18ed6: Difference between revisions
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{{ED intro}} | {{ED intro}} | ||
18ed6 is close to [[7edo]], but with the [[ | == Theory == | ||
18ed6 is close to [[7edo]], but with the 6th harmonic rather than the [[2/1|octave]] being just, which stretches octaves by about 6.32 [[cent]]s. | |||
=== Harmonics === | |||
{{Harmonics in equal|18|6|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|18|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18ed6 (continued)}} | |||
=== Subsets and supersets === | |||
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18ed6 contains subset ed6's {{EDs|equave=6| 2, 3, 6, and 9 }}. | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
== See also == | == See also == | ||
* [[7edo]] – relative | * [[7edo]] – relative edo | ||
* [[11edt]] – relative | * [[11edt]] – relative edt | ||
Latest revision as of 13:13, 23 May 2025
| ← 17ed6 | 18ed6 | 19ed6 → |
(semiconvergent)
(semiconvergent)
18 equal divisions of the 6th harmonic (abbreviated 18ed6) is a nonoctave tuning system that divides the interval of 6/1 into 18 equal parts of about 172 ¢ each. Each step represents a frequency ratio of 61/18, or the 18th root of 6.
Theory
18ed6 is close to 7edo, but with the 6th harmonic rather than the octave being just, which stretches octaves by about 6.32 cents.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.3 | -6.3 | +12.6 | -29.0 | +0.0 | +77.8 | +18.9 | -12.6 | -22.7 | -15.4 | +6.3 |
| Relative (%) | +3.7 | -3.7 | +7.3 | -16.8 | +0.0 | +45.1 | +11.0 | -7.3 | -13.2 | -8.9 | +3.7 | |
| Steps (reduced) |
7 (7) |
11 (11) |
14 (14) |
16 (16) |
18 (0) |
20 (2) |
21 (3) |
22 (4) |
23 (5) |
24 (6) |
25 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +40.1 | +84.1 | -35.3 | +25.3 | -79.7 | -6.3 | +72.4 | -16.4 | +71.5 | -9.1 | -86.0 | +12.6 |
| Relative (%) | +23.3 | +48.8 | -20.5 | +14.7 | -46.2 | -3.7 | +42.0 | -9.5 | +41.5 | -5.3 | -49.9 | +7.3 | |
| Steps (reduced) |
26 (8) |
27 (9) |
27 (9) |
28 (10) |
28 (10) |
29 (11) |
30 (12) |
30 (12) |
31 (13) |
31 (13) |
31 (13) |
32 (14) | |
Subsets and supersets
Since 18 factors into primes as 2 × 32, 18ed6 contains subset ed6's 2, 3, 6, and 9.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 172.3 | 10/9, 11/10, 21/19 |
| 2 | 344.7 | 11/9, 16/13 |
| 3 | 517 | 19/14, 23/17 |
| 4 | 689.3 | 3/2 |
| 5 | 861.7 | 18/11 |
| 6 | 1034 | 9/5, 11/6, 20/11 |
| 7 | 1206.3 | 2/1 |
| 8 | 1378.6 | 11/5, 20/9 |
| 9 | 1551 | 22/9 |
| 10 | 1723.3 | 19/7 |
| 11 | 1895.6 | 3/1 |
| 12 | 2068 | 10/3 |
| 13 | 2240.3 | 11/3 |
| 14 | 2412.6 | 4/1 |
| 15 | 2585 | |
| 16 | 2757.3 | |
| 17 | 2929.6 | |
| 18 | 3102 | 6/1 |