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

English Wikipedia has an article on:

34 equal temperament

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
Error	Absolute (¢)	+3.9	+1.9	-15.9	+7.9	+13.4	+6.5	+5.8	+0.9	-15.2	-12.0	+7.0
	Relative (%)	+11.1	+5.4	-45.0	+22.3	+37.9	+18.5	+16.6	+2.6	-43.0	-33.9	+19.9
Steps (reduced)		54 (20)	79 (11)	95 (27)	108 (6)	118 (16)	126 (24)	133 (31)	139 (3)	144 (8)	149 (13)	154 (18)

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	Gb	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	A#	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	Db	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.

JI approximation

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.)

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.

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

The following table shows how 15-odd-limit intervals are represented in 34edo. Prime harmonics are in bold.

As 34edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

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 logarithmic 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. (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.)

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

• List of 34edo rank two temperaments by badness
• List of edo-distinct 34d rank two temperaments

Rank-2 temperaments by period and generator

Periods per octave	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	Monzo	Cents	Color name	Name(s)
3	(18 digits)	[27 -17⟩	66.765	Sasawa	17-comma
5	20000/19683	[5 -9 4⟩	27.660	Saquadyo	Minimal diesis, 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, semicomma majeur
5	(20 digits)	[23 6 -14⟩	3.338	Sasepbigu	Vishnuzma, semisuper comma
7	1029/1000	[-3 1 -3 3⟩	49.492	Trizogu	Keega
7	49/48	[-4 -1 0 2⟩	35.697	Zozo	Septimal diesis, 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, septimal semicomma
11	100/99	[2 -2 2 0 -1⟩	17.399	Luyoyo	Ptolemisma, Ptolemy's comma
11	243/242	[-1 5 0 0 -2⟩	7.139	Lulu	Rastma, neutral third comma
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

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.

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.

Music

Amano Hideya

• Like refracted light (2023)

E8 Heterotic

• "Elements - Water" from Elements (2020)

Francium

• Travel to Stay (2023)

Peter Kosmorsky

• Ascension (2010)

luphoria

• look (2023)

Claudi Meneghin

• Semitone-Canon on The Mother's Malison Theme, for Cor Anglais & Violin (2022)

Ray Perlner

• A stroll through some retuned maqams (2020)

Robin Perry

• Uncomfortable In Crowds (extended) (2013)

Userminusone

• Perspective (2021)

Randy Wells

• 傘がなくても嬉しい (The Puddle Song) (2021)

Xotla

• from Lesser Groove (2020)
  • "Apparatus" – Spotify | Bandcamp | YouTube – sci-fi electro
  • "Electrostat" – Spotify | Bandcamp | YouTube – ambient electro, tetracot[13] in 34edo tuning
  • "Dreams Of Ambience" – Spotify | Bandcamp | YouTube – ambient electro
• from Xotla's Microtonal Funk & Blues Vol. 2 (2020)
  • "Chaparral" – Spotify | Bandcamp | YouTube – space rock
  • "Lull" – Spotify | Bandcamp | YouTube – ambient
  • "Vortices" – Spotify | Bandcamp | YouTube – electro-rock
  • "Escapade" – Spotify | Bandcamp | YouTube – electronic rock
• Between Space (2022) – ambient sci-fi

Zhea Erose

• Modal Studies in Tetracot (2021)

See also

• Lumatone mapping for 34edo
• Skip fretting system 34 2 9

External links

• 34 Equal Guitar by Larry Hanson ^{[dead link]}
• Websites of Neil Haverstick
• [1] – somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)