10ed7/3: Difference between revisions
Jump to navigation
Jump to search
m Marked page as stub |
m Table of harmonics |
||
| (5 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
10ed7/3 is essentially a tritave stretch of [[13edt]], the equalized [[Bohlen–Pierce]] scale, and as a result tempers out [[245/243]] and [[3125/3087]] in the [[3.5.7 subgroup]], as well as [[529/525]] and [[1127/1125]] when prime 23 is introduced. It fails to represent any other primes of note within 20 cents. Making this stretch makes [[5/1]] and [[23/1]] closer to just compared to 13edt, while both [[3/1]] and [[7/1]] are about 5 [[cents]] sharp of just. | |||
=== Harmonics === | |||
{{Harmonics in equal|10|7|3|columns=11}} | |||
{{Harmonics in equal|10|7|3|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 10ed7/3 (continued)}} | |||
== Intervals == | |||
{{Interval table}} | |||
Latest revision as of 18:12, 14 May 2026
| ← 9ed7/3 | 10ed7/3 | 11ed7/3 → |
(convergent)
10 equal divisions of 7/3 (abbreviated 10ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 10 equal parts of about 147 ¢ each. Each step represents a frequency ratio of (7/3)1/10, or the 10th root of 7/3.
Theory
10ed7/3 is essentially a tritave stretch of 13edt, the equalized Bohlen–Pierce scale, and as a result tempers out 245/243 and 3125/3087 in the 3.5.7 subgroup, as well as 529/525 and 1127/1125 when prime 23 is introduced. It fails to represent any other primes of note within 20 cents. Making this stretch makes 5/1 and 23/1 closer to just compared to 13edt, while both 3/1 and 7/1 are about 5 cents sharp of just.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -26.5 | +5.0 | -53.0 | +0.7 | -21.5 | +5.0 | +67.2 | +10.0 | -25.8 | -44.1 | -48.0 |
| Relative (%) | -18.1 | +3.4 | -36.1 | +0.5 | -14.7 | +3.4 | +45.8 | +6.8 | -17.6 | -30.0 | -32.7 | |
| Steps (reduced) |
8 (8) |
13 (3) |
16 (6) |
19 (9) |
21 (1) |
23 (3) |
25 (5) |
26 (6) |
27 (7) |
28 (8) |
29 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -39.9 | -21.5 | +5.7 | +40.7 | -64.3 | -16.5 | +36.5 | -52.3 | +10.0 | -70.6 | -0.9 | +72.2 |
| Relative (%) | -27.2 | -14.7 | +3.9 | +27.7 | -43.8 | -11.3 | +24.9 | -35.6 | +6.8 | -48.1 | -0.6 | +49.2 | |
| Steps (reduced) |
30 (0) |
31 (1) |
32 (2) |
33 (3) |
33 (3) |
34 (4) |
35 (5) |
35 (5) |
36 (6) |
36 (6) |
37 (7) |
38 (8) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 146.7 | 11/10, 12/11, 13/12, 14/13, 15/14, 21/19 |
| 2 | 293.4 | 6/5, 7/6, 13/11, 20/17 |
| 3 | 440.1 | 9/7, 13/10, 14/11, 17/13, 19/15, 22/17 |
| 4 | 586.7 | 7/5, 10/7, 17/12, 18/13 |
| 5 | 733.4 | 3/2, 14/9, 17/11, 20/13 |
| 6 | 880.1 | 5/3, 18/11, 22/13 |
| 7 | 1026.8 | 9/5, 11/6, 20/11 |
| 8 | 1173.5 | 2/1 |
| 9 | 1320.2 | 13/6, 15/7, 17/8, 19/9 |
| 10 | 1466.9 | 7/3 |