140edo: Difference between revisions
→Approximation to JI: "no-limit" (you can't claim anything about it unless you've actually done the research). Misc. cleanup and fix the tone to avoid suggesting this be objective |
→Theory: +a useful summary for JI approximation |
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== Theory == | == Theory == | ||
In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap|53 & | 140edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list. | ||
In the 5-limit, 140et [[tempering out|tempers out]] [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap| 53 & 87 }} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 140 factors into {{ | Since 140 factors into primes as {{nowrap| 2 × 5 × 7 }}, 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. | 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. | ||
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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|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. | 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|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 === | ||
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]]. | 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]]. | ||