34edo

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34edo divides the octave into 34 equal steps of approximately 35.29412 cents. 34edo contains two 17edo's and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than 31edo, but with a sharp fifth rather than a flat one, and supports hanson, srutal, tetracot, würschmidt and vishnu temperaments. It does less well in the 7-limit, with two mappings possible for 7/4: a flat one from the patent val, and a sharp one from the 34d val. By way of the patent val 34 supports keemun temperament, and 34d is an excellent alternative to 22edo for 7-limit pajara temperament. In the 11-limit, 34de supports 11-limit pajaric, and in fact is quite close to the POTE tuning; it adds 4375/4374 to the commas of 11-limit pajaric. On the other hand, the 34d val supports pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports semaphore on the 2.3.7 subgroup.

Approximations to Just Intonation

Like 17edo, 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11.* 34edo adds ratios of 5 into the mix -- including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions -- as well as 17 -- including 17/16, 18/17, 17/12, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the "syntonic comma" of 81/80, from 21.5 cents to 35.3 cents), it is suitable for 5-limit JI. It is not a meantone system. In layman's terms while no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds, technically will be the same pitch as one somewhere upon the cycle of seventeen fifths.

Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9. (Wikipedia)

  • The sharpening of ~13 cents of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.

Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustomed to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless 68edo (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just.

34edo and phi

As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval phi -- 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates Moment of Symmetry scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and | -6 2 6 0 0 -13 >. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and 36edo.

Rank two temperaments

List of 34edo rank two temperaments by badness

Periods

per octave

Generator Cents Linear temperaments
1 1\34 35.294
3\34 105.882
5\34 176.471 Tetracot/Bunya/Monkey
7\34 247.059 Immunity
9\34 317.647 Hanson/Keemun
11\34 388.235 Wuerschmidt/Worschmidt
13\34 458.824
15\34 529.412
2 1\34 35.294
2\34 70.588 Vishnu
3\34 105.882 Srutal/Pajara/Diaschismic
4\34 141.176 Fifive
5\34 176.471
6\34 211.765
7\34 247.059
8\34 282.353
17 1\34 35.294

