1448edo: Difference between revisions
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The 1448edo is a strong 13-limit system, and it is an excellent 2.3.5.7.11.13.19.23 [[subgroup]] system. It is a [[zeta peak edo]], and provides the [[optimal patent val]] for [[donar]]. A basis for the 13-limit [[comma]]s is {[[3025/3024]], [[4225/4224]], [[4375/4374]], 140625/140608, 823680/823543}. | |||
Notably, it is the first edo to be [[diamond monotone]] to the [[95-odd-limit]], completing the first five octaves and a fifth of the [[harmonic series]], in fact by the [[patent val]]. It is thus usable in the full [[89-limit]], where prime 89 is the start of a record {{W|prime gap}} from 89 to 97. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|1448}} | {{Harmonics in equal|1448|columns=12}} | ||
{{Harmonics in equal|1448|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 1448edo (continued)}} | |||
=== Subsets and supersets === | |||
Since 1448 factors into {{factorization|1448}}, it has subset edos 2, 4, 8, 181, 362, and 724. | |||
[[Category: | [[Category:Thor]] | ||
[[Category:Donar]] | |||
Latest revision as of 04:09, 27 March 2026
| ← 1447edo | 1448edo | 1449edo → |
1448 equal divisions of the octave (abbreviated 1448edo or 1448ed2), also called 1448-tone equal temperament (1448tet) or 1448 equal temperament (1448et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1448 equal parts of about 0.829 ¢ each. Each step represents a frequency ratio of 21/1448, or the 1448th root of 2.
The 1448edo is a strong 13-limit system, and it is an excellent 2.3.5.7.11.13.19.23 subgroup system. It is a zeta peak edo, and provides the optimal patent val for donar. A basis for the 13-limit commas is {3025/3024, 4225/4224, 4375/4374, 140625/140608, 823680/823543}.
Notably, it is the first edo to be diamond monotone to the 95-odd-limit, completing the first five octaves and a fifth of the harmonic series, in fact by the patent val. It is thus usable in the full 89-limit, where prime 89 is the start of a record prime gap from 89 to 97.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.021 | -0.126 | -0.041 | -0.213 | -0.196 | +0.293 | +0.001 | -0.098 | -0.295 | +0.268 | -0.239 |
| Relative (%) | +0.0 | -2.6 | -15.2 | -5.0 | -25.7 | -23.7 | +35.4 | +0.1 | -11.8 | -35.6 | +32.4 | -28.8 | |
| Steps (reduced) |
1448 (0) |
2295 (847) |
3362 (466) |
4065 (1169) |
5009 (665) |
5358 (1014) |
5919 (127) |
6151 (359) |
6550 (758) |
7034 (1242) |
7174 (1382) |
7543 (303) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.219 | -0.192 | -0.037 | -0.024 | -0.056 | +0.242 | +0.251 | +0.138 | +0.111 | +0.104 | -0.047 | +0.114 |
| Relative (%) | +26.5 | -23.1 | -4.5 | -2.9 | -6.7 | +29.2 | +30.3 | +16.6 | +13.4 | +12.5 | -5.7 | +13.8 | |
| Steps (reduced) |
7758 (518) |
7857 (617) |
8043 (803) |
8294 (1054) |
8518 (1278) |
8588 (1348) |
8784 (96) |
8905 (217) |
8963 (275) |
9128 (440) |
9231 (543) |
9377 (689) | |
Subsets and supersets
Since 1448 factors into 23 × 181, it has subset edos 2, 4, 8, 181, 362, and 724.