35edo: Difference between revisions

← 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

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Latest revision as of 14:42, 7 May 2026

35 equal divisions of the octave (abbreviated 35edo or 35ed2), also called 35-tone equal temperament (35tet) or 35 equal temperament (35et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 35 equal parts of about 34.3 ¢ each. Each step represents a frequency ratio of 2^1/35, or the 35th root of 2.

Theory

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 ¢. Since it has two approximations of the perfect fifth which are close to equally off, 35edo is a classic example of a dual-fifth system. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1 ¢ 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 the higher primes (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).

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 (unless 7edo is considered a diatonic scale).

Odd harmonics

Approximation of odd harmonics in 35edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-16.2	-9.2	-8.8	+1.8	-2.7	+16.6	+8.9	-2.1	+11.1	+9.2	-11.1
Error	Relative (%)	-47.4	-26.7	-25.7	+5.3	-8.0	+48.5	+25.9	-6.1	+32.3	+26.9	-32.5
Steps (reduced)		55 (20)	81 (11)	98 (28)	111 (6)	121 (16)	130 (25)	137 (32)	143 (3)	149 (9)	154 (14)	158 (18)

Dual-fifth harmony

35edo has two viable mappings of the perfect fifth, one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad 4:5:6, then we find that 35edo's best approximation of 5/4 is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for 6/5 to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686 ¢), and the best approximation of its inverse 1/(6:5:4), the classical minor triad, is 0–9–20 steps (0–309–686 ¢). Here, the 5/4 and 6/5 intervals are tuned fairly accurately, being about 7–9 ¢ flat each, while 3/2 is more damaged at about 16 ¢ flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned.

Amazingly, almost the exact same situation occurs with 7/4, for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate 7/5). If we wish to use the 4:6:7 chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960 ¢). Its inverse, the 1/(12:8:7) chord, is best mapped to 0–20–27 steps (0–686–926 ¢). Here the damage is split between 7/4 and 12/7, with both being around 7–9 ¢ flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord 4:5:6:7 is 0–11–20–28 steps (0–377–686–960 ¢), while the best approximation of the subharmonic sixth chord 1/(12:10:8:7) is 0–9–20–27 steps (0–309–686–926 ¢).

Overall, we find that 35edo's patent val is surprisingly accurate overall for the 7-odd-limit, with 3/2 being the only interval with high damage. However, this mapping does not work well in the 9-odd-limit, as 9/8 is tuned over 32 ¢ flat of just at 171 ¢, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206 ¢), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the 5edo fifth of 720 ¢, being around 18 ¢ sharp of just. There are two mappings of the perfect fifth, with some chords preferring the flat fifth, while other chords prefer the sharp fifth.

For example, suppose we want to use the 6:7:9 subminor triad. Then the closest approximation of 7/6 is 8 steps, and the closest approximation of 9/7 is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720 ¢). Here, we use the sharp fifth instead of the flat one, so that 7/6 and 9/7 are tuned more accurately, being around 7 ¢ and 11 ¢ sharp of just respectively. The best approximation of the supermajor triad 1/(9:7:6) is 0–13–21 steps (0–446–720 ¢), which also uses the sharp fifth. A similar situation occurs with 6:9:10 and its inverse 1/(9:6:5), where the best approximations of 5/3 and 9/5 are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891 ¢) and 0–21–30 steps (0–720–1029 ¢) respectively, with 5/3 and 9/5 being around 7 ¢ and 11 ¢ sharp respectively. This leads to an approximation of the 6:7:9:10 harmonic sixth chord (sometimes known as the subminor tetrad) at 0–8–21–26 steps (0–274–720–891 ¢), and an approximation of the 1/(9:7:6:5) subharmonic seventh chord (sometimes called the supermajor tetrad) at 0–13–21–30 steps (0–446–720–1029 ¢).

The best approximation of the harmonic ninth chord 4:5:6:7:9 is 0–11–20–28–41 steps (0–377–686–960–1406 ¢). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo that is a subset of the 9-odd-limit otonal or utonal pentad (up to octave equivalence) can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse 1/(9:7:6:5:4), that being 0–13–21–30–41 steps (0–446–720–1029–1406 ¢), where any interval more complex than the perfect fifth is no more than 11 ¢ out of tune.

Additionally, many triads are tuned very close to delta-rational tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads 4:5:6, 1/(6:5:4), 6:7:9, and 1/(9:7:6) are very close to DR tunings. Voicings of chords that divide the fourth, those being 6:7:8, 1/(8:7:6), 9:10:12, and 1/(12:10:9), are also tuned fairly close to DR.

Caveats of dual-fifth

However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the 10:12:15:18 and 12:14:18:21 chords and their inversions, known as anomalous saturated suspensions, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as 8:10:12:15, also cannot be mapped without a step being close to 3/4 of a 35edo step off.

Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several mos scale, such as the traditional diatonic scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an equalized version of the scale, while the 21\35 fifth generates a collapsed version of the scale. Since 35edo does not have a diatonic scale, chain-of-fifths notation also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or hobbling a 34edo or 36edo diatonic scale.

35edo is only one of many dual-fifth systems, with others including 18edo, 23edo, 25edo, 28edo, 30edo, 37edo, and 40edo, each with their own unique properties.

Subsets and supersets

Since 35 factors as 5 × 7, its nontrivial subsets are 5edo and 7edo. Its double 70edo corrects the perfect fifth, as well as the 13th harmonic, though the 5th and 7th harmonics become relatively inaccurate. The quadruple of 35edo, which is 140edo, additionally corrects the mappings of primes 5 and 7, and makes for an excellent 17-limit system and beyond.

Intervals

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

#	Cents value	Ratios in the 2.5.7.11.17 subgroup	Ratios with flat 3	Ratios with sharp 3	Ratios with best 9
0	0.000	1/1
1	34.286	50/49, 121/119, 33/32	36/35	25/24	64/63, 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	240.000	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	480.000			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	720.000			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	960.000	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
35	1200.000	2/1

Notation

The 7edo fifth is preferred as the diatonic generator for ups and downs notation due to being much easier to notate than the 5edo fifth (which involves E and F being enharmonic), as well as being closer to 3/2.

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	dup unison	^^1	^^D	diminished 2nd
3	102.857	dud 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	dup 2nd	^^2	^^E	supermajor 2nd
8	274.286	dud 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	dup 3rd	^^3	^^F	augmented 3rd
13	445.714	dud 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	dup 4th	^^4	^^G	minor tritone
18	617.143	dud 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	dup 5th	^^5	^^A	augmented 5th
23	788.571	dud 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	dup 6th	^^6	^^B	augmented 6th
28	960	dud 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	dup 7th	^^7	^^C	major 7th
33	1131.429	dud 8ve	vv8	vvD	augmented 7th
34	1165.714	down 8ve	v8	vD	diminished 8ve
35	1200	8ve	8	D	8ve

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as EDOs 30b and 40, and is a superset of the notation for 7-EDO.

Second-best fifth notation

This notation uses the same sagittal sequence as 42-EDO, and is a superset of the notation for 5-EDO.

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.

Approximation to JI

The following tables show how 15-odd-limit intervals are represented in 35edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 35edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/5, 10/7	0.345	1.0
13/12, 24/13	1.430	4.2
9/8, 16/9	1.804	5.3
11/8, 16/11	2.747	8.0
11/9, 18/11	4.551	13.3
11/7, 14/11	6.079	17.7
11/10, 20/11	6.424	18.7
5/3, 6/5	7.070	20.6
7/6, 12/7	7.415	21.6
15/13, 26/15	7.741	22.6
13/10, 20/13	8.500	24.8
7/4, 8/7	8.826	25.7
13/7, 14/13	8.845	25.8
15/8, 16/15	8.874	25.9
5/4, 8/5	9.171	26.7
9/7, 14/9	10.630	31.0
9/5, 10/9	10.975	32.0
15/11, 22/15	11.621	33.9
11/6, 12/11	13.494	39.4
13/9, 18/13	14.811	43.2
13/11, 22/13	14.924	43.5
3/2, 4/3	16.241	47.4
15/14, 28/15	16.586	48.4
13/8, 16/13	16.615	48.5

15-odd-limit intervals in 35edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
7/5, 10/7	0.345	1.0
11/8, 16/11	2.747	8.0
11/7, 14/11	6.079	17.7
11/10, 20/11	6.424	18.7
5/3, 6/5	7.070	20.6
7/6, 12/7	7.415	21.6
7/4, 8/7	8.826	25.7
5/4, 8/5	9.171	26.7
11/6, 12/11	13.494	39.4
3/2, 4/3	16.241	47.4
15/14, 28/15	16.586	48.4
13/8, 16/13	16.615	48.5
13/11, 22/13	19.362	56.5
15/11, 22/15	22.665	66.1
9/5, 10/9	23.311	68.0
9/7, 14/9	23.656	69.0
15/8, 16/15	25.412	74.1
13/7, 14/13	25.441	74.2
13/10, 20/13	25.786	75.2
11/9, 18/11	29.735	86.7
9/8, 16/9	32.481	94.7
13/12, 24/13	32.856	95.8
15/13, 26/15	42.027	122.6
13/9, 18/13	49.097	143.2

