35edo

Prime factorization

5 × 7

Step size

34.2857¢

Fifth

20\35 (685.714¢) (→4\7)

Semitones (A1:m2)

0:5 (0¢ : 171.4¢)

Dual sharp fifth

21\35 (720¢) (→3\5)

Dual flat fifth

20\35 (685.714¢) (→4\7)

Dual major 2nd

6\35 (205.714¢)
(semiconvergent)

Consistency limit

7

Distinct consistency limit

7

35 equal divisions of the octave (abbreviated 35edo or 35ed2), also called 35-tone equal temperament (35tet) or 35 equal temperament (35et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 35 equal parts of about 34.286 ¢ each. Each step represents a frequency ratio of 2^1/35, or the 35th root of 2.

Theory

As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic macrotonal edos: 5edo and 7edo. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71¢ and 5edo's wide fifth of 720¢. Because it includes 7edo, 35edo tunes the 29th harmonic with +1 cent of error. 35edo can also represent the 2.3.5.7.11.17 subgroup and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among whitewood tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore 22edo's more in-tune versions of 35edo MOS's and consistent representation of both subgroups. 35edo has the optimal patent val for greenwood and secund temperaments, as well as 11-limit muggles, and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a diatonic scale (unless 7edo is considered a diatonic scale).

Odd harmonics

Approximation of odd harmonics in 35edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-16.2	-9.2	-8.8	+1.8	-2.7	+16.6	+8.9	-2.1	+11.1	+9.2	-11.1
Error	relative (%)	-47	-27	-26	+5	-8	+48	+26	-6	+32	+27	-32
Steps (reduced)		55 (20)	81 (11)	98 (28)	111 (6)	121 (16)	130 (25)	137 (32)	143 (3)	149 (9)	154 (14)	158 (18)

Notation

The 7edo fifth is preferred as the diatonic generator for ups and downs notation due to being much easier to notate than the 5edo fifth (which involves E and F being enharmonic), as well as being closer to 3/2.

Degrees	Cents	Ups and downs notation			Dual-fifth notation based on closest 12edo interval
0	0.000	unison	1	D	1sn, prime
1	34.286	up unison	^1	^D	augmented 1sn
2	68.571	dup unison	^^1	^^D	diminished 2nd
3	102.857	dud 2nd	vv2	vvE	minor 2nd
4	137.143	down 2nd	v2	vE	neutral 2nd
5	171.429	2nd	2	E	submajor 2nd
6	205.714	up 2nd	^2	^E	major 2nd
7	240	dup 2nd	^^2	^^E	supermajor 2nd
8	274.286	dud 3rd	vv3	vvF	diminished 3rd
9	308.571	down 3rd	v3	vF	minor 3rd
10	342.857	3rd	3	F	neutral 3rd
11	377.143	up 3rd	^3	^F	major 3rd
12	411.429	dup 3rd	^^3	^^F	augmented 3rd
13	445.714	dud 4th	vv4	vvG	diminished 4th
14	480	down 4th	v4	vG	minor 4th
15	514.286	4th	4	G	major 4th
16	548.571	up 4th	^4	^G	augmented 4th
17	582.857	dup 4th	^^4	^^G	minor tritone
18	617.143	dud 5th	vv5	vvA	major tritone
19	651.429	down 5th	v5	vA	diminished 5th
20	685.714	5th	5	A	minor 5th
21	720	up 5th	^5	^A	major 5th
22	754.286	dup 5th	^^5	^^A	augmented 5th
23	788.571	dud 6th	vv6	vvB	diminished 6th
24	822.857	down 6th	v6	vB	minor 6th
25	857.143	6th	6	B	neutral 6th
26	891.429	up 6th	^6	^B	major 6th
27	925.714	dup 6th	^^6	^^B	augmented 6th
28	960	dud 7th	vv7	vvC	diminished 7th
29	994.286	down 7th	v7	vC	minor 7th
30	1028.571	7th	7	C	superminor 7th
31	1062.857	up 7th	^7	^C	neutral 7th
32	1097.143	dup 7th	^^7	^^C	major 7th
33	1131.429	dud 8ve	vv8	vvD	augmented 7th
34	1165.714	down 8ve	v8	vD	diminished 8ve
35	1200	8ve	8	D	8ve

Chord Names

Ups and downs can be used to name 35edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.

0-10-20 = C E G = C = C or C perfect

0-9-20 = C vE G = Cv = C down

0-11-20 = C ^E G = C^ = C up

0-10-19 = C E vG = C(v5) = C down-five

0-11-21 = C ^E ^G = C^(^5) = C up up-five

0-10-20-30 = C E G B = C7 = C seven

0-10-20-29 = C E G vB = C,v7 = C add down-seven

0-9-20-30 = C vE G B = Cv,7 = C down add-seven

0-9-20-29 = C vE G vB = Cv7 = C down seven

For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.

JI Intervals

(Bolded ratio indicates that the ratio is most accurately tuned by the given 35edo interval.)

