Ed12: Difference between revisions

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The twelfth harmonic, duodecuple, or dodecatave, is particularly wide as far as [[equivalence]]s go, as there are at absolute most about 3.1 instances of the 12th harmonic within the [[human hearing range]]. This width means that the listener probably will not hear the interval as an equivalence, but instead will hear the [[pseudo-octave]] or pseudo-tritave or similar as one – this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that [[13edo]] can be used to make melodies that sound like [[12edo]], until they suddenly do not.

However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between [[~~2/1|~~octave]] and [[3/1|twelfth]] based tunings, like an [[ed6]] – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of [[error]] on 3 and 4 and on 2 and 6.

However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between [[octave]] and [[3/1|twelfth]] based tunings, like an [[ed6]] – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of [[error]] on 3 and 4 and on 2 and 6.

As such, an ed12 sometimes gives you the right amount of [[stretched and compressed tuning|stretch]] for equal temperaments whose 3 is more inaccurate than its higher [[prime interval|primes]]. Here for example, you can choose how much you wish to stretch [[31edo]] depending on your harmonic style: [[80ed6]] vs [[111ed12]].

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; 200 and beyond

* [[258ed12|258]]

[[201ed12|201]], [[258ed12|258]]

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 The twelfth harmonic, duodecuple, or dodecatave, is particularly wide as far as [[equivalence]]s go, as there are at absolute most about 3.1 instances of the 12th harmonic within the [[human hearing range]]. This width means that the listener probably will not hear the interval as an equivalence, but instead will hear the [[pseudo-octave]] or pseudo-tritave or similar as one – this disconnect between period versus equivalence could be used by a composer to surprise their listener, in a similar way that [[13edo]] can be used to make melodies that sound like [[12edo]], until they suddenly do not.
-However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between [[2/1|octave]] and [[3/1|twelfth]] based tunings, like an [[ed6]] – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of [[error]] on 3 and 4 and on 2 and 6.
+However, using ed12's does not necessarily imply using the 12th harmonic as an interval of equivalence. The quintessential reason for using a 12th-harmonic based tuning is that it is a compromise between [[octave]] and [[3/1|twelfth]] based tunings, like an [[ed6]] – but ed12 leans more towards octaves than ed6 does. In fact, ed12's optimize for the 1:2:3:4:6:12 chord, with equal magnitudes and opposite signs of [[error]] on 3 and 4 and on 2 and 6.
 As such, an ed12 sometimes gives you the right amount of [[stretched and compressed tuning|stretch]] for equal temperaments whose 3 is more inaccurate than its higher [[prime interval|primes]]. Here for example, you can choose how much you wish to stretch [[31edo]] depending on your harmonic style: [[80ed6]] vs [[111ed12]].
@@ Line 234: / Line 234: @@
 ; 200 and beyond
-* [[258ed12|258]]
+[[201ed12|201]], [[258ed12|258]]
 <!-- Uncomment this when there are more pages