Regular temperament properties

Rank-2 temperaments

Periods per 8ve	Generator	Temperaments with flat 3/2 (patent val)	Temperaments with sharp 3/2 (35b val)	Mos scales
1	1\35
1	2\35			1L 16s, 17L 1s
1	3\35		Ripple	1L 10s, 11L 1s, 12L 11s
1	4\35	Secund		1L 7s, 8L 1s, 9L 8s, 9L 17s
1	6\35	Baldy (messed-up)^{[idiosyncratic term]}		1L 4s, 5L 1s, 6L 5s, 6L 11s, 6L 17s, 6L 23s
1	8\35		Orwell (messed-up)^{[idiosyncratic term]}	1L 3s, 4L 1s, 4L 5s, 9L 4s, 13L 9s
1	9\35	Myna		1L 3s, 4L 3s, 4L 7s, 4L 11s, 4L 15s, …, 4L 27s
1	11\35	Muggles		3L 1s, 3L 4s, 3L 7s 3L 10s, 3L 13s, 16L 3s
1	12\35		Roman	2L 1s, 3L 2s, 3L 5s, 3L 8s, 3L 11s, 3L 14s, 3L 17s, 3L 20s, …, 3L 29s
1	13\35	Inconsistent 2.5/3.9/7 sensi/sentry		2L 1s, 3L 2s, 3L 5s, 8L 3s, 8L 11s, 8L 19s
1	16\35			2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 11L 2s, 11L 13s
1	17\35			2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s, 2L 13s, 2L 15s, 2L 17s, 2L 19s, …, 2L 31s
5	1\35		Blackwood (favoring 7/6)	5L 5s, 5L 10s, 5L 15s, 5L 20s, 5L 25s
5	2\35		Blackwood (favoring 6/5 and 20/17)	5L 5s, 5L 10s, 15L 5s
5	3\35		Blackwood (favoring 5/4 and 17/14)	5L 5s, 10L 5s, 10L 15s
7	1\35	Whitewood / redwood		7L 7s, 7L 14s, 7L 21s
7	2\35	Greenwood		7L 7s, 14L 7s

Commas

35et 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	Whitewood comma, apotome, Pythagorean chroma
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	Magic comma, small diesis
7	405/392	[-3 4 1 -2⟩	56.48	Ruruyo	Greenwoodma
7	16807/16384	[-14 0 0 5⟩	44.13	Laquinzo	Cloudy comma
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal semicomma, starling comma
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

Octave stretch or compression

35edo's primes 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from octave stretching. Some stretched-octave 35edo tunings (least to most stretched) include 149zpi, 116ed10, 98ed7, 81ed5, 125ed12 or 90ed6.

Scales

Polymicrotonal scales

12-tone 7edo&5edo

The 12-tone 7edo&5edo scale is designed to be mapped to the key of C on a conventional piano keyboard, with 7edo on the white keys, and 5edo on black:

5 2 3 4 1 5 1 4 3 2 5 0

24-tone blackwood&greenwood

You can have two pianos/keyboards, one 68.6 cents sharp of the other, both tuned to the 12-tone 7edo&5edo scale. The combined black keys across both keyboards will be blackwood[10] and the white keys will be greenwood[14].

3 2 0 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 0 2 3 0 2 0

20-tone blackwood&greenwood

Removing the duplicates from the previous scale (perhaps for use on other instruments beside keyboard) gives this 20-tone scale, which includes both blackwood[10] and greenwood[14] as subsets.

3 2 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 2 3 2

MOS scales

Of the MOS scales available in 35edo, the muggles scales most closely approximate just intonation.

MOS scales

Greenwood[7]/whitewood[7]: 5 5 5 5 5 5 5 (a.k.a. 7edo; an equiheptatonic scale)
Greenwood[14]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2
Greenwood[21]: 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
Muggles[5] (a.k.a. sub-diatonic): 9 4 9 9 4
Muggles[13]: 2 2 5 2 2 2 5 2 2 2 5 2 2
Muggles[16]: 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2
Muggles[19]: 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2
Ripple[12]: 3 3 3 3 3 3 3 3 2 3 3 3
Ripple[23]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
Secund[17]: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
Whitewood[14]: 1 4 1 4 1 4 1 4 1 4 1 4 1 4
Whitewood[21]: 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1
Blackwood[5]: 7 7 7 7 7 (a.k.a. 5edo; an equipentatonic scale; slendro-like; works with all three blackwood tunings)
5/4-blackwood[10]: 4 3 4 3 4 3 4 3 4 3
5/4-blackwood[15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
5/4-blackwood[25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
6/5-blackwood[10]: 2 5 2 5 2 5 2 5 2 5
6/5-blackwood[15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
6/5-blackwood[20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
2L 9s (4:3) [11]: 3 3 4 3 3 3 3 3 4 3 3 --- A scale doing great job tempering the 2.9.11.17 subgroup near JI.

Ripple scales

Ripple[23]

The ripple[23] MOS scale makes maximum use of 35edo's dual-fifth nature, with both its sizes of fifth and fourth occurring frequently throughout the whole scale:

Symmetrical mode (has the most consonances): 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Mode that includes the clear pond^{[idiosyncratic term]} modmos: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1

Ripple[23] subsets approximated from 96edo
Contains idiosyncratic terms. Flattened major: 6 5 4 5 6 6 3 Sharpened minor: 6 3 6 5 4 6 5 Sharpened harmonic minor: 6 3 6 6 3 9 2 Flattened major pentatonic: 5 6 9 6 9 Sharpened minor pentatonic: 9 6 5 10 5 Evened minor hexatonic: 5 4 6 5 9 6 Roughened augmented: 10 2 9 2 10 2 Evened dominant pentatonic: 6 6 8 9 6 Sharpened Dorian: 6 3 6 6 6 3 5 Flattened Ionian pentatonic: 11 4 5 12 3 Sharpened Dorian harmonic: 6 3 9 3 6 3 5 Evened Mixolydian pentatonic: 11 4 6 8 6 Roughened Phrygian dominant: 2 10 2 6 3 6 6 Evened Phrygian dominant hexatonic: 3 8 4 6 8 6 Sharpened Phrygian pentatonic: 3 6 12 3 11 Sharpened blues Aeolian hexatonic: 9 6 3 2 3 12 Flattened blues Aeolian pentatonic I: 8 6 6 3 12 Sharpened blues Aeolian pentatonic II: 9 12 2 6 6 Roughened blues Dorian heptatonic: 9 6 3 2 7 2 6 Sharpened blues Dorian hexatonic: 9 6 6 5 4 5 Roughened blues Dorian pentatonic: 9 11 7 2 6 Roughened blues pentachordal: 6 3 5 4 2 15 Sharpened minor harmonic pentatonic I: 6 3 12 12 2 Sharpened minor harm. pent. II: 9 6 6 12 2 Evened hirajoshi: 6 3 11 4 11 Sharpened hirajoshi: 6 3 12 3 11 Roughened hirajoshi: 6 2 13 2 12 Evened akebono I: 6 3 11 6 9 Sharpened akebono I: 6 3 12 5 9 Roughened akebono I: 7 1 13 6 8 Roughened Javanese pentachordal: 2 7 9 2 15 Roughened cosmic: 14 6 2 7 6 Roughened cosmic II: 6 2 7 5 15 Lost spirit: 9 6 2 3 7 3 5 Moonbeam: 6 3 11 12 3 Palace: 5 4 6 5 5 4 6 Underpass: 9 11 7 3 5

Clear pond^{[idiosyncratic term]}

The clear pond scale^{[idiosyncratic term]}, a modmos of ripple[12], tries to sound close to the familiar 12edo:

3 3 3 2 3 3 3 4 2 3 3 3

Clear pond subsets
Contains idiosyncratic terms. Lydian: 6 5 6 3 6 6 3 Major: 6 5 3 6 6 6 3 Mixolydian: 6 5 3 6 6 3 6 Dorian: 6 3 5 6 6 3 6 Minor: 6 3 5 6 4 5 6 Phrygian: 3 6 5 6 4 5 6 Locrian: 3 6 5 3 7 5 6 Harmonic minor: 6 3 5 6 4 8 3 Melodic minor: 6 3 5 6 6 6 3 Major pentatonic: 6 8 6 6 9 Minor pentatonic: 9 5 6 9 6 Minor blues: 9 5 3 3 9 6 Minor blues heptatonic: 9 5 3 3 6 3 6 Akebono I: 6 3 11 6 9

Secund scales

Secund[17]

The secund[17] MOS scale includes a motley mix of quirky, quite xenharmonic subsets, suited for exploring those consonances very different to any found in 12edo.

