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). 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.

Approximation of prime harmonics in 46edo

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)

Thirds in 46edo

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)
• 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:

• 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)
• 51/50 (between 25/24 and 17/16)

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

• Rodan with 41edo (46edo is at the harder end)
• Diaschismic with 34edo
• Valentine with 31edo

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