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.

== Theory ==

60ed6 can be viewed as [[23edo]] with the [[2/1|octave]] being [[stretched and compressed tuning|compressed]] by 10.9 cents, and with the 6th harmonic being [[just]], instead of the octave being just.

~~== Harmonics ==~~

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.

~~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.

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.

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.

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

=== Harmonics ===

~~23edo for comparison:~~

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

== Intervals ==

@@ Line 2: / Line 2: @@
 {{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.
+== Theory ==
+ed6 can be viewed as [[23edo]] with the [[2/1|octave]] being [[stretched and compressed tuning|compressed]] by 10.9 cents, and with the 6th harmonic being [[just]], instead of the octave being just.
-== Harmonics ==
+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.
-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.
-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.
+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.
-{{Harmonics in equal|60|6|1|intervals=prime}}
+=== Harmonics ===
-edo for comparison:
+{{Harmonics in equal|60|6|1|intervals=integer|columns=11}}
-{{Harmonics in equal|23|2|1|intervals=prime}}
+{{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60ed6 (continued)}}
 == Intervals ==