2edt: Difference between revisions
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== Theory == | == Theory == | ||
As a temperament in the 3.5 | As a temperament in the [[3.5 subgroup]], it tempers out [[27/25]], equating 5/3 with 9/5. Because of this, it is related to the [[bug]] temperament. In fact, removing octaves from the [[bug]] temperament results in a 3.5 interpretation of 2edt. | ||
Since [[26/15]] is a [[convergent]] of sqrt(3), 26/15 (and its [[tritave complement]] 45/26) are good rational representations of the square root of 3. 2edt thus tempers out (26/15)<sup>2</sup> / (3/1) = [[676/675]], the island comma. | Since [[26/15]] is a [[convergent]] of sqrt(3), 26/15 (and its [[tritave complement]] 45/26) are good rational representations of the square root of 3. 2edt thus tempers out (26/15)<sup>2</sup> / (3/1) = [[676/675]], the island comma. | ||
Revision as of 04:53, 19 September 2026
| ← 1edt | 2edt | 3edt → |
(convergent)
2 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 2edt or 2ed3), is a nonoctave tuning system that divides the interval of 3/1 into 2 equal parts of about 951 ¢ each. Each step represents a frequency ratio of 31/2, or the square root of 3.
Theory
As a temperament in the 3.5 subgroup, it tempers out 27/25, equating 5/3 with 9/5. Because of this, it is related to the bug temperament. In fact, removing octaves from the bug temperament results in a 3.5 interpretation of 2edt.
Since 26/15 is a convergent of sqrt(3), 26/15 (and its tritave complement 45/26) are good rational representations of the square root of 3. 2edt thus tempers out (26/15)2 / (3/1) = 676/675, the island comma.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -249 | +0 | +453 | +67 | -249 | +435 | +204 | +0 | -182 | -347 | +453 | +314 | +186 | +67 |
| Relative (%) | -26.2 | +0.0 | +47.6 | +7.0 | -26.2 | +45.8 | +21.4 | +0.0 | -19.2 | -36.5 | +47.6 | +33.1 | +19.6 | +7.0 | |
| Steps (reduced) |
1 (1) |
2 (0) |
3 (1) |
3 (1) |
3 (1) |
4 (0) |
4 (0) |
4 (0) |
4 (0) |
4 (0) |
5 (1) |
5 (1) |
5 (1) |
5 (1) | |
Relationship to octave temperaments
One step of 2edt can represent the generator for any rank-2 octavated temperament which takes 2 generators to reach the 3rd harmonic, such as monzismic.