41edo

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

Special properties

Zeta peak
Zeta peak integer
Zeta integral

English Wikipedia has an article on:

41 equal temperament

41 equal divisions of the octave (abbreviated 41edo or 41ed2), also called 41-tone equal temperament (41tet) or 41 equal temperament (41et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 41 equal parts of about 29.3 ¢ each. Each step represents a frequency ratio of 2^1/41, or the 41st root of 2.

Theory

41edo 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. Anyway, it is consistent in the 15-odd-limit, or the no-17's 21-odd-limit. In fact, all of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 arguably manifests itself as 21/16 rather than 13/10. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the 11-odd-limit except for 11/10 and 20/11 is represented with more than 10 cents of error in it. Apart from the full 13-limit, it is even more prominent as a 2.3.5.7.11.19.29.31 subgroup temperament for its size, and perhaps the smallest edo with a satisfactory model of the 9-odd-limit, not only because it is the smallest one to tune the 9-odd-limit distinctly consistent, but it is also consistent in it to distance 2. In other words, all intervals in the 9-odd-limit are more in-tune than out of tune.

A step of 41edo is close and consistently mapped to 64/63, the septimal comma.

41edo can be seen as a tuning of the garibaldi temperament^[1]^[2], the magic temperament, the superkleismic temperament and multiple temperaments in the tetracot family. 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 also a great tetracot tuning, and works as an alternative to 34edo, providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting monkey, bunya and octacot simultaneously. All three of these extend to the 11-limit by way of interpreting the flat 10/9 as an 11/10 by tempering out 100/99. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of tetracot[7] can also be more accurately interpreted as 21/19 – which is equated with 32/29 above 31/28 below (both very near) — providing an explanation of the accuracy of primes 29 and 31 so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.

41et is used by the Kite Guitar, see below in #Instruments.

Prime harmonics

Approximation of prime harmonics in 41edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+0.5	-5.8	-3.0	+4.8	+8.3	+12.1	-4.8	-13.6	-5.2	-3.6
Error	Relative (%)	+0.0	+1.7	-19.9	-10.2	+16.3	+28.2	+41.4	-16.5	-46.6	-17.7	-12.2
Steps (reduced)		41 (0)	65 (24)	95 (13)	115 (33)	142 (19)	152 (29)	168 (4)	174 (10)	185 (21)	199 (35)	203 (39)

Subsets and supersets

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

205edo, which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the H-System, which uses the scale degrees of 41edo as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41edo circle in 205edo.

