41edo

← 40edo 41edo 42edo →
Prime factorization	41 (prime)
Step size	29.2683 ¢
Fifth	24\41 (702.439 ¢); (convergent)
Semitones (A1:m2)	4:3 (117.1 ¢ : 87.8 ¢)
Consistency limit	15
Distinct consistency limit	9

The 41 equal divisions of the octave (41edo), or 41(-tone) equal temperament (41tet, 41et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 41 equally-sized steps. Each step is about 29.3 cents, an interval close in size to 64/63, the septimal comma.

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13	prime 17	prime 19
Error	absolute (¢)	0.0	+0.5	-5.8	-3.0	+4.8	+8.3	+12.1	-4.8
Error	relative (%)	0	+2	-20	-10	+16	+28	+41	-17
Nearest edomapping		41	24	13	33	19	29	4	10
Fifthspan		0	+1	-8	-14	-18	+20	+7	-3

41edo can be seen as a tuning of the garibaldi temperament^[1]^[2]^[3], the magic temperament^[4] and the superkleismic (26&41) temperament. It is the second smallest equal division (after 29edo) whose perfect fifth is closer to just intonation than that of 12edo, and is the seventh zeta integral edo after 31; it is not, however, a zeta gap edo. This has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp. Various 13-limit magic extensions are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.

41edo is consistent in the 15-odd-limit. In fact, all of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, 31edo is only consistent up to the 11-odd-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.

41et forms the foundation of the H-System, which uses the scale degrees of 41et as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41et circle in 205edo. 41et is also used by the Kite Guitar, see below in #instruments.

41edo is the 13th prime edo, following 37edo and coming before 43edo.

↑ Schismic Temperaments at x31eq.com, the website of Graham Breed
↑ Lattices with Decimal Notation at x31eq.com
↑ Wikipedia: Schismatic temperament
↑ Wikipedia: Magic temperament

