34edo

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← 33edo34edo35edo →
Prime factorization 2 × 17
Step size 35.2941¢ 
Fifth 20\34 (705.882¢) (→10\17)
Semitones (A1:m2) 4:2 (141.2¢ : 70.59¢)
Consistency limit 5
Distinct consistency limit 5

34 equal divisions of the octave (abbreviated 34edo or 34ed2), also called 34-tone equal temperament (34tet) or 34 equal temperament (34et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 34 equal parts of about 35.3 ¢ each. Each step represents a frequency ratio of 21/34, or the 34th root of 2.

English Wikipedia has an article on:

Theory

34edo contains two 17edo's and the half-octave tritone of 600 cents. It excels in approximating harmonics 3, 5, 13, 17, and 23 (2.3.5.13.17.23 subgroup a.k.a. the no-7's no-11's no-19's 23-limit), with tuning even more accurate than 31edo in the 5-limit, but with a sharp tendency and fifth rather than a flat one, and not tempering out 81/80 unlike 31edo.

Odd harmonics

Approximation of odd harmonics in 34edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25
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
Relative (%) +11.1 +5.4 -45.0 +22.3 +37.9 +18.5 +16.6 +2.6 -43.0 -33.9 +19.9 +10.9
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)

Intervals

Cents Approx. Ratios of
2.3.5.11.13.17.23 subgroup 		Ratios of 7
Using the 34 Val 		Ratios of 7
Using the 34d Val 		Ups and Downs Notation Solfeges
0 0.000 1/1 P1 perfect unison D da do
1 35.294 81/80, 128/125, 51/50 28/27, 64/63 36/35 ^1, vm2 up 1sn, downminor 2nd ^D, vEb du/fro di
2 70.588 25/24, 26/25, 24/23, 27/26, 23/22, 648/625, 33/32 21/20, 36/35, 50/49 28/27, 49/48 ^^1, m2 dup 1sn, minor 2nd ^^D, Eb fra rih
3 105.882 17/16, 18/17, 16/15 14/13 15/14, 21/20 vA1, ^m2 downaug 1sn,
upminor 2nd 		vD#, ^Eb fru ra
4 141.176 13/12, 12/11, 25/23 15/14 14/13 A1, ~2 aug 1sn, mid 2nd D#, vvE ri ru
5 176.471 10/9, 11/10 vM2 downmajor 2nd vE ro reh
6 211.765 9/8, 17/15, 26/23 8/7 M2 major 2nd E ra re
7 247.059 15/13, 23/20 7/6, 8/7 ^M2, vm3 upmajor 2nd,
downminor 3rd 		^E, vF ru/no raw
8 282.353 20/17, 75/64, 27/23, 13/11 7/6 m3 minor 3rd F na meh
9 317.647 6/5 17/14 ^m3 upminor 3rd ^F nu me
10 352.941 16/13, 11/9, 27/22 17/14, 21/17 ~3 mid 3rd ^^F mi mu
11 388.235 5/4 14/11 21/17 vM3 downmajor 3rd vF# mo mi
12 423.529 51/40, 32/25, 23/18 9/7, 14/11 M3 major 3rd F# ma maa
13 458.824 13/10, 30/23, 17/13, 22/17 9/7, 21/16 ^M3, v4 upmajor 3rd, down 4th ^F#, vG mu/fo maw
14 494.118 4/3 21/16 P4 4th G fa fa
15 529.412 27/20, 34/25, 15/11, 23/17 ^4 up 4th ^G fu fih
16 564.706 25/18, 18/13, 11/8, 32/23 7/5 ~4, d5 mid 4th, dim 5th ^^G, Ab fi/sha fu
17 600.000 45/32, 64/45, 17/12, 24/17 7/5, 10/7 vA4, ^d5 downaug 4th, updim 5th vG#, ^Ab po/shu fi/se
18 635.294 36/25, 13/9, 16/11, 23/16 10/7 A4, ~5 aug 4th, mid 5th G#, vvA pa/si su
19 670.588 40/27, 25/17, 22/15, 34/23 v5 down 5th vA so sih
20 705.882 3/2 32/21 P5 perfect 5th A sa sol
21 741.176 20/13, 23/15, 26/17, 17/11 14/9, 32/21 ^5, vm6 up 5th, downminor 6th ^A, vBb su/flo saw
22 776.471 25/16, 80/51, 36/23 14/9, 11/7 m6 minor 6th Bb fla leh
23 811.765 8/5 11/7 34/21 ^m6 upminor 6th ^Bb flu le
24 847.059 13/8, 18/11, 44/27 28/17, 34/21 ~6 mid 6th vvB li lu
25 882.353 5/3 28/17 vM6 downmajor 6th vB lo la
26 917.647 17/10, 128/75, 46/27, 22/13 12/7 M6 major 6th B la laa
27 952.941 26/15, 40/23 7/4, 12/7 ^M6, vm7 upmajor 6th,
downminor 7th 		^B, vC lu/tho law
28 988.235 16/9, 30/17, 23/13 7/4 m7 minor 7th C tha teh
29 1023.529 9/5, 20/11 ^m7 upminor 7th ^C thu te
30 1058.824 24/13, 11/6, 46/25 28/15 13/7 ~7 mid 7th ^^C ti tu
31 1094.118 32/17, 17/9, 15/8 13/7 28/15, 40/21 vM7 downmajor 7th vC# to ti
32 1129.412 48/25, 25/13, 23/12, 625/324, 64/33 40/21, 35/18, 49/25 27/14, 96/49 M7 major 7th C# ta taa
33 1164.706 160/81, 125/64, 100/51 27/14, 63/32 35/18 ^M7, v8 upmajor 7th, down 8ve vD tu/do da
34 1200.000 2/1 P8 8ve D da do

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.

