541edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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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|Comma List]] | ||
| Line 25: | Line 26: | ||
| {{monzo| 1715 -541 }} | | {{monzo| 1715 -541 }} | ||
| {{mapping| 541 1715 }} | | {{mapping| 541 1715 }} | ||
| | | −0.0247 | ||
| 0.0247 | | 0.0247 | ||
| 1.11 | | 1.11 | ||
| Line 39: | Line 40: | ||
| 40500000/40353607, 43046721/43025920, 95703125/95551488 | | 40500000/40353607, 43046721/43025920, 95703125/95551488 | ||
| {{mapping| 541 1715 1256 1519 }} | | {{mapping| 541 1715 1256 1519 }} | ||
| | | −0.0171 | ||
| 0.1184 | | 0.1184 | ||
| 5.34 | | 5.34 | ||
| Line 46: | Line 47: | ||
| 4096/4095, 10985/10976, 2734375/2729376, 11390625/11361532 | | 4096/4095, 10985/10976, 2734375/2729376, 11390625/11361532 | ||
| {{mapping| 541 1715 1256 1519 2002 }} | | {{mapping| 541 1715 1256 1519 2002 }} | ||
| | | −0.0211 | ||
| 0.1062 | | 0.1062 | ||
| 4.79 | | 4.79 | ||
|} | |} | ||
Latest revision as of 17:38, 20 February 2025
| ← 540edo | 541edo | 542edo → |
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
| 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 |