27edt
| ← 26edt | 27edt | 28edt → |
(semiconvergent)
27edt means division of the tritave (3/1) into 27 equal parts.
Dividing the interval of 3/1 into 27 equal parts gives a scale with a basic step of 70.4428 cents, corresponding to 17.035 edo, which is nearly identical to one step of 17edo (70.59 cents). Hence it has similar melodic and harmonic properties as 17edo, with the difference that 27 is not a prime number. In fact, the prime edos that approximate Pythagorean tuning commonly become composite edts: e. g. 19edo > 30edt, 29edo > 46edt and 31edo > 49edt.
27 being the third power of 3, and the base interval being 3/1, 27edt is a tuning where the number 3 prevails. This property seems to predestine 27edt as base tuning for Klingon music (since the tradtional Klingon number system is also based on 3). The rather harsh harmonic character of 27edt would suit very well, too.
See, e.g., http://launch.dir.groups.yahoo.com/group/tuning/message/86909 and http://www.klingon.org/smboard/index.php?topic=1810.0.
This being said, such a proposal is rather short-sighted from a general cultural perspective, since any kind of living creature would most likely gravitate towards some form of low-complexity JI, and while 27edt will gain appreciation in base-3 cultures at some point, it may not be the first temperament they discover. That would be like aliens assuming dominant tuning in human music is 100ed10 (or 1000ed10 or variation thereof) just because we count in base 10.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.47 | +0.00 | -4.95 | +31.40 | -2.47 | +12.43 | -7.42 | +0.00 | +28.92 | +4.81 | -4.95 | -2.63 | +9.95 | +31.40 | -9.89 |
| Relative (%) | -3.5 | +0.0 | -7.0 | +44.6 | -3.5 | +17.6 | -10.5 | +0.0 | +41.1 | +6.8 | -7.0 | -3.7 | +14.1 | +44.6 | -14.0 | |
| Steps (reduced) |
17 (17) |
27 (0) |
34 (7) |
40 (13) |
44 (17) |
48 (21) |
51 (24) |
54 (0) |
57 (3) |
59 (5) |
61 (7) |
63 (9) |
65 (11) |
67 (13) |
68 (14) | |
Intervals
| Steps | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 70.443 | 22/21, 23/22, 24/23 |
| 2 | 140.886 | 12/11, 13/12 |
| 3 | 211.328 | 9/8, 17/15, 26/23 |
| 4 | 281.771 | 13/11, 20/17 |
| 5 | 352.214 | 11/9, 16/13 |
| 6 | 422.657 | 14/11, 23/18 |
| 7 | 493.099 | 4/3 |
| 8 | 563.542 | 18/13 |
| 9 | 633.985 | 13/9, 23/16 |
| 10 | 704.428 | 3/2 |
| 11 | 774.871 | 11/7, 14/9 |
| 12 | 845.313 | 13/8, 18/11 |
| 13 | 915.756 | 17/10, 22/13 |
| 14 | 986.199 | 16/9, 23/13 |
| 15 | 1056.642 | 11/6, 24/13 |
| 16 | 1127.084 | 21/11, 23/12 |
| 17 | 1197.527 | 2/1 |
| 18 | 1267.97 | 23/11 |
| 19 | 1338.413 | 13/6 |
| 20 | 1408.856 | 9/4 |
| 21 | 1479.298 | 26/11 |
| 22 | 1549.741 | 22/9 |
| 23 | 1620.184 | 23/9 |
| 24 | 1690.627 | 8/3 |
| 25 | 1761.069 | 11/4 |
| 26 | 1831.512 | 23/8, 26/9 |
| 27 | 1901.955 | 3/1 |