35edo

35-tET or 35-EDO refers to a tuning system which divides the octave into 35 steps of approximately 34.29¢ each.

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.

Notation

Degrees	Cents	Up/down Notation
0	0.000	unison	1	D
1	34.286	up unison	^1	D^
2	68.571	double-up unison	^^1	D^^
3	102.857	double-down 2nd	vv2	Evv
4	137.143	down 2nd	v2	Ev
5	171.429	2nd	2	E
6	205.714	up 2nd	^2	E^
7	240	double-up 2nd	^^2	E^^
8	274.286	double-down 3rd	vv3	Fvv
9	308.571	down 3rd	v3	Fv
10	342.857	3rd	3	F
11	377.143	up 3rd	^3	F^
12	411.429	double-up 3rd	^^3	F^^
13	445.714	double-down 4th	vv4	Gvv
14	480	down 4th	v4	Gv
15	514.286	4th	4	G
16	548.571	up 4th	^4	G^
17	582.857	double-up 4th	^^4	G^^
18	617.143	double-down 5th	vv5	Avv
19	651.429	down 5th	v5	Av
20	685.714	5th	5	A
21	720	up 5th	^5	A^
22	754.286	double-up 5th	^^5	A^^
23	788.571	double-down 6th	vv6	Bvv
24	822.857	down 6th	v6	Bv
25	857.143	6th	6	B
26	891.429	up 6th	^6	B^
27	925.714	double-up 6th	^^6	B^^
28	960	double-down 7th	vv7	Cvv
29	994.286	down 7th	v7	Cv
30	1028.571	7th	7	C
31	1062.857	up 7th	^7	C^
32	1097.143	double-up 7th	^^7	C^^
33	1131.429	double-down 8ve	vv8	Dvv
34	1165.714	down 8ve	v8	Dv
35	1200	8ve	8	D

Ups and downs for chords

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.

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

0-9-20 = C Ev G = C(v3) = C down-three

0-11-20 = C E^ G = C(^3) = C up-three

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

0-11-21 = C E^ G^ = C(^3,^5) = C up-three up-five

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

0-10-20-29 = C E G Bv = C(v7) = C down-seven

0-9-20-30 = C Ev G B = C7(v3) = C seven down-three

0-9-20-29 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Intervals

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

Degrees	Cents value	Ratios in2.5.7.11.17 subgroup	Ratios with flat 3	Ratios with sharp 3	Ratios with best 9
0	0	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

Rank two temperaments

Periods per octave	Generator	Temperaments with flat 3/2 (patent val)	Temperaments with sharp 3/2 (35b val)
1	1\35
1	2\35
1	3\35		Ripple
1	4\35	Secund
1	6\35	Messed-up Baldy
1	8\35		Messed-up Orwell
1	9\35	Myna
1	11\35	Muggles
1	12\35		Roman
1	13\35	Inconsistent 2.9'/7.5/3 Sensi
1	16\35
1	17\35
5	1\35		Blackwood (favoring 7/6)
5	2\35		Blackwood (favoring 6/5 and 20/17)
5	3\35		Blackwood (favoring 5/4 and 17/14)
7	1\35	Whitewood/Redwood
7	2\35	Greenwood

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

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

Comma	Monzo	Value (Cents)	Name 1	Name 2
2187/2048	| -11 7 >	113.69	Apotome	Whitewood comma
6561/6250	| -1 8 -5 >	84.07	Ripple comma
10077696/9765625	| 9 9 -10 >	54.46	Mynic comma
3125/3072	| -10 -1 5 >	29.61	Small diesis	Magic comma
405/392	| -3 4 1 -2 >	56.48	Greenwoodma
16807/16384	| -14 0 0 5 >	44.13
525/512	| -9 1 2 1 >	43.41	Avicenna
126/125	| 1 2 -3 1 >	13.79	Starling comma	Septimal semicomma
99/98	| -1 2 0 -2 1 >	17.58	Mothwellsma
66/65	| 1 1 -1 0 1 -1 >	26.43

Music

Little Prelude & Fugue, "The Bijingle" by Claudi Meneghin

Self-Destructing Mechanical Forest by Chuckles McGee (in Secund[9])