9ed4/3
 |
Todo: add source, research, cleanup |
| ← 8ed4/3 | 9ed4/3 | 10ed4/3 → |
9ed4/3, also known as Noleta, is a tuning system based on the division of the perfect fourth (4/3) into 9 equal parts, each 55.3383 cents in size; this corresponds to 21.6848edo, or approximately to every three steps of 65edo. The name ‘Noleta’ seems to be coined by Ron Sword: Nonoctave.com: Messages: 9197
Regular temperaments that divide 4/3 into 9 equal parts include:
Intervals
9ed4/3 has good approximations of 5/4 and 11/10 (the latter from 3ed4/3) and their fourth-complements 16/15 and 40/33. Treated as a 4/3.5/4.11/8 temperament, it tempers out the commas 4000/3993 and 5632/5625, a structure that is identical to escapade. 32/31 can additionally be added as a representation of a single step, tempering out S31 and S32.
All ratios in the below table are in the 88-integer-limit and in the 4/3.5/4.11/8.31/24 subgroup.
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 55.338 | 33/32, 32/31, 31/30 |
| 2 | 110.677 | 16/15, 33/31 |
| 3 | 166.015 | 11/10 |
| 4 | 221.353 | 25/22 |
| 5 | 276.692 | 75/64, 88/75 |
| 6 | 332.030 | 40/33 |
| 7 | 387.368 | 5/4 |
| 8 | 442.707 | 31/24, 40/31 |
| 9 | 498.045 | exact 4/3 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.4 | -20.5 | -20.5 | -19.4 | -3.0 | +6.8 | -3.0 | +14.4 | -2.0 | -0.9 | +14.4 |
| Relative (%) | +31.5 | -37.0 | -37.0 | -35.1 | -5.4 | +12.3 | -5.4 | +26.1 | -3.5 | -1.7 | +26.1 | |
| Steps (reduced) |
22 (4) |
34 (7) |
43 (7) |
50 (5) |
56 (2) |
61 (7) |
65 (2) |
69 (6) |
72 (0) |
75 (3) |
78 (6) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -13.5 | +24.3 | +15.5 | +14.4 | +20.2 | -23.5 | -6.4 | +15.5 | -13.6 | +16.5 | -5.1 |
| Relative (%) | -24.3 | +43.8 | +28.0 | +26.1 | +36.4 | -42.4 | -11.5 | +28.0 | -24.6 | +29.8 | -9.2 | |
| Steps (reduced) |
80 (8) |
83 (2) |
85 (4) |
87 (6) |
89 (8) |
90 (0) |
92 (2) |
94 (4) |
95 (5) |
97 (7) |
98 (8) | |