1051edo: 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]] (¢) | ||
| Line 27: | Line 28: | ||
| {{monzo| 1666 -1051 }} | | {{monzo| 1666 -1051 }} | ||
| {{mapping| 1051 1666 }} | | {{mapping| 1051 1666 }} | ||
| | | −0.0736 | ||
| 0.0736 | | 0.0736 | ||
| 6.45 | | 6.45 | ||
| Line 41: | Line 42: | ||
| {{monzo| 40 7 -22 }}, {{monzo| 63 -50 7 }} | | {{monzo| 40 7 -22 }}, {{monzo| 63 -50 7 }} | ||
| {{mapping| 1051 1666 2441 }} (1051c) | | {{mapping| 1051 1666 2441 }} (1051c) | ||
| | | −0.1562 | ||
| 0.1313 | | 0.1313 | ||
| 11.5 | | 11.5 | ||
| Line 48: | Line 49: | ||
=== 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 | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
--> | --> | ||
Latest revision as of 12:54, 21 February 2025
| ← 1050edo | 1051edo | 1052edo → |
1051 equal divisions of the octave (abbreviated 1051edo or 1051ed2), also called 1051-tone equal temperament (1051tet) or 1051 equal temperament (1051et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1051 equal parts of about 1.14 ¢ each. Each step represents a frequency ratio of 21/1051, or the 1051st root of 2.
Theory
1051edo only has a consistency limit of 3 and does poorly with approximating the harmonic 5. However, it has a reasonable representation of the 2.3.7.11.17.19 subgroup.
Assume the patent val, 1051et tempers out 2460375/2458624 in the 7-limit; 820125/819896, 2097152/2096325, 514714375/514434888, 180224/180075, 184549376/184528125, 43923/43904 and 20614528/20588575 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.233 | -0.396 | +0.537 | +0.467 | +0.157 | -0.185 | -0.162 | +0.087 | +0.489 | -0.372 | -0.301 |
| Relative (%) | +20.4 | -34.6 | +47.0 | +40.9 | +13.7 | -16.2 | -14.2 | +7.7 | +42.8 | -32.6 | -26.4 | |
| Steps (reduced) |
1666 (615) |
2440 (338) |
2951 (849) |
3332 (179) |
3636 (483) |
3889 (736) |
4106 (953) |
4296 (92) |
4465 (261) |
4616 (412) |
4754 (550) | |
Subsets and supersets
1051edo is the 177th prime edo. 2102edo, which doubles it, gives a good correction to the harmonic 5 and 7. 4212edo, which quadruples it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1666 -1051⟩ | [⟨1051 1666]] | −0.0736 | 0.0736 | 6.45 |
| 2.3.5 | [-68 18 17⟩, [-26 -29 31⟩ | [⟨1051 1666 2440]] (1051) | +0.0077 | 0.1298 | 11.4 |
| 2.3.5 | [40 7 -22⟩, [63 -50 7⟩ | [⟨1051 1666 2441]] (1051c) | −0.1562 | 0.1313 | 11.5 |
Music
- you have to run! (2023) – edson in 1051edo tuning