541edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|541}}
{{ED intro}}


== Theory ==
== Theory ==
541et is only consistent to the [[5-odd-limit]] and the [[harmonic]] 3 is about halfway between ist steps. It has a reasonable approximation to the 2.9.5.7.11.13 [[subgroup]].
541et is only [[consistent]] to the [[5-odd-limit]] and the [[harmonic]] [[3/1|3]] is about halfway between its steps. It has a reasonable approximation to the 2.9.5.7.13 [[subgroup]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
541edo is the 100th [[prime EDO]]. [[1082edo]], which doubles it, gives a good correction to the harmonic 3.  
541edo is the 100th [[prime edo]]. [[1082edo]], which doubles it, gives a good correction to the harmonic 3.


== 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" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|1715 -541}}
| {{monzo| 1715 -541 }}
|{{mapping|541 1715}}
| {{mapping| 541 1715 }}
| -0.0247
| −0.0247
| 0.0247
| 0.0247
| 1.11
| 1.11
|-
|-
|2.9.5
| 2.9.5
|{{monzo|-20 -12 25}}, {{monzo|63 -25 7}}
| {{monzo| -20 -12 25 }}, {{monzo| 63 -25 7 }}
|{{mapping|541 1715 1256}}
| {{mapping| 541 1715 1256 }}
| +0.0355
| +0.0355
| 0.0874
| 0.0874
| 3.94
| 3.94
|-
|-
|2.9.5.7
| 2.9.5.7
|95703125/95551488, 43046721/43025920, 1280000000/1275989841
| 40500000/40353607, 43046721/43025920, 95703125/95551488
|{{mapping|541 1715 1256 1519}}
| {{mapping| 541 1715 1256 1519 }}
| -0.0171
| −0.0171
| 0.1184
| 0.1184
| 5.34
| 5.34
|-
|-
|2.9.5.7.11
| 2.9.5.7.13
|6250/6237, 496125/495616, 46656/46585, 275653125/275365888
| 4096/4095, 10985/10976, 2734375/2729376, 11390625/11361532
|{{mapping|541 1715 1256 1519 1872}}
| {{mapping| 541 1715 1256 1519 2002 }}
| -0.0710
| −0.0211
| 0.1512
| 0.1062
| 6.82
| 4.79
|-
|2.9.5.7.11.13
|1575/1573, 4096/4095, 6250/6237, 67392/67375, 3247695/3246152
|{{mapping|541 1715 1256 1519 1872 2002}}
| -0.0654
| 0.1386
| 6.25
|}
|}

Latest revision as of 17:38, 20 February 2025

← 540edo 541edo 542edo →
Prime factorization 541 (prime)
Step size 2.21811 ¢ 
Fifth 316\541 (700.924 ¢)
Semitones (A1:m2) 48:43 (106.5 ¢ : 95.38 ¢)
Dual sharp fifth 317\541 (703.142 ¢)
Dual flat fifth 316\541 (700.924 ¢)
Dual major 2nd 92\541 (204.067 ¢)
Consistency limit 5
Distinct consistency limit 5

541 equal divisions of the octave (abbreviated 541edo or 541ed2), also called 541-tone equal temperament (541tet) or 541 equal temperament (541et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 541 equal parts of about 2.22 ¢ each. Each step represents a frequency ratio of 21/541, or the 541st root of 2.

Theory

541et is only consistent to the 5-odd-limit and the harmonic 3 is about halfway between its steps. It has a reasonable approximation to the 2.9.5.7.13 subgroup.

Odd harmonics

Approximation of odd harmonics in 541edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.031 -0.362 +0.490 +0.157 +0.993 +0.138 +0.826 -0.704 -0.286 -0.541 -0.548
Relative (%) -46.5 -16.3 +22.1 +7.1 +44.7 +6.2 +37.2 -31.7 -12.9 -24.4 -24.7
Steps
(reduced) 		857
(316) 		1256
(174) 		1519
(437) 		1715
(92) 		1872
(249) 		2002
(379) 		2114
(491) 		2211
(47) 		2298
(134) 		2376
(212) 		2447
(283)

Subsets and supersets

541edo is the 100th prime edo. 1082edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.9 [1715 -541⟩ [⟨541 1715]] −0.0247 0.0247 1.11
2.9.5 [-20 -12 25⟩, [63 -25 7⟩ [⟨541 1715 1256]] +0.0355 0.0874 3.94
2.9.5.7 40500000/40353607, 43046721/43025920, 95703125/95551488 [⟨541 1715 1256 1519]] −0.0171 0.1184 5.34
2.9.5.7.13 4096/4095, 10985/10976, 2734375/2729376, 11390625/11361532 [⟨541 1715 1256 1519 2002]] −0.0211 0.1062 4.79
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