14ed6: Difference between revisions
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{{Mathematical interest}} | |||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
14ed6 corresponds to 5.4159…[[edo]]. It is the generator chain for the [[hemisensi]] temperament. | |||
=== Harmonics === | |||
{{Harmonics in equal|14|6|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|14|6|1|intervals=integer|columns=12|start=12|collapsed=true}} | |||
=== Subsets and supersets === | |||
Since 14 factors into primes as {{nowrap| 2 × 7 }}, 14ed6 contains [[2ed6]] and [[7ed6]] as subset ed6's. | |||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
Latest revision as of 22:16, 10 August 2025
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 13ed6 | 14ed6 | 15ed6 → |
14 equal divisions of the 6th harmonic (abbreviated 14ed6) is a nonoctave tuning system that divides the interval of 6/1 into 14 equal parts of about 222 ¢ each. Each step represents a frequency ratio of 61/14, or the 14th root of 6.
Theory
14ed6 corresponds to 5.4159…edo. It is the generator chain for the hemisensi temperament.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -92.2 | +92.2 | +37.3 | +94.1 | +0.0 | -45.3 | -54.9 | -37.3 | +1.9 | +58.5 | -92.2 |
| Relative (%) | -41.6 | +41.6 | +16.8 | +42.5 | +0.0 | -20.4 | -24.8 | -16.8 | +0.9 | +26.4 | -41.6 | |
| Steps (reduced) |
5 (5) |
9 (9) |
11 (11) |
13 (13) |
14 (0) |
15 (1) |
16 (2) |
17 (3) |
18 (4) |
19 (5) |
19 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -9.2 | +84.1 | -35.3 | +74.5 | -30.5 | +92.2 | -1.4 | -90.2 | +46.9 | -33.7 | -110.6 | +37.3 |
| Relative (%) | -4.1 | +38.0 | -15.9 | +33.6 | -13.7 | +41.6 | -0.7 | -40.7 | +21.1 | -15.2 | -49.9 | +16.8 | |
| Steps (reduced) |
20 (6) |
21 (7) |
21 (7) |
22 (8) |
22 (8) |
23 (9) |
23 (9) |
23 (9) |
24 (10) |
24 (10) |
24 (10) |
25 (11) | |
Subsets and supersets
Since 14 factors into primes as 2 × 7, 14ed6 contains 2ed6 and 7ed6 as subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 221.6 | 23/20 |
| 2 | 443.1 | 13/10, 17/13, 22/17 |
| 3 | 664.7 | 19/13 |
| 4 | 886.3 | 5/3 |
| 5 | 1107.8 | 19/10, 21/11, 23/12 |
| 6 | 1329.4 | 13/6 |
| 7 | 1551 | 17/7 |
| 8 | 1772.5 | |
| 9 | 1994.1 | 19/6, 22/7 |
| 10 | 2215.7 | 18/5 |
| 11 | 2437.3 | |
| 12 | 2658.8 | |
| 13 | 2880.4 | |
| 14 | 3102 | 6/1 |