343edo

← 342edo

343edo

344edo →

Prime factorization	73
Step size	3.49854 ¢
Fifth	201\343 (703.207 ¢)
Semitones (A1:m2)	35:24 (122.4 ¢ : 83.97 ¢)
Dual sharp fifth	201\343 (703.207 ¢)
Dual flat fifth	200\343 (699.708 ¢)
Dual major 2nd	58\343 (202.915 ¢)
Consistency limit	3
Distinct consistency limit	3

343 equal divisions of the octave (abbreviated 343edo or 343ed2), also called 343-tone equal temperament (343tet) or 343 equal temperament (343et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 343 equal parts of about 3.5 ¢ each. Each step represents a frequency ratio of 2^1/343, or the 343rd root of 2.

Theory

343edo is only consistent to the 3-odd-limit since its errors of harmonics 3 and 5 are quite large. To start with, consider the 2.9.15.7 subgroup, where it tempers out 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.

For the full 7-limit, the 343c val tempers out 4375/4374 and 5120/5103, supporting amity (gen. 97\343, per. 343\343). The 343cdd val tempers out 16875/16807 and 65536/64827. The patent val tempers out 10976/10935 and 390625/387072.

Odd harmonics

Approximation of odd harmonics in 343edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+1.25	-1.47	+0.27	-0.99	+1.45	-0.88	-0.22	+0.00	-0.14	+1.52	+1.46
	Relative (%)	+35.8	-42.1	+7.7	-28.4	+41.5	-25.1	-6.3	+0.0	-3.9	+43.5	+41.8
Steps (reduced)		544 (201)	796 (110)	963 (277)	1087 (58)	1187 (158)	1269 (240)	1340 (311)	1402 (30)	1457 (85)	1507 (135)	1552 (180)

Subsets and supersets

Since 343 factors into 7³, 343edo has 7edo and 49edo as its subsets. 686edo, which doubles it, gives a good correction to the harmonics 3 and 5.

Use as a NEJI

Of all n-afdos where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, 476afdo (7x17x2x2) approximates 343edo with the least relative error. (See Approximating 343edo in afdos.)

343edo could be approximated into 476afdo as a neji scale. Doing so would make it an over-17-by-7 scale (when viewed through a primodal lens). (Scala file.)

It would make sense to use smaller over-17, over-7, or over-17-by-7 JI scales as subsets of this neji.

Regular temperament properties

343edo is on the optimal ET sequence of gammy temperament (343be, 10\343 generator, 2/1 period), protolangwidge temperament (343, 200\343 g, 2/1 p) and anthoine temperament (343dd, 110\343 g, 2/1 p).

343edo might potentially be useful for 49th-octave temperaments (see Fractional-octave temperaments), this is something which hasn't been explored yet.

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.9	[-1087 343⟩	[⟨343 1087]]	+0.1569	0.1569	4.48
2.9.5	[-27 -1 13⟩, [40 -28 21⟩	[⟨343 1087 796]]	+0.3162	0.2592	7.41
2.9.5.7	118098/117649, 7381125/7340032, 9765625/9680832	[⟨343 1087 796 963]]	+0.2130	0.2869	8.20

Octave stretch or compression

If one is using 343edo, it's probably either for a specific temperament, or because of its good primes 2, 7, 17 and 19, which will inform how one might want to octave stretch or compress it.

Using for a temperament

TE octave stretch:

• For 13-limit gammy
  • Octave size: 1200.437 ¢

• For 7-limit anthoine
  • Octave size: 1199.630 ¢

Using for primes 2, 7, 17, 19

If one is using 343 for its accurate 2.7.17.19 intervals, one will probably not want to use 343edo with warts a, d, g or h.

That leaves the following TE tunings for the 19-limit:

• 343cf
  • Octave size: 1199.643 ¢
  • TE error: 0.363 ¢/octave

• 343c
  • Octave size: 1199.761 ¢
  • TE error: 0.382 ¢/octave

• 343 (patent val)
  • Octave size: 1199.950 ¢
  • TE error: 0.395 ¢/octave

• 343e
  • Octave size: 1200.076 ¢
  • TE error: 0.418 ¢/octave

• 343f
  • Octave size: 1199.831 ¢
  • TE error: 0.431 ¢/octave

• 343ce
  • Octave size: 1199.888 ¢
  • TE error: 0.461 ¢/octave

Odd harmonics in TE-tuned 343cf

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.96	+1.59	-0.25	-1.58	+0.81	-1.57	-0.95	-0.76	-0.93	+0.71	+0.62
	Relative (%)	+27.4	+45.5	-7.2	-45.3	+23.1	-44.8	-27.1	-21.7	-26.5	+20.2	+17.8
Step		544	797	963	1087	1187	1269	1340	1402	1457	1507	1552

Scales

343edo includes every 49edo scale (see 49edo#Scales).