3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3

Secund[17] subsets
Contains idiosyncratic terms. Antipental blues: 8 7 1 4 8 7 Antipental blues maj 6th: 8 7 1 4 7 1 7 Antipental blues neutral 7th: 8 7 1 4 8 3 4 Antipental blues maj 7th: 8 7 1 4 8 4 3 Antipental blues harmonic: 8 7 1 4 3 9 3 Pelog-like heptatonic: 3 5 7 5 3 8 4 (Phrygian-like) Pelog-like pentatonic: 3 5 12 3 12 Secund chance (modmos of secund[8]): 4 7 4 1 4 4 7 4 Secund-tempered rotated 5afdo: 7 4 9 8 7 Secund-tempered 6afdo: 8 7 5 7 4 4 Undecimal Mixolydian: 7 4 4 5 7 1 7 Undecimal minor hexatonic: 7 1 7 5 8 7 Undecimal quasi-equipentatonic: 7 8 5 8 7 12 from secund[17]: 7 1 3 4 1 4 3 4 1 3 1 3

Blackwood scales

The three blackwood temperaments

There are actually three versions of the blackwood temperament available in 35edo. One optimises the subminor third 7/6, one optimises the minor third 6/5, the other optimises the major third 5/4. Try them each and see which one you prefer:

5/4-blackwood[15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
5/4-blackwood[25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
6/5-blackwood[15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
6/5-blackwood[20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
7/6-blackwood[15]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
7/6-blackwood[20]: 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1

6/5-blackwood[20] subsets
Contains idiosyncratic terms. Blackwood meta-Hirajoshi: 2 3 4 2 5 7 2 12 Blackwood pseudo-Akebono neutral: 5 9 7 2 12 Blackwood pseudo-Akebono supermajor: 7 7 7 2 12 Blackwood pseudo-Hirajoshi: 2 12 7 2 12 Blackwood pseudo-pelog: 5 4 12 2 12 Blackwood meta-partial: 4 3 2 2 3 7 7 7 Blackwood-tempered 5afdo: 7 4 10 7 7 Mechanical (from 16afdo): 9 2 10 7 7 Starship (from 68ifdo, see ifdo): 4 7 3 7 7 7 Volcanic (from 16afdo): 4 7 10 7 7 Meta-monsoon: 7 4 3 2 5 9 5 Monsoon (from 47zpi): 7 7 7 9 5 Monsoon otonal: 7 9 5 9 5 Monsoon major: 11 5 5 9 5 Blackwood neutral nonatonic: 4 7 3 2 5 4 5 2 3 Blackwood undecimal harmonic: 4 8 4 5 4 5 5 Dungeon (from 30afdo): 11 3 7 2 12 Moonbeam (from 16afdo): 7 2 12 12 2 Underpass (from 10afdo): 9 12 5 4 5 12 from 6/5-blackwood[20]: 4 3 2 2 3 7 2 3 2 2 3 2

Other scales

Amulet^{[idiosyncratic term]}, approximated from magic in 25edo: 3 1 3 3 1 3 4 3 3 1 3 4 3
Fourfourths^{[idiosyncratic term]} (modmos of 7/6-blackwood[20]): 3 1 1 2 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4
Near-just rotated 5afdo: 6 5 9 8 7
Near-just 6afdo: 8 7 5 6 5 4

Instruments

Lumatone

35edo can be played on the Lumatone. See Lumatone mapping for 35edo

Skip fretting

Skip fretting system 35 3 8 is a skip fretting system for 35edo. All examples on this page are for 7-string guitar.

Prime harmonics

1/1: string 2 open

2/1: string 3 fret 9 and string 6 fret 1

3/2: string 3 fret 4 and string 4 fret 13

5/4: string 3 fret 1, string 4 fret 10, and string 7 fret 2

7/4: string 4 fret 4

11/8: string 1 fret 8, string 4 open, and string 5 fret 9

13/8: string 1 fret 11, string 4 fret 3, and string 5 fret 12

17/16: string 2 fret 1 and string 3 fret 10

Music

Modern renderings

Frederick Chopin

CHOPIN Waltz op 64 #2 (1847) – rendered in 35-edo with alternating sharp and flat fifths by Claudi Meneghin (2025)

Carlo Gesualdo

Dolcissima mia vita – in three comparative tunings including 35edo (5:05–10:05), rendered by Chris Vaisvil (2025)

21st century

Bryan Deister

35edo (2025)
Whistling Like An Oberon - 35edo (2026) [short 1], [short 2], [full piece]

dotuXil

"Icebound Gallery of Refractions" from Collected Refractions (2024)

E8 Heterotic

G2 Manifold (2020) – uses a combination of 5edo and 7edo, which can be classified as a 35edo subset.

Francium

"What Kind Of Things" from TOTMC 2025 (2025) – Bandcamp | YouTube

groundfault

"Sakura Blade Minivan", from Souvenirs of the Affliction (2025) – Bandcamp | YouTube (23:56–27:58) – in part, the rest being in 27edo

JUMBLE

Penguins...? (2024)

Budjarn Lambeth

Lighting the Jack-o'-lanterns (2025, uses meta-monsoon scale^{[idiosyncratic term]} from 6/5-Blackwood[20])

Chuckles McGee

Self-Destructing Mechanical Forest – in Secund[9], 35edo tuning

Claudi Meneghin

No Clue Music

DarkSciFiThing (2024)

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

[1]

