5edf: Difference between revisions
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=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|5|3|2}} | {{Harmonics in equal|5|3|2|columns=15}} | ||
=== Subsets and supersets === | |||
5edf is the 3rd [[prime equal division|prime edf]], after [[3edf]] and before [[7edf]]. | |||
== Intervals == | == Intervals == | ||
| Line 13: | Line 16: | ||
! # | ! # | ||
! Cents | ! Cents | ||
! Approximate | ! Approximate ratios | ||
! colspan="2"| [[1L 3s (fifth-equivalent)|Neptunian]] | ! colspan="2"| [[1L 3s (fifth-equivalent)|Neptunian]] notation | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 23: | Line 26: | ||
|- | |- | ||
| 1 | | 1 | ||
| 140 | | 140 | ||
| [[13/12]], [[49/45]] | | [[13/12]], [[49/45]] | ||
| augmented unison, minor second | | augmented unison, minor second | ||
| Line 29: | Line 32: | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 281 | ||
| [[ | | [[13/11]], [[20/17]], [[75/64]] | ||
| major second, minor third | | major second, minor third | ||
| D, Eb | | D, Eb | ||
|- | |- | ||
| 3 | | 3 | ||
| 421 | | 421 | ||
| [[14/11]], [[23/18]] | | [[14/11]], [[23/18]] | ||
| major third, diminished fourth | | major third, diminished fourth | ||
| Line 41: | Line 44: | ||
|- | |- | ||
| 4 | | 4 | ||
| | | 562 | ||
| [[11/8]], [[18/13]], [[25/18]] | | [[11/8]], [[18/13]], [[25/18]] | ||
| perfect fourth | | perfect fourth | ||
| Line 47: | Line 50: | ||
|- | |- | ||
| 5 | | 5 | ||
| 702 | | 702 | ||
| [[3/2]] | | [[3/2]] | ||
| perfect fifth | | perfect fifth | ||
| Line 53: | Line 56: | ||
|- | |- | ||
| 6 | | 6 | ||
| 842 | | 842 | ||
| [[ | | [[13/8]], [[18/11]], [[21/13]] | ||
| augmented fifth, minor sixth | | augmented fifth, minor sixth | ||
| C#, Db | | C#, Db | ||
|- | |- | ||
| 7 | | 7 | ||
| | | 983 | ||
| [[7/4]], [[30/17]] | | [[7/4]], [[30/17]] | ||
| major sixth, minor seventh | | major sixth, minor seventh | ||
| Line 65: | Line 68: | ||
|- | |- | ||
| 8 | | 8 | ||
| 1123 | | 1123 | ||
| | | | ||
| major seventh, minor octave | | major seventh, minor octave | ||
| Line 71: | Line 74: | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 1264 | ||
| | | | ||
| major octave | | major octave | ||
| F | | F | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 1404 | ||
| [[9/4]] | |||
| | |||
| | | | ||
| C | | C | ||
|} | |} | ||
{{ | {{Todo|expand}} | ||
Revision as of 13:45, 23 February 2025
| ← 4edf | 5edf | 6edf → |
5 equal divisions of the perfect fifth (abbreviated 5edf or 5ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 5 equal parts of about 140 ¢ each. Each step represents a frequency ratio of (3/2)1/5, or the 5th root of 3/2. It corresponds to 8.5476 edo.
Theory
5edf is close to the bleu generator chain and every second step of 17edo. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is 112/81 instead of 7/4 since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +63.5 | +63.5 | -13.4 | +21.5 | -13.4 | +0.6 | +50.2 | -13.4 | -55.4 | +60.4 | +50.2 | +52.0 | +64.1 | -55.4 | -26.7 |
| Relative (%) | +45.2 | +45.2 | -9.5 | +15.3 | -9.5 | +0.4 | +35.7 | -9.5 | -39.4 | +43.0 | +35.7 | +37.0 | +45.6 | -39.4 | -19.0 | |
| Steps (reduced) |
9 (4) |
14 (4) |
17 (2) |
20 (0) |
22 (2) |
24 (4) |
26 (1) |
27 (2) |
28 (3) |
30 (0) |
31 (1) |
32 (2) |
33 (3) |
33 (3) |
34 (4) | |
Subsets and supersets
5edf is the 3rd prime edf, after 3edf and before 7edf.
Intervals
| # | Cents | Approximate ratios | Neptunian notation | |
|---|---|---|---|---|
| 0 | 0.0 | 1/1 | perfect unison | C |
| 1 | 140 | 13/12, 49/45 | augmented unison, minor second | C#, Db |
| 2 | 281 | 13/11, 20/17, 75/64 | major second, minor third | D, Eb |
| 3 | 421 | 14/11, 23/18 | major third, diminished fourth | E, Fb |
| 4 | 562 | 11/8, 18/13, 25/18 | perfect fourth | F |
| 5 | 702 | 3/2 | perfect fifth | C |
| 6 | 842 | 13/8, 18/11, 21/13 | augmented fifth, minor sixth | C#, Db |
| 7 | 983 | 7/4, 30/17 | major sixth, minor seventh | D, Eb |
| 8 | 1123 | major seventh, minor octave | E, Fb | |
| 9 | 1264 | major octave | F | |
| 10 | 1404 | 9/4 | C | |