313edo: Difference between revisions

Revision as of 07:50, 15 January 2023

313edo provides the optimal patent val for 11- and 13-limit hemischis temperament and the 13-limit rank-3 temperaments madagascar and hera. It tempers out the schisma, 32805/32768, in the 5-limit; 6144/6125 and 19683/19600 in the 7-limit; 540/539, 5632/5625, 8019/8000 and 16384/16335 in the 11-limit; 351/350, 676/675, 729/728, 1001/1000, 2080/2079 and 4096/4095 in the 13-limit.

Approximation of prime harmonics in 313edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.36	+0.91	+1.14	+0.76	-0.91	-1.44	+1.53	+0.48	+1.73	+1.29
Error	Relative (%)	+0.0	-9.3	+23.7	+29.8	+19.8	-23.8	-37.6	+39.9	+12.5	+45.2	+33.7
Steps (reduced)		313 (0)	496 (183)	727 (101)	879 (253)	1083 (144)	1158 (219)	1279 (27)	1330 (78)	1416 (164)	1521 (269)	1551 (299)

313edo is the 65th prime edo.

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-496 313⟩	[⟨313 496]]	+0.113	0.113	2.94
2.3.5	32805/32768, [-1 49 -33⟩	[⟨313 496 727]]	-0.055	0.254	6.64
2.3.5.7	6144/6125, 19683/19600, 40500000/40353607	[⟨313 496 727 879]]	-0.143	0.268	6.99
2.3.5.7.11	540/539, 5632/5625, 8019/8000, 43923/43904	[⟨313 496 727 879 1083]]	-0.158	0.242	6.30
2.3.5.7.11.13	351/350, 540/539, 676/675, 4096/4095, 43923/43904	[⟨313 496 727 879 1083 1158]]	-0.091	0.267	6.97

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	65\313	249.201	15/13	Hemischis
1	156\313	598.083	847/600	Vydubychi

See also: Category:313edo tracks

@@ Line 4: / Line 4: @@
 == Theory ==
 edo provides the [[optimal patent val]] for 11- and 13-limit [[hemischis]] temperament and the 13-limit rank-3 temperaments [[madagascar]] and [[hera]]. It tempers out the [[schisma]], 32805/32768, in the 5-limit; [[6144/6125]] and [[19683/19600]] in the 7-limit; [[540/539]], [[5632/5625]], [[8019/8000]] and [[16384/16335]] in the 11-limit; [[351/350]], [[676/675]], [[729/728]], [[1001/1000]], [[2080/2079]] and [[4096/4095]] in the 13-limit.
-edo is the 65th [[prime edo]].
 === Prime harmonics ===
 {{Harmonics in equal|313}}
+=== Divisors ===
+edo is the 65th [[prime edo]].
 == Regular temperament properties ==
@@ Line 15: / Line 16: @@
 ! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 59: / Line 60: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per Octave
+! Periods<br>per 8ve
 ! Generator<br>(Reduced)
 ! Cents<br>(Reduced)