3125edo: Difference between revisions
→Regular temperament properties: cleanup, "curious notations" is not notable |
→Rank-2 temperaments: high accuracy ones |
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>(Reduced) | ! Generator<br>(Reduced) | ||
! Cents<br>(Reduced) | ! Cents<br>(Reduced) | ||
| Line 24: | Line 24: | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|139\3125 | | 139\3125 | ||
|53.376 | | 53.376 | ||
|33/32 | | 33/32 | ||
|[[Prequartismic]] | | [[Prequartismic]] | ||
|- | |||
| 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 | | 1 | ||
Revision as of 01:10, 21 March 2023
| ← 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 | 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 |