60ed6: Difference between revisions

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~~It is similar to [[23edo]], but with the octave (2/1) being [[octave shrinking|compressed]] by 10.9 cents, and with the interval [[6/1]] being [[just]], instead of 2/1 being just.~~

== ~~Harmonics~~ ==

== Theory ==

~~23edo’s 3/1, 5/1, 7/1 and 11/1 are all more than 20 cents away from just, causing them to exhibit very little~~ [[~~consonance~~]]. ~~60ed6 improves upon all of their tunings, bringing all of them within 16~~ cents ~~of just~~, and ~~bringing 3~~, ~~5 and 7 within 11 cents~~ of just~~. This dramatically increases the number of consonant intervals and chords available in the tuning~~.

60ed6 can be viewed as [[23edo]] with the [[octave]] being [[stretched and compressed tuning|compressed]] by 10.9 cents, and with the 6th harmonic being [[just]], instead of the octave being just.

~~The trade-off is that 60ed6’s octave - 2~~/1 ~~- is significantly worse than 23edo. It has almost 11 cents of error~~, ~~compared to 0. For some composers~~, ~~11 cents error on the octave may be unacceptable. But for others, it may be considered still close enough for consonance~~ and [[~~octave equivalence~~]] ~~to be well preserved~~, ~~and~~ they ~~may see it a worthwhile sacrifice to unlock so many warm~~ [[~~11-limit~~]] ~~harmonies~~.

23edo's [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] are all more than 20 cents away from just, so they exhibit very little [[consonance]]. 60ed6 improves upon all of their tunings, bringing all of them within 16 cents of just, and bringing 3, 5 and 7 within 11 cents of just. This dramatically increases the number of consonant intervals and chords available in the tuning.

~~{{Harmonics~~ in ~~equal|60|6|1|intervals=prime}}~~

The trade-off is that 60ed6's octave is significantly worse than 23edo. It has almost 11 cents of error, compared to none. For some composers, 11 cents error on the octave may be unacceptable, but for others, it may be considered still close enough for consonance and [[octave equivalence]] to be well preserved, and they may see it a worthwhile sacrifice to unlock so many warm [[11-limit]] harmonies.

~~23edo for comparison:~~

=== Harmonics ===

{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}

=== Subsets and supersets ===

60ed6 is the 9th [[highly composite equal division|highly composite ed6]], with subset ed6's {{EDs|equave=6| 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 }}.

== Intervals ==

~~{{todo|expand}}~~

== See also ==

* [[37edt]] – relative edt

[[Category:23edo]]

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 {{Infobox ET}}
 {{ED intro}}
-It is similar to [[23edo]], but with the octave (2/1) being [[octave shrinking|compressed]] by 10.9 cents, and with the interval [[6/1]] being [[just]], instead of 2/1 being just.
-== Harmonics ==
+== Theory ==
-edo’s 3/1, 5/1, 7/1 and 11/1 are all more than 20 cents away from just, causing them to exhibit very little [[consonance]]. 60ed6 improves upon all of their tunings, bringing all of them within 16 cents of just, and bringing 3, 5 and 7 within 11 cents of just. This dramatically increases the number of consonant intervals and chords available in the tuning.
+ed6 can be viewed as [[23edo]] with the [[octave]] being [[stretched and compressed tuning|compressed]] by 10.9 cents, and with the 6th harmonic being [[just]], instead of the octave being just.
-The trade-off is that 60ed6’s octave - 2/1 - is significantly worse than 23edo. It has almost 11 cents of error, compared to 0. For some composers, 11 cents error on the octave may be unacceptable. But for others, it may be considered still close enough for consonance and [[octave equivalence]] to be well preserved, and they may see it a worthwhile sacrifice to unlock so many warm [[11-limit]] harmonies.
+edo's [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] are all more than 20 cents away from just, so they exhibit very little [[consonance]]. 60ed6 improves upon all of their tunings, bringing all of them within 16 cents of just, and bringing 3, 5 and 7 within 11 cents of just. This dramatically increases the number of consonant intervals and chords available in the tuning.
-{{Harmonics in equal|60|6|1|intervals=prime}}
+The trade-off is that 60ed6's octave is significantly worse than 23edo. It has almost 11 cents of error, compared to none. For some composers, 11 cents error on the octave may be unacceptable, but for others, it may be considered still close enough for consonance and [[octave equivalence]] to be well preserved, and they may see it a worthwhile sacrifice to unlock so many warm [[11-limit]] harmonies.
-edo for comparison:
+=== Harmonics ===
-{{Harmonics in equal|23|2|1|intervals=prime}}
+{{Harmonics in equal|60|6|1|intervals=integer|columns=11}}
+{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}
+=== Subsets and supersets ===
+ed6 is the 9th [[highly composite equal division|highly composite ed6]], with subset ed6's {{EDs|equave=6| 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 }}.
 == Intervals ==
 {{Interval table}}
-{{todo|expand}}
+== See also ==
+* [[37edt]] – relative edt
+[[Category:23edo]]