7edf: Difference between revisions
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== Theory == | == Theory == | ||
7edf is related to [[12edo]], but with the 3/2 rather than the 2/1 being just | 7edf is related to [[12edo]], but with the 3/2 rather than the 2/1 being just, which stretches the octave by 3.3514{{c}}. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13. It forms as a decent approximation to stretched-octave tuning on pianos, since pianos' strings have overtones that tend slightly sharp and are thus often tuned with stretched octaves. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal|7|3|2|prec=2|columns=15}} | {{Harmonics in equal|7|3|2|prec=2|columns=15}} | ||
=== Subsets and supersets === | |||
7edf is the 4th [[prime equal division|prime edf]], after [[5edf]] and before [[11edf]]. | |||
== Intervals == | == Intervals == | ||
| Line 15: | Line 16: | ||
! # | ! # | ||
! Cents | ! Cents | ||
! 12edo | ! Approximate ratios | ||
! 12edo notation | |||
|- | |- | ||
| 0 | | 0 | ||
| 0 | | 0 | ||
| exact 1/1 | |||
| C | | C | ||
|- | |- | ||
| 1 | | 1 | ||
| 100.3 | | 100.3 | ||
| 18/17, 17/16 | |||
| C#, Db | | C#, Db | ||
|- | |- | ||
| 2 | | 2 | ||
| 200.6 | | 200.6 | ||
| 9/8 | |||
| D | | D | ||
|- | |- | ||
| 3 | | 3 | ||
| 300.8 | | 300.8 | ||
| 19/16, 44/37 | |||
| D#, Eb | | D#, Eb | ||
|- | |- | ||
| 4 | | 4 | ||
| 401.1 | | 401.1 | ||
| 63/50 | |||
| E | | E | ||
|- | |- | ||
| 5 | | 5 | ||
| 501.4 | | 501.4 | ||
|4/3 | |||
| F | | F | ||
|- | |- | ||
| 6 | | 6 | ||
| 601.7 | | 601.7 | ||
| 64/45 | |||
| F#, Gb | | F#, Gb | ||
|- | |- | ||
| 7 | | 7 | ||
| 702.0 | | 702.0 | ||
| exact 3/2 | |||
| G | | G | ||
|- | |- | ||
| 8 | | 8 | ||
| 802.2 | | 802.2 | ||
| 100/63 | |||
| G#, Ab | | G#, Ab | ||
|- | |- | ||
| 9 | | 9 | ||
| 902.5 | | 902.5 | ||
| 27/16 | |||
| A | | A | ||
|- | |- | ||
| 10 | | 10 | ||
| 1002.8 | | 1002.8 | ||
| 16/9 | |||
| A#, Bb | | A#, Bb | ||
|- | |- | ||
| 11 | | 11 | ||
| 1103.1 | | 1103.1 | ||
| 17/9 | |||
| B | | B | ||
|- | |- | ||
| 12 | | 12 | ||
| 1203.4 | | 1203.4 | ||
| 2/1 | |||
| C | | C | ||
|- | |- | ||
| 13 | | 13 | ||
| 1303.6 | | 1303.6 | ||
| 17/8 | |||
| C#, Db | | C#, Db | ||
|- | |- | ||
| 14 | | 14 | ||
| 1403.9 | | 1403.9 | ||
| exact 9/4 | |||
| D | | D | ||
|} | |} | ||
{{ | == See also == | ||
* [[12edo]] – relative edo | |||
* [[19edt]] – relative edt | |||
* [[28ed5]] – relative ed5 | |||
* [[31ed6]] – relative ed6 | |||
* [[34ed7]] – relative ed7 | |||
* [[40ed10]] – relative ed10 | |||
* [[43ed12]] – relative ed12 | |||
* [[76ed80]] – close to the zeta-optimized tuning for 12edo | |||
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]] | |||
{{Todo|expand}} | |||
[[Category:12edo]] | |||
Latest revision as of 13:26, 10 June 2025
| ← 6edf | 7edf | 8edf → |
(convergent)
(convergent)
7 equal divisions of the perfect fifth (abbreviated 7edf or 7ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 7 equal parts of about 100 ¢ each. Each step represents a frequency ratio of (3/2)1/7, or the 7th root of 3/2.
Theory
7edf is related to 12edo, but with the 3/2 rather than the 2/1 being just, which stretches the octave by 3.3514 ¢. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13. It forms as a decent approximation to stretched-octave tuning on pianos, since pianos' strings have overtones that tend slightly sharp and are thus often tuned with stretched octaves.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.35 | +3.35 | +6.70 | +21.51 | +6.70 | +40.67 | +10.05 | +6.70 | +24.86 | -39.87 | +10.05 | -28.24 | +44.02 | +24.86 | +13.41 |
| Relative (%) | +3.3 | +3.3 | +6.7 | +21.4 | +6.7 | +40.6 | +10.0 | +6.7 | +24.8 | -39.8 | +10.0 | -28.2 | +43.9 | +24.8 | +13.4 | |
| Steps (reduced) |
12 (5) |
19 (5) |
24 (3) |
28 (0) |
31 (3) |
34 (6) |
36 (1) |
38 (3) |
40 (5) |
41 (6) |
43 (1) |
44 (2) |
46 (4) |
47 (5) |
48 (6) | |
Subsets and supersets
7edf is the 4th prime edf, after 5edf and before 11edf.
Intervals
| # | Cents | Approximate ratios | 12edo notation |
|---|---|---|---|
| 0 | 0 | exact 1/1 | C |
| 1 | 100.3 | 18/17, 17/16 | C#, Db |
| 2 | 200.6 | 9/8 | D |
| 3 | 300.8 | 19/16, 44/37 | D#, Eb |
| 4 | 401.1 | 63/50 | E |
| 5 | 501.4 | 4/3 | F |
| 6 | 601.7 | 64/45 | F#, Gb |
| 7 | 702.0 | exact 3/2 | G |
| 8 | 802.2 | 100/63 | G#, Ab |
| 9 | 902.5 | 27/16 | A |
| 10 | 1002.8 | 16/9 | A#, Bb |
| 11 | 1103.1 | 17/9 | B |
| 12 | 1203.4 | 2/1 | C |
| 13 | 1303.6 | 17/8 | C#, Db |
| 14 | 1403.9 | exact 9/4 | D |