34edo

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	49/48	P1	perfect unison	D
1	di	35.294	128/125 (diesis), 51/50, 81/80	64/63, 28/27	^1, vm2	up unison, downminor 2nd	D^, Ebv
2	rih	70.588	25/24, 648/625 (large diesis)	50/49, 36/35	m2	minor 2nd	Eb
3	ra	105.882	17/16, 18/17, 16/15	14/13	^m2	upminor 2nd	Eb^
4	ru	141.176	13/12	15/14, 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, 7/6	^M2, vm3	upmajor 2nd, downminor 3rd	E^, Fv
8	meh	282.353	20/17, 75/64	13/11	m3	minor 3rd	F
9	me	317.647	6/5		^m3	upminor 3rd	F^
10	mu	352.941	16/13	11/9, 17/14	~3	mid 3rd	F^^
11	mi	388.235	5/4	14/11	vM3	downmajor 3rd	F#v
12	maa	423.529	51/40, 32/25		M3	major 3rd	F#
13	maw	458.823	13/10, 17/13	22/17, 9/7	^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, 7/5	^^4, d5	double-up 4th, dim 5th	G^^, Ab
17	fi/se	600.000	17/12, 24/17		vA4, ^d5	downaug 4th, updim 5th	G#v, Ab^
18	su	635.294	13/9	16/11, 10/7	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, 14/9	^5, vm6	up 5th, downminor 6th	A^, Bbv
22	leh	776.471	25/16, 80/51		m6	minor 6th	Bb
23	le	811.765	8/5	11/7	^m6	upminor 6th	Bb^
24	lu	847.059	13/8	18/11, 28/17	~6	mid 6th	Bvv
25	la	882.353	5/3		vM6	downmajor 6th	Bv
26	laa	917.647	17/10, 128/75	22/13	M6	major 6th	B
27	law	952.941	26/15	7/4, 12/7	^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	28/15, 11/6	~7	mid 7th	C^^
31	ti	1094.118	32/17, 17/9, 15/8	13/7	vM7	downmajor 7th	C#v
32	taa	1129.412	48/25, 625/324	49/25, 35/18	M7	major 7th	C#
33	da	1164.706	125/64, 100/51, 160/81	63/32, 27/14	^M7, v8	upmajor 7th, down 8ve	C#^, Dv
34	do	1200.000	2/1	96/49	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
49/48	| -4 -1 0 2 >	35.697	slendro diesis
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

Listen

• Ascension
• Uncomfortable In Crowds (extended) by Robin Perry

Links

• 34 Equal Guitar by Larry Hanson
• http://microstick.net/ websites of Neil Haverstick
• https://myspace.com/microstick

See also=

• List of 34edo rank two temperaments by badness