243edo: Difference between revisions
→Regular temperament properties: + more data |
→Theory: + octave stretch, a brief discussion |
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== Theory == | == Theory == | ||
243edo is a strong higher-limit system, especially if we skip [[prime harmonic|prime]] [[11/1|11]]. It is [[consistent]] to the no-11 [[29-odd-limit]] tending flat, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], [[13/1|13]], [[17/1|17]], [[19/1|19]], [[23/1|23]], and [[29/1|29]] all tuned flat. | |||
As an equal temperament, it [[tempering out|tempers out]] the [[semicomma]] (2109375/2097152, the 5-limit orwell comma) and the [[ennealimma]] in the 5-limit, and [[2401/2400]] and [[4375/4374]] in the 7-limit. It [[support]]s [[ennealimmal]], [[quadrawell]], and [[sabric]]. | |||
Using the alternative val 243e {{val| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], 1375/1372, [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]]. | Using the [[patent val]], it tempers out [[243/242]], [[441/440]], and [[540/539]] in the 11-limit, and provides the [[optimal patent val]] for the [[Ragismic microtemperaments #Ennealimmal|ennealimnic]] temperament. In the 13-limit it tempers out [[364/363]], [[625/624]], [[729/728]], and [[2080/2079]], and provides the optimal temperament for 13-limit ennealimnic and the rank-3 [[Breed family #Jovial|jovial]] temperament, and in the 17-limit it tempers out [[375/374]] and [[595/594]] and provides the optimal patent val for 17-limit ennealimnic. | ||
Using the alternative val 243e {{val| 241 385 564 682 '''840''' }}, with an lower error, it tempers out [[385/384]], [[1375/1372]], [[8019/8000]], and [[14641/14580]], and in the 13-limit, 625/624, 729/728, [[847/845]], [[1001/1000]], and [[1716/1715]]. It provides a good tuning for [[fibo]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|243}} | {{Harmonics in equal|243}} | ||
=== Octave stretch === | |||
243edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[385edt]] or [[628ed6]]. This improves most of the approximated harmonics, including the 11 if we use the 243e val. | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 243 factors into | Since 243 factors into primes as 3<sup>5</sup>, 243edo has subset edos {{EDOs| 3, 9, 27, and 81 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||