Xenharmonic Wiki

35edo

Revision as of 14:42, 20 August 2022 by FloraC (talk | contribs) (+infobox)
← 34edo 35edo 36edo →
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

Theory

Approximation of odd harmonics in 35 EDO
Odd harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Error absolute (¢) -16.2 -9.2 -8.8 +1.8 -2.7 +16.6 +8.9 -2.1 +11.1 +9.2 -11.1 +15.9 -14.4 -1.0 -13.6
relative (%) -47 -27 -26 +5 -8 +48 +26 -6 +32 +27 -32 +47 -42 -3 -40
Steps (reduced) 55 (20) 81 (11) 98 (28) 111 (6) 121 (16) 130 (25) 137 (32) 143 (3) 149 (9) 154 (14) 158 (18) 163 (23) 166 (26) 170 (30) 173 (33)

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. 35edo is the largest edo with a lack of a diatonic scale.

Notation

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 double-up unison ^^1 ^^D diminished 2nd
3 102.857 double-down 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 double-up 2nd ^^2 ^^E supermajor 2nd
8 274.286 double-down 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 double-up 3rd ^^3 ^^F augmented 3rd
13 445.714 double-down 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 double-up 4th ^^4 ^^G minor tritone
18 617.143 double-down 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 double-up 5th ^^5 ^^A augmented 5th
23 788.571 double-down 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 double-up 6th ^^6 ^^B augmented 6th
28 960 double-down 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 double-up 7th ^^7 ^^C major 7th
33 1131.429 double-down 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
Interval, complement Error (abs., in cents)
7/5 10/7 0.3448
13/12 24/13 1.4296
9/8 16/9 1.8039
17/16 32/17 2.0984
11/8 16/11 2.7469
18/17 17/9 3.9024
11/9 18/11 4.5509
11/10 20/11 6.4247
14/11 11/7 6.0789
6/5 5/3 7.0703
7/6 12/7 7.4151
8/7 7/4 8.8259
14/13 13/7 8.8447
16/15 15/8 8.8742
5/4 8/5 9.1707
10/9 9/5 10.9747
12/11 11/6 13.494
3/2 4/3 16.241
15/14 28/15 16.5858

Rank-2 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

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
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Music

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