35edo: Difference between revisions

Latest revision as of 14:42, 7 May 2026

Contents

Theory

Odd harmonics

Dual-fifth harmony

Caveats of dual-fifth

Subsets and supersets

Intervals

Notation

Sagittal notation

Best fifth notation

Second-best fifth notation

Chord Names

Approximation to JI

Regular temperament properties

Rank-2 temperaments

Commas

Octave stretch or compression

Scales

Polymicrotonal scales

MOS scales

Ripple scales

Secund scales

Blackwood scales

Other scales

Instruments

Lumatone

Skip fretting

Music

Modern renderings

21st century

Navigation menu

@@ Line 1: / Line 1: @@
-{{User:IlL/Template:RTT restriction}}
+{{Infobox ET}}
-= Theory =
+{{ED intro}}
--tET or 35-[[EDO]] refers to a tuning system which divides the octave into 35 steps of approximately [[cent|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 [[Just_intonation_subgroups|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.
+== Theory ==
+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{{Cent}} and 5edo's wide fifth of 720{{C}}. Since it has two approximations of the perfect fifth which are close to equally off, 35edo is a classic example of a [[dual-fifth]] system. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.
-=Notation=
+edo 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 the higher primes ([[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).
-{| class="wikitable"
+edo 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]] (unless 7edo is considered a diatonic scale).
+=== Odd harmonics ===
+{{Harmonics in equal|35}}
+=== Dual-fifth harmony ===
+edo has two viable mappings of the [[3/2|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned.
+Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).
+Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. There are two mappings of the perfect fifth, with some chords preferring the flat fifth, while other chords prefer the sharp fifth.
+For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).
+The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo that is a subset of the 9-odd-limit otonal or utonal pentad (up to octave equivalence) can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315|1/(9:7:6:5:4)]], that being 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.
+Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15|1/(6:5:4)]], [[6:7:9]], and [[14:18:21|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28|1/(8:7:6)]], [[9:10:12]], and [[15:18:20|1/(12:10:9)]], are also tuned fairly close to DR.
+==== Caveats of dual-fifth ====
+However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
+Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.
+edo is only one of many dual-fifth systems, with others including [[18edo]], [[23edo]], [[25edo]], [[28edo]], [[30edo]], [[37edo]], and [[40edo]], each with their own unique properties.
+=== Subsets and supersets ===
+Since 35 factors as 5 × 7, its nontrivial subsets are [[5edo]] and [[7edo]]. Its double [[70edo]] corrects the perfect fifth, as well as the [[13/1|13th harmonic]], though the [[5/1|5th]] and [[7/1|7th]] harmonics become relatively inaccurate. The quadruple of 35edo, which is [[140edo]], additionally corrects the mappings of primes 5 and 7, and makes for an excellent [[17-limit]] system and beyond.
+== Intervals ==
+(Bolded ratio indicates that the ratio is most accurately tuned by the given 35edo interval.)
+{| class="wikitable center-1 right-2"
 |-
-! style="text-align:center;" | Degrees
+! #
-! style="text-align:center;" | Cents
+! Cents value
-! colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|Up/down]] [[Ups_and_Downs_Notation|Notation]]
+! Ratios in the<br>2.5.7.11.17 subgroup
+! Ratios with<br>flat 3
+! Ratios with<br>sharp 3
+! Ratios with<br>best 9
 |-
-| style="text-align:center;" | 0
+| 0
-| style="text-align:center;" | 0.000
+| 0.000
-| style="text-align:center;" | unison
+| '''1/1'''
-| style="text-align:center;" | 1
+|
-| style="text-align:center;" | D
+|
+|
 |-
-| style="text-align:center;" | 1
+| 1
-| style="text-align:center;" | 34.286
+| 34.286
-| style="text-align:center;" | up unison
+| '''50/49''', '''121/119''', 33/32
-| style="text-align:center;" | ^1
+| '''36/35'''
-| style="text-align:center;" | ^D
+| 25/24
+| '''64/63''', '''81/80'''
 |-
-| style="text-align:center;" | 2
+| 2
-| style="text-align:center;" | 68.571
+| 68.571
-| style="text-align:center;" | double-up unison
+| 128/125
-| style="text-align:center;" | ^^1
+| '''25/24'''
-| style="text-align:center;" | ^^D
+| 81/80
+|
 |-
-| style="text-align:center;" | 3
+| 3
-| style="text-align:center;" | 102.857
+| 102.857
-| style="text-align:center;" | double-down 2nd
+| '''17/16'''
-| style="text-align:center;" | vv2
+| '''15/14'''
-| style="text-align:center;" | vvE
+| '''16/15'''
+| '''18/17'''
 |-
-| style="text-align:center;" | 4
+| 4
-| style="text-align:center;" | 137.143
+| 137.143
-| style="text-align:center;" | down 2nd
+|
-| style="text-align:center;" | v2
+| '''12/11''', 16/15
-| style="text-align:center;" | vE
+|
+|
 |-
-| style="text-align:center;" | 5
+| 5
-| style="text-align:center;" |171.429
+| 171.429
-| style="text-align:center;" | 2nd
+| '''11/10'''
-| style="text-align:center;" | 2
+|
-| style="text-align:center;" | E
+| 12/11
+| '''10/9'''
 |-
-| style="text-align:center;" | 6
+| 6
-| style="text-align:center;" | 205.714
+| 205.714
-| style="text-align:center;" | up 2nd
+|
-| style="text-align:center;" | ^2
+|
-| style="text-align:center;" | ^E
+|
+| '''9/8'''
 |-
-| style="text-align:center;" | 7
+| 7
-| style="text-align:center;" | 240
+| 240.000
-| style="text-align:center;" | double-up 2nd
+| '''8/7'''
-| style="text-align:center;" | ^^2
+|
-| style="text-align:center;" | ^^E
+| 7/6
+|
 |-
-| style="text-align:center;" | 8
+| 8
-| style="text-align:center;" | 274.286
+| 274.286
-| style="text-align:center;" | double-down 3rd
+| '''20/17'''
-| style="text-align:center;" | vv3
+| '''7/6'''
-| style="text-align:center;" | vvF
+|
+|
 |-
-| style="text-align:center;" | 9
+| 9
-| style="text-align:center;" | 308.571
+| 308.571
-| style="text-align:center;" | down 3rd
+|
-| style="text-align:center;" | v3
+| '''6/5'''
-| style="text-align:center;" | vF
+|
+|
 |-
-| style="text-align:center;" | 10
+| 10
-| style="text-align:center;" |342.857
+| 342.857
-| style="text-align:center;" | 3rd
+| '''17/14'''
-| style="text-align:center;" | 3
+|
-| style="text-align:center;" | F
+| 6/5
+| '''11/9'''
 |-
-| style="text-align:center;" | 11
+| 11
-| style="text-align:center;" | 377.143
+| 377.143
-| style="text-align:center;" | up 3rd
+| '''5/4'''
-| style="text-align:center;" | ^3
+|
-| style="text-align:center;" | ^F
+|
+|
 |-
-| style="text-align:center;" | 12
+| 12
-| style="text-align:center;" | 411.429
+| 411.429
-| style="text-align:center;" | double-up 3rd
+| '''14/11'''
-| style="text-align:center;" | ^^3
+|
-| style="text-align:center;" | ^^F
+|
+|
 |-
-| style="text-align:center;" | 13
+| 13
-| style="text-align:center;" | 445.714
+| 445.714
-| style="text-align:center;" | double-down 4th
+| '''22/17''', 32/25
-| style="text-align:center;" | vv4
+|
-| style="text-align:center;" | vvG
+|
+| '''9/7'''
 |-
-| style="text-align:center;" | 14
+| 14
-| style="text-align:center;" | 480
+| 480.000
-| style="text-align:center;" | down 4th
+|
-| style="text-align:center;" | v4
