764edo: Difference between revisions

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-The ''764 equal division'' divides the octave into 764 equal parts of 1.571 cents each. It is a very strong 17-limit system distinctly consistent to the 17-limit, and is the fourteenth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral division]]. In the 5-limit it tempers out the hemithirds comma, {{monzo| 38 -2 -15 }}; in the 7-limit  4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the [[optimal patent val]] for [[Ragismic_microtemperaments#Abigail|abigail temperament]] in the 11-limit.
+{{Infobox ET}}
+{{ED intro}}
-{{Primes in edo|764|prec=3|columns=7}}
+== Theory ==
+edo is a very strong 17-limit system, [[consistent]] to the [[17-odd-limit]] or the no-19 no-29 [[41-odd-limit]]. It is the fourteenth [[zeta integral edo]]. In the 5-limit it [[tempering out|tempers out]] the hemithirds comma, {{monzo| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.
-[[Category:Equal divisions of the octave]]
+In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.
-[[Category:Todo:expand]]
+=== Prime harmonics ===
+{{Harmonics in equal|764|columns=15}}
+=== Subsets and supersets ===
+Since 764 factors into {{factorization|764}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo|22\16808]]).
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 1211 -764 }}
+| {{mapping| 764 1211 }}
+| −0.0439
+| 0.0439
+| 2.80
+|-
+| 2.3.5
+| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }}
+| {{mapping| 764 1211 1774 }}
+| −0.0399
+| 0.0363
+| 2.31
+|-
+| 2.3.5.7
+| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }}
+| {{mapping| 764 1211 1774 2145 }}
+| −0.0552
+| 0.0412
+| 2.62
+|-
+| 2.3.5.7.11
+| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
+| {{mapping| 764 1211 1774 2145 2643 }}
+| −0.0436
+| 0.0435
+| 2.77
+|-
+| 2.3.5.7.11.13
+| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
+| {{mapping| 764 1211 1774 2145 2643 2827 }}
+| −0.0267
+| 0.0548
+| 3.49
+|-
+| 2.3.5.7.11.13.17
+| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
+| {{mapping| 764 1211 1774 2145 2643 2827 3123 }}
+| −0.0327
+| 0.0528
+| 3.36
+|-
+| 2.3.5.7.11.13.17.23
+| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
+| {{mapping| 764 1211 1774 2145 2643 2827 3123 3456 }}
+| −0.0286
+| 0.0506
+| 3.22
+|}
+* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo|684]] and is only bettered by [[935edo|935]]. In the 17-limit it beats [[742edo|742]] and is only bettered by [[814edo|814]].
+* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo|494]] and before [[1578edo|1578]].
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 123\764
+| 193.19
+| 262144/234375
+| [[Lunatic]] (7-limit)
+|-
+| 1
+| 277\764
+| 435.08
+| 9/7
+| [[Supermajor]]
+|-
+| 2
+| 133\764
+| 208.90
+| 44/39
+| [[Abigail]]
+|-
+| 2
+| 277\764<br />(105\764)
+| 435.08<br />(164.92)
+| 9/7<br />(11/10)
+| [[Semisupermajor]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+[[Category:Abigail]]