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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
approx. ratios of
2.3.5.13.17 subgroup
additional ratios
of 7 and 11
ups and downs notation
0
do
0.000
1/1

P1
perfect unison
D
1
di
35.294
128/125 (diesis), 51/50
50/49, 49/48
^1, vm2
up unison, downminor 2nd
D^, Ebv
2
rih
70.588
25/24, 648/625 (large diesis)

m2
minor 2nd
Eb
3
ra
105.882
17/16, 18/17, 16/15
15/14
^m2
upminor 2nd
Eb^
4
ru
141.176
13/12
14/13, 12/11
~2
mid 2nd
Evv
5
reh
176.471
10/9
11/10
vM2
downmajor 2nd
Ev
6
re
211.765
9/8, 17/15

M2
major 2nd
E
7
raw
247.059
15/13
8/7
^M2, vm3
upmajor 2nd, downminor 3rd
E^, Fv
8
meh
282.353
20/17, 75/64
7/6, 13/11
m3
minor 3rd
F
9
me
317.647
6/5
17/14
^m3
upminor 3rd
F^
10
mu
352.941
16/13
11/9
~3
mid 3rd
F^^
11
mi
388.235
5/4

vM3
downmajor 3rd
F#v
12
maa
423.529
51/40, 32/25
14/11, 9/7
M3
major 3rd
F#
13
maw
458.823
13/10, 17/13
22/17
^M3, v4
upmajor 3rd,down 4th
F#^, Gv
14
fa
494.118
4/3

P4
4th
G
15
fih
529.412

15/11
^4
up 4th
G^
16
fu
564.706
18/13
11/8
^^4, d5
double-up 4th, dim 5th
G^^, Ab
17
fi/se
600.000
17/12, 24/17
7/5, 10/7
vA4, ^d5
downaug 4th, updim 5th
G#v, Ab^
18
su
635.294
13/9
16/11
A4, vv5
aug 4th, double-down 5th
G#, Avv
19
sih
670.588

22/15
v5
down 5th
Av
20
sol
705.882
3/2

P5
perfect 5th
A
21
saw
741.176
20/13, 26/17
17/11
^5, vm6
up 5th, downminor 6th
A^, Bbv
22
leh
776.471
25/16, 80/51
14/9
m6
minor 6th
Bb
23
le
811.765
8/5

^m6
upminor 6th
Bb^
24
lu
847.059
13/8
18/11
~6
mid 6th
Bvv
25
la
882.353
5/3
28/17
vM6
downmajor 6th
Bv
26
laa
917.647
17/10
12/7, 22/13
M6
major 6th
B
27
law
952.941
26/15
7/4
^M6, vm7
upmajor 6th, downminor 7th
B^, Cv
28
teh
988.235
16/9, 30/17

m7
minor 7th
C
29
te
1023.529
9/5
20/11
^m7
upminor 7th
C^
30
tu
1058.823
24/13
13/7, 11/6
~7
mid 7th
C^^
31
ti
1094.118
32/17, 17/9, 15/8
28/15
vM7
downmajor 7th
C#v
32
taa
1129.412
48/25, 625/324

M7
major 7th
C#
33
da
1164.706
125/64, 100/51
49/25, 96/49
^M7, v8
upmajor 7th, down 8ve
C#^, Dv
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.

Selected just intervals by error

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

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
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