Intervals

See also: 41edo solfege

#	Cents	Approximate Ratios*	Ups and Downs Notation			SKULO notation (K or S = 1, U = 2)			Kite's Solfege	Andrew's Solfege
0	0.00	1/1	perfect unison	P1	D	perfect unison	P1	D	Da	Do
1	29.27	81/80, 64/63, 49/48	up-unison	^1	^D	comma-wide unison, super unison	K1/S1	KD, SD	Du	Di
2	58.54	25/24, 28/27, 36/35, 33/32	dup-unison, downminor 2nd	^^1, vm2	^^D, vEb	subminor 2nd, classic aug unison, uber unison	sm2, kkA1, U1	sEb, kkD#, UD	Fro	Ro
3	87.80	21/20, 22/21, 19/18, 20/19	down-aug 1sn, minor 2nd	vA1, m2	vD#, Eb	minor 2nd, comma-narrow augmented unison	m2, kA1	Eb, kD#	Fra	Rih
4	117.07	16/15, 15/14, 14/13	augmented 1sn, upminor 2nd	A1, ^m2	D#, ^Eb	classic minor 2nd, augmented unison	Km2, A1	KEb, D#	Fru	Ra
5	146.34	12/11, 13/12	mid 2nd	~2	^D#, vvE	neutral second, super augmented unison	N2, SA1	UEb/uE, sD#	Ri	Ru
6	175.61	10/9, 11/10, 21/19	downmajor 2nd	vM2	vE	classic/comma-wide major 2nd	kM2	kE	Ro	Reh
7	204.88	9/8	major 2nd	M2	E	major 2nd	M2	E	Ra	Re
8	234.15	8/7, 15/13	upmajor 2nd	^M2	^E	supermajor 2nd	SM2	SE	Ru	Ri
9	263.41	7/6, 22/19	downminor 3rd	vm3	vF	subminor 3rd	sm3	sF	No	Ma
10	292.68	32/27, 13/11, 19/16	minor 3rd	m3	F	minor 3rd	m3	F	Na	Meh
11	321.95	6/5	upminor 3rd	^m3	^F	classicminor 3rd	Km3	KF	Nu	Me
12	351.22	11/9, 27/22, 16/13	mid 3rd	~3	^^F, vGb	neutral 3rd, sub diminished 4th	N3, sd4	UF/uF#, sGb	Mi	Mu
13	380.49	5/4, 26/21	downmajor 3rd	vM3	vF#, Gb	classic major 3rd, diminished 4th	kM3, d4	kF#, Gb	Mo	Mi
14	409.76	81/64, 14/11, 24/19, 19/15	major 3rd	M3	F#, ^Gb	major 3rd, comma-wide diminished 4th	M3, Kd4	F#, KGb	Ma	Maa
15	439.02	9/7, 32/25	upmajor 3rd	^M3	^F#, vvG	supermajor 3rd, classic diminished 4th	SM3, KKd4	SF#, KKGb	Mu	Mo
16	468.29	21/16, 13/10	down-4th	v4	vG	sub 4th	s4	sG	Fo	Fe
17	497.56	4/3	perfect 4th	P4	G	perfect 4th	P4	G	Fa	Fa
18	526.83	27/20, 15/11, 19/14	up-4th	^4	^G	comma-wide 4th	K4	KG	Fu	Fih
19	556.10	11/8, 18/13, 26/19	mid-4th, downdim 5th	~4, vd5	^^G, vAb	uber/neutral 4th, classic augmented 4th	U4/N4, kkA4	UG, kkG#	Fi/Sho	Fu
20	585.37	7/5, 45/32	downaug 4th, dim 5th	vA4, d5	vG#, Ab	comma-narrow augmented 4th, diminished 5th	kA4/d5	kG#, Ab	Po/Sha	Fi
21	614.63	10/7, 64/45	aug 4th, updim 5th	A4, ^d5	G#, ^Ab	augmented 4th, comma-wide diminished 5th	A4/Kd5	G#, KAb	Pa/Shu	Se
22	643.90	16/11, 13/9, 19/13	mid-5th, upaug 4th	~5, ^A4	^G#, vvA	unter/neutral 5th, classic diminished 5th	u5/N5, KKd5	uA, KKAb	Pu/Si	Su
23	673.17	40/27, 22/15, 28/19	down-5th	v5	vA	comma-narrow 5th	k5	kA	So	Sih
24	702.44	3/2	perfect 5th	P5	A	perfect 5th	P5	A	Sa	Sol
25	731.71	32/21, 20/13	up-5th	^5	^A	super 5th	S5	SA	Su	Si
26	760.98	14/9, 25/16	downminor 6th	vm6	^^A, vBb	subminor 6th, classic augmented 5th	sm6	sBb, kkA#	Flo	Lo
27	790.24	128/81, 11/7, 19/12, 30/19	minor 6th	m6	vA#, Bb	minor 6th, comma-narrow augmented 5th	m6	Bb, kA#	Fla	Leh
28	819.51	8/5, 21/13	upminor 6th	^m6	A#, ^Bb	classic minor 6th, augmented 5th	Km6, A5	KBb, A#	Flu	Le
29	848.78	18/11, 44/27, 13/8	mid 6th	~6	^A#, vvB	neutral 6th, super augmented 5th	N6	UBb/uB, sA#	Li	Lu
30	878.05	5/3	downmajor 6th	vM6	vB	classic major 6th	kM6	kB	Lo	La
31	907.32	27/16, 22/13, 32/19	major 6th	M6	B	major 6th	M6	B	La	Laa
32	936.59	12/7, 19/11	upmajor 6th	^M6	^B	supermajor 6th	SM6	SB	Lu	Li
33	965.85	7/4, 26/15	downminor 7th	vm7	vC	subminor 7th	sm7	sC	Tho	Ta
34	995.12	16/9	minor 7th	m7	C	minor 7th	m7	C	Tha	Teh
35	1024.39	9/5, 20/11, 38/21	upminor 7th	^m7	^C	classic/comma-wide minor seventh	Km7	KC	Thu	Te
36	1053.66	11/6, 24/13	mid 7th	~7	^^C, vDb	neutral 7th, sub diminished 8ve	N7	UC/uC#, sDb	Ti	Tu
37	1082.93	15/8, 28/15, 13/7	downmajor 7th	vM7	vC#, Db	classic major 7th, diminished 8ve	kM7, d8	kC#, Db	To	Ti
38	1112.20	40/21, 21/11, 36/19, 19/10	major 7th	M7	C#, ^Db	major 7th, comma-wide diminished 8ve	M7, Kd8	C#, KDb	Ta	Taa
39	1141.46	48/25, 27/14, 35/18, 64/33	upmajor 7th	^M7	C#^, vvD	supermajor 7th, classic dim 8ve, unter 8ve	SM7, KKd8, U8	SC#, KKDb, u8	Tu	To
40	1170.73	160/81, 63/32, 96/49	dim 8ve	v8	vD	comma-narrow 8ve, sub 8ve	k8/s8	kD, sD	Do	Da
41	1200.00	2/1	perfect 8ve	P8	D	perfect 8ve	P8	D	Da	Do

