552edo: Difference between revisions

Revision as of 19:27, 15 January 2025

552edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through 13 all tuned sharp. The equal temperament tempers out [8 14 -3⟩ (parakleisma) in the 5-limit; 250047/250000 (landscape comma), 589824/588245 (hewuermera comma), 26873856/26796875, and 33554432/33480783 (garischisma) in the 7-limit; 5632/5625, 9801/9800, 46656/46585, 151263/151250, and 161280/161051 in the 11-limit; and 1716/1715, 2080/2079, 10648/10647, and 20480/20449 in the 13-limit. It supports sextile and gives a good tuning for it.

It is also consistent in the no-17 23-odd-limit and the no-17 no-25 33-odd-limit. In the 2.3.5.7.11.13.19 subgroup, it tempers out 1216/1215, 2376/2375, 2926/2925, 3136/3135, 3328/3325, 3971/3969 among other commas.

Approximation of prime harmonics in 552edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.219	+0.643	+0.739	+0.856	+0.777	-0.608	+0.313	-0.013	+0.858	+0.617
Error	Relative (%)	+0.0	+10.1	+29.6	+34.0	+39.4	+35.7	-27.9	+14.4	-0.6	+39.4	+28.4
Steps (reduced)		552 (0)	875 (323)	1282 (178)	1550 (446)	1910 (254)	2043 (387)	2256 (48)	2345 (137)	2497 (289)	2682 (474)	2735 (527)

Since 552 factors into 2³ × 3 × 23, 552edo has subset edos 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, and 276.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[875 -552⟩	[⟨552 875]]	−0.0691	0.0691	3.18
2.3.5	[8 14 -13⟩, [71 -36 -6⟩	[⟨552 875 1282]]	−0.1383	0.1130	5.20
2.3.5.7	250047/250000, 589824/588245, 33554432/33480783	[⟨552 875 1282 1550]]	−0.1696	0.1118	5.15
2.3.5.7.11	5632/5625, 9801/9800, 151263/151250, 161280/161051	[⟨552 875 1282 1550 1910]]	−0.1851	0.1048	4.82
2.3.5.7.11.13	1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449	[⟨552 875 1282 1550 1910 2043]]	−0.1892	0.0961	4.42
2.3.5.7.11.13.19	1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379	[⟨552 875 1282 1550 1910 2043 2345]]	−0.1727	0.0977	4.50

552et is notable for being the first equal temperament to beat 270 in the 2.3.5.7.11.13.19 subgroup in terms of absolute error. The next equal temperament that does better in this subgroup is 581.

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	145\552	315.22	6/5	Parakleismic (5-limit)
1	229\552	497.83	4/3	Gary (2.3.7 subgroup)
6	229\552 (45\552)	497.83 (97.83)	4/3 (128/121)	Sextile
24	232\552 (2\552)	504.348 (4/348)	7/5 (?)	Chromium
46	229\552 (1\552)	497.83 (97.83)	4/3 (?)	Palladium (5-limit)

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

@@ Line 28: / Line 28: @@
 | {{monzo| 875 -552 }}
 | {{mapping| 552 875 }}
-| &minus;0.0691
+| −0.0691
 | 0.0691
 | 3.18
@@ Line 35: / Line 35: @@
 | {{monzo| 8 14 -13 }}, {{monzo| 71 -36 -6 }}
 | {{mapping| 552 875 1282 }}
-| &minus;0.1383
+| −0.1383
 | 0.1130
 | 5.20
@@ Line 42: / Line 42: @@
 | 250047/250000, 589824/588245, 33554432/33480783
 | {{mapping| 552 875 1282 1550 }}
-| &minus;0.1696
+| −0.1696
 | 0.1118
 | 5.15
@@ Line 49: / Line 49: @@
 | 5632/5625, 9801/9800, 151263/151250, 161280/161051
 | {{mapping| 552 875 1282 1550 1910 }}
-| &minus;0.1851
+| −0.1851
 | 0.1048
 | 4.82
@@ Line 56: / Line 56: @@
 | 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449
 | {{mapping| 552 875 1282 1550 1910 2043 }}
-| &minus;0.1892
+| −0.1892
 | 0.0961
 | 4.42
@@ Line 63: / Line 63: @@
 | 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379
 | {{mapping| 552 875 1282 1550 1910 2043 2345 }}
-| &minus;0.1727
+| −0.1727
 | 0.0977
 | 4.50
@@ Line 109: / Line 109: @@
 | [[Palladium]] (5-limit)
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct