9edf: Difference between revisions
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== Theory == | == Theory == | ||
9edf corresponds to 15.385602edo. It is closely related to [[Carlos Alpha]], and can be used as a temperament of the 3/2.5/4.8/7.11/8 subgroup. [[Valentine]] can be seen as 9edf with an independent dimension for 2. | 9edf corresponds to 15.385602edo. It is closely related to [[Carlos Alpha]] (though the Carlos Alpha generator is 0.03{{C}} narrower at 77.965{{C}}), and can be used as a temperament of the 3/2.5/4.8/7.11/8 subgroup. [[Valentine]] can be seen as 9edf with an independent dimension for 2. | ||
=== Harmonics === | === Harmonics === | ||
Latest revision as of 19:21, 20 March 2026
| ← 8edf | 9edf | 10edf → |
9 equal divisions of the perfect fifth (abbreviated 9edf or 9ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 9 equal parts of about 78 ¢ each. Each step represents a frequency ratio of (3/2)1/9, or the 9th root of 3/2.
Theory
9edf corresponds to 15.385602edo. It is closely related to Carlos Alpha (though the Carlos Alpha generator is 0.03 ¢ narrower at 77.965 ¢), and can be used as a temperament of the 3/2.5/4.8/7.11/8 subgroup. Valentine can be seen as 9edf with an independent dimension for 2.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -30.1 | -30.1 | +17.8 | +21.5 | +17.8 | -15.0 | -12.2 | +17.8 | -8.6 | -17.6 | -12.2 |
| Relative (%) | -38.6 | -38.6 | +22.9 | +27.6 | +22.9 | -19.3 | -15.7 | +22.9 | -11.0 | -22.5 | -15.7 | |
| Steps (reduced) |
15 (6) |
24 (6) |
31 (4) |
36 (0) |
40 (4) |
43 (7) |
46 (1) |
49 (4) |
51 (6) |
53 (8) |
55 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | +32.9 | -8.6 | +35.7 | +8.7 | -12.2 | -27.8 | -38.6 | +32.9 | +30.3 | +31.4 | +35.7 |
| Relative (%) | +6.7 | +42.2 | -11.0 | +45.8 | +11.2 | -15.7 | -35.7 | -49.5 | +42.2 | +38.9 | +40.2 | +45.8 | |
| Steps (reduced) |
57 (3) |
59 (5) |
60 (6) |
62 (8) |
63 (0) |
64 (1) |
65 (2) |
66 (3) |
68 (5) |
69 (6) |
70 (7) |
71 (8) | |
Intervals
| # | Cents | Approximate ratios | Neptunian notation using 7\9edf |
|---|---|---|---|
| 0 | 0.0 | 1/1 | C |
| 1 | 78.0 | 21/20, 22/21, 25/24 | C# |
| 2 | 156.0 | 11/10, 12/11 | Db |
| 3 | 234.0 | 8/7 | D |
| 4 | 312.0 | 6/5 | D#, Eb |
| 5 | 390.0 | 5/4 | E |
| 6 | 468.0 | 21/16 | E#, Fb |
| 7 | 546.0 | 11/8, 15/11 | F |
| 8 | 624.0 | 10/7, 36/25 | F#, Cb |
| 9 | 702.0 | 3/2 | C |
Scales
|
This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community. |
| Name | Step pattern within 1 period | Interval of repetition | First described by | ||
|---|---|---|---|---|---|
| (In 9edf steps) | (In cents) | (Nearby JI) | |||
| Molten slendro | 3 | 3 | 234.0 | 8/7 | Budjarn Lambeth (2023) |
| Pylon | 6 3 | 9 | 702.0 | 3/2 | Budjarn Lambeth (2023) |
| Quest | 7 2 | 9 | 702.0 | 3/2 | Budjarn Lambeth (2023) |
| Snowcone | 3 6 | 9 | 702.0 | 3/2 | Budjarn Lambeth (2023) |
| Molten pelog | 2 2 5 2 5 | 16 | 1247.9 | 33/16 | Budjarn Lambeth (2023) |
| Swan | 4 5 4 5 4 5 4 | 31 | 2417.8 | 4/1 | Budjarn Lambeth (2023) |
| Livewire | 1 8 1 8 1 8 1 8 1 8 1 | 46 | 3587.8 | 8/1 | Budjarn Lambeth (2023) |
| Purgatory | 8 1 8 1 8 1 8 1 8 1 8 1 8 | 62 | 4835.7 | 16/1 | Budjarn Lambeth (2023) |
| Cloudscape | 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 | 72 | 5615.6 | 128/5 | Budjarn Lambeth (2023) |
| Corrugated | 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 | 135 | 10529.3 | 437/1 | Budjarn Lambeth (2023) |
None of these scales are approximated from another tuning unless specified otherwise. For exact cents values see User:BudjarnLambeth/Longer versions of scales#9edf.
Music
See Carlos Alpha #Music. Many are tuned to exactly 9edf.