13ed8/3: Difference between revisions
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== Related tunings == | == Related tunings == | ||
13ed8/3 is a bit like [[9edo]] but with extreme [[octave shrinking]] (the octave is about 24 [[cents]] flat). As a [[23-limit]] equal [[temperament]] | 13ed8/3 is a bit like [[9edo]] but with extreme [[octave shrinking]] (the octave is about 24 [[cents]] flat). As a [[23-limit]] equal [[temperament]], 13ed8/3 could be considered a tuning for 9bdefhi (''see [[wart]]s''). | ||
13ed8/3 is quite close to [[23zpi]] (''see [[ZPI]]'') but 13ed8/3 has a much better [[2/1]] than 23zpi, at the expense of most other [[prime]]s being a little worse than 23zpi. | 13ed8/3 is quite close to [[23zpi]] (''see [[ZPI]]'') but 13ed8/3 has a much better [[2/1]] than 23zpi, at the expense of most other [[prime]]s being a little worse than 23zpi. | ||
Revision as of 07:44, 5 October 2025
| ← 12ed8/3 | 13ed8/3 | 14ed8/3 → |
13 equal divisions of 8/3 (abbreviated 13ed8/3) is a nonoctave tuning system that divides the interval of 8/3 into 13 equal parts of about 131 ¢ each. Each step represents a frequency ratio of (8/3)1/13, or the 13th root of 8/3.
The first person to explore 13ed8/3 was Maeve Gutierrez in early October 2025, who documented her findings in the Xenharmonic Alliance Discord server.
Properties
Gutierrez noted that:
- 13ed8/3 is like 9edo with the octave compressed by 24 cents
- It has a good approximation of the 7/6 subminor third (one of her favourite intervals)
- It has a major third close to the familiar 12edo major third
- It has no perfect fifth, instead steps 5 (653 ¢) and 6 (784 ¢) may be treated as a quarter flat fifth and a sharp fifth for triads
- It has a 1176 ¢ suboctave which she described as a "very fun shimmery interval"
- It has a near-12edo minor 9th interval
- Its 1437 ¢ major 9th interval is about an octave above the 8/7 septimal whole tone (so it approximates 16/7)
- The sharp perfect fourth of 522 ¢ is very dissonant in a triad with either fifth, but with the major 6th instead it makes a very pretty chord
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -24.4 | +57.3 | -48.9 | -43.3 | +32.9 | +27.3 | +57.3 | -16.0 | +62.9 | +28.5 | +8.5 |
| Relative (%) | -18.7 | +43.9 | -37.4 | -33.2 | +25.2 | +20.9 | +43.9 | -12.2 | +48.1 | +21.8 | +6.5 | |
| Steps (reduced) |
9 (9) |
15 (2) |
18 (5) |
21 (8) |
24 (11) |
26 (0) |
28 (2) |
29 (3) |
31 (5) |
32 (6) |
33 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.5 | +2.8 | +14.0 | +32.9 | +58.6 | -40.4 | -3.4 | +38.4 | -46.0 | +4.1 | +57.7 |
| Relative (%) | +0.4 | +2.2 | +10.7 | +25.2 | +44.8 | -30.9 | -2.6 | +29.4 | -35.2 | +3.1 | +44.2 | |
| Steps (reduced) |
34 (8) |
35 (9) |
36 (10) |
37 (11) |
38 (12) |
38 (12) |
39 (0) |
40 (1) |
40 (1) |
41 (2) |
42 (3) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 130.6 | 12/11, 13/12, 14/13, 15/14 |
| 2 | 261.2 | 7/6, 15/13, 22/19 |
| 3 | 391.9 | 5/4, 14/11, 19/15 |
| 4 | 522.5 | 15/11, 19/14, 23/17 |
| 5 | 653.1 | 19/13, 22/15 |
| 6 | 783.7 | 11/7, 19/12 |
| 7 | 914.3 | 12/7, 22/13 |
| 8 | 1045 | 11/6 |
| 9 | 1175.6 | 2/1 |
| 10 | 1306.2 | 15/7 |
| 11 | 1436.8 | |
| 12 | 1567.4 | 5/2 |
| 13 | 1698 |
Related tunings
13ed8/3 is a bit like 9edo but with extreme octave shrinking (the octave is about 24 cents flat). As a 23-limit equal temperament, 13ed8/3 could be considered a tuning for 9bdefhi (see warts).
13ed8/3 is quite close to 23zpi (see ZPI) but 13ed8/3 has a much better 2/1 than 23zpi, at the expense of most other primes being a little worse than 23zpi.
13ed8/3 is identical (octave less than 1 ¢ different) to every second step of 62zpi (see ZPI).
Scales
- Described by Budjarn Lambeth