Intervals

#	Cents	Approximate Ratios*	Ups and Downs Notation			Andrew's Solfege	Kite's Solfege
0	0.00	1/1	perfect unison	P1	D	do	do
1	29.27	81/80, 64/63, 49/48	up-unison	^1	^D	di	da
2	58.54	25/24, 28/27, 36/35, 33/32	double-up 1sn, downminor 2nd	^^1, vm2	^^D, vEb	ro	ru
3	87.80	21/20, 22/21, 19/18, 20/19	down-aug 1sn, minor 2nd	vA1, m2	vD#, Eb	rih	ro
4	117.07	16/15, 15/14, 14/13	augmented 1sn, upminor 2nd	A1, ^m2	D#, ^Eb	ra	ra
5	146.34	12/11, 13/12	mid 2nd	~2	^D#, vvE	ru	ruh
6	175.61	10/9, 11/10, 21/19	downmajor 2nd	vM2	vE	reh	reh
7	204.88	9/8	major 2nd	M2	E	re	rih
8	234.15	8/7, 15/13	upmajor 2nd	^M2	^E	ri	ri
9	263.41	7/6, 22/19	downminor 3rd	vm3	vF	ma	mu
10	292.68	32/27, 13/11, 19/16	minor 3rd	m3	F	meh	mo
11	321.95	6/5	upminor 3rd	^m3	^F	me	ma
12	351.22	11/9, 27/22, 16/13	mid 3rd	~3	^^F, vGb	mu	muh
13	380.49	5/4, 26/21	downmajor 3rd	vM3	vF#, Gb	mi	meh
14	409.76	81/64, 14/11, 24/19, 19/15	major 3rd	M3	F#, ^Gb	maa	mih
15	439.02	9/7, 32/25	upmajor 3rd	^M3	^F#, vvG	mo	mi
16	468.29	21/16, 13/10	down-4th	v4	vG	fe	fu
17	497.56	4/3	perfect 4th	P4	G	fa	fo
18	526.83	27/20, 15/11, 19/14	up-4th	^4	^G	fih	fa
19	556.10	11/8, 18/13, 26/19	mid-4th	~4	^^G, vAb	fu	fuh
20	585.37	7/5	downaug 4th, dim 5th	vA4, d5	vG#, Ab	fi	feh / so
21	614.63	10/7	aug 4th, updim 5th	A4, ^d5	G#, ^Ab	se	fih / sa
22	643.90	16/11, 13/9, 19/13	mid-5th	~5	vvA	su	suh
23	673.17	40/27, 22/15, 28/19	down-5th	v5	vA	sih	seh
24	702.44	3/2	perfect 5th	P5	A	sol	sih
25	731.71	32/21, 20/13	up-5th	^5	^A	si	si
26	760.98	14/9, 25/16	downminor 6th	vm6	^^A, vBb	lo	lu
27	790.24	128/81, 11/7, 19/12, 30/19	minor 6th	m6	vA#, Bb	leh	lo
28	819.51	8/5, 21/13	upminor 6th	^m6	A#, ^Bb	le	la
29	848.78	18/11, 44/27, 13/8	mid 6th	~6	^A#, vvB	lu	luh
30	878.05	5/3	downmajor 6th	vM6	vB	la	leh
31	907.32	27/16, 22/13, 32/19	major 6th	M6	B	laa	lih
32	936.59	12/7, 19/11	upmajor 6th	^M6	^B	li	li
33	965.85	7/4, 26/15	downminor 7th	vm7	vC	ta	tu
34	995.12	16/9	minor 7th	m7	C	teh	to
35	1024.39	9/5, 20/11, 38/21	upminor 7th	^m7	^C	te	ta
36	1053.66	11/6, 24/13	mid 7th	~7	^^C, vDb	tu	tuh
37	1082.93	15/8, 28/15, 13/7	downmajor 7th	vM7	vC#, Db	ti	teh
38	1112.20	40/21, 21/11, 36/19, 19/10	major 7th	M7	C#, ^Db	taa	tih
39	1141.46	48/25, 27/14, 35/18, 64/33	upmajor 7th	^M7	C#^, vvD	to	ti
40	1170.73	160/81, 63/32, 96/49	dim 8ve	v8	vD	da	du
41	1200.00	2/1	perfect 8ve	P8	D	do	do

* Based on treating 41-edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

quality	color	monzo format	examples
downminor	zo	(a, b, 0, 1)	7/6, 7/4
minor	fourthward wa	(a, b) with b < -1	32/27, 16/9
upminor	gu	(a, b, -1)	6/5, 9/5
mid	ilo	(a, b, 0, 0, 1)	11/9, 11/6
"	lu	(a, b, 0, 0, -1)	12/11, 18/11
downmajor	yo	(a, b, 1)	5/4, 5/3
major	fifthward wa	(a, b) with b > 1	9/8, 27/16
upmajor	ru	(a, b, 0, -1)	9/7, 12/7

Chord names

All 41edo chords can be named using ups and downs. An up, down or mid 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. Here are the zo, gu, ilo, yo and ru triads:

color of the 3rd	JI chord	notes as edosteps	notes of C chord	written name	spoken name
zo (7-over)	6:7:9	0-9-24	C vEb G	Cvm	C downminor
gu (5-under)	10:12:15	0-11-24	C ^Eb G	C^m	C upminor
ilo (11-over)	18:22:27	0-12-24	C vvE G	C~	C mid
yo (5-over)	4:5:6	0-13-24	C vE G	Cv	C downmajor or C down
ru (7-under)	14:18:21	0-15-24	C ^E G	C^	C upmajor or C up

0-10-20 = D F Ab = Dd = D dim

0-10-21 = D F ^Ab = Dd(^5) = D dim up-five

0-10-22 = D F vvA = Dm(~5) = D minor mid-five

0-10-23 = D F vA = Dm(v5) = D minor down-five

0-10-24 = D F A = Dm = D minor

0-14-24 = D F# A = D = D or D major

0-14-25 = D F# ^A = D(^5) = D up-five

0-14-26 = D F# ^^A = D(^^5) = D half-aug

0-14-27 = D F# vA# = Da(v5) = D aug down-five or perhaps D(v#5) = D downsharp-five

0-14-28 = D F# A# = Da = D aug

For a more complete list, see 41edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.

