[21ed4]], [[31ed4]], [[17edo]], [[21edo]]/[[41ed4]], [[26edo]], [[34edo]] and [[51edo]]/[[103ed4]].

The simplest supersets of 60edf are [[120edf]] and [[180edf]].

== Intervals ==

== Instruments ==

A [[Lumatone mapping for 60edf]] is now available.

== Music ==

Line 17:

Line 40:

== See also ==

* [[103edo]] – relative ~~edo~~

* [[102edo]], [[103edo]] – relative edos

~~{{todo|complete section|inline=1}}~~

* [[162edt]], [[163edt]] - relative edts

* [[205ed4]] – relative ed4

* [[265ed6]] - relative ed6

~~{{Stub}}~~

[[Category:Nonoctave]]

@@ Line 3: / Line 3: @@
 == Theory ==
-{{todo|complete section|inline=1}}
+edf can be thought of as a very [[octave stretch]]ed version of [[103edo]], or a very compressed version of [[102edo]], but it actually inherits few properties from either.
+It makes available [[dual-n|dual]] versions of [[prime]]s 2 and 3 from both systems. Yet its mappings of primes 5, 7, 11, 13 and up are actually all different from either of those edos. For example mapping prime 5 to the 238th step (not 237 as in 102edo, nor 239 as in 103edo).
+edf is very similar to [[205ed4]].
 === Harmonics ===
-{{Harmonics in equal|60|3|2|intervals=integer|columns=11}}
+edf's approximations of primes are strange. Because of its small step size, it's difficult not to hear primes 2, 3, or even 13, even though they have a lot of [[relative error]].
-{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edf (continued)}}
+edf is much more accurate on higher primes than on smaller primes. It approximates all primes from 17 through 31 with less than 29% relative error, but has over 43% rel. err. on 2, 3 and 13.
+So perhaps a reasonable - if clunky - way to interpret 60edf, is as a [[dual-n|dual]]-2, dual-3, dual-13 [[31-limit]] tuning. Extending it to the [[37-limit]] could also be an option.
+{{Harmonics in equal|60|3|2|intervals=prime|columns=13|title=Approximation of primes in 60edf}}
+{{Harmonics in equal|60|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of integers in 60edf }}
+{{Harmonics in equal|60|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of integers in 60edf (continued)}}
 === Subsets and supersets ===
-{{todo|complete section|inline=1}}
+As a highly composite edf, 60edf includes as subsets: [[2edf]], [[3edf]], [[5edf]], [[6edf]], [[9edf]], [[10edf]], [[12edf]], [[15edf]], [[20edf]] and [[30edf]].
+This makes it potentially a good [[polymicrotonal]] system for using multiple edfs (or their stretched/compressed relative edos/ed4s) simultaneously.
+The relative edos/ed4s of its subsets are: [[7ed4]], [[5edo]], [[9edo]]/[[17ed4]], [[21ed4]], [[31ed4]], [[17edo]], [[21edo]]/[[41ed4]], [[26edo]], [[34edo]] and [[51edo]]/[[103ed4]].
+The simplest supersets of 60edf are [[120edf]] and [[180edf]].
+== Intervals ==
+{{Interval table}}
+== Instruments ==
+A [[Lumatone mapping for 60edf]] is now available.
 == Music ==
@@ Line 17: / Line 40: @@
 == See also ==
-* [[103edo]] – relative edo
+* [[102edo]], [[103edo]] – relative edos
-{{todo|complete section|inline=1}}
+* [[162edt]], [[163edt]] - relative edts
+* [[205ed4]] – relative ed4
+* [[265ed6]] - relative ed6
-{{Stub}}
+[[Category:Nonoctave]]