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 pions 7mus Up/down Notation
0 0 unison 1 D
1 34.286 36.343 43.886 (2B.E2C16) up unison ^1 D^
2 68.571 72.686 87.771 (57.C5816) double-up unison ^^1 D^^
3 102.857 109.029 131.657 (83.A8416) double-down 2nd vv2 Evv
4 137.143 145.371 175.543 (AF.8B16) down 2nd v2 Ev
5 171.429 181.714 219.429 (DB.6DB16) 2nd 2 E
6 205.714 218.057 263.314 (107.50716) up 2nd ^2 E^
7 240 254.4 307.2 (133.33316) double-up 2nd ^^2 E^^
8 274.286 290.743 351.086 (15F.15F16) double-down 3rd vv3 Fvv
9 308.571 327.086 394.971 (18A.F8B16) down 3rd v3 Fv
10 342.857 363.429 438.857 (1B6.DB716) 3rd 3 F
11 377.143 399.771 482.743 (1E2.BE316) up 3rd ^3 F^
12 411.429 436.114 526.629 (20E.A0F16) double-up 3rd ^^3 F^^
13 445.714 472.457 570.514 (23A.83A816) double-down 4th vv4 Gvv
14 480 508.8 614.4 (266.66616) down 4th v4 Gv
15 514.286 545.143 658.286 (292.49216) 4th 4 G
16 548.571 581.486 702.171 (2BE.2BE16) up 4th ^4 G^
17 582.857 617.829 746.057 (2EA.0EA16) double-up 4th ^^4 G^^
18 617.143 654.171 789.943 (315.F1616) double-down 5th vv5 Avv
19 651.429 690.514 833.829 (341.D4216) down 5th v5 Av
20 685.714 726.857 877.714 (36D.B6E16) 5th 5 A
21 720 763.2 921.6 (399.99A16) up 5th ^5 A^
22 754.286 799.443 965.486 (3C5.7C5816) double-up 5th ^^5 A^^
23 788.571 835.886 1009.371 (3F1.5F116) double-down 6th vv6 Bvv
24 822.857 872.229 1053.257 (40B.21B16) down 6th v6 Bv
25 857.143 908.571 1097.143 (449.24916) 6th 6 B
26 891.429 944.914 1141.029 (475.07316) up 6th ^6 B^
27 925.714 981.257 1184.914 (1A0.EA116) double-up 6th ^^6 B^^
28 960 1017.6 1228.8 (4CC.CCD16) double-down 7th vv7 Cvv
29 994.286 1053.943 1272.686 (4F8.AF916) down 7th v7 Cv
30 1028.571 1090.286 1316.571 (524.92516) 7th 7 C
31 1062.857 1126.629 1360.467 (550.7316). up 7th ^7 C^
32 1097.143 1162.971 1404.343 (57C.57C16) double-up 7th ^^7 C^^
33 1131.429 1199.314 1448.229 (5A8.3A816) double-down 8ve vv8 Dvv
34 1165.714 1235.657 1492.114 (5D4.1D416) down 8ve v8 Dv
35 1200 1272 1536 (60016) 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 pions 7mus 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 34.286 36.343 43.886 (2B.E2C16) 50/49 , 121/119 , 33/32 36/35 25/24 81/80
2 68.571 72.686 87.771 (57.C5816) 128/125 25/24 81/80
3 102.857 109.029 131.657 (83.A8416) 17/16 15/14 16/15 18/17
4 137.143 145.371 175.543 (AF.8B16) 12/11 , 16/15
5 171.429 181.714 219.429 (DB.6DB16) 11/10 12/11 10/9
6 205.714 218.057 263.314 (107.50716) 9/8
7 240 254.4 307.2 (133.33316) 8/7 7/6
8 274.286 290.743 351.086 (15F.15F16) 20/17 7/6
9 308.571 327.086 394.971 (18A.F8B16) 6/5
10 342.857 363.429 438.857 (1B6.DB716) 17/14 6/5 11/9
11 377.143 399.771 482.743 (1E2.BE316) 5/4
12 411.429 436.114 526.629 (20E.A0F16) 14/11
13 445.714 472.457 570.514 (23A.83A816) 22/17 , 32/25 9/7
14 480 508.8 614.4 (266.66616) 4/3, 21/16
15 514.286 545.143 658.286 (292.49216) 4/3
16 548.571 581.486 702.171 (2BE.2BE16) 11/8
17 582.857 617.829 746.057 (2EA.0EA16) 7/5 24/17 17/12
18 617.143 654.171 789.943 (315.F1616) 10/7 17/12 24/17
19 651.429 690.514 833.829 (341.D4216) 16/11
20 685.714 726.857 877.714 (36D.B6E16) 3/2
21 720 763.2 921.6 (399.99A16) 3/2, 32/21
22 754.286 799.443 965.486 (3C5.7C5816) 17/11 , 25/16 14/9
23 788.571 835.886 1009.371 (3F1.5F116) 11/7
24 822.857 872.229 1053.257 (40B.21B16) 8/5
25 857.143 908.571 1097.143 (449.24916) 28/17 5/3 18/11
26 891.429 944.914 1141.029 (475.07316) 5/3
27 925.714 981.257 1184.914 (1A0.EA116) 17/10 12/7
28 960 1017.6 1228.8 (4CC.CCD16) 7/4
29 994.286 1053.943 1272.686 (4F8.AF916) 16/9
30 1028.571 1090.286 1316.571 (524.92516) 20/11 9/5
31 1062.857 1126.629 1360.467 (550.7316). 11/6 , 15/8
32 1097.143 1162.971 1404.343 (57C.57C16) 32/17 28/15 15/8 17/9
33 1131.429 1199.314 1448.229 (5A8.3A816)
34 1165.714 1235.657 1492.114 (5D4.1D416)
3 1200 1272 1536 (60016)

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])