3edt: Difference between revisions
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== | == Theory == | ||
3edt can be thought of as [[2edo]] with the 3/1 made just, by [[Stretched tuning|stretching]] the octave by 67.97{{c}}. | |||
Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63{{c}} flat of 13/1. One step of 3edt has two good 13-limit [[Nearest just interval|rational approximations]], [[13/9]] and 75/52, both which are [[convergent]]s. 3edt thus tempers out {{nowrap|(13/9)<sup>3</sup> / (3/1) {{=}} [[2197/2187]]}}, the threedie, and {{nowrap|(75/52)<sup>3</sup> / (3/1) {{=}} [[140625/140608]]}}, the catasma. The good approximation for 13/9 and 75/52 also implies a good approximation for 25/4, or ([[5/2]])<sup>2</sup>. | |||
[[ | === Harmonics === | ||
[[ | {{Harmonics in equal|3|3|1|columns=15}} | ||
== Relationship to octave temperaments == | |||
One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are: | |||
=== Simple octave temperaments === | |||
* [[Liese]] | |||
* [[Triton]] | |||
* [[Alphatricot]] | |||
=== Fractional-octave temperaments === | |||
* [[Augene]], [[augmented (temperament)|augmented]], [[august]] – can be seen as a superset of [[3edo]] and 3edt | |||
* [[Soviet ferris wheel]] – [[20edo]] and 3edt | |||
* [[Akjayland]] – [[21edo]] and 3edt | |||
* [[Oganesson]] – [[118edo]] and 3edt | |||
== See also == | |||
* [[Alpha, beta, and gamma family of equal divisions]] | |||
Latest revision as of 09:48, 26 February 2026
| ← 2edt | 3edt | 4edt → |
(convergent)
3 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 3edt or 3ed3), is a nonoctave tuning system that divides the interval of 3/1 into 3 equal parts of about 634 ¢ each. Each step represents a frequency ratio of 31/3, or the cube root of 3.
Theory
3edt can be thought of as 2edo with the 3/1 made just, by stretching the octave by 67.97 ¢.
Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63 ¢ flat of 13/1. One step of 3edt has two good 13-limit rational approximations, 13/9 and 75/52, both which are convergents. 3edt thus tempers out (13/9)3 / (3/1) = 2197/2187, the threedie, and (75/52)3 / (3/1) = 140625/140608, the catasma. The good approximation for 13/9 and 75/52 also implies a good approximation for 25/4, or (5/2)2.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68 | +0 | +136 | -250 | +68 | -199 | +204 | +0 | -182 | +287 | +136 | -3 | -131 | -250 | +272 |
| Relative (%) | +10.7 | +0.0 | +21.4 | -39.5 | +10.7 | -31.4 | +32.2 | +0.0 | -28.8 | +45.2 | +21.4 | -0.4 | -20.7 | -39.5 | +42.9 | |
| Steps (reduced) |
2 (2) |
3 (0) |
4 (1) |
4 (1) |
5 (2) |
5 (2) |
6 (0) |
6 (0) |
6 (0) |
7 (1) |
7 (1) |
7 (1) |
7 (1) |
7 (1) |
8 (2) | |
Relationship to octave temperaments
One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are:
Simple octave temperaments
Fractional-octave temperaments
- Augene, augmented, august – can be seen as a superset of 3edo and 3edt
- Soviet ferris wheel – 20edo and 3edt
- Akjayland – 21edo and 3edt
- Oganesson – 118edo and 3edt