65edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide. | 65et can be characterized as the temperament which [[tempering out|tempers out]] 32805/32768 ([[schisma]]), 78732/78125 ([[sensipent comma]]), 393216/390625 ([[würschmidt comma]]), and {{monzo| -13 17 -6 }} ([[graviton]]). In the [[7-limit]], there are two different maps; the first is {{val| 65 103 151 '''182''' }} (65), tempering out [[126/125]], [[245/243]] and [[686/675]], so that it [[support]]s [[sensi]], and the second is {{val| 65 103 151 '''183''' }} (65d), tempering out [[225/224]], [[3125/3087]], [[4000/3969]] and [[5120/5103]], so that it supports [[garibaldi]]. In both cases, the tuning privileges the [[5-limit]] over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit [[würschmidt]] temperament (wurschmidt and worschmidt) these two mappings provide. | ||
65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects | 65edo approximates the intervals [[3/2]], [[5/4]], [[11/8]], [[19/16]], [[23/16]], [[31/16]] and [[47/32]] well, so that it does a good job representing the 2.3.5.11.19.23.31.47 [[just intonation subgroup]]. To this one may want to add [[17/16]], [[29/16]] and [[43/32]], giving the [[47-limit]] no-7's no-13's no-37's no-41's subgroup 2.3.5.11.17.19.23.29.31.43.47. In this sense it is a tuning of [[schismic]]/[[nestoria]] that focuses on the very primes that [[53edo]] neglects (which instead elegantly connects primes 7, 13, 37, and 41 to nestoria). Also of interest is the [[19-limit]] [[k*N subgroups|2*65 subgroup]] 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the [[zeta]] edo [[130edo]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
65edo contains [[13edo]] as | 65edo contains [[5edo]] and [[13edo]] as subsets. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see [[Andrew Heathwaite]]'s composition [https://soundcloud.com/andrew_heathwaite/rubble-a-xenuke-unfolded ''Rubble: a Xenuke Unfolded'']. | ||
[[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13. | [[130edo]], which doubles its, corrects its approximation to harmonics 7 and 13. | ||
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| D | | D | ||
|} | |} | ||
<references group="note" /> | |||
== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
65edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc. | |||
{{Sharpness-sharp6a}} | |||
Half-sharps and half-flats can be used to avoid triple arrows: | |||
{{Sharpness-sharp6b}} | |||
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have arrows borrowed from extended [[Helmholtz–Ellis notation]]: | |||
{{Sharpness-sharp6}} | {{Sharpness-sharp6}} | ||
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals: | |||
{{Sharpness-sharp6-qt}} | {{Sharpness-sharp6-qt}} | ||
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== Approximation to JI == | == Approximation to JI == | ||
=== Zeta peak index === | === Zeta peak index === | ||
{ | {{ZPI | ||
| | | zpi = 334 | ||
| steps = 65.0158450885860 | |||
| step size = 18.4570391781413 | |||
| tempered height = 7.813349 | |||
| pure height = 7.642373 | |||
| integral = 1.269821 | |||
| gap = 16.514861 | |||
| octave = 1199.70754657919 | |||
| consistent = 6 | |||
| distinct = 6 | |||
}} | |||
| 18.4570391781413 | |||
| 7.813349 | |||
| 1.269821 | |||
| 16.514861 | |||
| | |||
| 6 | |||
| 6 | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 498.46 | | 498.46 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] / [[nestoria]] / [[photia]] | | [[Helmholtz (temperament)|Helmholtz]] / [[nestoria]] / [[photia]] | ||
|- | |- | ||
| 1 | | 1 | ||
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[[Lumatone mapping for 65edo]] | [[Lumatone mapping for 65edo]] | ||
== | == Music == | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/W5PXWFduPco ''microtonal improvisation in 65edo''] (2025). | |||
[[Category:Listen]] | [[Category:Listen]] | ||