Notations

Red-Blue Notation

A red-note/blue-note system, similar to the one proposed for 36edo, is one option for notating 41edo. (This is separate from and not compatible with Kite's color notation.) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:

A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.

Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.

The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.

If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as ups and downs notation. The only difference is the use of minor tritone and major tritone.

Sagittal

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 41-EDO in the Revo flavor of Sagittal:

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 41edo. Prime harmonics are in bold.

Interval, complement	Error (abs, ¢)
4/3, 3/2	0.484
9/8, 16/9	0.968
15/14, 28/15	2.370
7/5, 10/7	2.854
8/7, 7/4	2.972
7/6, 12/7	3.456
13/11, 22/13	3.473
11/9, 18/11	3.812
9/7, 14/9	3.940
12/11, 11/6	4.296
11/8, 16/11	4.780
16/15, 15/8	5.342
5/4, 8/5	5.826
6/5, 5/3	6.310
10/9, 9/5	6.794
18/13, 13/9	7.285
14/11, 11/7	7.752
13/12, 24/13	7.769
16/13, 13/8	8.253
15/11, 22/15	10.122
11/10, 20/11	10.606
14/13, 13/7	11.225
15/13, 26/15	13.595
13/10, 20/13	14.079

Relationship to 12-edo

Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo. The two innermost and two outermost intervals on the spiral are duplicates.

The same spiral, but with notes not intervals:

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[65 -41⟩	[⟨41 65]]	-0.153	0.15	0.52
2.3.5	3125/3072, 20000/19683	[⟨41 65 95]]	+0.734	1.26	4.31
2.3.5.7	225/224, 245/243, 1029/1024	[⟨41 65 95 115]]	+0.815	1.10	3.76
2.3.5.7.11	100/99, 225/224, 243/242, 245/242	[⟨41 65 95 115 142]]	+0.375	1.32	4.51
2.3.5.7.11.13	100/99, 105/104, 144/143, 196/195, 243/242	[⟨41 65 95 115 142 152]]	-0.060	1.55	5.29
2.3.5.7.11.13.19	100/99, 105/104, 133/132, 144/143, 171/169, 196/195	[⟨41 65 95 115 142 152 174]]	+0.111	1.49	5.10

41et is lower in relative error than any previous equal temperaments in the 3-, 13- and 19-limit. The next ETs better in these subgroups are 53, 53, and 46, respectively. It is even more prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next ETs better in these subgroups are 72 and 53, respectively.

