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			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	Pure octave	45ed(7φ+6)\(5φ^2)	2ed25/24	Pure octave	45ed(7φ+6)\(5φ^2)	2ed25/24
0	do	0.000									1/1		P1	perfect unison	D
1	di	35.294	35.296	35.336	37.412	37.414	37.456	45.1765 (2D.2D216)	45.179 (2D.2DE16)	45.23 (2D.3AF816)	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	111.235	112.242	112.369	135.529 (87.878816)	135.537 (87.89916)	135.691 (87.B0F16)	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	187.059	187.07	187.282	225.882 (E1.E1E16)	225.896 (E1.E5416)	226.152 (E2.26E16)	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	261.882	261.898	262.195	326.235 (13C.3C316)	316.254 (13C.40F816)	316.6125 (13C.7EA16)	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	336.706	336.726	337.1075	406.588 (196.96916)	406.612 (196.9CB16)	407.073 (197.12C16)	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	411.529	411.5535	412.02	496.941 (1F0.F0F16)	496.97 (1F0.F8616)	497.534 (1F1.88B16)	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	486.353	486.381	486.933	587.294 (24B.4B416)	587.3285 (24B.54116)	587.995 (24B.FEA16)	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 (279.399A16)	4/3		P4	4th	G
15	fih	529.412	529.443	530.043	561.1765	561.209	561.846	677.647 (2A5.A5A16)	677.687 (2A5.AFD16)	678.455 (2A6.74916)	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	636	636.037	636.758	768 (30016)	768.045 (300.0B816)	768.916 (300.EA816,)	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	710.8235	710.865	711.671	858.353 (35A.5A516)	858.403 (35A.67316)	859.377 (35B.60716)	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	785.647	785.693	786.584	948.706 (3B4.B4B16)	948.761 (3B4.C2E816)	949.837 (3B5.D6616)	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	860.471	860.521	861.497	1039.059 (40F.0F116)	1039.12 (40F.1EA16)	1040.298 (410.4C516)	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	935.294	935.349	936.41	1129.412 (469.69616)	1129.478 (469.7A516)	1130.759 (46A.C4216)	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	1010.118	1010.177	1011.322	1219.765 (4C3.C3C16)	1219.836 (4C3.D616)	1221.2195 (4C5.38316)	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	1,023.589	1024.75	1084.941	1085.005	1086.235	1310.118 (51E.1E116)	1310.194 (51E.31C16)	1311.68 (51F.AE216)	9/5	20/11	^m7	upminor 7th	C^
30	tu	1058.8235	1,058.885	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	1,094.182	1095.423	1149.765	1159.8325	1161.148	1400.471 (578.787816)	1400.5525 (578.8D716)	1402.141 (57A.24116)	32/17, 17/9, 15/8	28/15	vM7	downmajor 7th	C#v
32	taa	1129.412	1,129.478	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	1,164.774	1166.095	1234.588	1234.66	1236.061	1490.8235 (5D2.D2D16)	1490.911 (5D2.E9216)	1492.602 (5D4.9A116)	125/64, 100/51	49/25, 96/49	^M7, v8	upmajor 7th, down 8ve	C#^, Dv
34	do	1200.000	1,200.07	1201.431	1272	1272.074	1273.52	1536 (60016)	1536.09 (600.1716)	1537.832 (601.D56F)	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

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