Scales listed below are rotated to modes with lots of consonances.

MOS scales

• Amity[7]: 52 52 45 52 45 52 45
• Amity[11]: 45 7 45 45 7 45 7 45 45 7 42
• Amity[18]: 7 38 7 38 7 7 38 7 7 38 7 38 7 7 38 7 38 7
• Amity[25]: 7 31 7 7 7 31 7 7 31 7 7 7 31 7 7 7 31 7 7 31 7 7 7 31 7
• Amity[32]: 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7
• Amity[39]: 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7
• Amity[53]: 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7
• Amity[99]: 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3

Gordon Wery's 23-tone MOS

A 23-tone MOS scale discovered by Gordon Wery in October 2025:

• 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13

It is very similar to 23edo and can be used as a well temperament of 23edo.

In his post on Discord describing it, Wery said of the scale:

"Basically a more complicated version of 23edo, centered around a more minor (less neutral) anti-diatonic scale.

This scale has a sort of glassy quality, ample neutral seconds, and two sets of dual fifths--a true dual dual fifth scale."

It is generated by 30\343.

Scales approximated from JI

• 4 of 7-17-19-21-51 pentany: 96 50 55 96 46 (sounds like minor pentatonic)
• 4 of 7-17-19-21-51 by 3/2 tetrapentany: 3 9 46 4 34 21 29 8 47 3 9 46 38 46
• 7-17-19-21 hexany: 50 46 50 55 87 55 (sounds like minor hexatonic)
• 7-17-19-21 by 3/2 trihexany: 3 47 8 38 12 38 8 47 3 46 9 29 9 46
• 9afdo: 40 37 34 32 30 28 52 47 43
• 18afdo: 21 19 19 18 17 17 16 16 15 15 14 14 27 25 24 23 22 21
• 36afdo: 11 10 9 10 9 10 9 9 8 9 8 9 8 8 8 8 7 8 7 8 7 7 7 7 14 13 13 12 12 12 12 11 11 11 11 10
• 72afdo: 5 6 5 5 4 5 5 5 5 4 5 5 4 5 4 5 4 4 5 4 4 4 5 4 4 4 4 4 4 4 4 4 4 3 4 4 4 3 4 4 3 4 4 3 4 3 4 3 7 7 6 7 6 7 6 6 6 6 6 6 6 6 6 5 6 5 6 5 6 5 5 5

343ed16

343ed16 is contained within 343edo (it is every 4th step of 343edo). It is like 86edo with the octave stretched by 3.5 cents.

It is quite similar to 136edt.

Compared to 86edo it improves harmonics 3, 5, 7 and 11. Its mappings of multiple-of-2 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having octave equivalence fall on a scale step one might not expect.

Many temperaments and scales from 86edo can be used here in 343ed16 too.

Approximation of harmonics in 343ed16

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13
Error	Absolute (¢)	+3.50	+1.25	+7.00	-1.47	+4.75	+3.77	-3.50	+2.50	+2.02	+4.95	-5.75	-4.38
	Relative (%)	+25.0	+8.9	+50.0	-10.5	+33.9	+26.9	-25.0	+17.9	+14.5	+35.4	-41.1	-31.3
Steps (reduced)		86 (86)	136 (136)	172 (172)	199 (199)	222 (222)	241 (241)	257 (257)	272 (272)	285 (285)	297 (297)	307 (307)	317 (317)

Approximation of harmonics in 86edo

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13
Error	Absolute (¢)	+0.00	-4.28	+0.00	+4.38	-4.28	-6.04	+0.00	+5.39	+4.38	+6.82	-4.28	-3.32
	Relative (%)	+0.0	-30.7	+0.0	+31.4	-30.7	-43.3	+0.0	+38.6	+31.4	+48.9	-30.7	-23.8
Steps (reduced)		86 (0)	136 (50)	172 (0)	200 (28)	222 (50)	241 (69)	258 (0)	273 (15)	286 (28)	298 (40)	308 (50)	318 (60)

Others

• Equiheptatonic (as from 7edo): 49 49 49 49 49 49 49
• 49edo: 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7