10edt: Difference between revisions
→Harmonics: integer and prime harmonics side-by-side is nonstandard and potentially confusing |
"Very accurate 5-limit harmony" what? |
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== Theory == | == Theory == | ||
10edt | 10edt most notably inherits the [[5/4]] from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. | ||
10edt also splits the | 10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone. | ||
10edt can serve as the generator chain for the [[pocus]] temperament, a [[temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup. | |||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|10|3|1}} | {{Harmonics in equal|10|3|1|columns=11}} | ||
{{Harmonics in equal|10|3|1|start=12}} | {{Harmonics in equal|10|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 10edt (continued)}} | ||
== Interval table == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
Revision as of 07:12, 31 May 2026
| ← 9edt | 10edt | 11edt → |
10 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 10edt or 10ed3), is a nonoctave tuning system that divides the interval of 3/1 into 10 equal parts of about 190 ¢ each. Each step represents a frequency ratio of 31/10, or the 10th root of 3.
Theory
10edt most notably inherits the 5/4 from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as 7/5. We can use this to readily construct chords such as 4:5:7:12, although the 7/4, being 18 cents flat, does considerable damage to the consonance of this chord.
10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone.
10edt can serve as the generator chain for the pocus temperament, a merge of sensamagic and 2.3.5.7.13 catakleismic, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -58.8 | +0.0 | +72.5 | +66.6 | -58.8 | +54.7 | +13.7 | +0.0 | +7.8 | +33.0 | +72.5 |
| Relative (%) | -30.9 | +0.0 | +38.1 | +35.0 | -30.9 | +28.8 | +7.2 | +0.0 | +4.1 | +17.3 | +38.1 | |
| Steps (reduced) |
6 (6) |
10 (0) |
13 (3) |
15 (5) |
16 (6) |
18 (8) |
19 (9) |
20 (0) |
21 (1) |
22 (2) |
23 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -66.0 | -4.1 | +66.6 | -45.1 | +40.1 | -58.8 | +37.8 | -51.0 | +54.7 | -25.8 | +87.4 | +13.7 |
| Relative (%) | -34.7 | -2.2 | +35.0 | -23.7 | +21.1 | -30.9 | +19.9 | -26.8 | +28.8 | -13.6 | +46.0 | +7.2 | |
| Steps (reduced) |
23 (3) |
24 (4) |
25 (5) |
25 (5) |
26 (6) |
26 (6) |
27 (7) |
27 (7) |
28 (8) |
28 (8) |
29 (9) |
29 (9) | |
Interval table
| Degrees | Cents | Hekts | Approximate Ratios |
|---|---|---|---|
| 0 | 1/1 | ||
| 1 | 190.196 | 130 | 10/9, 28/25 |
| 2 | 380.391 | 260 | 5/4 |
| 3 | 570.587 | 390 | 7/5 |
| 4 | 760.782 | 520 | 14/9 |
| 5 | 950.978 | 650 | 45/26, 26/15 |
| 6 | 1141.173 | 780 | 27/14 |
| 7 | 1331.369 | 910 | 15/7 (15/14 plus an octave) |
| 8 | 1521.564 | 1040 | 12/5 (6/5 plus an octave) |
| 9 | 1711.760 | 1170 | 27/10 |
| 10 | 1901.955 | 1300 | 3/1 |