Approximation to JI

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Selected 19-limit intervals approximated in 34edo

Like 17edo, 34edo contains good approximations of just intervals involving 3, 11, and 13 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7 given its step size. 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 (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.)

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 34edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 34edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/13, 26/15 0.682 1.9
13/9, 18/13 1.324 3.8
5/4, 8/5 1.922 5.4
5/3, 6/5 2.006 5.7
13/12, 24/13 2.604 7.4
3/2, 4/3 3.927 11.1
13/10, 20/13 4.610 13.1
11/9, 18/11 5.533 15.7
15/8, 16/15 5.849 16.6
9/5, 10/9 5.933 16.8
11/7, 14/11 6.021 17.1
13/8, 16/13 6.531 18.5
13/11, 22/13 6.857 19.4
15/11, 22/15 7.539 21.4
9/8, 16/9 7.855 22.3
11/6, 12/11 9.461 26.8
11/10, 20/11 11.466 32.5
9/7, 14/9 11.555 32.7
13/7, 14/13 12.878 36.5
11/8, 16/11 13.388 37.9
15/14, 28/15 13.560 38.4
7/6, 12/7 15.482 43.9
7/4, 8/7 15.885 45.0
7/5, 10/7 17.488 49.5
15-odd-limit intervals in 34edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/13, 26/15 0.682 1.9
13/9, 18/13 1.324 3.8
5/4, 8/5 1.922 5.4
5/3, 6/5 2.006 5.7
13/12, 24/13 2.604 7.4
3/2, 4/3 3.927 11.1
13/10, 20/13 4.610 13.1
11/9, 18/11 5.533 15.7
15/8, 16/15 5.849 16.6
9/5, 10/9 5.933 16.8
13/8, 16/13 6.531 18.5
13/11, 22/13 6.857 19.4
15/11, 22/15 7.539 21.4
9/8, 16/9 7.855 22.3
11/6, 12/11 9.461 26.8
11/10, 20/11 11.466 32.5
11/8, 16/11 13.388 37.9
7/4, 8/7 15.885 45.0
7/5, 10/7 17.806 50.5
7/6, 12/7 19.812 56.1
15/14, 28/15 21.734 61.6
13/7, 14/13 22.416 63.5
9/7, 14/9 23.739 67.3
11/7, 14/11 29.273 82.9
15-odd-limit intervals by 34d val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/13, 26/15 0.682 1.9
13/9, 18/13 1.324 3.8
5/4, 8/5 1.922 5.4
5/3, 6/5 2.006 5.7
13/12, 24/13 2.604 7.4
3/2, 4/3 3.927 11.1
13/10, 20/13 4.610 13.1
11/9, 18/11 5.533 15.7
15/8, 16/15 5.849 16.6
9/5, 10/9 5.933 16.8
11/7, 14/11 6.021 17.1
13/8, 16/13 6.531 18.5
13/11, 22/13 6.857 19.4
15/11, 22/15 7.539 21.4
9/8, 16/9 7.855 22.3
11/6, 12/11 9.461 26.8
11/10, 20/11 11.466 32.5
9/7, 14/9 11.555 32.7
13/7, 14/13 12.878 36.5
11/8, 16/11 13.388 37.9
15/14, 28/15 13.560 38.4
7/6, 12/7 15.482 43.9
7/5, 10/7 17.488 49.5
7/4, 8/7 19.409 55.0

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%.

Approximation to irrational intervals

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.)

Counterpoint

34edo has such an excellent sqrt(25/24) that the next edo to have a better one is 441. Every sequence of intervals available in 17edo are reachable by strict contrary motion in 34edo.

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

Rank-2 temperaments by period and generator
Periods
per 8ve 		Generator Cents Mosses 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 		Stratosphere
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 		Tobago
8\34 282.353 2L 2s
4L 2s
4L 6s
4L 10s
4L 14s
4L 18s
4L 22s
Pathological 4L 26s 		Bikleismic

Commas

34edo 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
3 (18 digits) [27 -17⟩ 66.765 Sasawa 17-comma
5 20000/19683 [5 -9 4⟩ 27.660 Saquadyo 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
5 (20 digits) [23 6 -14⟩ 3.338 Sasepbigu Vishnuzma
7 1029/1000 [-3 1 -3 3⟩ 49.492 Trizogu Keega
7 49/48 [-4 -1 0 2⟩ 35.697 Zozo 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
11 100/99 [2 -2 2 0 -1⟩ 17.399 Luyoyo Ptolemisma
11 243/242 [-1 5 0 0 -2⟩ 7.139 Lulu Rastma
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

Notation

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

Modern renderings

Scott Joplin
Marco Uccellini

21st century

Flora Canou
E8 Heterotic
Francium
Adam Freese
Hideya
Peter Kosmorsky
luphoria
Claudi Meneghin
Ray Perlner
Robin Perry
Tapeworm Saga
Cam Taylor
Userminusone
Randy Wells
Xotla
Zhea Erose

See also

External links

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