2711edo: Difference between revisions
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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 | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 25: | Line 26: | ||
| {{monzo| 4297 -2711 }} | | {{monzo| 4297 -2711 }} | ||
| {{mapping| 2711 4297 }} | | {{mapping| 2711 4297 }} | ||
| | | −0.0233 | ||
| 0.0233 | | 0.0233 | ||
| 5.26 | | 5.26 | ||
| Line 32: | Line 33: | ||
| {{monzo| 77 -31 -12 }}, {{monzo| 18 -89 53 }} | | {{monzo| 77 -31 -12 }}, {{monzo| 18 -89 53 }} | ||
| {{mapping| 2711 4297 6295 }} | | {{mapping| 2711 4297 6295 }} | ||
| | | −0.0316 | ||
| 0.0223 | | 0.0223 | ||
| 5.04 | | 5.04 | ||
| Line 39: | Line 40: | ||
| {{monzo| 3 -13 10 -2 }}, {{monzo| 37 -9 -11 1 }}, {{monzo| 0 -11 -7 12 }} | | {{monzo| 3 -13 10 -2 }}, {{monzo| 37 -9 -11 1 }}, {{monzo| 0 -11 -7 12 }} | ||
| {{mapping| 2711 4297 6295 7611 }} | | {{mapping| 2711 4297 6295 7611 }} | ||
| | | −0.0340 | ||
| 0.0198 | | 0.0198 | ||
| 4.47 | | 4.47 | ||
| Line 46: | Line 47: | ||
| 151263/151250, 14348907/14348180, 2359296/2358125, 21437500/21434787 | | 151263/151250, 14348907/14348180, 2359296/2358125, 21437500/21434787 | ||
| {{mapping| 2711 4297 6295 7611 9379 }} | | {{mapping| 2711 4297 6295 7611 9379 }} | ||
| | | −0.0395 | ||
| 0.0209 | | 0.0209 | ||
| 4.72 | | 4.72 | ||
| Line 53: | Line 54: | ||
| 4096/4095, 43940/43923, 67392/67375, 151263/151250, 4429568/4428675 | | 4096/4095, 43940/43923, 67392/67375, 151263/151250, 4429568/4428675 | ||
| {{mapping| 2711 4297 6295 7611 9379 10032 }} | | {{mapping| 2711 4297 6295 7611 9379 10032 }} | ||
| | | −0.0351 | ||
| 0.0215 | | 0.0215 | ||
| 4.86 | | 4.86 | ||
Latest revision as of 12:53, 21 February 2025
| ← 2710edo | 2711edo | 2712edo → |
2711 equal divisions of the octave (abbreviated 2711edo or 2711ed2), also called 2711-tone equal temperament (2711tet) or 2711 equal temperament (2711et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2711 equal parts of about 0.443 ¢ each. Each step represents a frequency ratio of 21/2711, or the 2711th root of 2.
Theory
2711edo is distinctly consistent to the 15-odd-limit, or the no-11 19-odd-limit. The equal temperament tempers out 78125000/78121827 in the 7-limit; 35156250/35153041, 14348907/14348180, 21437500/21434787, 151263/151250, 2359296/2358125, 5767168/5764801 and 199297406/199290375 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.074 | +0.112 | +0.115 | +0.213 | +0.048 | -0.049 | -0.058 | -0.167 | +0.006 | +0.077 |
| Relative (%) | +0.0 | +16.7 | +25.3 | +26.1 | +48.1 | +10.8 | -11.2 | -13.1 | -37.6 | +1.4 | +17.4 | |
| Steps (reduced) |
2711 (0) |
4297 (1586) |
6295 (873) |
7611 (2189) |
9379 (1246) |
10032 (1899) |
11081 (237) |
11516 (672) |
12263 (1419) |
13170 (2326) |
13431 (2587) | |
Subsets and supersets
2711edo is the 395th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [4297 -2711⟩ | [⟨2711 4297]] | −0.0233 | 0.0233 | 5.26 |
| 2.3.5 | [77 -31 -12⟩, [18 -89 53⟩ | [⟨2711 4297 6295]] | −0.0316 | 0.0223 | 5.04 |
| 2.3.5.7 | [3 -13 10 -2⟩, [37 -9 -11 1⟩, [0 -11 -7 12⟩ | [⟨2711 4297 6295 7611]] | −0.0340 | 0.0198 | 4.47 |
| 2.3.5.7.11 | 151263/151250, 14348907/14348180, 2359296/2358125, 21437500/21434787 | [⟨2711 4297 6295 7611 9379]] | −0.0395 | 0.0209 | 4.72 |
| 2.3.5.7.11.13 | 4096/4095, 43940/43923, 67392/67375, 151263/151250, 4429568/4428675 | [⟨2711 4297 6295 7611 9379 10032]] | −0.0351 | 0.0215 | 4.86 |