+|
-| style="text-align:center;" | vG
+| 4/3, '''21/16'''
+|
 |-
-| style="text-align:center;" | 15
+| 15
-| style="text-align:center;" |514.286
+| 514.286
-| style="text-align:center;" | 4th
+|
-| style="text-align:center;" | 4
+| '''4/3'''
-| style="text-align:center;" | G
+|
+|
 |-
-| style="text-align:center;" | 16
+| 16
-| style="text-align:center;" | 548.571
+| 548.571
-| style="text-align:center;" | up 4th
+| '''11/8'''
-| style="text-align:center;" | ^4
+|
-| style="text-align:center;" | ^G
+|
+|
 |-
-| style="text-align:center;" | 17
+| 17
-| style="text-align:center;" | 582.857
+| 582.857
-| style="text-align:center;" | double-up 4th
+| '''7/5'''
-| style="text-align:center;" | ^^4
+| '''24/17'''
-| style="text-align:center;" | ^^G
+| 17/12
+|
 |-
-| style="text-align:center;" | 18
+| 18
-| style="text-align:center;" | 617.143
+| 617.143
-| style="text-align:center;" | double-down 5th
+| '''10/7'''
-| style="text-align:center;" | vv5
+| '''17/12'''
-| style="text-align:center;" | vvA
+| 24/17
+|
 |-
-| style="text-align:center;" | 19
+| 19
-| style="text-align:center;" | 651.429
+| 651.429
-| style="text-align:center;" | down 5th
+| '''16/11'''
-| style="text-align:center;" | v5
+|
-| style="text-align:center;" | vA
+|
+|
 |-
-| style="text-align:center;" | 20
+| 20
-| style="text-align:center;" |685.714
+| 685.714
-| style="text-align:center;" | 5th
+|
-| style="text-align:center;" | 5
+| '''3/2'''
-| style="text-align:center;" | A
+|
+|
 |-
-| style="text-align:center;" | 21
+| 21
-| style="text-align:center;" | 720
+| 720.000
-| style="text-align:center;" | up 5th
+|
-| style="text-align:center;" | ^5
+|
-| style="text-align:center;" | ^A
+| 3/2, '''32/21'''
+|
 |-
-| style="text-align:center;" | 22
+| 22
-| style="text-align:center;" | 754.286
+| 754.286
-| style="text-align:center;" | double-up 5th
+| '''17/11''', 25/16
-| style="text-align:center;" | ^^5
+|
-| style="text-align:center;" | ^^A
+|
+| '''14/9'''
 |-
-| style="text-align:center;" | 23
+| 23
-| style="text-align:center;" | 788.571
+| 788.571
-| style="text-align:center;" | double-down 6th
+| '''11/7'''
-| style="text-align:center;" | vv6
+|
-| style="text-align:center;" | vvB
+|
+|
 |-
-| style="text-align:center;" | 24
+| 24
-| style="text-align:center;" | 822.857
+| 822.857
-| style="text-align:center;" | down 6th
+| '''8/5'''
-| style="text-align:center;" | v6
+|
-| style="text-align:center;" | vB
+|
+|
 |-
-| style="text-align:center;" | 25
+| 25
-| style="text-align:center;" |857.143
+| 857.143
-| style="text-align:center;" | 6th
+| '''28/17'''
-| style="text-align:center;" | 6
+|
-| style="text-align:center;" | B
+| 5/3
+| '''18/11'''
 |-
-| style="text-align:center;" | 26
+| 26
-| style="text-align:center;" | 891.429
+| 891.429
-| style="text-align:center;" | up 6th
+|
-| style="text-align:center;" | ^6
+| '''5/3'''
-| style="text-align:center;" | ^B
+|
+|
 |-
-| style="text-align:center;" | 27
+| 27
-| style="text-align:center;" | 925.714
+| 925.714
-| style="text-align:center;" | double-up 6th
+| '''17/10'''
-| style="text-align:center;" | ^^6
+| '''12/7'''
-| style="text-align:center;" | ^^B
+|
+|
 |-
-| style="text-align:center;" | 28
+| 28
-| style="text-align:center;" | 960
+| 960.000
-| style="text-align:center;" | double-down 7th
+| '''7/4'''
-| style="text-align:center;" | vv7
+|
-| style="text-align:center;" | vvC
+|
+|
 |-
-| style="text-align:center;" | 29
+| 29
-| style="text-align:center;" | 994.286
+| 994.286
-| style="text-align:center;" | down 7th
+|
-| style="text-align:center;" | v7
+|
-| style="text-align:center;" | vC
+|
+| '''16/9'''
 |-
-| style="text-align:center;" | 30
+| 30
-| style="text-align:center;" |1028.571
+| 1028.571
-| style="text-align:center;" | 7th
+| '''20/11'''
-| style="text-align:center;" | 7
+|
-| style="text-align:center;" | C
+|
+| '''9/5'''
 |-
-| style="text-align:center;" | 31
+| 31
-| style="text-align:center;" | 1062.857
+| 1062.857
-| style="text-align:center;" | up 7th
+|
-| style="text-align:center;" | ^7
+| '''11/6''', 15/8
-| style="text-align:center;" | ^C
+|
+|
 |-
-| style="text-align:center;" | 32
+| 32
-| style="text-align:center;" | 1097.143
+| 1097.143
-| style="text-align:center;" | double-up 7th
+| '''32/17'''
-| style="text-align:center;" | ^^7
+| '''28/15'''
-| style="text-align:center;" | ^^C
+| '''15/8'''
+| '''17/9'''
 |-
-| style="text-align:center;" | 33
+| 33
-| style="text-align:center;" | 1131.429
+| 1131.429
-| style="text-align:center;" | double-down 8ve
+|
-| style="text-align:center;" | vv8
+|
-| style="text-align:center;" | vvD
+|
+|
 |-
-| style="text-align:center;" | 34
+| 34
-| style="text-align:center;" | 1165.714
+| 1165.714
-| style="text-align:center;" | down 8ve
+|
-| style="text-align:center;" | v8
+|
-| style="text-align:center;" | vD
+|
+|
 |-
-| style="text-align:center;" | 35
+| 35
-| style="text-align:center;" | 1200
+| 1200.000
-| style="text-align:center;" | 8ve
+| 2/1
-| style="text-align:center;" | 8
+|
-| style="text-align:center;" | D
+|
+|
 |}
-==Chord Names==
+== Notation ==
-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.
+The 7edo fifth is preferred as the diatonic generator for ups and downs notation due to being much easier to notate than the 5edo fifth (which involves E and F being enharmonic), as well as being closer to 3/2.
--10-20 = C E G = C = C or C perfect
--9-20 = C vE G = Cv = C down
--11-20 = C ^E G = C^ = C up
--10-19 = C E vG = C(v5) = C down-five
--11-21 = C ^E ^G = C^(^5) = C up up-five
--10-20-30 = C E G B = C7 = C seven
--10-20-29 = C E G vB = C,v7 = C add down-seven
--9-20-30 = C vE G B = Cv,7 = C down add-seven
--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|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.)
 {| class="wikitable"
 |-
-! | Degrees
+! Degrees
-! | Cents value
+! Cents
-! | Ratios in 2.5.7.11.17 subgroup
+! colspan="3" | [[Ups and downs notation]]
-! | Ratios with flat 3
+! [[Dual-fifth tuning|Dual-fifth]] notation
-! | Ratios with sharp 3
+<small>based on closest 12edo interval</small>
-! | Ratios with best 9
 |-
-| | 0
+| 0
 | 0.000
-| | '''1/1'''
+| unison
-| | (see comma table)
+| 1
-| |
+| D
-| |
+| 1sn, prime
 |-
-| | 1
+| 1
-| | 34.286
+| 34.286
-| | '''50/49''', '''121/119''', 33/32
+| up unison
-| | '''36/35'''
+| ^1
-| | 25/24
+| ^D
-| | '''81/80'''
+| augmented 1sn
 |-
-| | 2
+| 2
-| | 68.571
+| 68.571
-| | 128/125
+| dup unison
-| | '''25/24'''
+| ^^1
-| | 81/80
+| ^^D
-| |
+| diminished 2nd
 |-
-| | 3
+| 3
-| | 102.857
+| 102.857
-| | '''17/16'''
+| dud 2nd
-| | '''15/14'''
+| vv2
-| | '''16/15'''
+| vvE
-| | '''18/17'''
+| minor 2nd
 |-
-| | 4
+| 4
-| | 137.143
+| 137.143
-| |
+| down 2nd
-| | '''12/11''', 16/15
+| v2
-| |
+| vE
-| |
+| neutral 2nd
 |-
-| | 5
+| 5
-| |171.429
+| 171.429
-| | '''11/10'''
+| 2nd
-| |
+| 2
-| | 12/11
+| E
-| | '''10/9'''
+| submajor 2nd
 |-
-| | 6
+| 6
-| | 205.714
+| 205.714
-| |
+| up 2nd
-| |
+| ^2
-| |
+| ^E
-| | '''9/8'''
+| major 2nd
 |-
-| | 7
+| 7
-| | 240
+| 240
-| | '''8/7'''
+| dup 2nd
-| |
+| ^^2
-| | 7/6
+| ^^E
-| |
+| supermajor 2nd
 |-
-| | 8
+| 8
-| | 274.286
+| 274.286
-| | '''20/17'''
+| dud 3rd
-| | '''7/6'''
+| vv3
-| |
+| vvF
-| |
+| diminished 3rd
 |-
-| | 9
+| 9
-| | 308.571
+| 308.571
-| |
+| down 3rd
-| | '''6/5'''
+| v3
-| |
+| vF
-| |
+| minor 3rd
 |-
-| | 10
+| 10
-| |342.857
+| 342.857
