777edo: Difference between revisions

Revision as of 12:07, 2 November 2023

Prime factorization

3 × 7 × 37

Step size

1.5444 ¢

Fifth

455\777 (702.703 ¢) (→ 65\111)

Semitones (A1:m2)

77:56 (118.9 ¢ : 86.49 ¢)

Dual sharp fifth

455\777 (702.703 ¢) (→ 65\111)

Dual flat fifth

454\777 (701.158 ¢)

Dual major 2nd

132\777 (203.861 ¢) (→ 44\259)

Consistency limit

3

Distinct consistency limit

3

777edo is inconsistent to 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it is excellent in approximating harmonics 5, 7, 9, 11, 13, and 17, making it suitable for a 2.9.5.7.11.13.17 subgroup interpretation. A comma basis for the 2.9.5.7.11.13 subgroup is {4459/4455, 41503/41472, 496125/495616, 105644/105625, 123201/123200}. In addition, it tempers out the landscape comma in the 2.9.5.7 subgroup.

Approximation of odd harmonics in 777edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.748	-0.213	-0.486	-0.049	+0.033	-0.373	+0.534	+0.064	+0.556	+0.262	+0.297
Error	Relative (%)	+48.4	-13.8	-31.5	-3.2	+2.2	-24.2	+34.6	+4.1	+36.0	+16.9	+19.2
Steps (reduced)		1232 (455)	1804 (250)	2181 (627)	2463 (132)	2688 (357)	2875 (544)	3036 (705)	3176 (68)	3301 (193)	3413 (305)	3515 (407)

Since 777 factors into 3 × 7 × 37, 777edo has subset edos 3, 7, 21, 37, 111, and 333.

@@ Line 2: / Line 2: @@
 {{EDO intro|777}}
-edo is a [[dual-fifth system]] with a [[consistency|consistency limit]] of only 3, but otherwise it is excellent in approximating harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], and [[13/1|13]], making it suitable for a 2.9.5.7.11.13 [[subgroup]] interpretation with the comma basis {4459/4455, 41503/41472, 496125/495616, 123201/123200, 105644/105625}. In addition, it tempers out the [[landscape comma]] in the 2.9.5.7 subgroup.
+edo is in[[consistent]] to [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise it is excellent in approximating harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], and [[17/1|17]], making it suitable for a 2.9.5.7.11.13.17 [[subgroup]] interpretation. A [[comma basis]] for the 2.9.5.7.11.13 subgroup is {4459/4455, [[41503/41472]], 496125/495616, 105644/105625, [[123201/123200]]}. In addition, it [[tempering out|tempers out]] the [[landscape comma]] in the 2.9.5.7 subgroup.
 === Odd harmonics ===
@@ Line 8: / Line 8: @@
 === Subsets and supersets ===
-Since 777 factors into 3 × 7 × 37, 777edo has subset edos {{EDOs| 3, 7, 21, 37, 111, and 333 }}.
+Since 777 factors into {{factorization|777}}, 777edo has subset edos {{EDOs| 3, 7, 21, 37, 111, and 333 }}.