8539edo: Difference between revisions

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-The '''8539 equal temperament''' divides the octave into 8539 equal parts of 0.1405 cents each. While it may strike many people as too large to be practical, it's seen actual use as a bookeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the [[tina|tina]] (see [http://www.tonalsoft.com/enc/t/tina.aspx http://www.tonalsoft.com/enc/t/tina.aspx].) This is because it is a very strong higher limit system, distinctly consistent through the 27 limit, and is both a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak]] and zeta integral tuning. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203. Some 17-limit commas it tempers out are 28561/28560, 31213/31212 and 37180/37179; in the 19-limit it tempers out are 27456/27455 and 43681/43680. 8539 is a prime number, and the tina as a unit of measure could be criticized on that basis; however some people prefer primes for this sort of job, as they don't imply a preference for one smaller edo over another.
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
+While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists|strict zeta]] tuning, and is also the first [[trivial temperament|non-trivial]] edo to be consistent in the [[odd prime sum limit|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo|20203]].
+Some of the simpler commas it [[tempering out|tempers out]] include the [[senior|senior comma]] in the [[5-limit]]; [[very high accuracy temperaments #Sepbizo-asegu (171 & 3566 & 4973)|{{monzo| -3 2 -17 14 }}]] in the [[7-limit]]; [[3294225/3294172]] in the [[11-limit]]; [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the [[23-limit]].
+Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].
+=== Prime harmonics ===
+{{Harmonics in equal|8539|columns=11}}
+{{Harmonics in equal|8539|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 8539edo (continued)}}
+=== Subsets and supersets ===
+edo is the 1065th [[prime edo]]. On that basis, the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.
+== Notation ==
+edo is special in the [[Sagittal notation]], as it has been the model for the Magrathean set, which offers "insane" precision. This is, and will be, the highest precision available in Sagittal notation. The diacritics are independent of the sagittals. Scroll the table to see accidentals for use in Revo flavor (503\8539 onwards).
+<div style="overflow-x:auto;">
+{| class="wikitable" style="text-align:center"
+|-
+!'''Steps'''
+! 0 !! 24 !! 41
+!62
+!69
+!86
+!105
+!118
+!124
+!143
+!153
+!177
+!194
+!218
+!226
+!236
+!254
+!271
+!277
+!285
+!306
+!330
+!347
+!357
+!379
+!389
+!402
+!420
+!430
+!452
+!462
+!479
+!503
+!524
+!532
+!538
+!555
+!573
+!583
+!591
+!615
+!632
+!656
+!666
+!671
+!691
+!704
+!723
+!740
+!747
+!768
+!785
+!809
+|-
+|Symbol
+|<big>{{sagittal|h}}</big>
+|<big>{{sagittal|)|}}</big>
+|<big>{{sagittal||(}}</big>
+|<big>{{sagittal|~|}}</big>
+|<big>{{sagittal|)|(}}</big>
+|<big>{{sagittal|)~|}}</big>
+|<big>{{sagittal|~|(}}</big>
+|<big>{{sagittal||~}}</big>
+|<big>{{sagittal|~~|}}</big>
+|<big>{{sagittal|)|~}}</big>
+|<big>{{sagittal|/|}}</big>
+|<big>{{sagittal|)/|}}</big>
+|<big>{{sagittal||)}}</big>
+|<big>{{sagittal|)|)}}</big>
+|<big>{{sagittal||\}}</big>
+|<big>{{sagittal|(|}}</big>
+|<big>{{sagittal|~|)}}</big>
+|<big>{{sagittal|/|~}}</big>
+|<big>{{sagittal|(|(}}</big>
+|<big>{{sagittal|~|\}}</big>
+|<big>{{sagittal|//|}}</big>
+|<big>{{sagittal|)//|}}</big>
+|<big>{{sagittal|/|)}}</big>
+|<big>{{sagittal|(|~}}</big>
+|<big>{{sagittal|/|\}}</big>
+|<big>{{sagittal|(/|}}</big>
+|<big>{{sagittal|)/|\}}</big>
+|<big>{{sagittal||\)}}</big>
+|<big>{{sagittal|(|)}}</big>
+|<big>{{sagittal||\\}}</big>
+|<big>{{sagittal|(|\}}</big>
+|<big>{{sagittal|)|\\}}</big>
+|<big>{{sagittal|)||(}}</big>
+|<big>{{sagittal|)~||}}</big>
+|<big>{{sagittal|~||(}}</big>
+|<big>{{sagittal|||~}}</big>
+|<big>{{sagittal|~~||}}</big>
+|<big>{{sagittal|)||~}}</big>
+|<big>{{sagittal|/||}}</big>
+|<big>{{sagittal|)/||}}</big>
+|<big>{{sagittal|||)}}</big>
+|<big>{{sagittal|)||)}}</big>
+|<big>{{sagittal|||\}}</big>
+|<big>{{sagittal|(||}}</big>
+|<big>{{sagittal|~||)}}</big>
+|<big>{{sagittal|/||~}}</big>
+|<big>{{sagittal|(||(}}</big>
+|<big>{{sagittal|~||\}}</big>
+|<big>{{sagittal|//||}}</big>
+|<big>{{sagittal|)//||}}</big>
+|<big>{{sagittal|/||)}}</big>
+|<big>{{sagittal|(||~}}</big>
+|<big>{{sagittal|/||\}}</big>
+|}
+</div>
+{| class="wikitable data-darkreader-inline-color="
+|+Magrathean diacritics
+!'''Steps'''
+!1
+!2
+!3
+!4
+!5
+!6
+!7
+!8
+!9
+!10
+!11
+!12
+!13
+!14
+!15
+!16
+!17
+!18
+!19
+!20
+!21
+!22
+!23
+|-
+|Symbol
+|<big>{{sagittal|@1}}</big>
+|<big>{{sagittal|@2}}</big>
+|<big>{{sagittal|@3}}</big>
+|<big>{{sagittal|@4}}</big>
+|<big>{{sagittal|@5}}</big>
+|<big>{{sagittal|@6}}</big>
+|<big>{{sagittal|@7}}</big>
+|<big>{{sagittal|@8}}</big>
+|<big>{{sagittal|@9}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|l4}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|l3}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|l2}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|l1}}</big>
+|<big>{{sagittal|'}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@1}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@3}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@3}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@4}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@5}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@6}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@7}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@8}}</big>
+|<big>{{sagittal|'}}</big><big>{{sagittal|@9}}</big>
+|}
+== External links ==
+* [http://www.tonalsoft.com/enc/t/tina.aspx Tina] on [[Tonalsoft Encyclopedia]]
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
 [[Category:Tina]]