Intervals

Degree Solfege Cents pions 7mus approx. ratios of

2.3.5.13.17 subgroup

additional ratios

of 7 and 11

ups and downs notation
Pure octave 45ed(7φ+6)\(5φ^2) 2ed25/24 48:49:50-WT Pure octave 45ed(7φ+6)\(5φ^2) 2ed25/24 48:49:50-WT Pure octave 45ed(7φ+6)\(5φ^2) 2ed25/24 48:49:50-WT
0 do 0.000 1/1 P1 perfect unison D
1 di 35.294 35.296 35.336 35.697 37.412 37.414 37.456 37.839 45.1765 (2D.2D216) 45.179 (2D.2DE16) 45.23 (2D.3AF816) 45.691 (2D.B1216) 128/125 (diesis), 51/50 50/49, 49/48 ^1, vm2 up unison, downminor 2nd D^, Ebv
2 rih 70.588 70.592 70.672 74.8235 74.828 74.913 90.353 (5A.5A516) 90.358 (5A.5BB16) 90.461 (5A.75F16) 25/24, 648/625 (large diesis) m2 minor 2nd Eb
3 ra 105.882 105.8885 106.008 106.369 111.235 112.242 112.369 112.751 135.529 (87.878816) 135.537 (87.89916) 135.691 (87.B0F16) 136.153 (88.27116) 17/16, 18/17, 16/15 15/14 ^m2 upminor 2nd Eb^
4 ru 141.1765 141.185 141.345 149.647 149.656 149.8255 180.706 (B4.B4B16) 180.716 (B4.B7716) 180.921 (B4.EBE16) 13/12 14/13, 12/11 ~2 mid 2nd Evv
5 reh 176.471 176.481 176.681 177.042 187.059 187.07 187.282 187.664 225.882 (E1.E1E16) 225.896 (E1.E5416) 226.152 (E2.26E16) 226.613 (E2.9D16) 10/9 11/10 vM2 downmajor 2nd Ev
6 re 211.765 211.777 212.017 224.471 224.484 224.738 271.059 (10F.0F116) 271.075 (10F.13216) 271.382 (10F.61D16) 9/8, 17/15 M2 major 2nd E
7 raw 247.059 247.073 247.3535 247.714 261.882 261.898 262.195 262.577 326.235 (13C.3C316) 316.254 (13C.40F816) 316.6125 (13C.7EA16) 317.074 (13D.12F16) 15/13 8/7 ^M2, vm3 upmajor 2nd, downminor 3rd E^, Fv
8 meh 282.353 282.3695 282.69 299.294 299.312 299.651 361.412 (169.69616) 361.433 (169.6ED16) 361.843 (169.D7C16) 20/17, 75/64 7/6, 13/11 m3 minor 3rd F
9 me 317.647 317.666 318.026 318.3865 336.706 336.726 337.1075 337.49 406.588 (196.96916) 406.612 (196.9CB16) 407.073 (197.12C16) 407.535 (197.88E16) 6/5 17/14 ^m3 upminor 3rd F^
10 mu 352.941 352.962 353.362 374.118 374.1395 374.564 451.765 (1C3.C3C16) 451.791 (1C3.CA816) 452.3035 (1C4.4DB16) 16/13 11/9 ~3 mid 3rd F^^
11 mi 388.235 388.258 388.698 389.059 411.529 411.5535 412.02 412.4025 496.941 (1F0.F0F16) 496.97 (1F0.F8616) 497.534 (1F1.88B16) 497.9955 (1F1.FED16) 5/4 vM3 downmajor 3rd F#v
12 maa 423.529 423.554 424.035 448.941 448.967 449.477 542.118 (21E.1E116) 542.149 (21E.26416) 542.764 (21E.C3A16) 51/40, 32/25 14/11, 9/7 M3 major 3rd F#
13 maw 458.8235 458.85 459.371 459.731 486.353 486.381 486.933 487.315 587.294 (24B.4B416) 587.3285 (24B.54116) 587.995 (24B.FEA16) 588.456 (24C.74C16) 13/10, 17/13 22/17 ^M3, v4 upmajor 3rd,down 4th F#^, Gv
14 fa 494.118 494.1465 494.707 523.765 523.795 524.389 632.471 (278.787816) 632.508 (278.81F16) 633.225 (279.39A16) 4/3 P4 4th G
15 fih 529.412 529.443 530.043 530.40 561.1765 561.209 561.846 562.228 677.647 (2A5.A5A16) 677.687 (2A5.AFD16) 678.455 (2A6.74916) 678.917 (2A6.EAB816) 512/375, 34/25 15/11 ^4 up 4th G^
16 fu 564.706 564.739 565.379 598.588 598.623 599.302 722.8235 (2D2.D2D16) 722.866 (2D2.DDA16) 723.686 (2D3.AF816) 36/25, 18/13 11/8 ^^4, d5 double-up 4th, dim 5th G^^, Ab
17 fi/se 600.000 600.035 600.716 601.076 636 636.037 636.758 637.141 768 (30016) 768.045 (300.0B816) 768.916 (300.EA816) 769.378 (301.60A816) 17/12, 24/17 7/5, 10/7 vA4, ^d5 downaug 4th, updim 5th G#v, Ab^
18 su 635.294 635.331 636.052 673.412 673.451 674.215 813.1765 (32D.2D216) 813.224 (32D.39616) 814.146 (32E.257816) 25/18, 13/9 16/11 A4, vv5 aug 4th, double-down 5th G#, Avv
19 sih 670.588 670.627 671.388 671.749 710.8235 710.865 711.671 712.054 858.353 (35A.5A516) 858.403 (35A.67316) 859.377 (35B.60716) 859.838 (35B.D6A16) 375/256, 25/17 22/15 v5 down 5th Av
20 sol 705.882 705.924 706.724 748.235 748.279 749.128 903.529 (387.878816) 903.582 (387.95116) 904.607 (388.9B716) 3/2 P5 perfect 5th A
21 saw 741.1765 741.22 742.0605 742.421 785.647 785.693 786.584 786.966 948.706 (3B4.B4B16) 948.761 (3B4.C2E816) 949.837 (3B5.D6616) 950.299 (3B6.4C916) 20/13, 26/17 17/11 ^5, vm6 up 5th, downminor 6th A^, Bbv
22 leh 776.471 776.516 777.397 823.059 823.107 824.0405 993.882 (3E1.E1E16) 993.9405 (3E1.F0C16) 995.068 (3E3.11616) 25/16, 80/51 14/9 m6 minor 6th Bb
23 le 811.765 811.812 812.733 813.0935 860.471 860.521 861.497 861.879 1039.059 (40F.0F116) 1039.12 (40F.1EA16) 1040.298 (410.4C516) 1040.76 (410.C2816) 8/5 ^m6 upminor 6th Bb^
24 lu 847.059 847.108 848.069 897.882 897.935 898.953 1084.235 (43C.3C316) 1084.299 (43C.4C716) 1085.5285 (43D.87416) 13/8 18/11 ~6 mid 6th Bvv
25 la 882.353 882.4045 883.405 883.766 935.294 935.349 936.41 936.792 1129.412 (469.69616) 1129.478 (469.7A516) 1130.759 (46A.C4216) 1131.22 (46B.38716) 5/3 28/17 vM6 downmajor 6th Bv
26 laa 917.647 917.701 918.7415 972.706 972.763 973.866 1174.588 (496.96916) 1174.657 (496.A8316) 1175.989 (497.FD416) 17/10 12/7, 22/13 M6 major 6th B
27 law 952.941 952.997 954.078 954.438 1010.118 1010.177 1011.322 1011.705 1219.765 (4C3.C3C16) 1219.836 (4C3.D616) 1221.2195 (4C5.38316) 1221.681 (4C5.AE616) 26/15 7/4 ^M6, vm7 upmajor 6th, downminor 7th B^, Cv
28 teh 988.235 988.293 989.414 1047.529 1047.591 1048.779 1264.941 (4F0.F0F16) 1265.015 (4F1.03E16) 1266.45 (4F2.73316) 16/9, 30/17 m7 minor 7th C
29 te 1023.529 1023.589 1024.75 1025.111 1084.941 1085.005 1086.235 1086.617 1310.118 (51E.1E116) 1310.194 (51E.31C16) 1311.68 (51F.AE216) 1312.142 (520.24516) 9/5 20/11 ^m7 upminor 7th C^
30 tu 1058.8235 1058.885 1060.086 1122.353 1122.419 1123.692 1355.294 (54B.4B416) 1355.373 (54B.5F916) 1356.911 (54C.E9116) 24/13 13/7, 11/6 ~7 mid 7th C^^
31 ti 1094.118 1094.182 1095.423 1095.783 1149.765 1159.8325 1161.148 1161.53 1400.471 (578.787816) 1400.5525 (578.8D716) 1402.141 (57A.24116) 1402.6025 (57A.9A416) 32/17, 17/9, 15/8 28/15 vM7 downmajor 7th C#v
32 taa 1129.412 1129.478 1130.759 1197.1765 1197.2465 1198.604 1445.647 (5A5.A5A16) 1445.732 (5A5.BB516) 1447.371 (5A7.5F116) 48/25, 625/324 M7 major 7th C#
33 da 1164.706 1164.774 1166.095 1166.456 1234.588 1234.66 1236.061 1236.443 1490.8235 (5D2.D2D16) 1490.911 (5D2.E9216) 1492.602 (5D4.9A116) 1493.063 (5D5.10316) 125/64, 100/51 49/25, 96/49 ^M7, v8 upmajor 7th, down 8ve C#^, Dv
34 do 1200.000 1200.07 1201.431 1272 1272.074 1273.52 1536 (60016) 1536.09 (600.1716) 1537.832 (601.D516) 2/1 P8 8ve D