Degrees	Cents value	Ratios in 2.5.7.11.17 subgroup	Ratios with flat 3	Ratios with sharp 3	Ratios with best 9
0	0.000	1/1	(see comma table)
1	34.286	50/49, 121/119, 33/32	36/35	25/24	81/80
2	68.571	128/125	25/24	81/80
3	102.857	17/16	15/14	16/15	18/17
4	137.143		12/11, 16/15
5	171.429	11/10		12/11	10/9
6	205.714				9/8
7	240	8/7		7/6
8	274.286	20/17	7/6
9	308.571		6/5
10	342.857	17/14		6/5	11/9
11	377.143	5/4
12	411.429	14/11
13	445.714	22/17, 32/25			9/7
14	480			4/3, 21/16
15	514.286		4/3
16	548.571	11/8
17	582.857	7/5	24/17	17/12
18	617.143	10/7	17/12	24/17
19	651.429	16/11
20	685.714		3/2
21	720			3/2, 32/21
22	754.286	17/11, 25/16			14/9
23	788.571	11/7
24	822.857	8/5
25	857.143	28/17		5/3	18/11
26	891.429		5/3
27	925.714	17/10	12/7
28	960	7/4
29	994.286				16/9
30	1028.571	20/11			9/5
31	1062.857		11/6, 15/8
32	1097.143	32/17	28/15	15/8	17/9
33	1131.429
34	1165.714
3	1200

15-odd-limit intervals by patent val mapping
Interval, complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/5, 10/7	0.345	1.0
11/8, 16/11	2.747	8.0
11/7, 14/11	6.079	17.7
11/10, 20/11	6.424	18.7
5/3, 6/5	7.070	20.6
7/6, 12/7	7.415	21.6
7/4, 8/7	8.826	25.7
5/4, 8/5	9.171	26.7
11/6, 12/11	13.494	39.4
3/2, 4/3	16.241	47.4
15/14, 28/15	16.586	48.4
13/8, 16/13	16.615	48.5
13/11, 22/13	19.362	56.5
15/11, 22/15	22.665	66.1
9/5, 10/9	23.311	68.0
9/7, 14/9	23.656	69.0
15/8, 16/15	25.412	74.1
13/7, 14/13	25.441	74.2
13/10, 20/13	25.786	75.2
11/9, 18/11	29.735	86.7
9/8, 16/9	32.481	94.7
13/12, 24/13	32.856	95.8
15/13, 26/15	42.027	122.6
13/9, 18/13	49.097	143.2

Rank-2 temperaments

Periods per octave	Generator	Temperaments with flat 3/2 (patent val)	Temperaments with sharp 3/2 (35b val)	Mos Scales
1	1\35
1	2\35			1L 16s, 17L 1s
1	3\35		Ripple	1L 10s, 11L 1s, 12L 11s
1	4\35	Secund		1L 7s, 8L 1s, 9L 8s, 9L 17s
1	6\35	Messed-up Baldy		1L 4s, 5L 1s, 6L 5s, 6L 11s, 6L 17s, 6L 23s
1	8\35		Messed-up Orwell	1L 3s, 4L 1s, 4L 5s, 9L 4s, 13L 9s
1	9\35	Myna		1L 3s, 4L 3s, 4L 7s, 4L 11s, 4L 15s, etc ... 4L 27s
1	11\35	Muggles		3L 1s, 3L 4s, 3L 7s 3L 10s, 3L 13s, 16L 3s
1	12\35		Roman	2L 1s, 3L 2s, 3L 5s, 3L 8s, 3L 11s, 3L 14s, 3L 17s, 3L 20s, etc ... 3L 29s
1	13\35	Inconsistent 2.9'/7.5/3 Sensi		2L 1s, 3L 2s, 3L 5s, 8L 3s, 8L 11s, 8L 19s
1	16\35			2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 11L 2s, 11L 13s
1	17\35			2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, 2L 13s, 2L 15s, 2L 17s, 2L 19s etc ... 2L 31s
5	1\35		Blackwood (favoring 7/6)	5L 5s, 5L 10s, 5L 15s, 5L 20s, 5L 25s
5	2\35		Blackwood (favoring 6/5 and 20/17)	5L 5s, 5L 10s, 15L 5s
5	3\35		Blackwood (favoring 5/4 and 17/14)	5L 5s, 10L 5s, 10L 15s
7	1\35	Whitewood/Redwood		7L 7s, 7L 14s, 7L 21s
7	2\35	Greenwood		7L 7s, 14L 7s

Scales

A good beginning for start to play 35-EDO is with the Sub-diatonic scale, that is a MOS of 3L2s: 9 4 9 9 4.

Commas

35 EDO tempers out the following commas. (Note: This assumes the val ⟨35 55 81 98 121 130].)

Prime limit	Ratio^[1]	Monzo	Cents	Color name	Name(s)
3	2187/2048	[-11 7⟩	113.69	Lawa	Apotome, Whitewood comma
5	6561/6250	[-1 8 -5⟩	84.07	Quingu	Ripple comma
5	(15 digits)	[9 9 -10⟩	54.46	Quinbigu	Mynic comma
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Small diesis, Magic comma
7	405/392	[-3 4 1 -2⟩	56.48	Ruruyo	Greenwoodma
7	16807/16384	[-14 0 0 5⟩	44.13	Laquinzo	Cloudy
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicenna
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma, Septimal semicomma
11	99/98	[-1 2 0 -2 1⟩	17.58	Loruru	Mothwellsma
13	66/65	[1 1 -1 0 1 -1⟩	26.43	Thulogu	Winmeanma