User:Francium/5113edo: Difference between revisions

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Created page with "{{Infobox ET}} {{ED intro}} == Theory == 5113edo is consistent to the 11-limit, tempering out 21437500/21434787, 47265625/47258883, 184549376/184528125 and 246071287/246037500 in the 11-limit; 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928 and 1063348/1063125 in the 13-limit; and 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893 and 7109375/7108992 in the 17-limit. {{Harmonics in equal|5113}}"
 
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== Theory ==
== Theory ==
5113edo is [[consistent]] to the [[11-limit]], [[tempering out]] 21437500/21434787, 47265625/47258883, 184549376/184528125 and 246071287/246037500 in the 11-limit; [[123201/123200]], 196625/196608, 1664000/1663893, 5175625/5174928 and 1063348/1063125 in the [[13-limit]]; and [[12376/12375]], 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893 and 7109375/7108992 in the [[17-limit]].
5113edo is [[consistent]] to the [[11-limit]], [[tempering out]] 21437500/21434787, 47265625/47258883, 184549376/184528125 and 246071287/246037500 in the 11-limit; [[123201/123200]], 196625/196608, 1664000/1663893, 5175625/5174928 and 1063348/1063125 in the [[13-limit]]; and [[12376/12375]], 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893 and 7109375/7108992 in the [[17-limit]]. It is strong in the 2.3.5.7.11.17.23.29.31 [[subgroup]], tempering out 21505/21504, 126225/126224, 30625/30624, 150920/150903, 15625/15624, 53361/53360, 750200/750141 and 69632/69629.


=== Prime harmonics ===
{{Harmonics in equal|5113}}
{{Harmonics in equal|5113}}
=== Subsets and supersets ===
5113edo is the 684th [[prime edo]].
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo|8104 -5113}}
| {{mapping|5113 8104}}
| −0.0064
| 0.0064
| 2.73
|-
| 2.3.5
| {{monzo|56 -91 38}}, {{monzo|144 -22 -47}}
| {{mapping|5113 8104 11872}}
| −0.0037
| 0.0065
| 2.77
|-
| 2.3.5.7
| 24414062500/24407490807, 13841287201/13839609375, 281484423828125/281474976710656
| {{mapping|5113 8104 11872 14354}}
| −0.0026
| 0.0059
| 2.51
|-
| 2.3.5.7.11
| 21437500/21434787, 47265625/47258883, 184549376/184528125, 246071287/246037500
| {{mapping|5113 8104 11872 14354 17688}}
| −0.0011
| 0.0061
| 2.60
|-
| 2.3.5.7.11.13
| 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928, 1063348/1063125
| {{mapping|5113 8104 11872 14354 17688 18920}}
| +0.0028
| 0.0103
| 4.39
|-
| 2.3.5.7.11.13.17
| 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893, 7109375/7108992
| {{mapping|5113 8104 11872 14354 17688 18920 20899}}
| +0.0040
| 0.0100
| 4.26
|}

Latest revision as of 19:18, 3 June 2026

← 5112edo 5113edo 5114edo →
Prime factorization 5113 (prime)
Step size 0.234696 ¢ 
Fifth 2991\5113 (701.975 ¢)
Semitones (A1:m2) 485:384 (113.8 ¢ : 90.12 ¢)
Consistency limit 11
Distinct consistency limit 11

5113 equal divisions of the octave (abbreviated 5113edo or 5113ed2), also called 5113-tone equal temperament (5113tet) or 5113 equal temperament (5113et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 5113 equal parts of about 0.235 ¢ each. Each step represents a frequency ratio of 21/5113, or the 5113th root of 2.

Theory

5113edo is consistent to the 11-limit, tempering out 21437500/21434787, 47265625/47258883, 184549376/184528125 and 246071287/246037500 in the 11-limit; 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928 and 1063348/1063125 in the 13-limit; and 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893 and 7109375/7108992 in the 17-limit. It is strong in the 2.3.5.7.11.17.23.29.31 subgroup, tempering out 21505/21504, 126225/126224, 30625/30624, 150920/150903, 15625/15624, 53361/53360, 750200/750141 and 69632/69629.

Prime harmonics

Approximation of prime harmonics in 5113edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.020 -0.004 -0.001 -0.017 -0.082 -0.046 +0.081 +0.007 +0.034 +0.046
Relative (%) +0.0 +8.7 -1.8 -0.6 -7.4 -34.8 -19.8 +34.7 +2.8 +14.3 +19.4
Steps
(reduced) 		5113
(0) 		8104
(2991) 		11872
(1646) 		14354
(4128) 		17688
(2349) 		18920
(3581) 		20899
(447) 		21720
(1268) 		23129
(2677) 		24839
(4387) 		25331
(4879)

Subsets and supersets

5113edo is the 684th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [8104 -5113⟩ [⟨5113 8104]] −0.0064 0.0064 2.73
2.3.5 [56 -91 38⟩, [144 -22 -47⟩ [⟨5113 8104 11872]] −0.0037 0.0065 2.77
2.3.5.7 24414062500/24407490807, 13841287201/13839609375, 281484423828125/281474976710656 [⟨5113 8104 11872 14354]] −0.0026 0.0059 2.51
2.3.5.7.11 21437500/21434787, 47265625/47258883, 184549376/184528125, 246071287/246037500 [⟨5113 8104 11872 14354 17688]] −0.0011 0.0061 2.60
2.3.5.7.11.13 123201/123200, 196625/196608, 1664000/1663893, 5175625/5174928, 1063348/1063125 [⟨5113 8104 11872 14354 17688 18920]] +0.0028 0.0103 4.39
2.3.5.7.11.13.17 12376/12375, 123201/123200, 221221/221184, 4685824/4685625, 1664000/1663893, 7109375/7108992 [⟨5113 8104 11872 14354 17688 18920 20899]] +0.0040 0.0100 4.26
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