2048edo: Difference between revisions
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{{ | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | == Theory == | ||
2048edo has an excellent [[3/2|perfect fifth]], which derives from [[1024edo]]. It [[tempering out|tempers out]] the breedsma ([[2401/2400]]) in the 7-limit, and supports the rank-3 breed temperament. | |||
2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13's limit patent val is unambiguous and has an error of only 0.019 cents/octave. | |||
In the 5-limit, it supports [[monzismic]] temperament. In the 11-limit, 2048edo tempers out the [[quartisma]]. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|2048}} | {{Harmonics in equal|2048}} | ||
2048edo is | === Subsets and supersets === | ||
2048edo is the 11th-power-of-two edo, with subset edos {{EDOs| 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024 }}. | |||
== 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{monzo| 54 -37 2 }}, {{monzo| -111 -12 56 }} | |||
| {{mapping| 2048 3246 4755 }} | |||
| +0.0264 | |||
| 0.0364 | |||
| 6.22 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, {{monzo| -18 -13 13 3 }}, {{monzo| 49 -38 0 4 }} | |||
| {{mapping| 2048 3246 4755 5749 }} | |||
| +0.0439 | |||
| 0.0438 | |||
| 7.48 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179 | |||
| {{mapping| 2048 3246 4755 5749 7085 }} | |||
| +0.0323 | |||
| 0.0456 | |||
| 7.78 | |||
|} | |||
== Scales == | |||
* [[MET-24]] | |||
Latest revision as of 23:06, 20 February 2025
| ← 2047edo | 2048edo | 2049edo → |
2048 equal divisions of the octave (abbreviated 2048edo or 2048ed2), also called 2048-tone equal temperament (2048tet) or 2048 equal temperament (2048et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2048 equal parts of about 0.586 ¢ each. Each step represents a frequency ratio of 21/2048, or the 2048th root of 2.
Theory
2048edo has an excellent perfect fifth, which derives from 1024edo. It tempers out the breedsma (2401/2400) in the 7-limit, and supports the rank-3 breed temperament.
2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13's limit patent val is unambiguous and has an error of only 0.019 cents/octave.
In the 5-limit, it supports monzismic temperament. In the 11-limit, 2048edo tempers out the quartisma.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.002 | -0.181 | -0.271 | +0.049 | +0.293 | -0.073 | +0.143 | -0.149 | -0.085 | -0.114 |
| Relative (%) | +0.0 | -0.3 | -30.9 | -46.3 | +8.4 | +49.9 | -12.4 | +24.4 | -25.5 | -14.5 | -19.4 | |
| Steps (reduced) |
2048 (0) |
3246 (1198) |
4755 (659) |
5749 (1653) |
7085 (941) |
7579 (1435) |
8371 (179) |
8700 (508) |
9264 (1072) |
9949 (1757) |
10146 (1954) | |
Subsets and supersets
2048edo is the 11th-power-of-two edo, with subset edos 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [54 -37 2⟩, [-111 -12 56⟩ | [⟨2048 3246 4755]] | +0.0264 | 0.0364 | 6.22 |
| 2.3.5.7 | 2401/2400, [-18 -13 13 3⟩, [49 -38 0 4⟩ | [⟨2048 3246 4755 5749]] | +0.0439 | 0.0438 | 7.48 |
| 2.3.5.7.11 | 2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179 | [⟨2048 3246 4755 5749 7085]] | +0.0323 | 0.0456 | 7.78 |