3125edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
3125et is notable for being an extremely strong [[7-limit]] system. It is also [[consistent]] through the [[15-odd-limit]], and except for [[17/11]], [[19/17]] and their [[octave complement]]s, it is consistent to the [[35-odd-limit]]. | |||
In the | 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 [[quartisma|117440512/117406179]] are tempered out—it should be noted this edo is so far the only one [https://en.xen.wiki/index.php?title=3125edo&oldid=7860 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]] and 1399680/1399489 are all tempered out. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|3125|columns=11}} | {{Harmonics in equal|3125|columns=11}} | ||
{{Harmonics in equal|3125|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 3125edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
{{nowrap| 3125 {{=}} 5<sup>5</sup> }}, and as such 3125edo is the 5th edo of the form ''n''<sup>''n''</sup>. It has subset edos {{EDOs| 5, 25, 125, and 625 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
3125et is | 3125et is the first equal temperament past [[171edo|171]] with a lower [[Tenney–Euclidean temperament measures #TE simple badness|relative error]]. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 68: | Line 69: | ||
| [[Revopent]] | | [[Revopent]] | ||
|} | |} | ||
<nowiki />* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
Latest revision as of 11:02, 28 May 2026
| ← 3124edo | 3125edo | 3126edo → |
3125 equal divisions of the octave (abbreviated 3125edo or 3125ed2), also called 3125-tone equal temperament (3125tet) or 3125 equal temperament (3125et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3125 equal parts of exactly 0.384 ¢ each. Each step represents a frequency ratio of 21/3125, or the 3125th root of 2.
Theory
3125et is notable for being an extremely strong 7-limit system. It is also consistent through the 15-odd-limit, and except for 17/11, 19/17 and their octave complements, it is consistent to the 35-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 and 1399680/1399489 are all tempered out.
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) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.176 | -0.134 | -0.030 | -0.035 | +0.095 | -0.100 | +0.171 | +0.181 | +0.015 | -0.077 | -0.121 |
| Relative (%) | +45.8 | -35.0 | -7.7 | -9.0 | +24.9 | -26.0 | +44.6 | +47.1 | +4.0 | -20.2 | -31.5 | |
| Steps (reduced) |
16280 (655) |
16742 (1117) |
16957 (1332) |
17358 (1733) |
17900 (2275) |
18383 (2758) |
18534 (2909) |
18957 (207) |
19218 (468) |
19343 (593) |
19699 (949) | |
Subsets and supersets
3125 = 55, and as such 3125edo is the 5th edo of the form nn. It has subset edos 5, 25, 125, and 625.
Regular temperament properties
3125et is the first equal temperament past 171 with a lower relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct