Ed6: Difference between revisions

'''Equal divisions of the 6th harmonic''' ('''ed6''') are [[tuning]]s obtained by dividing the [[6/1|6th harmonic]] in a certain number of [[equal]] steps.  

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]]. In fact, ed6's optimize for the 1:2:3:6 chord, with equal magnitudes and opposite signs of [[error]] on 2 and 3.

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)

* [[6ed6]] – [[harry]] generator (with octaves)

* [[7ed6]] – [[sensi]] generator (with octaves)

* [[8ed6]] – [[würschmidt]] generator (with octaves)

* [[10ed6]] – [[myna]] generator (with octaves)

* [[14ed6]] – [[hemisensi]] generator (with octaves)

* [[16ed6]] – [[hemiwürschmidt]] generator (with octaves)

* [[17ed6]] – [[minortone]] generator (with octaves)

* [[26ed6]] – [[septidiasemi]] generator (with octaves)

| [[6ed6|6]]

| [[11ed6|11]]

| [[13ed6|13]]

| [[31ed6|31]]

| [[43ed6|43]]

| [[44ed6|44]]

| [[153ed6|153]]

 

* [[256ed6|256]], [[269ed6|269]], [[287ed6|287]], [[323ed6|323]], [[512ed6|512]]

== Ed6–edo correspondence ==

{| class="wikitable center-1 center-2"

| 13ed6 is 5edo with ~6.9 cent compressed octaves. Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents. Patent vals match through the 13-limit.

<references />

[[Category:Ed6's| ]]

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