140edo: Difference between revisions

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28edo and 35edo are both interesting subsets worth exploring as a precursory assessment of the tuning qualities of 140edo; despite the fact that their [[11-limit]] is for the most part highly damaged, their sound (depending on who you ask) may be heard to cohere in unexpected nuanced ways, which could be explained as part of the high-limit behavior of 140edo. The same is true for 70edo, which interestingly provides a dual-5's and dual-7's system of at least the [[17-limit]]. These are also interesting because their sound is generally rather unlike that of 7edo which is the subset edo common to all of them.

140edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth or semisixth despite admitting [[interseptimal interval]]s. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log2(3/2)/log2(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.

140edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth, neutral third or semisixth despite admitting both [[interseptimal interval]]s and [[neutral]] intervals. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log2(3/2)/log2(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.

=== Miscellany ===

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 edo and 35edo are both interesting subsets worth exploring as a precursory assessment of the tuning qualities of 140edo; despite the fact that their [[11-limit]] is for the most part highly damaged, their sound (depending on who you ask) may be heard to cohere in unexpected nuanced ways, which could be explained as part of the high-limit behavior of 140edo. The same is true for 70edo, which interestingly provides a dual-5's and dual-7's system of at least the [[17-limit]]. These are also interesting because their sound is generally rather unlike that of 7edo which is the subset edo common to all of them.
-edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth or semisixth despite admitting [[interseptimal interval]]s. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log<sub>2</sub>(3/2)/log<sub>2</sub>(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.
+edo can be seen as a [[hemipyth]] analogue of 70edo, which has no exact semifourth, neutral third or semisixth despite admitting both [[interseptimal interval]]s and [[neutral]] intervals. The slightly sharpened approximation of [[Pythagorean tuning]] given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after [[53edo]] when approximating {{nowrap|log<sub>2</sub>(3/2)/log<sub>2</sub>(4/3) {{=}} ~1.409}} as {{nowrap|√2 {{=}} ~1.414…}}, though the theoretical significance is unclear.
 === Miscellany ===