46edo
46edo, or 46 equal divisions of the octave, is an equal tuning with a step size of approximately 26.1 cents. It is known for its relatively good approximation of 13-limit just intonation, about as accurate as 41edo's approximation of the 11-limit.
Theory
JI approximation
46edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to 41edo in the 2.3.5.7.11.13 subgroup, though it has 41edo's Rodan characteristic of sharp prime 3 and flat prime 7 and exhibits it more extremely (and indeed their sum 87edo is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13). 46edo has extremely accurate approximations of 14/11 and 10/9.
Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a neogothic MOS diatonic of 8-8-3-8-8-8-3.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | +2.4 | +5.0 | -3.6 | -3.5 | -5.7 | -0.6 | -10.6 | -2.2 | -12.2 | +2.8 |
| Relative (%) | 0.0 | +9.2 | +19.1 | -13.8 | -13.4 | -22.0 | -2.3 | -40.5 | -8.4 | -46.7 | +10.7 | |
| Steps
(reduced) |
46
(0) |
73
(27) |
107
(15) |
129
(37) |
159
(21) |
170
(32) |
188
(4) |
195
(11) |
208
(24) |
223
(39) |
228
(44) | |
| Quality | Subminor | Farminor | Nearminor | Supraminor | Submajor | Nearmajor | Farmajor | Supermajor |
|---|---|---|---|---|---|---|---|---|
| Cents | 261 | 287 | 313 | 339 | 365 | 391 | 417 | 443 |
| Just interpretation | 7/6 | 13/11 | 6/5 | 11/9, 17/14 | 16/13, 21/17 | 5/4 | 14/11 | 9/7, 13/10 |
| Steps | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
Thirds available in the diatonic scale generated by stacking the perfect fifth are bolded.
Edostep interpretations
One step of 46edo represents the following ratios of 2.3.5.7.11.13.17.23:
- 81/80 (between 5/4 and 81/64)
- 49/48 (between 7/6 and 8/7)
- 64/63 (between 7/4 and 16/9)
- 65/64 (between 16/13 and 5/4)
- 66/65 (between 13/11 and 6/5)
- 69/68 (between 17/12 and 23/16)
- 70/69 (between 23/16 and 35/24)
- 51/50 (between 25/24 and 17/16)
- 52/51 (between 17/16 and 13/12)
- 55/54 (between 27/20 and 11/8)
- 56/55 (between 5/4 and 14/11)
- 85/84 (betweem 6/5 and 17/14)
- 99/98 (between 14/11 and 9/7)
- 100/99 (between 10/9 and 11/10)
- 144/143 (between 11/9 and 16/13)
- 225/224 (between 9/7 and 32/25)
- 256/253 (between 23/16 and 16/11)
Two steps of 46edo represent the following ratios:
- 26/25 (between 25/16 and 13/8)
- 28/27 (between 9/8 and 7/6)
- 33/32 (between 4/3 and 11/8)
- 34/33 (between 11/8 and 17/12)
- 35/34 (between 35/32 and 17/16)
- 36/35 (between 7/4 and 9/5)
- 50/49 (between 7/5 and 10/7)
Chords
46edo has four different flavors of minor and major intervals but lacks true neutrals and interordinals. Its subminor and supermajor intervals approximate simpler septimal ratios such as 7/4 and 9/7, while its nearminor and nearmajor intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its farmajor and farminor intervals approximate more complex neogothic triads 22:28:33 and 22:26:33. Its supraminor triad approximates 14:17:21.
As a result, 46edo has eight qualities of tertian, fifth-bounded triads: supermajor, farmajor, nearmajor, submajor, supraminor, nearminor, farminor, subminor. 46edo lacks true interordinal intervals, so as for latal fourth-bounded triads, there are only four qualities.
Scales
46edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being a triple-diminished fifth and 5/4 a triple-augmented unison. 46edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-2-3-3-2-3-3-3-2-3-3-2-3-3-2, however it is close to 17edo in that it lacks nearmajor and nearminor thirds, with supraminor and submajor thirds instead.
11-limit penslen is available in 46edo (because of Slendric and 385/384 tempering): 2-1-6-1-2-6-1-2-6-1-2-1-6-2-1-6.
Regular temperaments
46edo shares
- Hemifamity with 41edo and 53edo
- Rodan with 41edo (46edo is at the harder end)
- Diaschismic with 34edo
- Valentine with 31edo
- Sensi with 27edo
- Amity with 53edo
Notation
46edo can be notated with diatonic notation plus ups and downs, which naturally reflect 46edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub".)
Practice
Isomorphic keyboard layouts
Rodan isomorphic layout for 46edo: +1\46 up-left, +10\46 up, +9\46 up-right