Commas

41 EDO tempers out the following commas using its patent val, ⟨41 65 95 115 142 152 168 174 185 199 203].

Prime limit	Ratio^[1]	Cents	Monzo	Color name		Name(s)
3	(40 digits)	19.84	[65 -41⟩	Wa-41	41-edo	41-comma
5	(14 digits)	57.27	[-5 -10 9⟩	Tritriyo	y⁹	Shibboleth comma
5	(16 digits)	31.57	[-25 7 6⟩	Lala-tribiyo	LLy³	Ampersand
5	3125/3072	29.61	[-10 -1 5⟩	Laquinyo	Ly⁵	Magic comma
5	(10 digits)	27.66	[5 -9 4⟩	Saquadyo	sy⁴	Tetracot comma
5	(18 digits)	25.71	[20 -17 3⟩	Sasa-triyo	ssy³	Roda
5	(10 digits)	1.95	[-15 8 1⟩	Layo	Ly	Schisma
7	(10 digits)	35.37	[0 -7 6 -1⟩	Rutribiyo	ry⁶	Great BP diesis
7	(18 digits)	22.41	[-10 7 8 -7⟩	Lasepru-aquadbiyo	Lr⁷y⁸	Blackjackisma
7	875/864	21.90	[-5 -3 3 1⟩	Zotriyo	zy³	Keema
7	3125/3087	21.18	[0 -2 5 -3⟩	Triru-aquinyo	r³y⁵	Gariboh comma
7	(12 digits)	19.95	[10 -11 2 1⟩	Sazoyoyo	szyy	Tolerma
7	(10 digits)	16.14	[-15 3 2 2⟩	Labizoyo	Lzzyy	Mirwomo comma
7	245/243	14.19	[0 -5 1 2⟩	Zozoyo	zzy	Sensamagic comma
7	4000/3969	13.47	[5 -4 3 -2⟩	Rurutriyo	rry³	Octagar comma
7	(12 digits)	9.15	[-15 0 -2 7⟩	Lasepzo-agugu	Lz⁷gg	Quince comma
7	1029/1024	8.43	[-10 1 0 3⟩	Latrizo	Lz³	Gamelisma
7	225/224	7.71	[-5 2 2 -1⟩	Ruyoyo	ryy	Marvel comma
7	(10 digits)	6.99	[0 3 4 -5⟩	Quinru-aquadyo	r⁵y⁴	Mirkwai comma
7	(10 digits)	6.48	[5 -7 -1 3⟩	Satrizo-agu	sz³g	Hemimage comma
7	5120/5103	5.76	[10 -6 1 -1⟩	Saruyo	sry	Hemifamity comma
7	(16 digits)	3.80	[25 -14 0 -1⟩	Sasaru	ssr	Garischisma
7	2401/2400	0.72	[-5 -1 -2 4⟩	Bizozogu	z⁴gg	Breedsma
11	(12 digits)	29.72	[15 0 1 0 -5⟩	Saquinlu-ayo	s1u⁵y	Thuja comma
11	245/242	21.33	[-1 0 1 2 -2⟩	Luluzozoyo	1uuzzy	Cassacot
11	100/99	17.40	[2 -2 2 0 -1⟩	Luyoyo	1uyy	Ptolemisma
11	1344/1331	16.83	[6 1 0 1 -3⟩	Trilu-azo	1u³z	Hemimin
11	896/891	9.69	[7 -4 0 1 -1⟩	Saluzo	s1uz	Pentacircle
11	(10 digits)	8.39	[16 0 0 -2 -3⟩	Satrilu-aruru	s1u³rr	Orgonisma
11	243/242	7.14	[-1 5 0 0 -2⟩	Lulu	1uu	Rastma
11	385/384	4.50	[-7 -1 1 1 1⟩	Lozoyo	1ozg	Keenanisma
11	441/440	3.93	[-3 2 -1 2 -1⟩	Luzozogu	1uzzg	Werckisma
11	1375/1372	3.78	[-2 0 3 -3 1⟩	Lotriruyo	1or³y	Moctdel
11	540/539	3.21	[2 3 1 -2 -1⟩	Lururuyo	1urry	Swetisma
11	3025/3024	0.57	[-4 -3 2 -1 2⟩	Loloruyoyo	1ooryy	Lehmerisma
11	(12 digits)	0.15	[-1 2 -4 5 -2⟩	Luluquinzo-aquadgu	1uuz⁵g⁴	Odiheim
13	343/338	25.42	[-1 0 0 3 0 -2⟩	Thuthutrizo	3uuz³
13	105/104	16.57	[-3 1 1 1 0 -1⟩	Thuzoyo	3uzy	Animist
13	(10 digits)	14.61	[12 -7 0 1 0 -1⟩	Sathuzo	s3uz	Secorian
13	275/273	12.64	[0 -1 2 -1 1 -1⟩	Thuloruyoyo	3u1oryy	Gassorma
13	144/143	12.06	[4 2 0 0 -1 -1⟩	Thulu	3u1u	Grossma
13	196/195	8.86	[2 -1 -1 2 0 -1⟩	Thuzozogu	3uzzg	Mynucuma
13	640/637	8.13	[7 0 1 -2 0 -1⟩	Thururuyo	3urry	Huntma
13	1188/1183	7.30	[2 3 0 -1 1 -2⟩	Thuthuloru	3uu1or	Kestrel comma
13	325/324	5.34	[-2 -4 2 0 0 1⟩	Thoyoyo	3oyy	Marveltwin comma
13	352/351	4.93	[5 -3 0 0 1 -1⟩	Thulo	3u1o	Minthma
13	364/363	4.76	[2 -1 0 1 -2 1⟩	Tholuluzo	3o1uuz	Gentle comma
13	847/845	4.09	[0 0 -1 1 2 -2⟩	Thuthulolozogu	3uu1oozg	Cuthbert
13	729/728	2.38	[-3 6 0 -1 0 -1⟩	Lathuru	L3ur	Squbema
13	4096/4095	0.42	[12 -2 -1 -1 0 -1⟩	Sathurugu	s3urg	Schismina
13	(10 digits)	0.16	[3 -2 0 -1 3 -2⟩	Thuthutrilo-aru	3uu1o³r	Harmonisma
17	2187/2176	8.73	[-7 7 0 0 0 0 -1⟩	Lasu	L17u	Septendecimal schisma
17	256/255	6.78	[8 -1 -1 0 0 0 -1⟩	Sugu	17ug	Septendecimal kleisma
17	715/714	2.42	[-1 -1 1 -1 1 1 -1⟩	Sutholoruyo	17u3o1ory	Septendecimal bridge comma
19	210/209	8.26	[1 1 1 1 -1 0 0 -1⟩	Nuluzoyo	19u1uzy	Spleen comma
19	361/360	4.80	[-3 -2 -1 0 0 0 0 2⟩	Nonogu	19oog2	Go comma
19	513/512	3.38	[-9 3 0 0 0 0 0 1⟩	Lano	L19o	Boethius' comma
19	1216/1215	1.42	[6 -5 -1 0 0 0 0 1⟩	Sanogu	s19og	Eratosthenes' comma
23	736/729	16.54	[5 -6 0 0 0 0 0 0 1⟩	Satwetho	s23o	Vicesimotertial comma
29	145/144	11.98	[-4 -2 1 0 0 0 0 0 0 1⟩	Twenoyo	29oy	29th-partial chroma

