35edo

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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 unison 1 D
1 34.29 up unison ^1 D^
2 68.57 double-up unison ^^1 D^^
3 102.86 double-down 2nd vv2 Evv
4 137.14 down 2nd v2 Ev
5 171.43 2nd 2 E
6 205.71 up 2nd ^2 E^
7 240 double-up 2nd ^^2 E^^
8 274.29 double-down 3rd vv3 Fvv
9 308.57 down 3rd v3 Fv
10 342.86 3rd 3 F
11 377.14 up 3rd ^3 F^
12 411.43 double-up 3rd ^^3 F^^
13 445.71 double-down 4th vv4 Gvv
14 480 down 4th v4 Gv
15 514.29 4th 4 G
16 548.57 up 4th ^4 G^
17 582.86 double-up 4th ^^4 G^^
18 617.14 double-downv 5th vv5 Avv
19 651.43 down 5th v5 Av
20 685.71 5th 5 A
21 720 up 5th ^5 A^
22 754.29 double-up 5th ^^5 A^^
23 788.57 double-down 6th vv6 Bvv
24 822.86 down 6th v6 Bv
25 857.15 6th 6 B
26 891.43 up 6th ^6 B^
27 925.71 double-up 6th ^^6 B^^
28 960 double-down 7th vv7 Cvv
29 994.29 down 7th v7 Cv
30 1028.57 7th 7 C
31 1062.86 up 7th ^7 C^
32 1097.14 double-up 7th ^^7 C^^
33 1131.43 double-down 8ve vv8 Dvv
34 1165.71 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 patent 9
0 0 1/1 (see comma table)
1 34.29 50/49 , 121/119 , 33/32 36/35 25/24 81/80
2 68.57 128/125 25/24 81/80
3 102.86 17/16 15/14 16/15 18/17
4 137.14 12/11 , 16/15
5 171.43 11/10 12/11 10/9
6 205.71 9/8
7 240 8/7 7/6
8 274.29 20/17 7/6
9 308.57 6/5
10 342.86 17/14 6/5 11/9
11 377.14 5/4
12 411.43 14/11
13 445.71 22/17 , 32/25 9/7
14 480 4/3, 21/16
15 514.29 4/3
16 548.57 11/8
17 582.86 7/5 24/17 17/12
18 617.14 10/7 17/12 24/17
19 651.43 16/11
20 685.71 3/2
21 720 3/2, 32/21
22 754.29 17/11 , 25/16 14/9
23 788.57 11/7
24 822.86 8/5
25 857.14 28/17 5/3 18/11
26 891.43 5/3
27 925.71 17/10 12/7
28 960 7/4
29 994.29 16/9
30 1028.57 20/11 9/5
31 1062.86 11/6 , 15/8
32 1097.14 32/17 28/15 15/8 17/9
33 1131.43
34 1165.71


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])