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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}


== Theory ==
== Theory ==
552edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out|tempers out]] {{monzo| 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 [[support]]s [[sextile]] and gives a good tuning for it.  
552edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out|tempers out]] {{monzo| 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 [[support]]s [[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 ===
=== Prime harmonics ===
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
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| {{monzo| 875 -552 }}
| {{monzo| 875 -552 }}
| {{mapping| 552 875 }}
| {{mapping| 552 875 }}
| -0.0691
| −0.0691
| 0.0691
| 0.0691
| 3.18
| 3.18
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| {{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
| −0.1383
| 0.1130
| 0.1130
| 5.20
| 5.20
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| 250047/250000, 589824/588245, 33554432/33480783
| 250047/250000, 589824/588245, 33554432/33480783
| {{mapping| 552 875 1282 1550 }}
| {{mapping| 552 875 1282 1550 }}
| -0.1696
| −0.1696
| 0.1118
| 0.1118
| 5.15
| 5.15
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| 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
| −0.1851
| 0.1048
| 0.1048
| 4.82
| 4.82
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| 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
| −0.1892
| 0.0961
| 0.0961
| 4.42
| 4.42
|-
| 2.3.5.7.11.13.19
| 1216/1215, 1716/1715, 2080/2079, 2376/2375, 9633/9625, 15390/15379
| {{mapping| 552 875 1282 1550 1910 2043 2345 }}
| −0.1727
| 0.0977
| 4.50
|}
|}
* 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 ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 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
| 232\552<br />(2\552)
| 504.348<br />(4/348)
| 7/5<br />(?)
| [[Chromium]]
|-
| 46
| 229\552<br />(1\552)
| 497.83<br />(97.83)
| 4/3<br />(?)
| [[Palladium]] (5-limit)
|}
|}
<nowiki>*</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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
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