Ed6: Difference between revisions

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The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1|10th]], and to a lesser extent, the [[12/1|12th]] share this property).

However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1|octave]] and [[3/1|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. ~~This is not to say~~ ed6's ~~not supporting this should be dismissed out of hand as entirely worthless,~~ for ~~to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking~~ the ~~''n''-th root of~~ 6 ~~is itself an approach to finding temperaments like squares, tritonic~~, and ~~sensi. This approach can~~ of ~~course be used indiscriminately~~.

However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1|octave]] and [[3/1|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. In fact, ed6's optimize for the 1:2:3:6 chord, with equal magnitudes and opposite signs of [[error]] on 2 and 3.

~~Some equal divisions~~ of ~~the 6th harmonic~~ serve as ~~generators~~ for ~~octave~~ temperaments:

== As generator chains for temperaments ==

Taking the ''n''-th root of 6 is itself an approach to finding [[regular temperament|temperaments]] like [[squares]], [[tritonic]], and [[sensi]]. This approach can of course be used indiscriminately. The ed6's serve as generator chains for the temperaments:

* [[4ed6]] – [[squares]] generator (with octaves)

* [[5ed6]] – [[tritonic]] generator (with octaves)

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 The 6th harmonic, sextuple, or hexatave, is particularly wide as far as [[interval of equivalence|equivalence]]s go, and there are only about 3.9 instances of the 6th harmonic in the [[human hearing range]]. If one does indeed deal with equivalence of the 6th harmonic, this will alter one's musical approach dramatically. Even so, the 6th harmonic is one of the three particularly interesting composite harmonics whereof there are enough within the human hearing range to fill three periods of keyboard (the [[10/1|10th]], and to a lesser extent, the [[12/1|12th]] share this property).
-However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1|octave]] and [[3/1|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. This is not to say ed6's not supporting this should be dismissed out of hand as entirely worthless, for to do that would shut off all non-patent musical approaches to this equivalence. In fact, taking the ''n''-th root of 6 is itself an approach to finding temperaments like squares, tritonic, and sensi. This approach can of course be used indiscriminately.
+However, using ed6's does not necessarily imply using the 6th harmonic as an interval of equivalence. The quintessential reason for using a 6th-harmonic based tuning is that it will split the difference between [[2/1|octave]] and [[3/1|twelfth]] based tunings, which is a potentially very desirable thing for a tuning to do given the importance of these harmonics in the musics of much of the world. For example, [[44ed6]] gives us an excellent compromise between [[17edo]] and [[27edt]], and [[49ed6]] achieves the same with respect to [[19edo]] and [[30edt]]. In fact, ed6's optimize for the 1:2:3:6 chord, with equal magnitudes and opposite signs of [[error]] on 2 and 3.
-Some equal divisions of the 6th harmonic serve as generators for octave temperaments:
+== As generator chains for temperaments ==
+Taking the ''n''-th root of 6 is itself an approach to finding [[regular temperament|temperaments]] like [[squares]], [[tritonic]], and [[sensi]]. This approach can of course be used indiscriminately. The ed6's serve as generator chains for the temperaments:
 * [[4ed6]] – [[squares]] generator (with octaves)
 * [[5ed6]] – [[tritonic]] generator (with octaves)