-| | '''17/14'''
+| 3rd
-| |
+| 3
-| | 6/5
+| F
-| | '''11/9'''
+| neutral 3rd
 |-
-| | 11
+| 11
-| | 377.143
+| 377.143
-| | '''5/4'''
+| up 3rd
-| |
+| ^3
-| |
+| ^F
-| |
+| major 3rd
 |-
-| | 12
+| 12
-| | 411.429
+| 411.429
-| | '''14/11'''
+| dup 3rd
-| |
+| ^^3
-| |
+| ^^F
-| |
+| augmented 3rd
 |-
-| | 13
+| 13
-| | 445.714
+| 445.714
-| | '''22/17''', 32/25
+| dud 4th
-| |
+| vv4
-| |
+| vvG
-| | '''9/7'''
+| diminished 4th
 |-
-| | 14
+| 14
-| | 480
+| 480
-| |
+| down 4th
-| |
+| v4
-| | 4/3, '''21/16'''
+| vG
-| |
+| minor 4th
 |-
-| | 15
+| 15
-| |514.286
+| 514.286
-| |
+| 4th
-| | '''4/3'''
+| 4
-| |
+| G
-| |
+| major 4th
 |-
-| | 16
+| 16
-| | 548.571
+| 548.571
-| | '''11/8'''
+| up 4th
-| |
+| ^4
-| |
+| ^G
-| |
+| augmented 4th
 |-
-| | 17
+| 17
-| | 582.857
+| 582.857
-| | '''7/5'''
+| dup 4th
-| | '''24/17'''
+| ^^4
-| | 17/12
+| ^^G
-| |
+| minor tritone
 |-
-| | 18
+| 18
-| | 617.143
+| 617.143
-| | '''10/7'''
+| dud 5th
-| | '''17/12'''
+| vv5
-| | 24/17
+| vvA
-| |
+| major tritone
 |-
-| | 19
+| 19
-| | 651.429
+| 651.429
-| | '''16/11'''
+| down 5th
-| |
+| v5
-| |
+| vA
-| |
+| diminished 5th
 |-
-| | 20
+| 20
-| |685.714
+| 685.714
-| |
+| 5th
-| | '''3/2'''
+| 5
-| |
+| A
-| |
+| minor 5th
 |-
-| | 21
+| 21
-| | 720
+| 720
-| |
+| up 5th
-| |
+| ^5
-| | 3/2, '''32/21'''
+| ^A
-| |
+| major 5th
 |-
-| | 22
+| 22
-| | 754.286
+| 754.286
-| | '''17/11''', 25/16
+| dup 5th
-| |
+| ^^5
-| |
+| ^^A
-| | '''14/9'''
+| augmented 5th
 |-
-| | 23
+| 23
-| | 788.571
+| 788.571
-| | '''11/7'''
+| dud 6th
-| |
+| vv6
-| |
+| vvB
-| |
+| diminished 6th
 |-
-| | 24
+| 24
-| | 822.857
+| 822.857
-| | '''8/5'''
+| down 6th
-| |
+| v6
-| |
+| vB
-| |
+| minor 6th
 |-
-| | 25
+| 25
-| |857.143
+| 857.143
-| | '''28/17'''
+| 6th
-| |
+| 6
-| | 5/3
+| B
-| | '''18/11'''
+| neutral 6th
 |-
-| | 26
+| 26
-| | 891.429
+| 891.429
-| |
+| up 6th
-| | '''5/3'''
+| ^6
-| |
+| ^B
-| |
+| major 6th
 |-
-| | 27
+| 27
-| | 925.714
+| 925.714
-| | '''17/10'''
+| dup 6th
-| | '''12/7'''
+| ^^6
-| |
+| ^^B
-| |
+| augmented 6th
 |-
-| | 28
+| 28
-| | 960
+| 960
-| | '''7/4'''
+| dud 7th
-| |
+| vv7
-| |
+| vvC
-| |
+| diminished 7th
 |-
-| | 29
+| 29
-| | 994.286
+| 994.286
-| |
+| down 7th
-| |
+| v7
-| |
+| vC
-| | '''16/9'''
+| minor 7th
 |-
-| | 30
+| 30
-| |1028.571
+| 1028.571
-| | '''20/11'''
+| 7th
-| |
+| 7
-| |
+| C
-| | '''9/5'''
+| superminor 7th
 |-
-| | 31
+| 31
-| | 1062.857
+| 1062.857
-| |
+| up 7th
-| | '''11/6''', 15/8
+| ^7
-| |
+| ^C
-| |
+| neutral 7th
 |-
-| | 32
+| 32
-| | 1097.143
+| 1097.143
-| | '''32/17'''
+| dup 7th
-| | '''28/15'''
+| ^^7
-| | '''15/8'''
+| ^^C
-| | '''17/9'''
+| major 7th
 |-
-| | 33
+| 33
-| | 1131.429
+| 1131.429
-| |
+| dud 8ve
-| |
+| vv8
-| |
+| vvD
-| |
+| augmented 7th
 |-
-| | 34
+| 34
-| | 1165.714
+| 1165.714
-| |
+| down 8ve
-| |
+| v8
-| |
+| vD
-| |
+| diminished 8ve
 |-
-|35
+| 35
-|1200
+| 1200
-|
+| 8ve
-|
+| 8
-|
+| D
-|
+| 8ve
 |}
-{| class="wikitable" style="text-align:center;"
+===Sagittal notation===
-|-
+====Best fifth notation====
-! Interval, complement
+This notation uses the same sagittal sequence as EDOs [[30edo#Second-best fifth notation|30b]] and [[40edo#Sagittal notation|40]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
-! Error (abs., in [[cent]]s)
-|-
+<imagemap>
-|[[7/5]] [[10/7]]
+File:35-EDO_Sagittal.svg
-|0.3448
+desc none
-|-
+rect 80 0 300 50 [[Sagittal_notation]]
-|[[13/12]] [[24/13]]
+rect 415 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-|1.4296
+rect 20 80 415 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
-|-
+default [[File:35-EDO_Sagittal.svg]]
-|[[9/8]] [[16/9]]
+</imagemap>
-|1.8039
-|-
+====Second-best fifth notation====
-|[[17/16]] [[32/17]]
+This notation uses the same sagittal sequence as [[42edo#Sagittal notation|42-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
-|2.0984
-|-
+<imagemap>
-|[[11/8]] [[16/11]]
+File:35b_Sagittal.svg
-|2.7469
+desc none
-|-
+rect 80 0 300 50 [[Sagittal_notation]]
-|[[18/17]] [[17/9]]
+rect 391 0 551 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-|3.9024
+rect 20 80 391 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
-|-
+default [[File:35b_Sagittal.svg]]
-|[[11/9]] [[18/11]]
+</imagemap>
-|4.5509
-|-
+=== Chord Names ===
-|[[11/10]] [[20/11]]
+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.
-|6.4247
-|-
+-10-20 = C E G = C = C or C perfect
-|[[14/11]] [[11/7]]
-|6.0789
+-9-20 = C vE G = Cv = C down
-|-
-|[[6/5]] [[5/3]]
+-11-20 = C ^E G = C^ = C up
-|7.0703
-|-
+-10-19 = C E vG = C(v5) = C down-five
-|[[7/6]] [[12/7]]
-|7.4151
+-11-21 = C ^E ^G = C^(^5) = C up up-five
-|-
-|[[8/7]] [[7/4]]
+-10-20-30 = C E G B = C7 = C seven
-|8.8259
-|-
+-10-20-29 = C E G vB = C,v7 = C add down-seven
-|[[14/13]] [[13/7]]
-|8.8447
+-9-20-30 = C vE G B = Cv,7 = C down add-seven
-|-
-|[[16/15]] [[15/8]]
+-9-20-29 = C vE G vB = Cv7 = C down seven
-|8.8742
-|-
+For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions|Ups and downs notation - Chords and Chord Progressions]].
-|[[5/4]] [[8/5]]
-|9.1707
+== Approximation to JI ==
-|-
-|[[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=
+{{Q-odd-limit intervals|35}}
+== Regular temperament properties ==
+=== Rank-2 temperaments ===
 {| class="wikitable"
 |-
-! | Periods
+! Periods<br>per 8ve
+! Generator
-per octave
+! Temperaments with<br>flat 3/2 (patent val)
-! | Generator
+! Temperaments with sharp 3/2 (35b val)
-! | Temperaments with
+! [[Mos scale]]s
-flat 3/2 (patent val)
-! | <span style="display: block; text-align: center;">Temperaments with sharp 3/2 (35b val)</span>
 |-
-| | 1
+| 1
-| | 1\35
+| 1\35
-| |
+|
-| |
+|
+|
 |-
-| | 1
+| 1
-| | 2\35
+| 2\35
-| |
+|
-| |
+|
+| [[1L 16s]], [[17L 1s]]
 |-
-| | 1
+| 1
-| | 3\35
+| 3\35
-| |
+|
-| | [[Ripple|Ripple]]
+| [[Ripple]]
+| [[1L 10s]], [[11L 1s]], [[12L 11s]]
 |-
-| | 1
+| 1
-| | 4\35
+| 4\35
-| | [[Greenwoodmic_temperaments#Secund|Secund]]
+| [[Secund]]
-| |
+|
+| [[1L 7s]], [[8L 1s]], [[9L 8s]], [[9L 17s]]
 |-
-| | 1
+| 1
-| | 6\35
+| 6\35
-| colspan="2" | Messed-up [[Chromatic_pairs#Baldy|Baldy]]
+| colspan="2" | [[Baldy]] (messed-up){{idiosyncratic}}
+| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[6L 11s]], [[6L 17s]], [[6L 23s]]
 |-
-| | 1
+| 1
-| | 8\35
+| 8\35
-| |
+|
-| | Messed-up [[Orwell]]
+| [[Orwell]] (messed-up){{idiosyncratic}}
+| [[1L 3s]], [[4L 1s]], [[4L 5s]], [[9L 4s]], [[13L 9s]]
 |-
-| | 1
+| 1
-| | 9\35
+| 9\35
-| | [[Myna|Myna]]
+| [[Myna]]
-| |
+|
+| [[1L 3s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], …, [[4L 27s]]
 |-
-| | 1
+| 1
-| | 11\35
+| 11\35
-| | [[Muggles]]
+| [[Muggles]]
-| |
+|
+| [[3L 1s]], [[3L 4s]], [[3L 7s]] [[3L 10s]], [[3L 13s]], [[16L 3s]]
 |-
-| | 1
+| 1
-| | 12\35
+| 12\35
-| |
+|
-| | [[Avicennmic_temperaments#Roman|Roman]]
+| [[Roman]]
+| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[3L 8s]], [[3L 11s]], [[3L 14s]], [[3L 17s]], [[3L 20s]], …, [[3L 29s]]
 |-
-| | 1
+| 1
-| | 13\35
+| 13\35
-| colspan="2" | Inconsistent 2.9'/7.5/3 [[Sensi]]
+| colspan="2" | Inconsistent 2.5/3.9/7 [[sensi]]/[[subgroup_temperaments#Sentry|sentry]]
+| [[2L 1s]], [[3L 2s]], [[3L 5s]], [[8L 3s]], [[8L 11s]], [[8L 19s]]
 |-
-| | 1
+| 1
-| | 16\35
+| 16\35
-| |
+|
-| |
+|
+| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[11L 2s]], [[11L 13s]]
 |-
-| | 1
+| 1
-| | 17\35
+| 17\35
-| |
+|
-| |
+|
+| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]], [[2L 17s]], [[2L 19s]], …, [[2L 31s]]
 |-
-| | 5
+| 5
-| | 1\35
+| 1\35
-| |
+|
-| | [[Blackwood]] (favoring 7/6)
+| [[Blackwood]] (favoring 7/6)
+| [[5L 5s]], [[5L 10s]], [[5L 15s]], [[5L 20s]], [[5L 25s]]
 |-
-| | 5
+| 5
-| | 2\35
+| 2\35
-| |
+|
-| | [[Blackwood]] (favoring 6/5 and 20/17)
+| [[Blackwood]] (favoring 6/5 and 20/17)
+| [[5L 5s]], [[5L 10s]], [[15L 5s]]
 |-
-| | 5
+| 5
-| | 3\35
+| 3\35
-| |
+|
-| | [[Blackwood]] (favoring 5/4 and 17/14)
+| [[Blackwood]] (favoring 5/4 and 17/14)
+| [[5L 5s]], [[10L 5s]], [[10L 15s]]
 |-
-| | 7
+| 7
-| | 1\35
+| 1\35
-| | [[Apotome_family|Whitewood]]/[[Apotome_family#Redwood|Redwood]]
+| [[Whitewood]] / [[redwood]]
-| |
+|
+| [[7L 7s]], [[7L 14s]], [[7L 21s]]
 |-
-| | 7
+| 7
-| | 2\35
+| 2\35
-| | [[Greenwoodmic_temperaments#Greenwood|Greenwood]]
+| [[Greenwood]]
-| |
+|
+| [[7L 7s]], [[14L 7s]]
 |}
-=Scales=
+=== Commas ===
-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.
+et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 35 55 81 98 121 130 }}.)
-=Commas=
-EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 35 55 81 98 121 130 }}.)
 {| class="commatable wikitable center-1 center-2 right-4 center-5"
 |-
-! [[Harmonic limit|Prime <br> limit]]
+! [[Harmonic limit|Prime<br>limit]]
 ! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
 ! [[Monzo]]
@@ Line 701: / Line 743: @@
 | 113.69
 | Lawa
-| Apotome, Whitewood comma
+| Whitewood comma, apotome, Pythagorean chroma
 |-
 | 5
@@ Line 722: / Line 764: @@
 | 29.61
 | Laquinyo
-| Small diesis, Magic comma
+| Magic comma, small diesis
 |-
 | 7
@@ Line 736: / Line 778: @@
 | 44.13
 | Laquinzo
-| Cloudy
+| Cloudy comma
 |-
 | 7
@@ Line 742: / Line 784: @@
 | {{monzo| -9 1 2 1 }}
 | 43.41
-| Zoyoyo
+| Lazoyoyo
-| Avicenna
+| Avicennma
 |-
 | 7
@@ Line 750: / Line 792: @@
 | 13.79
 | Zotrigu
-| Starling comma, Septimal semicomma
+| Septimal semicomma, starling comma
 |-
 | 11
@@ Line 768: / Line 810: @@
 <references/>
-=Music=
+== Octave stretch or compression ==
-[http://soonlabel.com/xenharmonic/archives/2348 Little Prelude &amp; Fugue, "The Bijingle" by Claudi Meneghin]
+edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]]. Some stretched-octave 35edo tunings (least to most stretched) include [[149zpi]], [[equal tuning|116ed10]], [[ed7|98ed7]], [[ed5|81ed5]], [[ed12|125ed12]] or [[ed6|90ed6]].
+== Scales ==
+=== Polymicrotonal scales ===
+; 12-tone 7edo&5edo
+The ''12-tone 7edo&5edo scale'' is designed to be mapped to the key of C on a conventional piano keyboard, with [[7edo]] on the white keys, and [[5edo]] on black:
+* 5 2 3 4 1 5 1 4 3 2 5 0
+; 24-tone blackwood&greenwood
+You can have two pianos/keyboards, one 68.6 [[cents]] sharp of the other, both tuned to the 12-tone 7edo&5edo scale. The combined black keys across both keyboards will be ''[[blackwood]][10]'' and the white keys will be ''[[greenwood]][14]''.
+* 3 2 0 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 0 2 3 0 2 0
+; 20-tone blackwood&greenwood
+Removing the duplicates from the previous scale (perhaps for use on other instruments beside keyboard) gives this ''20-tone scale,'' which includes both blackwood[10] and greenwood[14] as subsets.
+* 3 2 2 1 2 2 1 1 1 3 1 1 1 2 2 1 2 2 3 2
+=== MOS scales ===
+Of the [[MOS scale]]s available in 35edo, the [[muggles]] scales most closely approximate [[just intonation]].
+; MOS scales
+* [[Greenwood]][7]/[[whitewood]][7]: 5 5 5 5 5 5 5 (''a.k.a. [[7edo]]; an [[equiheptatonic]] scale'')
+* [[Greenwood]][14]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2
+* [[Greenwood]][21]: 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
+* [[Muggles]][5] (a.k.a. sub-diatonic): 9 4 9 9 4
+* [[Muggles]][13]: 2 2 5 2 2 2 5 2 2 2 5 2 2
+* [[Muggles]][16]: 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2
+* [[Muggles]][19]: 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2
+* [[Ripple]][12]: 3 3 3 3 3 3 3 3 2 3 3 3
+* [[Ripple]][23]: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
+* [[Secund]][17]: 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
+* [[Whitewood]][14]: 1 4 1 4 1 4 1 4 1 4 1 4 1 4
+* [[Whitewood]][21]: 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1
+* [[Blackwood]][5]: 7 7 7 7 7 (''a.k.a. [[5edo]]; an [[equipentatonic]] scale; [[slendro]]-like; works with all three blackwood tunings'')
+* [[Blackwood|5/4-blackwood]][10]: 4 3 4 3 4 3 4 3 4 3
+* [[Blackwood|5/4-blackwood]][15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
+* [[Blackwood|5/4-blackwood]][25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
+* [[Blackwood|6/5-blackwood]][10]: 2 5 2 5 2 5 2 5 2 5
+* [[Blackwood|6/5-blackwood]][15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
+* [[Blackwood|6/5-blackwood]][20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
+* 2L 9s (4:3) [11]: 3 3 4 3 3 3 3 3 4 3 3 --- A scale doing great job tempering the 2.9.11.17 subgroup near JI.
+=== Ripple scales ===
+; Ripple[23]
+The ''[[ripple]][23]'' [[MOS scale]] makes maximum use of 35edo's dual-fifth nature, with both its sizes of fifth and fourth occurring frequently throughout the whole scale:
+* Symmetrical mode (has the most consonances): 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
+* Mode that includes the clear pond{{idio}} modmos: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
+{| class="wikitable mw-collapsible mw-collapsed"
+|+Ripple[23] subsets approximated from [[96edo]]
+|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
+* Flattened major: 6 5 4 5 6 6 3
+* Sharpened minor: 6 3 6 5 4 6 5
+* Sharpened harmonic minor: 6 3 6 6 3 9 2
+* Flattened major pentatonic: 5 6 9 6 9
+* Sharpened minor pentatonic: 9 6 5 10 5
+* Evened minor hexatonic: 5 4 6 5 9 6
+* Roughened augmented: 10 2 9 2 10 2
+* Evened dominant pentatonic: 6 6 8 9 6
+* Sharpened Dorian: 6 3 6 6 6 3 5
+* Flattened Ionian pentatonic: 11 4 5 12 3
+* Sharpened Dorian harmonic: 6 3 9 3 6 3 5
+* Evened Mixolydian pentatonic: 11 4 6 8 6
+* Roughened Phrygian dominant: 2 10 2 6 3 6 6
+* Evened Phrygian dominant hexatonic: 3 8 4 6 8 6
+* Sharpened Phrygian pentatonic: 3 6 12 3 11
+* Sharpened blues Aeolian hexatonic: 9 6 3 2 3 12
+* Flattened blues Aeolian pentatonic I: 8 6 6 3 12
+* Sharpened blues Aeolian pentatonic II: 9 12 2 6 6
+* Roughened blues Dorian heptatonic: 9 6 3 2 7 2 6
+* Sharpened blues Dorian hexatonic: 9 6 6 5 4 5
+* Roughened blues Dorian pentatonic: 9 11 7 2 6
+* Roughened blues pentachordal: 6 3 5 4 2 15
+* Sharpened minor harmonic pentatonic I: 6 3 12 12 2
+* Sharpened minor harm. pent. II: 9 6 6 12 2
+* Evened hirajoshi: 6 3 11 4 11
+* Sharpened hirajoshi: 6 3 12 3 11
+* Roughened hirajoshi: 6 2 13 2 12
+* Evened akebono I: 6 3 11 6 9
+* Sharpened akebono I: 6 3 12 5 9
+* Roughened akebono I: 7 1 13 6 8
+* Roughened Javanese pentachordal: 2 7 9 2 15
+* Roughened cosmic: 14 6 2 7 6
+* Roughened cosmic II: 6 2 7 5 15
+* [[Lost spirit]]: 9 6 2 3 7 3 5
+* Moonbeam: 6 3 11 12 3
+* Palace: 5 4 6 5 5 4 6
+* Underpass: 9 11 7 3 5
+|}
+; Clear pond{{idio}}
+The ''clear pond scale''{{idio}}, a [[modmos]] of ripple[12], tries to sound close to the familiar [[12edo]]:
+* 3 3 3 2 3 3 3 4 2 3 3 3
+{| class="wikitable mw-collapsible mw-collapsed"
+|+Clear pond subsets
+|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
+* Lydian: 6 5 6 3 6 6 3
