34edo

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34edo
Prime factorization 2 × 17
Step size 35.294 ¢
Fifth 20\34 = 705.88¢ (→10\17)
Major 2nd 6\34 = 212¢
Minor 2nd 2\34 = 71¢
Augmented 1sn 4\34 = 141¢

34edo divides the octave into 34 equal steps of approximately 35.3 cents.

Introduction

Approximation of odd harmonics in 34 EDO
Odd harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Error absolute (¢) +3.9 +1.9 -15.9 +7.9 +13.4 +6.5 +5.8 +0.9 -15.2 -12.0 +7.0 +3.8 +11.8 -6.0 -15.6
relative (%) +11 +5 -45 +22 +38 +19 +17 +3 -43 -34 +20 +11 +33 -17 -44
Steps (reduced) 54 (20) 79 (11) 95 (27) 108 (6) 118 (16) 126 (24) 133 (31) 139 (3) 144 (8) 149 (13) 154 (18) 158 (22) 162 (26) 165 (29) 168 (32)

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 not tempering out 81/80 unlike 31edo.

Intervals

Degree Solfege Cents Approx. Ratios of
2.3.5.11.13.17 Subgroup
Additional Ratios of 7
Tending Flat (34 Val)
Additional Ratios of 7
Tending Sharp (34d Val)
Ups and Downs Notation
0 do 0.000 1/1 P1 perfect unison D
1 di 35.294 81/80, 128/125, 51/50 28/27, 64/63 36/35 ^1, vm2 up 1sn, downminor 2nd ^D, vEb
2 rih 70.588 25/24, 648/625, 33/32 21/20, 36/35, 50/49 28/27, 49/48 ^^1, m2 double-up 1sn, minor 2nd ^^D, Eb
3 ra 105.882 17/16, 18/17, 16/15 14/13 15/14, 21/20 vA1, ^m2 downaug 1sn, upminor 2nd vD#, ^Eb
4 ru 141.176 13/12, 12/11 15/14 14/13 A1, ~2 aug 1sn, mid 2nd D#, vvE
5 reh 176.471 10/9, 11/10 vM2 downmajor 2nd vE
6 re 211.765 9/8, 17/15 8/7 M2 major 2nd E
7 raw 247.059 15/13 7/6, 8/7 ^M2, vm3 upmajor 2nd, downminor 3rd ^E, vF
8 meh 282.353 20/17, 75/64, 13/11 7/6 m3 minor 3rd F
9 me 317.647 6/5 17/14 ^m3 upminor 3rd ^F
10 mu 352.941 16/13, 11/9, 27/22 17/14, 21/17 ~3 mid 3rd ^^F
11 mi 388.235 5/4 14/11 21/17 vM3 downmajor 3rd vF#
12 maa 423.529 51/40, 32/25 9/7, 14/11 M3 major 3rd F#
13 maw 458.824 13/10, 17/13, 22/17 9/7, 21/16 ^M3, v4 upmajor 3rd,down 4th ^F#, vG
14 fa 494.118 4/3 21/16 P4 4th G
15 fih 529.412 27/20, 34/25, 15/11 ^4 up 4th ^G
16 fu 564.706 25/18, 18/13, 11/8 7/5 ~4, d5 mid 4th, dim 5th ^^G, Ab
17 fi/se 600.000 45/32, 64/45, 17/12, 24/17 7/5, 10/7 vA4, ^d5 downaug 4th, updim 5th vG#, ^Ab
18 su 635.294 36/25, 13/9, 16/11 10/7 A4, ~5 aug 4th, mid 5th G#, vvA
19 sih 670.588 40/27, 25/17, 22/15 v5 down 5th vA
20 sol 705.882 3/2 32/21 P5 perfect 5th A
21 saw 741.176 20/13, 26/17, 17/11 14/9, 32/21 ^5, vm6 up 5th, downminor 6th ^A, vBb
22 leh 776.471 25/16, 80/51 14/9, 11/7 m6 minor 6th Bb
23 le 811.765 8/5 11/7 34/21 ^m6 upminor 6th ^Bb
24 lu 847.059 13/8, 18/11, 44/27 28/17, 34/21 ~6 mid 6th vvB
25 la 882.353 5/3 28/17 vM6 downmajor 6th vB
26 laa 917.647 17/10, 128/75, 22/13 12/7 M6 major 6th B
27 law 952.941 26/15 7/4, 12/7 ^M6, vm7 upmajor 6th, downminor 7th ^B, vC
28 teh 988.235 16/9, 30/17 7/4 m7 minor 7th C
29 te 1023.529 9/5, 20/11 ^m7 upminor 7th ^C
30 tu 1058.824 24/13, 11/6 28/15 13/7 ~7 mid 7th ^^C
31 ti 1094.118 32/17, 17/9, 15/8 13/7 28/15, 40/21 vM7 downmajor 7th vC#
32 taa 1129.412 48/25, 625/324, 64/33 40/21, 35/18, 49/25 27/14, 96/49 M7 major 7th C#
33 da 1164.706 160/81, 125/64, 100/51 27/14, 63/32 35/18 ^M7, v8 upmajor 7th, down 8ve ^C#, vD
34 do 1200.000 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.

