17ed4
| ← 15ed4 | 17ed4 | 19ed4 → |
17 equal divisions of the 4th harmonic (abbreviated 17ed4) is a nonoctave tuning system that divides the interval of 4/1 into 17 equal parts of about 141 ¢ each. Each step represents a frequency ratio of 41/17, or the 17th root of 4. It corresponds to 8.5edo or every second step of 17edo.
Theory
17ed4 is the smallest ED4 to contain a diatonic fifth, in this case 17edo's sharp fifth, and it can be used to generate heptatonic (3L 4s<4/1>) and decatonic (7L 3s<4/1>) MOS scales with a period of 4/1. The decatonic scale is the more usable of these two scales, corresponding to an octave-repeating pentatonic scale in terms of step sizes, while the heptatonic scale has too large step sizes, corresponding to an octave-repeating tritonic or tetratonic scale in terms of step sizes.
Intervals
| # | Cents | Approximate ratios | 17edo notation | Diaquadic notation (J = 1/1) |
|---|---|---|---|---|
| 0 | 0.00 | 1/1 | C | J |
| 1 | 141.18 | 13/12, 12/11, 14/13, 25/23 | C# | J#, Kb |
| 2 | 282.36 | 13/11, 7/6 | Eb | K |
| 3 | 423.54 | 32/25, 9/7, 14/11, 33/26, 23/18 | E | K#, Lb |
| 4 | 564.72 | 11/8, 18/13, 32/23 | ^F | L |
| 5 | 705.90 | 3/2, 32/21 | G | M |
| 6 | 847.08 | 13/8, 18/11, 23/14 | G#, vA | M#, Nb |
| 7 | 988.26 | 16/9, 7/4, 25/14, 44/25, 23/13 | Bb | N |
| 8 | 1129.44 | 25/13, 48/25, 27/14, 64/33, 23/12 | B | N#, Ob |
| 9 | 1270.62 | 15/7 | ^C | O |
| 10 | 1411.80 | 9/4, 16/7 | D | O#, Pb |
| 11 | 1552.98 | 12/5, 5/2 | vE | P |
| 12 | 1694.16 | 8/3 | F | Q |
| 13 | 1835.34 | 3/1 | F# | Q#, Rb |
| 14 | 1976.52 | 16/5 | ^G, Ab | R |
| 15 | 2117.70 | 10/3 | A | R#, Sb |
| 16 | 2258.88 | 11/3 | vB | S |
| 17 | 2400.00 | 4/1 | C | J |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +70.6 | -66.7 | +0.0 | +37.2 | +3.9 | +19.4 | +70.6 | +7.9 | -33.4 | -57.2 | -66.7 |
| Relative (%) | +50.0 | -47.2 | +0.0 | +26.4 | +2.8 | +13.7 | +50.0 | +5.6 | -23.6 | -40.5 | -47.2 | |
| Steps (reduced) |
9 (9) |
13 (13) |
17 (0) |
20 (3) |
22 (5) |
24 (7) |
26 (9) |
27 (10) |
28 (11) |
29 (12) |
30 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -64.1 | -51.2 | -29.4 | +0.0 | +36.2 | -62.7 | -15.2 | +37.2 | -47.3 | +13.4 | -63.6 |
| Relative (%) | -45.4 | -36.3 | -20.9 | +0.0 | +25.7 | -44.4 | -10.7 | +26.4 | -33.5 | +9.5 | -45.0 | |
| Steps (reduced) |
31 (14) |
32 (15) |
33 (16) |
34 (0) |
35 (1) |
35 (1) |
36 (2) |
37 (3) |
37 (3) |
38 (4) |
38 (4) | |