552edo: Difference between revisions

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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
| 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]]
| [[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

Latest revision as of 13:31, 13 March 2026

← 551edo 552edo 553edo →
Prime factorization 23 × 3 × 23
Step size 2.17391 ¢ 
Fifth 323\552 (702.174 ¢)
Semitones (A1:m2) 53:41 (115.2 ¢ : 89.13 ¢)
Consistency limit 15
Distinct consistency limit 15

552 equal divisions of the octave (abbreviated 552edo or 552ed2), also called 552-tone equal temperament (552tet) or 552 equal temperament (552et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 552 equal parts of about 2.17 ¢ each. Each step represents a frequency ratio of 21/552, or the 552nd root of 2.

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

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
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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
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.

Rank-2 temperaments

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

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