40ed10
| ← 39ed10 | 40ed10 | 41ed10 → |
40 equal divisions of the 10th harmonic (abbreviated 40ed10) is a nonoctave tuning system that divides the interval of 10/1 into 40 equal parts of about 99.7 ¢ each. Each step represents a frequency ratio of 101/40, or the 40th root of 10.
Theory
40ed10 is related to 12edo, but with 10/1 instead of 2/1 being just. The octave, which comes from 10ed10, is compressed from pure by about 4.1 cents.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.1 | -8.5 | -8.2 | +4.1 | -12.6 | +19.5 | -12.3 | -16.9 | +0.0 | +34.3 | -16.7 |
| Relative (%) | -4.1 | -8.5 | -8.2 | +4.1 | -12.6 | +19.6 | -12.4 | -17.0 | +0.0 | +34.4 | -16.7 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (31) |
34 (34) |
36 (36) |
38 (38) |
40 (0) |
42 (2) |
43 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +44.1 | +15.4 | -4.4 | -16.4 | -21.7 | -21.0 | -15.0 | -4.1 | +11.1 | +30.2 | -46.8 | -20.8 |
| Relative (%) | +44.2 | +15.5 | -4.4 | -16.5 | -21.8 | -21.1 | -15.0 | -4.1 | +11.1 | +30.3 | -46.9 | -20.8 | |
| Steps (reduced) |
45 (5) |
46 (6) |
47 (7) |
48 (8) |
49 (9) |
50 (10) |
51 (11) |
52 (12) |
53 (13) |
54 (14) |
54 (14) |
55 (15) | |
Subsets and supersets
Since 40 factors into 23 × 5, 40ed10 has subset ed10's 2, 4, 5, 8, 10, and 20.
Miscellany
It is possible to call this division a form of kilobyte tuning, as
[math]2^{10} \approx 10^{3} = 1024 \approx 1000[/math];
which lies in the basis of using a "decimal" prefix to an otherwise binary unit of information.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 99.7 | 18/17 |
| 2 | 199.3 | 9/8 |
| 3 | 299.0 | 6/5 |
| 4 | 398.6 | 5/4 |
| 5 | 498.3 | 4/3 |
| 6 | 597.9 | 7/5 |
| 7 | 697.6 | 3/2 |
| 8 | 797.3 | 8/5 |
| 9 | 896.9 | 5/3 |
| 10 | 996.6 | 7/4 |
| 11 | 1096.2 | 15/8 |
| 12 | 1195.9 | 2/1 |
| 13 | 1295.6 | 17/8 |
| 14 | 1395.2 | 9/4 |
| 15 | 1494.9 | 12/5 |
| 16 | 1594.5 | 5/2 |
| 17 | 1694.2 | 8/3 |
| 18 | 1793.8 | 14/5 |
| 19 | 1893.5 | 3/1 |
| 20 | 1993.2 | 16/5 |
| 21 | 2092.8 | 10/3 |
| 22 | 2192.5 | 7/2 |
| 23 | 2292.1 | 15/4 |
| 24 | 2391.8 | 4/1 |
| 25 | 2491.4 | 17/4 |
| 26 | 2591.1 | 9/2 |
| 27 | 2690.8 | 19/4 |
| 28 | 2790.4 | 5/1 |
| 29 | 2890.1 | 16/3 |
| 30 | 2989.7 | 17/3 |
| 31 | 3089.4 | 6/1 |
| 32 | 3189.1 | 19/3 |
| 33 | 3288.7 | 20/3 |
| 34 | 3388.4 | 7/1 |
| 35 | 3488.0 | 15/2 |
| 36 | 3587.7 | 8/1 |
| 37 | 3687.3 | 17/2 |
| 38 | 3787.0 | 9/1 |
| 39 | 3886.7 | 19/2 |
| 40 | 3986.3 | 10/1 |
Regular temperaments
40ed10 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a cluster temperament with 12 clusters of notes in an octave (quintilischis temperament). This temperament is supported by 12-, 253-, 265-, 277-, 289-, 301-, 313-, and 325edo.
Tempering out 400/399 (equating 20/19 and 21/20) leads to quintilipyth (12 & 253), and tempering out 476/475 (equating 19/17 with 28/25) leads to quintaschis (12 & 289).