JI approximation

Like 17edo, 34edo contains good approximations of just intervals involving 13, 11, and 3 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7. 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, 17/11, 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 quasi-5-limit JI but is not a meantone system. While no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.

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 is 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 especially in kleismic or 4L 3s contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. (68edo contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)

Selected just intervals by error

The following table shows how 15-odd-limit intervals are represented in 34edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
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, ¢)
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

Tuning by ear

In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen 25/24 chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%.

34edo and logarithmic phi

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic 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. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and 36edo.)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3.5 2048/2025, 15625/15552 [34 54 79]] -1.10 1.03 2.92
2.3.5.7 50/49, 64/63, 4375/4374 [34 54 79 96]] (34d) -2.56 2.66 7.57
2.3.5.7.11 50/49, 64/63, 99/98, 243/242 [34 54 79 96 118]] (34d) -2.82 2.44 6.93
2.3.5.7.11.13 50/49, 64/63, 78/77, 99/98, 144/143 [34 54 79 96 118 126]] (34d) -2.64 2.26 6.42
2.3.5.7.11.13.17 50/49, 64/63, 78/77, 85/84, 99/98, 144/143 [34 54 79 96 118 126 139]] (34d) -2.30 2.26 6.41

In the 5-limit, 34edo 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.

Rank-2 temperaments

34et supports the following MOSes and rank-2 temperaments:

Rank-2 temperaments by period and generator
Periods
per octave
Generator Cents MOSes Temperaments
1 1\34 35.294 Gammic
" 3\34 105.88 11L 1s
11L 12s
" 5\34 176.471 6L 1s
7L 6s
7L 13s
7L 20s
Tetracot / bunya (34d) / modus (34d) / monkey (34) / wollemia (34)
" 7\34 247.059 5L 4s
5L 9s
5L 14s
5L 19s
Pathological 5L 24s
Immunity (34) / immunized (34d)
" 9\34 317.647 4L 3s
4L 7s
4L 11s
15L 4s
Hanson / keemun (34) / catalan (34d) / catakleismic (34d)
" 11\34 388.235 3L 7s
3L 10s
3L 13s
3L 16s
3L 19s
3L 22s
Pathological 3L 25s
Pathological 3L 28s
Würschmidt (34d) / worschmidt (34)
" 13\34 458.824 3L 2s
5L 3s
8L 5s
13L 8s
Petrtri
" 15\34 529.412 2L 3s
2L 5s
7L 2s
9L 7s
9L 16s
Mabila
2 2\34 70.588 16L 2s Vishnu
" 3\34 105.882 2L 6s
2L 8s
10L 2s
12L 10s
Srutal (34d) / pajara (34d) / diaschismic (34)
" 4\34 141.176 2L 6s
8L 2s
8L 10s
8L 16s
Fifive / crepuscular (34d) / fifives (34)
" 5\34 176.471 6L 2s
6L 8s
14L 6s
" 6\34 211.765 4L 2s
6L 4s
6L 10s
6L 16s
Pathological 6L 22s
Antikythera
" 7\34 247.059 4L 2s
4L 6s
10L 4s
10L 14s
" 8\34 282.353 2L 2s
4L 2s
4L 6s
4L 10s
4L 14s
4L 18s
4L 22s
Pathological 4L 26s

Commas

34-EDO tempers out the following commas. This assumes the patent val 34 54 79 95 118 126].

Prime
Limit
Ratio[1] Monzo Cents Color name Name(s)
3 (18 digits) [27 -17 66.765 Sasawa 17-comma
5 20000/19683 [5 -9 4 27.660 Saquadyo Minimal diesis, tetracot comma
5 2048/2025 [11 -4 -2 19.553 Sagugu Diaschisma
5 (12 digits) [17 1 -8 11.445 Saquadbigu Würschmidt comma
5 15625/15552 [-6 -5 6 8.107 Tribiyo Kleisma, semicomma majeur
5 (20 digits) [23 6 -14 3.338 Sasepbigu Vishnuzma, semisuper comma
7 1029/1000 [-3 1 -3 3 49.492 Trizogu Keega
7 49/48 [-4 -1 0 2 35.697 Zozo Septimal diesis, slendro diesis
7 875/864 [-5 -3 3 1 21.902 Zotriyo Keema
7 126/125 [1 2 -3 1 13.795 Zotrigu Starling comma, septimal semicomma
11 100/99 [2 -2 2 0 -1 17.399 Luyoyo Ptolemisma, Ptolemy's comma
11 243/242 [-1 5 0 0 -2 7.139 Lulu Rastma, neutral third comma
11 385/384 [-7 -1 1 1 1 4.503 Lozoyo Keenanisma
13 91/90 [-1 -2 -1 1 0 1 19.120 Thozogu Superleap
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Notations

Kosmorsky's thoughts

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. Users can construct their 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.

Music

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