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

Interval quality and chord names in color notation

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

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

Other common triads are

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 a special case of Kite's color notation, treating 41edo as a temperament of the 2,3,7 subgroup. 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 , A , B♭ , B♭ , A♯ , A♯ , B , B , B , C , C , C , D♭ , D♭ , C♯ , C♯ , D , D , D , E♭ , E♭ , D♯ , D♯ , E , E , E , F , F , F , G♭ , G♭ , F♯ , F♯ , G , G , G , A♭ , A♭ , G♯ , G♯ , 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:

Ups and downs notation

41edo can also be notated with quarter-tone accidentals and ups and downs. This can be done by combining sharps and flats with arrows borrowed from extended Helmholtz-Ellis notation:

Step Offset	0	1	2	3	4	5	6	7	8	9
Sharp Symbol
Flat Symbol

The notes within an octave from A are thus:

A, B , A , B♭, A♯, B , A , B, C , B , C, D , C , D♭, C♯, D , C , D, E , D , E♭, D♯, E , D , E, F , E , F, G , F , G♭, F♯, G , F , G, A , G , A♭, G♯, A , G , A

Approximation to JI

Interval mappings

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

As 41edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 41edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
3/2, 4/3	0.484	1.7
9/8, 16/9	0.968	3.3
15/14, 28/15	2.370	8.1
7/5, 10/7	2.854	9.8
7/4, 8/7	2.972	10.2
7/6, 12/7	3.456	11.8
13/11, 22/13	3.473	11.9
11/9, 18/11	3.812	13.0
9/7, 14/9	3.940	13.5
11/6, 12/11	4.296	14.7
11/8, 16/11	4.780	16.3
15/8, 16/15	5.342	18.3
5/4, 8/5	5.826	19.9
5/3, 6/5	6.310	21.6
9/5, 10/9	6.794	23.2
13/9, 18/13	7.285	24.9
11/7, 14/11	7.752	26.5
13/12, 24/13	7.769	26.5
13/8, 16/13	8.253	28.2
15/11, 22/15	10.122	34.6
11/10, 20/11	10.606	36.2
13/7, 14/13	11.225	38.4
15/13, 26/15	13.595	46.4
13/10, 20/13	14.079	48.1

Relationship to 12-edo

Whereas 12edo has a circle of twelve 5ths, 41edo has a spiral of twelve 5ths, since 24\41 is on the 7\12 kite in the scale tree. (See Chapter 5.7 of Kite's book for an explanation of kites.) This spiral of 5th shows 41edo in a 12edo-friendly format. Excellent for introducing 41edo 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 doing 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 doing better in these subgroups are 72 and 53, respectively.

Commas

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

Prime Limit	Ratio^[3]	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 comma
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	31213/31104	6.06	[-7 -5 0 4 0 1⟩	Thoquadzo	3oz⁴3	Praveensma
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	2080/2079	0.83	[5 -3 1 -1 -1 1⟩	Tholuruyo	3o1ury	Ibnsinma
13	4096/4095	0.42	[12 -2 -1 -1 0 -1⟩	Sathurugu	s3urg	Schismina
13	6656/6655	0.26	[9 0 -1 0 -3 1⟩	Thotrilo-agu	3u1o³g2	Jacobin comma
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

Rank-2 temperaments

Table of temperaments by generator
Degree	Cents	Temperament(s)	Pergen	MOS Scales
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

Main article: List of MOS scales in 41edo

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

MOS scales

See List of MOS scales in 41edo

Instruments

Guitars

The first 41edo guitar was probably this one, built by Erv Wilson in the 1960's:

Note the new bridge, several inches below the original bridge. The new bridge increases the scale length and spreads the frets out, making the guitar more playable. Erv numbered the frets as seen here, with the 3-limit dorian scale in enlarged numbers.

Several more modern guitars:

Melle Weijters' 10-string guitar (Melleweijters.com)

41edo electric guitar, by Gregory Sanchez.

41edo classical guitar, by Ron Sword.

The Kite Guitar (see also KiteGuitar.com and Kite Tuning) is a guitar fretting using every other step of 41edo, i.e. 41ed4 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 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.

For more photos of Kite guitars, see Kite Guitar Photographs.

Keyboards

A possible 41edo keyboard design:

See also 41-edo Keyboards for Lumatone, Linnstrument and Harpejji options, as well as DIY options.

