3125edo: Difference between revisions
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Revision as of 04:31, 9 July 2023
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This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
| ← 3124edo | 3125edo | 3126edo → |
Theory
3125edo is distinctly consistent through the 15-odd-limit. A basis for its 7-limit commas is 78125000/78121827, 645700815/645657712 and 281484423828125/281474976710656. In the 11-limit, 151263/151250, 820125/819896, 21437500/21434787 and 117440512/117406179 are tempered out – it should be noted this edo is so far the only one known to have been confirmed as tempering out 117440512/117406179 prior to the independent discovery of this comma's significance as the difference between a stack of five 33/32 quartertones and one 7/6 subminor third. In the 13-limit, 6656/6655, 123201/123200, 140625/140608, 151263/151250 and 1399680/1399489 are all tempered out.
In the 2.5.11.13.19.23.29.31 subgroup, it supports a temperament called estates general, described as 1789 & 3125.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.003 | -0.010 | +0.006 | +0.106 | +0.048 | -0.123 | +0.087 | -0.050 | -0.073 | +0.052 |
| Relative (%) | +0.0 | -0.8 | -2.5 | +1.6 | +27.6 | +12.6 | -32.1 | +22.7 | -13.1 | -19.1 | +13.7 | |
| Steps (reduced) |
3125 (0) |
4953 (1828) |
7256 (1006) |
8773 (2523) |
10811 (1436) |
11564 (2189) |
12773 (273) |
13275 (775) |
14136 (1636) |
15181 (2681) |
15482 (2982) | |
Subsets and supersets
3125 = 55 , and as such it is the 5th edo of the form x^x. It hhas subset edos 5, 25, 125, and 625.
Regular temperament properties
3125et is notable for being an extremely strong 7-limit system, being the first equal division past 171edo with a lower relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 139\3125 | 53.376 | 33/32 | Prequartismic |
| 1 | 411\3125 | 157.824 | 36756909/33554432 | Hemiegads |
| 1 | 577\3125 | 221.568 | 8388608/7381125 | Fortune |
| 1 | 822\3125 | 315.648 | 6/5 | Egads |
| 1 | 894\3125 | 343.296 | 8000/6561 | Raider |
| 1 | 1359\3125 | 521.856 | 80275/59392 | Estates general |
| 1 | 1412\3125 | 542.208 | 16807/12288 | Revopent |