Revision as of 18:58, 29 November 2021

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
Error	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	Up/down 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 35-edo 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 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

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

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

Music

Little Prelude & Fugue, "The Bijingle" by Claudi Meneghin
Self-Destructing Mechanical Forest by Chuckles McGee (in Secund[9])

"G2 Manifold" by E8 Heterotic (uses a combination of 5-EDO and 7-EDO, which can be classified as a 35-EDO subset.)

[1] Ratios longer than 10 digits are presented by placeholders with informative hints

[1]

35edo: Difference between revisions

Revision as of 18:58, 29 November 2021

Contents

Theory

Notation

Chord Names

JI Intervals

Rank two temperaments

Scales

Commas

Music

Navigation menu

@@ Line 12: / Line 12: @@
 ! style="text-align:center;" | Cents
 ! colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Up/down]] [[Ups_and_Downs_Notation|Notation]]
+![[Dual-fifth tuning|Dual-fifth]] notation
+<small>based on closest 12edo interval</small>
 |-
 | style="text-align:center;" | 0
@@ Line 18: / Line 20: @@
 | style="text-align:center;" | 1
 | style="text-align:center;" | D
+|1sn, prime
 |-
 | style="text-align:center;" | 1
@@ Line 24: / Line 27: @@
 | style="text-align:center;" | ^1
 | style="text-align:center;" | ^D
+|augmented 1sn
 |-
 | style="text-align:center;" | 2
@@ Line 30: / Line 34: @@
 | style="text-align:center;" | ^^1
 | style="text-align:center;" | ^^D
+|diminished 2nd
 |-
 | style="text-align:center;" | 3
@@ Line 36: / Line 41: @@
 | style="text-align:center;" | vv2
 | style="text-align:center;" | vvE
+|minor 2nd
 |-
 | style="text-align:center;" | 4
@@ Line 42: / Line 48: @@
 | style="text-align:center;" | v2
 | style="text-align:center;" | vE
+|neutral 2nd
 |-
 | style="text-align:center;" | 5
@@ Line 48: / Line 55: @@
 | style="text-align:center;" | 2
 | style="text-align:center;" | E
+|submajor 2nd
 |-
 | style="text-align:center;" | 6
@@ Line 54: / Line 62: @@
 | style="text-align:center;" | ^2
 | style="text-align:center;" | ^E
+|major 2nd
 |-
 | style="text-align:center;" | 7
@@ Line 60: / Line 69: @@
 | style="text-align:center;" | ^^2
 | style="text-align:center;" | ^^E
+|supermajor 2nd
 |-
 | style="text-align:center;" | 8
@@ Line 66: / Line 76: @@
 | style="text-align:center;" | vv3
 | style="text-align:center;" | vvF
+|diminished 3rd
 |-
 | style="text-align:center;" | 9
@@ Line 72: / Line 83: @@
 | style="text-align:center;" | v3
 | style="text-align:center;" | vF
+|minor 3rd
 |-
 | style="text-align:center;" | 10
@@ Line 78: / Line 90: @@
 | style="text-align:center;" | 3
 | style="text-align:center;" | F
+|neutral 3rd
 |-
 | style="text-align:center;" | 11
@@ Line 84: / Line 97: @@
 | style="text-align:center;" | ^3
 | style="text-align:center;" | ^F
+|major 3rd
 |-
 | style="text-align:center;" | 12
@@ Line 90: / Line 104: @@
 | style="text-align:center;" | ^^3
 | style="text-align:center;" | ^^F
+|augmented 3rd
 |-
 | style="text-align:center;" | 13
@@ Line 96: / Line 111: @@
 | style="text-align:center;" | vv4
 | style="text-align:center;" | vvG
+|diminished 4th
 |-
 | style="text-align:center;" | 14
@@ Line 102: / Line 118: @@
 | style="text-align:center;" | v4
 | style="text-align:center;" | vG
