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)

== Individual pages for ed6's ==

| [[6ed6|6]]

| [[10ed6|10]]

| [[11ed6|11]]

| [[31ed6|31]]

| [[43ed6|43]]

| [[44ed6|44]]

| [[112ed6|112]]

 

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

== 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>.

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

! Ed6

! Edo

! Comments

| [[13ed6]]

| [[5edo]]

| 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]]

| [[7edo]]

| 18ed6 is 7edo with ~6.3 cent stretched octaves. Patent vals match through the 13-limit.

| [[21ed6]]

| [[8edo]]

| 21ed6 is 8edo with ~18 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 7.

| [[23ed6]]

| [[9edo]]

| 23ed6 is 9edo with ~14 cent stretched octaves. Patent vals match through the 13-limit.

| [[26ed6]]

| [[10edo]]

| Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 17-limit.

| [[31ed6]]

| [[12edo]]

| 31ed6 is 12edo with ~0.76 cent stretched octaves. Patent vals match through the 31-limit, with the exception of 11.

| [[36ed6]]

| [[14edo]]

| Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 43-limit, with the exception of 5.

| [[39ed6]]

| [[15edo]]

| Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals match through the 13-limit.

| [[44ed6]]

| [[17edo]]

| 44ed6 is 17edo with ~1.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 5.

| [[49ed6]]

| [[19edo]]

| 49ed6 is 19edo with ~2.8 cent stretched octaves. Patent vals match through the 13-limit.

| [[52ed6]]

| [[20edo]]

| Same ~6.9 cent octave compression as 13ed6~5edo. Patent vals differ in the 5-limit.

| [[54ed6]]

| [[21edo]]

| Same ~6.3 cent octave stretch as 18ed6~7edo. Patent vals match through the 7-limit.

| [[57ed6]]

| [[22edo]]

| 57ed6 is 22edo with ~2.8 cent compressed octaves. Patent vals match through the 17-limit, with the exception of 13.

| [[62ed6]]

| [[24edo]]

| Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.

| [[67ed6]]

| [[26edo]]

| 67ed6 is 26edo with ~3.7 cent stretched octaves. Patent vals match through the 19-limit.

| [[70ed6]]

| [[27edo]]

| 70ed6 is 27edo with ~3.5 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 11.

| [[75ed6]]

| [[29edo]]

| 75ed6 is 29edo with ~0.58 cent compressed octaves. Patent vals match through the 137-limit.

| [[80ed6]]

| [[31edo]]

| 80ed6 is 31edo with ~2.0 cent stretched octaves. Patent vals match through the 13-limit.

| [[83ed6]]

| [[32edo]]

| 83ed6 is 32edo with ~4.1 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 5.

| [[85ed6]]

| [[33edo]]

| 85ed6 is 33edo with ~4.3 cent stretched octaves. Patent vals differ in the 5 and 7-limit.  

| [[88ed6]]

| [[34edo]]

| Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 17-limit, with the exception of 7.

| [[93ed6]]

| [[36edo]]

| Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.

| [[98ed6]]

| [[38edo]]

| Same ~2.8 cent octave stretch as 49ed6~19edo. Patent vals match through the 11-limit, with the exception of 7.

| [[101ed6]]

| [[39edo]]

| 101ed6 is 39edo with ~2.2 cent compressed octaves. Patent vals match through the 11-limit, with the exception of 7.

| [[106ed6]]

| [[41edo]]

| 106ed6 is 41edo with ~0.19 cent compressed octaves. Patent vals match through the 41-limit.

| [[111ed6]]

| [[43edo]]

| 111ed6 is 43edo with ~1.7 cent stretched octaves. Patent vals match through the 17-limit.

| [[114ed6]]

| [[44edo]]

| Same ~2.8 cent octave compression as 57ed6~22edo. Patent vals match through the 31-limit, with the exception of 11.

| [[116ed6]]

| [[45edo]]

| 116ed6 is 45edo with ~3.3 cent stretched octaves. Patent vals match through the 7-limit.

| [[119ed6]]

| [[46edo]]

| 119ed6 is 46edo with ~0.92 cent compressed octaves. Patent vals match through the 17-limit.

| [[124ed6]]

| [[48edo]]

| Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 43-limit.

| [[129ed6]]

| [[50edo]]

| 129ed6 is 50edo with ~2.3 cent stretched octaves. Patent vals match through the 23-limit.

| [[132ed6]]

| [[51edo]]

| Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 13-limit, with the exceptions of 5 and 11.

| [[137ed6]]

| [[53edo]]

| 137ed6 is 53edo with ~0.026 cent stretched octaves. Patent vals match through the 61-limit.

| [[142ed6]]

| [[55edo]]

| 142ed6 is 55edo with ~1.46 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 13.

| [[145ed6]]

| [[56edo]]

| 145ed6 is 56edo with ~2.0 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 13.

| [[150ed6]]

| [[58edo]]

| Same ~0.58 cent octave compression as 75ed6~29edo. Patent vals match through the 113-limit.

| [[155ed6]]

| [[60edo]]

| Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 31-limit, with the exception of 11.

| [[158ed6]]

| [[61edo]]

| 158ed6 is 61edo with ~2.4 cent compressed octaves. Patent vals match through the 13-limit, with the exception of 7.

| [[160ed6]]

| [[62edo]]

| Same ~2.0 cent octave stretch as 80ed6~31edo. Patent vals match through the 31-limit.

| [[163ed6]]

| [[63edo]]

| 163ed6 is 63edo with ~1.1 cent compressed octaves. Patent vals match through the 61-limit.

| [[168ed6]]

| [[65edo]]

| 168ed6 is 65edo with ~0.16 cent stretched octaves. Patent vals match through the 157-limit, with the exception of 13.

| [[173ed6]]

| [[67edo]]

| 173ed6 is 67edo with ~1.3 cent stretched octaves. Patent vals match through the 17-limit, with the exception of 5.

| [[176ed6]]

| [[68edo]]

| Same ~1.5 cent octave compression as 44ed6~17edo. Patent vals match through the 23-limit, with the exception of 11.

| [[181ed6]]

| [[70edo]]

| 181ed6 is 70edo with ~0.35 cent compressed octaves. Patent vals match through the 67-limit.

| [[186ed6]]

| [[72edo]]

| Same ~0.76 cent octave stretch as 31ed6~12edo. Patent vals match through the 19-limit.

| [[189ed6]]

| [[73edo]]

| 189ed6 is 73edo with ~1.9 cent compressed octaves. Patent vals match through the 11-limit.

| [[191ed6]]

| [[74edo]]

| 191ed6 is 74edo with ~1.8 cent stretched octaves. Patent vals match through the 11-limit, with the exception of 7.

| [[194ed6]]

| [[75edo]]

| 194ed6 is 75edo with ~0.79 cent compressed octaves. Patent vals match through the 23-limit, with the exception of 11.

| [[199ed6]]

| [[77edo]]

| 199ed6 is 77edo with ~0.25 cent stretched octaves. Patent vals match through the 37-limit.

<references />

[[Category:Ed6's| ]]

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[[Category:Lists of scales]]