14edo
14edo, or 14 equal divisions of the octave, is the equal tuning featuring steps of (1200/14) ~≃ 85.714 cents, 14 of which stack to the perfect octave 2/1. While it approximates the 5:7:9:11:17:19 harmony relatively well for its size, it lacks a convincing realization of other low-complexity just intervals. Consequently, DR-based approaches may be more practically useful.
Theory
Edostep interpretations
The edostep of 14edo allows several interpretations within the 19-limit:
- 20/19 (the third part of dividing 7/6)
- 21/20 (the difference between 4/3 and 7/5)
- 256/243 (the difference between 4/3 and two 9/8)
- 28/27 (the difference between 15/14 and 10/9)
- 22/21 (the difference between 12/11 and 8/7)
JI approximation
Some recognizably approximated low-complexity just intervals include 7/5, 7/6, 9/7, 10/7, 10/9, 11/7, 11/9, and 11/10, which can allow to interpret 14edo as high damage 11-limit temperament.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -16.2 | +42.3 | -26.0 | -37.0 | +16.6 | -19.2 | -40.4 | -28.3 | -1.0 | -30.7 |
| Relative (%) | 0.0 | -18.9 | +49.3 | -30.3 | -43.2 | +19.4 | -22.4 | -47.1 | -33.0 | -1.2 | -35.9 | |
| Steps
(reduced) |
14
(0) |
22
(8) |
33
(5) |
39
(11) |
48
(6) |
52
(10) |
57
(1) |
59
(3) |
63
(7) |
68
(12) |
69
(13) | |
| Quality | Subminor | Neutral | Supermajor |
|---|---|---|---|
| Cents | 257.143 | 342.857 | 428.571 |
| Just interpretation | 22/19 | 17/14 | 22/17 |
Chords
As a superset of the popular 7edo scale, it offers recognizable triadic harmonies built on subminor, neutral, and supermajor thirds; however, its poor approximation of perfect fourths and fifths gives it a distinctly xenharmonic character.
Its equally-spaced MOS diatonic scale (equivalent to 7edo) allows all intervals to have a "minor", "neutral/perfect", and "major" variant, wherein (for instance) a minor third is the same pitch as a major second. 21edo provides distinctions between these categories.
14edo can also support Western functional harmony. The most stable chords are [0 3 8] and [0 4 8], including extensions with the 11th and 12th steps.
Intervals
Here is a table of 14edo's intervals:
| Name | Degree | Cents | Approximate Ratios |
|---|---|---|---|
| Unison | 0 | 0 | 1/1 |
| Narrow minor second | 1 | 85.714 | 20/19, 21/20 |
| Neutral second | 2 | 171.429 | 11/10, 10/9 |
| Subminor third | 3 | 257.143 | 7/6, 22/19, 15/13 |
| Neutral third | 4 | 342.857 | 11/9, 17/14 |
| Supermajor third | 5 | 428.571 | 9/7, 22/17, 14/11 |
| Wide fourth | 6 | 514.286 | 4/3, 19/14 |
| Tritone | 7 | 600.000 | 7/5, 10/7 |
| Narrow fifth | 8 | 685.714 | 3/2, 28/19 |
| Subminor sixth | 9 | 771.429 | 14/9, 11/7 |
| Neutral sixth | 10 | 857.143 | 18/11, 28/17 |
| Supermajor sixth | 11 | 942.857 | 12/7, 17/10, 19/11 |
| Neutral seventh | 12 | 1028.571 | 9/5, 20/11 |
| Wide Major 7th | 13 | 1114.286 | 17/9, 36/19, 21/11 |
| Octave | 14 | 1200 | 2/1 |
| View • Talk • EditEqual temperaments | |
|---|---|
| EDOs | |
| Macrotonal | 5 • 7 • 8 • 9 • 10 • 11 |
| 12-23 | 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 |
| 24-35 | 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35 |
| 36-47 | 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47 |
| 48-59 | 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58 |
| 60-71 | 60 • 63 • 64 • 65 • 67 • 70 |
| 72-83 | 72 • 77 • 80 • 81 |
| 84-95 | 84 • 87 • 89 • 90 • 93 • 94 |
| Large EDOs | 99 • 104 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665 |
| Nonoctave equal temperaments | |
| Tritave | 4 • 9 • 13 • 17 • 26 • 39 |
| Fifth | 8 • 9 • 11 • 20 |
| Other | |
