209edo: Difference between revisions

Revision as of 13:11, 16 November 2024

209edo is only consistent to the 5-odd-limit. The equal temperament tempers out 129140163/128000000 (graviton) and 1220703125/1207959552 (ditonma) in the 5-limit. Using the patent val, it tempers out 225/224, 2125764/2100875, and 2500000/2470629 in the 7-limit; 243/242, 3025/3024, 4000/3993, and 16896/16807 in the 11-limit; 351/350, 625/624, 1573/1568, 1625/1617, and 15379/15360 in the 13-limit, so that it provides the optimal patent val for the 13-limit marvo temperament. It also supports the 13-limit spectacle temperament.

Approximation of odd harmonics in 209edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.48	-1.62	+1.51	+2.79	-0.12	-2.25	+2.64	-1.61	+1.05	+0.03	-2.44
Error	Relative (%)	-25.7	-28.3	+26.3	+48.6	-2.1	-39.2	+46.0	-28.0	+18.3	+0.6	-42.4
Steps (reduced)		331 (122)	485 (67)	587 (169)	663 (36)	723 (96)	773 (146)	817 (190)	854 (18)	888 (52)	918 (82)	945 (109)

Since 209 factors into 11 × 19, 209edo contains 11edo and 19edo as its subsets. 627edo, which triples it, gives a good correction to the harmonic 7.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-331 209⟩	[⟨209 331]]	+0.4658	0.4660	8.12
2.3.5	[-13 17 -6⟩, [-27 -2 13⟩	[⟨209 331 485]]	+0.5439	0.3962	6.90

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	71\209	407.66	15625/12288	Ditonic
1	90\209	516.75	27/20	Larry / marvo (209) / zarvo (209d)
19	122\209 (1\209)	700.48 (5.74)	3/2 (225/224)	Enneadecal (209d)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 27: / Line 36: @@
 | 0.3962
 | 6.90
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 49: / Line 65: @@
 | 3/2<br>(225/224)
 | [[Enneadecal]] (209d)
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Marvo]]