552edo: Difference between revisions
Notable as a 2.3.5.7.11.13.19 subgroup temp |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 875 -552 }} | | {{monzo| 875 -552 }} | ||
| {{mapping| 552 875 }} | | {{mapping| 552 875 }} | ||
| | | −0.0691 | ||
| 0.0691 | | 0.0691 | ||
| 3.18 | | 3.18 | ||
| Line 34: | Line 26: | ||
| {{monzo| 8 14 -13 }}, {{monzo| 71 -36 -6 }} | | {{monzo| 8 14 -13 }}, {{monzo| 71 -36 -6 }} | ||
| {{mapping| 552 875 1282 }} | | {{mapping| 552 875 1282 }} | ||
| | | −0.1383 | ||
| 0.1130 | | 0.1130 | ||
| 5.20 | | 5.20 | ||
| Line 41: | Line 33: | ||
| 250047/250000, 589824/588245, 33554432/33480783 | | 250047/250000, 589824/588245, 33554432/33480783 | ||
| {{mapping| 552 875 1282 1550 }} | | {{mapping| 552 875 1282 1550 }} | ||
| | | −0.1696 | ||
| 0.1118 | | 0.1118 | ||
| 5.15 | | 5.15 | ||
| Line 48: | Line 40: | ||
| 5632/5625, 9801/9800, 151263/151250, 161280/161051 | | 5632/5625, 9801/9800, 151263/151250, 161280/161051 | ||
| {{mapping| 552 875 1282 1550 1910 }} | | {{mapping| 552 875 1282 1550 1910 }} | ||
| | | −0.1851 | ||
| 0.1048 | | 0.1048 | ||
| 4.82 | | 4.82 | ||
| Line 55: | Line 47: | ||
| 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449 | | 1716/1715, 2080/2079, 5632/5625, 10648/10647, 20480/20449 | ||
| {{mapping| 552 875 1282 1550 1910 2043 }} | | {{mapping| 552 875 1282 1550 1910 2043 }} | ||
| | | −0.1892 | ||
| 0.0961 | | 0.0961 | ||
| 4.42 | | 4.42 | ||
| Line 62: | Line 54: | ||
| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379 | | 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379 | ||
| {{mapping| 552 875 1282 1550 1910 2043 2345 }} | | {{mapping| 552 875 1282 1550 1910 2043 2345 }} | ||
| | | −0.1727 | ||
| 0.0977 | | 0.0977 | ||
| 4.50 | | 4.50 | ||
{{comma basis end}} | |||
* 552et is notable for being the first equal temperament to beat [[270edo|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 [[581edo|581]]. | * 552et is notable for being the first equal temperament to beat [[270edo|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 [[581edo|581]]. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 90: | Line 76: | ||
|- | |- | ||
| 6 | | 6 | ||
| 229\552<br>(45\552) | | 229\552<br />(45\552) | ||
| 497.83<br>(97.83) | | 497.83<br />(97.83) | ||
| 4/3<br>(128/121) | | 4/3<br />(128/121) | ||
| [[Sextile]] | | [[Sextile]] | ||
|- | |- | ||
| 24 | | 24 | ||
| 232\552<br>(2\552) | | 232\552<br />(2\552) | ||
| 504.348<br>(4/348) | | 504.348<br />(4/348) | ||
| 7/5<br>(?) | | 7/5<br />(?) | ||
| [[Chromium]] | | [[Chromium]] | ||
|- | |- | ||
| 46 | | 46 | ||
| 229\552<br>(1\552) | | 229\552<br />(1\552) | ||
| 497.83<br>(97.83) | | 497.83<br />(97.83) | ||
| 4/3<br>(?) | | 4/3<br />(?) | ||
| [[Palladium]] (5-limit) | | [[Palladium]] (5-limit) | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 04:55, 16 November 2024
| ← 551edo | 552edo | 553edo → |
Theory
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.
Prime harmonics
| 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 |
| 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) | |
Subsets and supersets
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.
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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)
Template:Rank-2 end
Template:Orf