+* Major: 6 5 3 6 6 6 3
+* Mixolydian: 6 5 3 6 6 3 6
+* Dorian: 6 3 5 6 6 3 6
+* Minor: 6 3 5 6 4 5 6
+* Phrygian: 3 6 5 6 4 5 6
+* Locrian: 3 6 5 3 7 5 6
+* Harmonic minor: 6 3 5 6 4 8 3
+* Melodic minor: 6 3 5 6 6 6 3
+* Major pentatonic: 6 8 6 6 9
+* Minor pentatonic: 9 5 6 9 6
+* Minor blues: 9 5 3 3 9 6
+* Minor blues heptatonic: 9 5 3 3 6 3 6
+* Akebono I: 6 3 11 6 9
+|}
+=== Secund scales ===
+; Secund[17]
+The ''secund[17]'' MOS scale includes a motley mix of quirky, quite [[xenharmonic]] subsets, suited for exploring those consonances very different to any found in [[12edo]].
+* 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
+{| class="wikitable mw-collapsible mw-collapsed"
+|+Secund[17] subsets
+|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
+*[[Antipental blues]]: 8 7 1 4 8 7
+* Antipental blues maj 6th: 8 7 1 4 7 1 7
+* Antipental blues neutral 7th: 8 7 1 4 8 3 4
+* Antipental blues maj 7th: 8 7 1 4 8 4 3
+* Antipental blues harmonic: 8 7 1 4 3 9 3
+* [[Pelog]]-like heptatonic: 3 5 7 5 3 8 4 (''Phrygian-like'')
+* Pelog-like pentatonic: 3 5 12 3 12
+* Secund chance ([[modmos]] of secund[8]): 4 7 4 1 4 4 7 4
+* Secund-tempered rotated [[5afdo]]: 7 4 9 8 7
+* Secund-tempered [[6afdo]]: 8 7 5 7 4 4
+* Undecimal Mixolydian: 7 4 4 5 7 1 7
+* Undecimal minor hexatonic: 7 1 7 5 8 7
+* Undecimal quasi-equipentatonic: 7 8 5 8 7
+* 12 from secund[17]: 7 1 3 4 1 4 3 4 1 3 1 3
+|}
+=== Blackwood scales ===
+; The three blackwood temperaments
+There are actually three versions of the ''[[blackwood]] temperament'' available in 35edo. One optimises the subminor third [[7/6]], one optimises the minor third [[6/5]], the other optimises the major third [[5/4]]. Try them each and see which one you prefer:
+* [[Blackwood|5/4-blackwood]][15]: 3 1 3 3 1 3 3 1 3 3 1 3 3 1 3
+* [[Blackwood|5/4-blackwood]][25]: 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
+* [[Blackwood|6/5-blackwood]][15]: 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2
+* [[Blackwood|6/5-blackwood]][20]: 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2
+* [[Blackwood|7/6-blackwood]][15]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
+* [[Blackwood|7/6-blackwood]][20]: 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1
+{| class="wikitable mw-collapsible mw-collapsed"
+|+6/5-blackwood[20] subsets
+|''Contains [[Template:Idiosyncratic|idiosyncratic terms]].''
+*Blackwood meta-Hirajoshi: 2 3 4 2 5 7 2 12
+** ''Blackwood pseudo-Akebono neutral: 5 9 7 2 12''
+** ''Blackwood pseudo-Akebono supermajor: 7 7 7 2 12''
+** ''Blackwood pseudo-Hirajoshi: 2 12 7 2 12''
+** ''Blackwood pseudo-[[pelog]]: 5 4 12 2 12''
+* Blackwood meta-partial: 4 3 2 2 3 7 7 7
+** ''Blackwood-tempered [[5afdo]]: 7 4 10 7 7''
+** ''Mechanical (from [[16afdo]]): 9 2 10 7 7''
+** ''Starship (from [[68ifdo]]'', see [[ifdo]]''): 4 7 3 7 7 7''
+** ''Volcanic (from [[16afdo]]): 4 7 10 7 7''
+* Meta-monsoon: 7 4 3 2 5 9 5
+** ''Monsoon (from [[47zpi]]): 7 7 7 9 5''
+** ''Monsoon otonal: 7 9 5 9 5''
+** ''Monsoon major: 11 5 5 9 5''
+* Blackwood neutral nonatonic: 4 7 3 2 5 4 5 2 3
+* Blackwood undecimal harmonic: 4 8 4 5 4 5 5
+* Dungeon (from [[30afdo]]): 11 3 7 2 12
+* Moonbeam (from [[16afdo]]): 7 2 12 12 2
+* Underpass (from [[10afdo]]): 9 12 5 4 5
+* 12 from 6/5-blackwood[20]: 4 3 2 2 3 7 2 3 2 2 3 2
+|}
+=== Other scales ===
+* Amulet{{idiosyncratic}}, approximated from [[magic]] in [[25edo]]: 3 1 3 3 1 3 4 3 3 1 3 4 3
+* Fourfourths{{idio}} ([[modmos]] of 7/6-blackwood[20]): 3 1 1 2 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4
+* Near-just rotated [[5afdo]]: 6 5 9 8 7
+* Near-just [[6afdo]]: 8 7 5 6 5 4
+== Instruments ==
+=== Lumatone ===
+edo can be played on the [[Lumatone]]. See [[Lumatone mapping for 35edo]]
+=== Skip fretting ===
+'''Skip fretting system 35 3 8''' is a [[skip fretting]] system for [[35edo]]. All examples on this page are for 7-string [[guitar]].
+; Prime harmonics
+/1: string 2 open
+/1: string 3 fret 9 and string 6 fret 1
+/2: string 3 fret 4 and string 4 fret 13
+/4: string 3 fret 1, string 4 fret 10, and string 7 fret 2
+/4: string 4 fret 4
+/8: string 1 fret 8, string 4 open, and string 5 fret 9
+/8: string 1 fret 11, string 4 fret 3, and string 5 fret 12
+/16: string 2 fret 1 and string 3 fret 10
+== Music ==
+=== Modern renderings ===
+; {{W|Frederick Chopin}}
+* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847) – rendered in 35-edo with alternating sharp and flat fifths by [[Claudi Meneghin]] (2025)
+; {{W|Carlo Gesualdo}}
+* [https://www.youtube.com/watch?v=idUG-x8kT3o&t=305 ''Dolcissima mia vita''] – in three comparative tunings including 35edo (5:05–10:05), rendered by [[Chris Vaisvil]] (2025)
+=== 21st century ===
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/6X1-692axAk ''35edo''] (2025)
+* ''Whistling Like An Oberon - 35edo'' (2026) [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>], [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>], [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]
+; [[dotuXil]]
+* [https://www.youtube.com/watch?v=61ssLv9H6rk "Icebound Gallery of Refractions"] from [https://dotuxil.bandcamp.com/album/collected-refractions ''Collected Refractions''] (2024)
+; [[E8 Heterotic]]
+* [https://youtu.be/07-wj6BaTOw ''G2 Manifold''] (2020) – uses a combination of 5edo and 7edo, which can be classified as a 35edo subset.
+; [[Francium]]
+* "What Kind Of Things" from ''TOTMC 2025'' (2025) – [https://francium223.bandcamp.com/track/what-kind-of-things Bandcamp] | [https://www.youtube.com/watch?v=WaRm0dlUqQU YouTube]
+; [[groundfault]]
+* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 27edo
+; [[JUMBLE]]
+* [https://www.youtube.com/watch?v=2qpsI26JfjY ''Penguins...?''] (2024)
+; [[Budjarn Lambeth]]
+* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[Blackwood]][20])
+; [[Chuckles McGee]]
+* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 ''Self-Destructing Mechanical Forest''] – in Secund[9], 35edo tuning
+; [[Claudi Meneghin]]
+* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude & Fugue, "The Bijingle"''] (2014)
+* [https://www.youtube.com/watch?v=JPie2YDwA8I ''MicroFugue on Happy Birthday for Baroque Ensemble''] (2023)
+* [https://www.youtube.com/shorts/c9rCrQwF1HI ''NEOBAROQUE CANON, 3-in-1 without Bass in 35-edo for Baroque Consort: Oboe, Recorder, Violin''] (2025)
-[http://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 Self-Destructing Mechanical Forest] by Chuckles McGee (in Secund[9])
+; [[No Clue Music]]
+* [https://www.youtube.com/watch?v=zMUQWdFRGao ''DarkSciFiThing''] (2024)
-[[Category:35edo]]
+[[Category:Listen]]
-[[Category:Equal divisions of the octave]]
-[[Category:listen]]
-[[Category:theory]]

35edo: Difference between revisions

Latest revision as of 14:42, 7 May 2026

Theory

Odd harmonics

Dual-fifth harmony

Caveats of dual-fifth

Subsets and supersets

Intervals

Notation

Sagittal notation

Best fifth notation

Second-best fifth notation

Chord Names

Approximation to JI

Regular temperament properties

Rank-2 temperaments

Commas

Octave stretch or compression

Scales

Polymicrotonal scales

MOS scales

Ripple scales

Secund scales

Blackwood scales

Other scales

Instruments

Lumatone

Skip fretting

Music

Modern renderings

21st century

Navigation menu

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