5edf: Difference between revisions
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== Theory == | == Theory == | ||
5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] | 5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size. | ||
5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in [[12edo]] due to the laxer subtritone. | |||
=== 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 17: | ||
! # | ! # | ||
! 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 27: | ||
|- | |- | ||
| 1 | | 1 | ||
| 140 | | 140 | ||
| [[13/12]], [[49/45]] | | [[13/12]], [[49/45]] | ||
| augmented unison, minor second | | augmented unison, minor second | ||
| Line 29: | Line 33: | ||
|- | |- | ||
| 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 45: | ||
|- | |- | ||
| 4 | | 4 | ||
| | | 562 | ||
| [[11/8]], [[18/13]], [[25/18]] | | [[11/8]], [[18/13]], [[25/18]] | ||
| perfect fourth | | perfect fourth | ||
| Line 47: | Line 51: | ||
|- | |- | ||
| 5 | | 5 | ||
| 702 | | 702 | ||
| [[3/2]] | | [[3/2]] | ||
| perfect fifth | | perfect fifth | ||
| Line 53: | Line 57: | ||
|- | |- | ||
| 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 69: | ||
|- | |- | ||
| 8 | | 8 | ||
| 1123 | | 1123 | ||
| | | 44/23 | ||
| major seventh, minor octave | | major seventh, minor octave | ||
| E, Fb | | E, Fb | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 1264 | ||
| | | 83/40 | ||
| major octave | | major octave | ||
| F | | F | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 1404 | ||
| [[9/4]] | |||
| major ninth | |||
| | |||
| | |||
| C | | C | ||
|} | |} | ||
{{Todo|expand}} | |||
Latest revision as of 00:38, 10 August 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 (also known as 17ed4) 5edf has an extremely accurate approximation of the seventh harmonic for its size.
5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in 12edo due to the laxer subtritone.
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 | 44/23 | major seventh, minor octave | E, Fb |
| 9 | 1264 | 83/40 | major octave | F |
| 10 | 1404 | 9/4 | major ninth | C |