34edo

← 33edo

34edo

35edo →

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

34edo divides the octave into 34 equal steps of approximately 35.29412 cents.

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17
Error	absolute (¢)	0.0	+3.9	+1.9	-15.9	+13.4	+6.5	+0.9
	relative (%)	0	+11	+5	-45	+38	+19	+3
nearest edomapping		34	20	11	27	16	24	3

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.

Intervals

Degree	Solfege	Cents	Approx. Ratios of 2.3.5.13.17 Subgroup	Additional Ratios of 7, 11 Tending Flat (Patent Val)	Additional Ratios of 7, 11 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	21/20, 36/35, 50/49, 33/32	28/27, 49/48, 33/32	^^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	15/14, 12/11	12/11, 14/13	A1, ~2	aug 1sn, mid 2nd	D#, vvE
5	reh	176.471	10/9	11/10	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, 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, 27/22, 17/14, 21/17	11/9, 27/22	~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	9/7, 21/16, 22/17	22/17	^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	15/11	^4	up 4th	^G
16	fu	564.706	25/18, 18/13	11/8, 7/5	11/8	~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	16/11	A4, ~5	aug 4th, mid 5th	G#, vvA
19	sih	670.588	40/27, 25/17	22/15	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	17/11	^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	18/11, 44/27	~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, 22/13	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	20/11	^m7	upminor 7th	^C
30	tu	1058.824	24/13	28/15, 11/6	11/6, 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	40/21, 35/18, 49/25, 64/33	27/14, 96/49, 64/33	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.

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 × 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form… 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

Temperament measures

The following table shows TE temperament measures (RMS normalized by the rank) of 34et.

Note: the 34d val is used for lower error.

		3-limit	5-limit	7-limit	11-limit	13-limit	17-limit	2.3.5.13.17	2.3.5.11.13.17
Octave stretch (¢)		-1.24	-1.10	-2.56	-2.82	-2.64	-2.30	-1.06	-1.53
Error	absolute (¢)	1.24	1.03	2.66	2.44	2.26	2.26	0.94	1.35
	relative (%)	3.51	2.92	7.57	6.93	6.42	6.41	2.65	3.83

• 34et has a lower relative error than any previous ETs in the 5-limit. The next ET that does better in this subgroup is 53.
• 34et is most prominent in the 2.3.5.13.17 and 2.3.5.11.13.17 subgroups. The next ET that does better in these subgroups is 217 and 87, respectively.

34edo and 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. 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
• List of edo-distinct 34d rank two temperaments

Temperaments sorted by generator

Periods per octave	Generator	Cents	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	Petrtri
"	15\34	529.412	Mabila
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

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. 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.

Commas

34-EDO tempers out the following commas. (Note: This assumes the val ⟨34 54 79 95 118 126].)

Prime Limit	Ratio	Monzo	Cents	Color Name	Name(s)
3	134217728/129140163	| 27 -17 >	66.765	Sasawa	17-comma
5	20000/19683	| 5 -9 4 >	27.660	Saquadyo	Minimal diesis, tetracot comma
"	2048/2025	| 11 -4 -2 >	19.553	Sagugu	Diaschisma
"	393216/390625	| 17 1 -8 >	11.445	Saquadbigu	Würschmidt comma
"	15625/15552	| -6 -5 6 >	8.107	Tribiyo	Kleisma, semicomma majeur
"	1212717/1210381	| 23 6 -14 >	3.338	Sasepbigu	Vishnuzma, semisuper
7	1029/1000	| -3 1 -3 3 >	49.492	Trizogu	Keega
"	49/48	| -4 -1 0 2 >	35.697	Zozo	Septimal diesis, slendro diesis
"	875/864	| -5 -3 3 1 >	21.902	Zotriyo	Keema
"	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
"	243/242	| -1 5 0 0 -2 >	7.139	Lulu	Rastma, neutral third comma
"	385/384	| -7 -1 1 1 1 >	4.503	Lozoyo	Keenanisma
13	91/90	| -1 -2 -1 1 0 1 >	19.120	Thozogu	Superleap

Listen

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

Links

• 34 Equal Guitar by Larry Hanson ^{[dead link]}
• http://microstick.net/ websites of Neil Haverstick
• https://myspace.com/microstick -- somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)