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 34edo (ordered by absolute error).

Best direct mapping, even if inconsistent

Interval, complement Error (abs., in cents)
15/13, 26/15 0.682
18/13, 13/9 1.324
5/4, 8/5 1.922
6/5, 5/3 2.006
13/12, 24/13 2.604
4/3, 3/2 3.927
13/10, 20/13 4.610
11/9, 18/11 5.533
16/15, 15/8 5.849
10/9, 9/5 5.933
14/11, 11/7 6.021
16/13, 13/8 6.531
13/11, 22/13 6.857
15/11, 22/15 7.539
9/8, 16/9 7.855
12/11, 11/6 9.461
11/10, 20/11 11.466
9/7, 14/9 11.555
14/13, 13/7 12.878
11/8, 16/11 13.388
15/14, 28/15 13.560
7/6, 12/7 15.482
8/7, 7/4 15.885
7/5, 10/7 17.488

Patent val mapping

Interval, complement Error (abs., in cents)
15/13, 26/15 0.682
18/13, 13/9 1.324
5/4, 8/5 1.922
6/5, 5/3 2.006
13/12, 24/13 2.604
4/3, 3/2 3.927
13/10, 20/13 4.610
11/9, 18/11 5.533
16/15, 15/8 5.849
10/9, 9/5 5.933
16/13, 13/8 6.531
13/11, 22/13 6.857
15/11, 22/15 7.539
9/8, 16/9 7.855
12/11, 11/6 9.461
11/10, 20/11 11.466
11/8, 16/11 13.388
8/7, 7/4 15.885
7/5, 10/7 17.806
7/6, 12/7 19.812
15/14, 28/15 21.734
14/13, 13/7 22.416
9/7, 14/9 23.739
14/11, 11/7 29.273

Notations

The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away - the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which may not be most efficient expressions of some musical purposes. The reader can construct his own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C ... F, instead of seven, might be waste - of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.

Commas

34-EDO tempers out the following commas. (Note: This assumes the val < 34 54 79 95 118 126 |.)

Rational Monzo Size (Cents) Names
134217728/129140163 | 27 -17 > 66.765 17-comma
20000/19683 | 5 -9 4 > 27.660 Minimal Diesis, Tetracot Comma
2048/2025 | 11 -4 -2 > 19.553 Diaschisma
393216/390625 | 17 1 -8 > 11.445 Würschmidt comma
15625/15552 | -6 -5 6 > 8.107 Kleisma, Semicomma Majeur
1212717/1210381 | 23 6 -14 > 3.338 Vishnuzma, Semisuper
1029/1000 | -3 1 -3 3 > 49.492 Keega
50/49 | 1 0 2 -2 > 34.976 Jubilisma
875/864 | -5 -3 3 1 > 21.902 Keema
126/125 | 1 2 -3 1 > 13.795 Starling comma, Septimal semicomma
100/99 | 2 -2 2 0 -1> 17.399 Ptolemisma, Ptolemy's comma
243/242 | -1 5 0 0 -2 > 7.139 Rastma, Neutral third comma
385/384 | -7 -1 1 1 1 > 4.503 Keenanisma
91/90 | -1 -2 -1 1 0 1 > 19.120 Superleap

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