140edo: Difference between revisions

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It is very strong as a high-limit/no-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|1\140 {{=}} ~8.57{{c}}}} of error), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].

~~=== Other info ===~~

[[35edo]] and [[28edo]] 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/smallest 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.

If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].

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=== Subsets and supersets ===

Since 140 factors into {{factorisation|140}}, 140edo has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.

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.

== Approximation to JI ==

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 It is very strong as a high-limit/no-limit system, performing generally very well for its size in extremely high [[odd-limit]]s like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than {{nowrap|1\140 {{=}} ~8.57{{c}}}} of error), even though there is far more that it gets right. It is especially notable as a tuning of [[degrees]] (with 1\20 period), [[decoid]] (with 1\10 period) and [[thunderclysmic]] (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of [[5edo]].
-=== Other info ===
-[[35edo]] and [[28edo]] 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/smallest 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.
 If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
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 === Subsets and supersets ===
 Since 140 factors into {{factorisation|140}}, 140edo has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.
+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.
 == Approximation to JI ==