40ed10: Difference between revisions
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|- | |- | ||
! # | ! # | ||
! Cents | ! Cents | ||
! Approximate | ! Approximate ratios | ||
|- | |- | ||
| 0 | | 0 | ||
| 0. | | 0.0 | ||
| [[1/1]] | | [[1/1]] | ||
|- | |- | ||
| 1 | | 1 | ||
| 99. | | 99.7 | ||
| [[18/17]] | | [[18/17]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 199. | | 199.3 | ||
| | | [[9/8]] | ||
|- | |- | ||
| 3 | | 3 | ||
| | | 299.0 | ||
| [[ | | [[6/5]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 398. | | 398.6 | ||
| | | [[5/4]] | ||
|- | |- | ||
| 5 | | 5 | ||
| 498. | | 498.3 | ||
| [[4/3]] | | [[4/3]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 597. | | 597.9 | ||
| [[ | | [[7/5]] | ||
|- | |- | ||
| 7 | | 7 | ||
| 697. | | 697.6 | ||
| | | [[3/2]] | ||
|- | |- | ||
| 8 | | 8 | ||
| 797. | | 797.3 | ||
| | | [[8/5]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 896. | | 896.9 | ||
| | | [[5/3]] | ||
|- | |- | ||
| 10 | | 10 | ||
| 996. | | 996.6 | ||
| [[ | | [[7/4]] | ||
|- | |- | ||
| 11 | | 11 | ||
| 1096. | | 1096.2 | ||
| [[ | | [[15/8]] | ||
|- | |- | ||
| 12 | | 12 | ||
| 1195. | | 1195.9 | ||
| [[2/1]] | | [[2/1]] | ||
|- | |- | ||
| 13 | | 13 | ||
| 1295. | | 1295.6 | ||
| | | [[17/8]] | ||
|- | |- | ||
| 14 | | 14 | ||
| 1395. | | 1395.2 | ||
| [[ | | [[9/4]] | ||
|- | |- | ||
| 15 | | 15 | ||
| 1494. | | 1494.9 | ||
| | | [[12/5]] | ||
|- | |- | ||
| 16 | | 16 | ||
| 1594. | | 1594.5 | ||
| | | [[5/2]] | ||
|- | |- | ||
| 17 | | 17 | ||
| 1694. | | 1694.2 | ||
| | | [[8/3]] | ||
|- | |- | ||
| 18 | | 18 | ||
| 1793. | | 1793.8 | ||
| | | [[14/5]] | ||
|- | |- | ||
| 19 | | 19 | ||
| 1893. | | 1893.5 | ||
| [[ | | [[3/1]] | ||
|- | |- | ||
| 20 | | 20 | ||
| 1993. | | 1993.2 | ||
| | | [[16/5]] | ||
|- | |- | ||
| 21 | | 21 | ||
| 2092. | | 2092.8 | ||
| | | [[10/3]] | ||
|- | |- | ||
| 22 | | 22 | ||
| 2192. | | 2192.5 | ||
| | | [[7/2]] | ||
|- | |- | ||
| 23 | | 23 | ||
| 2292. | | 2292.1 | ||
| | | [[15/4]] | ||
|- | |- | ||
| 24 | | 24 | ||
| 2391. | | 2391.8 | ||
| | | [[4/1]] | ||
|- | |- | ||
| 25 | | 25 | ||
| 2491. | | 2491.4 | ||
| | | [[17/4]] | ||
|- | |- | ||
| 26 | | 26 | ||
| 2591. | | 2591.1 | ||
| | | [[9/2]] | ||
|- | |- | ||
| 27 | | 27 | ||
| 2690. | | 2690.8 | ||
| | | 19/4 | ||
|- | |- | ||
| 28 | | 28 | ||
| 2790. | | 2790.4 | ||
| | | [[5/1]] | ||
|- | |- | ||
| 29 | | 29 | ||
| 2890. | | 2890.1 | ||
| | | [[16/3]] | ||
|- | |- | ||
| 30 | | 30 | ||
| 2989. | | 2989.7 | ||
| | | 17/3 | ||
|- | |- | ||
| 31 | | 31 | ||
| 3089. | | 3089.4 | ||
| | | [[6/1]] | ||
|- | |- | ||
| 32 | | 32 | ||
| 3189. | | 3189.1 | ||
| | | 19/3 | ||
|- | |- | ||
| 33 | | 33 | ||
| 3288. | | 3288.7 | ||
| | | 20/3 | ||
|- | |- | ||
| 34 | | 34 | ||
| 3388. | | 3388.4 | ||
| | | [[7/1]] | ||
|- | |- | ||
| 35 | | 35 | ||
| 3488. | | 3488.0 | ||
| [[15/2]] | | [[15/2]] | ||
|- | |- | ||
| 36 | | 36 | ||
| 3587. | | 3587.7 | ||
| | | [[8/1]] | ||
|- | |- | ||
| 37 | | 37 | ||
| 3687. | | 3687.3 | ||
| | | [[17/2]] | ||
|- | |- | ||
| 38 | | 38 | ||
| | | 3787.0 | ||
| | | [[9/1]] | ||
|- | |- | ||
| 39 | | 39 | ||
| 3886. | | 3886.7 | ||
| | | 19/2 | ||
|- | |- | ||
| 40 | | 40 | ||
| 3986. | | 3986.3 | ||
| [[10/1]] | | [[10/1]] | ||
|} | |} | ||
Revision as of 14:39, 15 January 2025
| ← 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) | |
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 |
Miscellany
It is possible to call this division a form of kilobyte tuning, as
[math]\displaystyle{ 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.
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).