EDT: Difference between revisions

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== EDT-EDO correspondences ==

It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).

=== Multiples of 13EDT which approximate EDO ===

Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the [[8L 3s (3/1-equivalent)|8L 3s]] scale against the tritave, therefore being between 5\8edt and 7\11edt.

~~Also, on~~ the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 65EDT is practically identical to [[41edo|41EDO]].

EDTs with this property include [[19edt|19]], [[27edt|27]], [[30edt|30]], [[35edt|35]], [[38edt|38]], [[41edt|41]], [[43edt|43]], [[46edt|46]], [[49edt|49]], [[51edt|51]], [[52edt|52]], [[54edt|54]], [[57edt|57]], [[59edt|59]], [[60edt|60]], [[62edt|62]], [[63edt|63]], [[65edt|65]], [[67edt|67]], [[68edt|68]], [[70edt|70]], [[71edt|71]], [[73edt|73]] to [[76edt|76]], [[78edt|78]], [[79edt|79]], [[81edt|81]] to [[87edt|87]], and all greater than [[88edt|88]].

EDTs ''without'' a diatonic val are [[1edt|1]] to [[7edt|7]], [[9edt|9]], [[10edt|10]], [[12edt|12]] to [[15edt|15]], [[17edt|17]], [[18edt|18]], [[20edt|20]], [[21edt|21]], [[23edt|23]], [[25edt|25]], [[26edt|26]], [[28edt|28]], [[29edt|29]], [[31edt|31]], [[34edt|34]], [[36edt|36]], [[37edt|37]], [[39edt|39]], [[42edt|42]], [[45edt|45]], [[47edt|47]], [[50edt|50]], [[53edt|53]], [[58edt|58]], [[61edt|61]], and [[69edt|69]].

Borderline cases (i.e. EDTs corresponding to a heptatonic or pentatonic fifth) are [[8edt|8]], [[11edt|11]], [[16edt|16]], [[22edt|22]], [[24edt|24]], [[32edt|32]], [[33edt|33]], [[40edt|40]], [[44edt|44]], [[48edt|48]], [[55edt|55]], [[56edt|56]], [[64edt|64]], [[66edt|66]], [[72edt|72]], [[77edt|77]], [[80edt|80]], and [[88edt|88]].

Correspondences are explained in more detail in the table below.

==== Multiples of 13EDT which approximate EDO ====

On the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 65EDT is practically identical to [[41edo|41EDO]].

=== Table of correspondences ===

@@ Line 401: / Line 401: @@
 == EDT-EDO correspondences ==
+It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).
-=== Multiples of 13EDT which approximate EDO ===
+Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the [[8L 3s (3/1-equivalent)|8L 3s]] scale against the tritave, therefore being between 5\8edt and 7\11edt.
-Also, on the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 65EDT is practically identical to [[41edo|41EDO]].
+EDTs with this property include [[19edt|19]], [[27edt|27]], [[30edt|30]], [[35edt|35]], [[38edt|38]], [[41edt|41]], [[43edt|43]], [[46edt|46]], [[49edt|49]], [[51edt|51]], [[52edt|52]], [[54edt|54]], [[57edt|57]], [[59edt|59]], [[60edt|60]], [[62edt|62]], [[63edt|63]], [[65edt|65]], [[67edt|67]], [[68edt|68]], [[70edt|70]], [[71edt|71]], [[73edt|73]] to [[76edt|76]], [[78edt|78]], [[79edt|79]], [[81edt|81]] to [[87edt|87]], and all greater than [[88edt|88]].
+EDTs ''without'' a diatonic val are [[1edt|1]] to [[7edt|7]], [[9edt|9]], [[10edt|10]], [[12edt|12]] to [[15edt|15]], [[17edt|17]], [[18edt|18]], [[20edt|20]], [[21edt|21]], [[23edt|23]], [[25edt|25]], [[26edt|26]], [[28edt|28]], [[29edt|29]], [[31edt|31]], [[34edt|34]], [[36edt|36]], [[37edt|37]], [[39edt|39]], [[42edt|42]], [[45edt|45]], [[47edt|47]], [[50edt|50]], [[53edt|53]], [[58edt|58]], [[61edt|61]], and [[69edt|69]].
+Borderline cases (i.e. EDTs corresponding to a heptatonic or pentatonic fifth) are [[8edt|8]], [[11edt|11]], [[16edt|16]], [[22edt|22]], [[24edt|24]], [[32edt|32]], [[33edt|33]], [[40edt|40]], [[44edt|44]], [[48edt|48]], [[55edt|55]], [[56edt|56]], [[64edt|64]], [[66edt|66]], [[72edt|72]], [[77edt|77]], [[80edt|80]], and [[88edt|88]].
+Correspondences are explained in more detail in the table below.
+==== Multiples of 13EDT which approximate EDO ====
+On the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo|33EDO]] and 78EDT is very nearly [[49edo|49EDO]], while 65EDT is practically identical to [[41edo|41EDO]].
 === Table of correspondences ===