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343edo

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 21/343, or the 343rd root of 2.

← 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

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).

Lucite[23]

Lucite[23] is 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
Properties

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 (104.956 ¢).

Lucite[23] can be generalised into a 17-limit regular temperament called lucite temperament.

Naming

Lucite is another name for acrylic glass.

Wery named the temperament "lucite" because musically, it sounds like frosted glass (perhaps to do with the timbre/partials of struck glass).

Some coincidences that make the name "lucite" particularly fitting

  • Lucite is often installed in double layers in building, and lucite temperament has two sizes of perfect fifth-like interval.
  • Lucite[23] is close to ripple[23], but turned inside out; and lucite is reflective and clear like water, but solid instead of liquid
  • Lucite is an especially lightweight material, and lucite temperament is lightweight in the way it only needs 18 generators to reach every 17-limit prime.
Subsets
  • Modmos of lucite[6]: 60 60 30 40 93 60

Other 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
    • Try approximating scales fron 53edo (53edo#Scales) within the amity[53] scale
  • 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
    • Try approximating scales fron 99edo (99edo#Scales) within the amity[99] scale
  • Lucite[23]: 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13
    • Try approximating scales fron 23edo (23edo#Scales) within the lucite[23] scale
  • Lucite[34]: 13 4 13 13 4 13 13 4 13 13 4 13 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13
    • Try approximating scales fron 34edo (34edo#Scales) within the lucite[34] scale

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)
86edo for comparison
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)

34.3edo

34.3edo is contained within 343edo (it is every 10th step of 343edo). It is like 34edo with the octave compressed by 11.51 cents.

It has a step size of 34.985 ¢.

It was discovered by chaseofspades513 and YoVariable on the Xenharmonic Alliance Discord server and further described by Gordon Wery.

Compared to 34edo it improves harmonics 7, 11 and 13, at the expense of 2, 3 and 5. Its mappings of multiple-of-2 and multiple-of-3 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 34edo can be used here in 34.3edo too.

Approximation of harmonics in 34.3edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -10.5 -12.8 +14.0 +12.5 +11.7 -10.3 +3.5 +9.5 +2.0 +11.9 +1.2 +2.6
Relative (%) -30.0 -36.5 +39.9 +35.7 +33.5 -29.3 +9.9 +27.0 +5.6 +34.0 +3.4 +7.3
Step 34 54 69 80 89 96 103 109 114 119 123 127
34edo for comparison
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +3.9 +0.0 +1.9 +3.9 -15.9 +0.0 +7.9 +1.9 +13.4 +3.9 +6.5
Relative (%) +0.0 +11.1 +0.0 +5.4 +11.1 -45.0 +0.0 +22.3 +5.4 +37.9 +11.1 +18.5
Steps
(reduced) 		34
(0) 		54
(20) 		68
(0) 		79
(11) 		88
(20) 		95
(27) 		102
(0) 		108
(6) 		113
(11) 		118
(16) 		122
(20) 		126
(24)

Scales approximated from JI

Other scales

  • 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

Music

Budjarn Lambeth
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