27edt
| ← 26edt | 27edt | 28edt → |
(semiconvergent)
27 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 27edt or 27ed3), is a nonoctave tuning system that divides the interval of 3/1 into 27 equal parts of about 70.4 ¢ each. Each step represents a frequency ratio of 31/27, or the 27th root of 3.
Theory
27edt corresponds 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 the 3-limit well often correspond to composite edts: e.g. 19edo → 30edt, 29edo → 46edt and 31edo → 49edt.
Compared to 17edo, 27edt approximates the primes 7, 11, and 13 better; it approximates prime 5 equally poorly, but maps it to 40 steps rather than 39 in the patent val, corresponding to the 17c val, often considered the better mapping as it equates 5/4 and 6/5 to major and minor thirds rather than to a neutral third, and 5 has the same sharp tendency as 7 and 11.
From a purely tritave-based perspective, it supports the minalzidar temperament, but otherwise it can be used as a retuning of 17edo with closer-to-just harmonic properties in the no-fives 2.3.7.11.13 subgroup.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.5 | +0.0 | -4.9 | +31.4 | -2.5 | +12.4 | -7.4 | +0.0 | +28.9 | +4.8 | -4.9 |
| Relative (%) | -3.5 | +0.0 | -7.0 | +44.6 | -3.5 | +17.6 | -10.5 | +0.0 | +41.1 | +6.8 | -7.0 | |
| Steps (reduced) |
17 (17) |
27 (0) |
34 (7) |
40 (13) |
44 (17) |
48 (21) |
51 (24) |
54 (0) |
57 (3) |
59 (5) |
61 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | +10.0 | +31.4 | -9.9 | +26.0 | -2.5 | -25.6 | +26.5 | +12.4 | +2.3 | -4.2 | -7.4 |
| Relative (%) | -3.7 | +14.1 | +44.6 | -14.0 | +37.0 | -3.5 | -36.4 | +37.6 | +17.6 | +3.3 | -5.9 | -10.5 | |
| Steps (reduced) |
63 (9) |
65 (11) |
67 (13) |
68 (14) |
70 (16) |
71 (17) |
72 (18) |
74 (20) |
75 (21) |
76 (22) |
77 (23) |
78 (24) | |
Subsets and supersets
Since 27 factors into primes as 33, 27edt contains 3edt and 9edt as subset edts.
Miscellany
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[1][2].
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.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 70.4 | 48.1 | 22/21, 23/22, 24/23 |
| 2 | 140.9 | 96.3 | 12/11, 13/12 |
| 3 | 211.3 | 144.4 | 9/8, 17/15, 26/23 |
| 4 | 281.8 | 192.6 | 13/11, 20/17 |
| 5 | 352.2 | 240.7 | 11/9, 16/13 |
| 6 | 422.7 | 288.9 | 14/11, 23/18 |
| 7 | 493.1 | 337 | 4/3 |
| 8 | 563.5 | 385.2 | 18/13 |
| 9 | 634 | 433.3 | 13/9, 23/16 |
| 10 | 704.4 | 481.5 | 3/2 |
| 11 | 774.9 | 529.6 | 11/7, 14/9 |
| 12 | 845.3 | 577.8 | 13/8, 18/11 |
| 13 | 915.8 | 625.9 | 17/10, 22/13 |
| 14 | 986.2 | 674.1 | 16/9, 23/13 |
| 15 | 1056.6 | 722.2 | 11/6, 24/13 |
| 16 | 1127.1 | 770.4 | 21/11, 23/12 |
| 17 | 1197.5 | 818.5 | 2/1 |
| 18 | 1268 | 866.7 | 23/11 |
| 19 | 1338.4 | 914.8 | 13/6 |
| 20 | 1408.9 | 963 | 9/4 |
| 21 | 1479.3 | 1011.1 | 26/11 |
| 22 | 1549.7 | 1059.3 | 22/9 |
| 23 | 1620.2 | 1107.4 | 23/9 |
| 24 | 1690.6 | 1155.6 | 8/3 |
| 25 | 1761.1 | 1203.7 | 11/4 |
| 26 | 1831.5 | 1251.9 | 23/8, 26/9 |
| 27 | 1902 | 1300 | 3/1 |