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'''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.  
'''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.  


== Introduction ==
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).  
The 6th harmonic, or ''hexatave'', is particularly wide as far as equivalences go, and there are only about 3.9 hexataves in the [[human hearing range]]. If one does indeed deal with hexatave equivalence, this will alter one's musical approach dramatically. Even so, the hexatave 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 10th, and to a lesser extent, the 12th share this property). Following this, the quintessential reason for using a hexatave based tuning is that it will split the difference between octave and tritave 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 (see [[44ed6]] and [[49ed6]]). However, this is not to say of ed6's not supporting this important 13 & 18 temperament that they can 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.


Some equal divisions of the hexatave serve as generators for octave temperaments:
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.


* [[4ed6]] [[squares]] generator (with octaves)
== As generator chains for temperaments ==
* [[5ed6]] [[tritonic]] generator (with octaves)
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:
* [[6ed6]] [[harry]] generator (with octaves)
* [[4ed6]] – [[squares]] generator (with octaves)
* [[7ed6]] [[sensi]] generator (with octaves)
* [[5ed6]] – [[tritonic]] generator (with octaves)
* [[8ed6]] [[würschmidt]] generator (with octaves)
* [[6ed6]] – [[harry]] generator (with octaves)
* [[10ed6]] [[myna]] generator (with octaves)
* [[7ed6]] – [[sensi]] generator (with octaves)
* [[14ed6]] [[Sensipent family|hemisensi]] generator (with octaves)
* [[8ed6]] – [[würschmidt]] generator (with octaves)
* [[16ed6]] [[Würschmidt family|hemiwürschimdt]] generator (with octaves)
* [[10ed6]] – [[myna]] generator (with octaves)
* [[17ed6]] [[Minortonic family|Minortonic]] generator (with octaves)
* [[14ed6]] – [[hemisensi]] generator (with octaves)
* [[19ed6]] [[Porcupine]] generator (with octaves)
* [[16ed6]] – [[hemiwürschmidt]] generator (with octaves)
* [[21ed6]] [[progression]] generator (with octaves)
* [[17ed6]] – [[minortone]] generator (with octaves)
* [[24ed6]] [[twothirdtonic]] generator (with octaves)
* [[26ed6]] – [[septidiasemi]] generator (with octaves)
* [[26ed6]] [[septidiasemi]] generator (with octaves)
* [[35ed6]] [[octacot]] generator (with octaves)
* [[40ed6]] [[valentine]] generator (with octaves)


== Individual pages for ed6's ==
== Individual pages for ed6's ==
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* [[256ed6|256]], [[269ed6|269]], [[287ed6|287]], [[323ed6|323]], [[512ed6|512]]
* [[256ed6|256]], [[269ed6|269]], [[287ed6|287]], [[323ed6|323]], [[512ed6|512]]


== Ed6-edo correspondence ==
== Ed6–edo correspondence ==
Following ed6's (up to 200) contain good correspondences to edo tunings<ref>Edo with relative error of 6th harmonic below 1/3</ref>.
Following ed6's (up to 200) contain good correspondences to edo tunings<ref>Edo with relative error of 6th harmonic below 1/3</ref>.


{| class="wikitable"
{| class="wikitable center-1 center-2"
|-
|-
! Ed6
! Ed6
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| [[13ed6]]
| [[13ed6]]
| [[5edo]]
| [[5edo]]
| 13ed6 is 5edo with ~6.9 cent compressed octaves.<br />Equivalently, 5edo is 13ed6 with hexataves stretched by ~18 cents.<br />Patent vals match through the 13-limit.
| 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.
|-
|-
| [[18ed6]]
| [[18ed6]]
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* [[Ed12]]
* [[Ed12]]


[[Category:Ed6| ]] <!-- main article -->
[[Category:Ed6's| ]]
[[Category:Edonoi]]
<!-- main article -->
[[Category:Lists of scales]]
[[Category:Lists of scales]]
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