26edt: Difference between revisions

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26edt is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].

== Theory ==

26edt corresponds to 16.404…[[edo]]. It is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].

A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

~~== Theory ==~~

While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl|8L 1s}} and {{sl|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

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 {{ED intro}}
-edt is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].
+== Theory ==
+edt corresponds to 16.404…[[edo]]. It is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes|zeta peak tritave division]].
 A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).
-== Theory ==
 While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl|8L 1s}} and {{sl|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.