+|minor 4th
 |-
 | style="text-align:center;" | 15
@@ Line 108: / Line 125: @@
 | style="text-align:center;" | 4
 | style="text-align:center;" | G
+|major 4th
 |-
 | style="text-align:center;" | 16
@@ Line 114: / Line 132: @@
 | style="text-align:center;" | ^4
 | style="text-align:center;" | ^G
+|augmented 4th
 |-
 | style="text-align:center;" | 17
@@ Line 120: / Line 139: @@
 | style="text-align:center;" | ^^4
 | style="text-align:center;" | ^^G
+|minor tritone
 |-
 | style="text-align:center;" | 18
@@ Line 126: / Line 146: @@
 | style="text-align:center;" | vv5
 | style="text-align:center;" | vvA
+|major tritone
 |-
 | style="text-align:center;" | 19
@@ Line 132: / Line 153: @@
 | style="text-align:center;" | v5
 | style="text-align:center;" | vA
+|diminished 5th
 |-
 | style="text-align:center;" | 20
@@ Line 138: / Line 160: @@
 | style="text-align:center;" | 5
 | style="text-align:center;" | A
+|minor 5th
 |-
 | style="text-align:center;" | 21
@@ Line 144: / Line 167: @@
 | style="text-align:center;" | ^5
 | style="text-align:center;" | ^A
+|major 5th
 |-
 | style="text-align:center;" | 22
@@ Line 150: / Line 174: @@
 | style="text-align:center;" | ^^5
 | style="text-align:center;" | ^^A
+|augmented 5th
 |-
 | style="text-align:center;" | 23
@@ Line 156: / Line 181: @@
 | style="text-align:center;" | vv6
 | style="text-align:center;" | vvB
+|diminished 6th
 |-
 | style="text-align:center;" | 24
@@ Line 162: / Line 188: @@
 | style="text-align:center;" | v6
 | style="text-align:center;" | vB
+|minor 6th
 |-
 | style="text-align:center;" | 25
@@ Line 168: / Line 195: @@
 | style="text-align:center;" | 6
 | style="text-align:center;" | B
+|neutral 6th
 |-
 | style="text-align:center;" | 26
@@ Line 174: / Line 202: @@
 | style="text-align:center;" | ^6
 | style="text-align:center;" | ^B
+|major 6th
 |-
 | style="text-align:center;" | 27
@@ Line 180: / Line 209: @@
 | style="text-align:center;" | ^^6
 | style="text-align:center;" | ^^B
+|augmented 6th
 |-
 | style="text-align:center;" | 28
@@ Line 186: / Line 216: @@
 | style="text-align:center;" | vv7
 | style="text-align:center;" | vvC
+|diminished 7th
 |-
 | style="text-align:center;" | 29
@@ Line 192: / Line 223: @@
 | style="text-align:center;" | v7
 | style="text-align:center;" | vC
+|minor 7th
 |-
 | style="text-align:center;" | 30
@@ Line 198: / Line 230: @@
 | style="text-align:center;" | 7
 | style="text-align:center;" | C
+|superminor 7th
 |-
 | style="text-align:center;" | 31
@@ Line 204: / Line 237: @@
 | style="text-align:center;" | ^7
 | style="text-align:center;" | ^C
+|neutral 7th
 |-
 | style="text-align:center;" | 32
@@ Line 210: / Line 244: @@
 | style="text-align:center;" | ^^7
 | style="text-align:center;" | ^^C
+|major 7th
 |-
 | style="text-align:center;" | 33
@@ Line 216: / Line 251: @@
 | style="text-align:center;" | vv8
 | style="text-align:center;" | vvD
+|augmented 7th
 |-
 | style="text-align:center;" | 34
@@ Line 222: / Line 258: @@
 | style="text-align:center;" | v8
 | style="text-align:center;" | vD
+|diminished 8ve
 |-
 | style="text-align:center;" | 35
@@ Line 228: / Line 265: @@
 | style="text-align:center;" | 8
 | style="text-align:center;" | D
+|8ve
 |}

35edo: Difference between revisions

Revision as of 18:58, 29 November 2021

Theory

Notation

Chord Names

JI Intervals

Rank two temperaments

Scales

Commas

Music

Navigation menu

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