↑ Ratios with more than 8 digits are presented by placeholders with informative hints

Rank-2 temperaments

Table of temperaments by generator
Degree	Cents	Temperament(s)	Pergen	Some MOS Scales Available
1	29.27	Slendi	(P8, P4/17)	Pathological 38-tone MOS
2	58.54	Hemimiracle Dodecacot	(P8, P5/12)	21-tone MOS
3	87.80	Octacot	(P8, P5/8)	14-tone MOS: 3 3 3 3 3 3 3 3 3 3 3 3 3 2
4	117.07	Miracle	(P8, P5/6)	11-tone MOS: 4 4 4 4 4 4 4 4 4 4 1
5	146.34	BPS / bohpier	(P8, P12/13)	20-tone MOS
6	175.61	Tetracot / bunya / monkey Sesquiquartififths / sesquart	(P8, P5/4)	13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
7	204.88	Baldy Quadrimage	(P8, c³P4/20)	11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8	234.15	Slendric / rodan / guiron	(P8, P5/3)	11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
9	263.41	Septimin	(P8, ccP4/11)	9-tone MOS: 5 4 5 5 4 5 4 5 4
10	292.68	Quasitemp	(P8, c³P4/14)	29-tone MOS
11	321.95	Superkleismic	(P8, ccP4/9)	11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
12	351.22	Hemif / hemififths / salsa Karadeniz	(P8, P5/2)	10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13	380.49	Magic / witchcraft Quanharuk	(P8, P12/5)	10-tone MOS: 2 9 2 2 9 2 2 9 2 2
14	409.76	Hocum hocus	(P8, c³P4/10)	32-tone MOS
15	439.02	Superthird	(P8, c⁶P5/18)	11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16	468.29	Barbad	(P8, c⁷P4/19)	8-tone MOS: 7 2 7 7 2 7 7 2
17	497.56	Helmholtz / garibaldi / cassandra / andromeda Kwai	(P8, P5)	12-tone MOS: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
18	526.83	Trismegistus	(P8, c⁶P5/15)	9-tone MOS: 5 5 3 5 5 5 5 3 5
19	556.10	Alphorn	(P8, c⁷P4/16)	9-tone MOS: 3 3 3 10 3 3 3 3 10
20	585.37	Pluto Merman	(P8, c³P4/7)	Pathological 35-tone MOS

