14ed5
The 14 equal divisions of 5/1 system divides the 5th harmoinc or pentave into 14 equal steps of 199.022 cents each. It is essentially a compressed 6edo whole tone scale with a tempered octave of 1194.13 cents.
Theory
14ed5 is the first ED5 which can reasonably be described as a 5.7.11 subgroup temperament, excluding harmonics 2 and 3 as implied by pentaves, with the mapping ⟨14 17 21]. 7/5 is mapped to near 2edo's tritone at 597 cents, while 11/5 mapped to the square root of five, a major ninth of 1393 cents. The approximation of the 11/5 is off by 28 cents, so it is a very basic system for this subgroup, but nonetheless the 14-form is very important to the 5.7.11 subgroup in much the same way the 7-form is to the 5-limit. The pentadacus temperament is based on the 14-form and is basically a compressed 14ed5 generator chain. 14ed5 represents the intersection of the juggernaut and pentadacus temperaments.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 199 | 17/15, 19/17 |
| 2 | 398 | 5/4, 14/11, 19/15 |
| 3 | 597.1 | 7/5, 10/7, 17/12 |
| 4 | 796.1 | 8/5, 11/7, 19/12 |
| 5 | 995.1 | 23/13 |
| 6 | 1194.1 | 2/1 |
| 7 | 1393.2 | |
| 8 | 1592.2 | 5/2 |
| 9 | 1791.2 | 14/5, 17/6 |
| 10 | 1990.2 | 16/5, 19/6, 22/7 |
| 11 | 2189.2 | 7/2 |
| 12 | 2388.3 | 4/1 |
| 13 | 2587.3 | 22/5 |
| 14 | 2786.3 | 5/1 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.9 | +88.3 | -11.7 | +0.0 | +82.4 | +14.6 | -17.6 | -22.5 | -5.9 | +28.2 | +76.5 |
| Relative (%) | -2.9 | +44.4 | -5.9 | +0.0 | +41.4 | +7.3 | -8.8 | -11.3 | -2.9 | +14.1 | +38.5 | |
| Steps (reduced) |
6 (6) |
10 (10) |
12 (12) |
14 (0) |
16 (2) |
17 (3) |
18 (4) |
19 (5) |
20 (6) |
21 (7) |
22 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -62.0 | +8.7 | +88.3 | -23.5 | +70.6 | -28.3 | +77.1 | -11.7 | -96.2 | +22.3 | -54.7 |
| Relative (%) | -31.2 | +4.4 | +44.4 | -11.8 | +35.5 | -14.2 | +38.7 | -5.9 | -48.3 | +11.2 | -27.5 | |
| Steps (reduced) |
22 (8) |
23 (9) |
24 (10) |
24 (10) |
25 (11) |
25 (11) |
26 (12) |
26 (12) |
26 (12) |
27 (13) |
27 (13) | |