Scales and modes

A list of 41edo modes (MOS and others). See also Kite Guitar Scales and Kite Giedraitis's Categorizations of 41edo Scales.

Harmonic scale

41edo is the first edo to do some justice to Mode 8 of the harmonic series, which Dante Rosati calls the "Diatonic Harmonic Series Scale," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).

Overtones in "Mode 8":	8	9	10	11	12	13	14	15	16
… as JI Ratio from 1/1:	1/1	9/8	5/4	11/8	3/2	13/8	7/4	15/8	2/1
… in cents:	0	203.9	386.3	551.3	702.0	840.5	968.8	1088.3	1200.0
Nearest degree of 41edo:	0	7	13	19	24	29	33	37	41
… in cents:	0	204.9	380.5	556.1	702.4	848.8	965.9	1082.9	1200.0

While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)

7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) – a close match.
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).

The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

Nonoctave temperaments

Taking every third degree of 41edo produces a scale extremely close to 88cET or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered Bohlen-Pierce Scale (or the 13th root of 3). See Relationship between Bohlen-Pierce and octave-ful temperaments, and see this chart:

3 degrees of 41edo near 88cET			overlap	5 degrees of 41edo near BP
41edo	88cET	cents	cents	cents	BP	41edo
0	0		0		0	0
3	1	87.8
				146.3	1	5
6	2	175.6
9	3	263.4
				292.7	2	10
12	4	351.2
15	5		439.0		3	15
18	6	526.8
				585.4	4	20
21	7	614.6
24	8	702.4
				731.7	5	25
27	9	790.2
30	10		878.0		6	30
33	11	965.9
				1024.4	7	35
36	12	1053.7
39	13	1141.5
				1170.7	8	40
[ second octave ]
1	14	29.2
4	15		117.1		9	4
7	16	204.9
				263.4	10	9
10	17	292.7
13	18	380.5
				409.8	11	14
16	19	468.3
19	20		556.1		12	19
22	21	643.9
				702.4	13	24
25	22	731.7
28	23	819.5
				848.8	14	29
31	24	907.3
34	25		995.1		15	34
37	26	1082.9
				1141.5	16	39
40	27	1170.7
[ third octave ]
2	28	58.5
				87.8	17	3
5	29	146.3
8	30		234.1		18	8
11	31	322.0
				380.5	19	13
14	32	409.8
17	33	497.6
				526.8	20	18
20	34	585.3
23	35		673.2		21	23
26	36	761.0
				819.5	22	28
29	37	848.8
32	38	936.6
				965.9	23	33
35	39	1024.4
38	40		1112.2		24	38

Instruments

41-EDO Electric guitar, by Gregory Sanchez.

41-EDO Classical guitar, by Ron Sword.

The Kite Guitar (see also Kite Tuning) is a guitar fretting using every other step of 41-edo, i.e. 41-ED4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41-edo, but the full edo can be found on every pair of adjacent strings.The Kite Tuning makes 41-edo about as playable as 19-edo or 22-edo, although there are certain trade-offs. If the interval between strings is 13\41, 25 of the 41 intervals are in easy reach: vm2, ^m2, vM2, M2, ^M2, vm3, ^m3, vM3, ^M3, P4, ~4, d5, A4, ~5, P5, vm6, ^m6, vM6, ^M6, vm7, m7, ^m7, vM7, ^M7, P8.