41edo as a Universal Tuning

41's claim to fame as a "universal tuning" is the fact that it approximates scales present in many important world music traditions, and thus is good for both combining and exploring cultural playstyles. It makes no claim to perfectly and faithfully represent the musical cultures listed, as doing so would require far more notes and small details than are present in 41. That being said, it has certain attributes that allow it to approximate common scales in these cultures with far more accuracy than most comparable EDOs.

Western

Due to 41edo's extremely accurate perfect fifth, it makes a good tuning for schismatic temperament and the 12-note MOS, which in turn is a good approximation of the standard 12edo scale, and when arranged as a Bbb-D gamut, approximates the 12-note roughly Pythagorean tuning known as Kirnberger I. This extends the Ptolemy Diatonic Scale (7 6 4 7 6 7 4), which 41 approximates excellently, by completing the circle of fifths with pure 3/2s. By using this system and occasionally substituting in alternate major seconds and sixths when necessary, it becomes quite reminiscent of (and can improve on) 12edo harmony. Additionally, the Pythagorean Pentatonic scale can be used for melodies overtop due to the strong quartal nature of the scale. The Pythagorean diatonic scale exists as an option as well, but use may be limited unless Gentle triads are ideal. An alternate option is approximating a Just Intonation scale such as the Asymmetric scale, a common option for a 5-limit JI scale, or Centaur, a 7-limit JI scale using "blue" or subminor intervals for the accidental notes. There exist other options for 5-limit JI scales, all of which have some reasonable approximation in 41 due to its relative excellence in the 5-limit.

Middle Eastern

See also: Arabic, Turkish, Persian

While the Hemif[7] scale itself and MODMOSes related to it give the middle eastern sound well, 41 has other interesting properties that make it an ideal system for Arabic and Turkish music. It is considered a "Level 2 EDO" due to the fact that it has neutral seconds and thirds as well as submajor and supraminor ones added to a Pythagorean skeleton, with small semitones as minor seconds and major whole tones as major seconds. The submajor third is great for Turkish Rast, around Ozan Yarman’s ideal size, and is sharp enough to sound close to a 5/4, while the neutral third exists as half of a 3/2 and works well for Arabic Rast and some Persian scales. Additionally, a large apotome exists for the Hijaz maqam.

Indonesian

Gamelan music is mainly based on two scales, the older Slendro and newer Pelog, though these scales are expanded on extensively through octave stretching, extensions and combination of the scales, and more. Slendro is excellently approximated by the 8\41 generator. Pelog is approximated quite well also, this time by mavila temperament, using the "grave" fifth of 41 as the generator (23\41).

Indian

See also: Magic22 as srutis

Carnatic music, which is normally based on a 22-note unequal scale, has found some use from 22edo as a good approximation, but 41 offers another option with Magic[22], which not only represents 22edo closely, but preserves accurate perfects fifths and the unequal quality of a more typical carnatic scale. Like any EDO system with an accurate 5-limit and essentially pure fifth, 41 can also approximate a system of Just Shrutis.

Japanese

Japanese classical music known as Gagaku is largely built around winds, strings, and percussion, and the melodies, like many Asian cultures, are built around Pythagorean pentatonic scales, alongside chromaticism with narrow semitones, which are well represented by Pythagorean limmas.

Blues

Due to its pure sounding major thirds and approximations of standard western harmony, 41 naturally is good for jazz and blues music, though a great strength of this system as opposed to many others is its excellent harmonic seventh, alongside MOS scales that supply them, Magic and Miracle in particular.

Coltrane changes can be represented with two Pythagorean major thirds and a pental one, or a temperament like Magic, whose MOSes are characterized by circles of major thirds, giving options for rotating major and minor triads within one scale. Similarly, the Whole Tone scale is represented by Baldy[6], with two pental major thirds and four Pythagorean. This scale can be extended to an 11-note MOS, including a single 4:5:7:9:11 chord and numerous subsets.

Superkleismic presents another option for this purpose, featuring circles of minor thirds, and generating harmonic sevenths with very low complexity.

Blue notes, rather than being considered inflections, can be notated as accidentals instead, such as the "blue third" which is represented by a neutral third, or any number of septimal intervals that are useful in a blues context.

Other

Georgian Polyphonic singing can be done in a 41edo context due to its excellent approximations of prime harmonics and neutral third, as well as Pythagorean seconds and sevenths. Asian musical traditions built around pentatonic scales can use both Pythagorean and Barbad[5].

Music

Modern renderings

Johann Sebastian Bach

"Contrapunctus 4" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)
"Contrapunctus 11" from The Art of Fugue, BWV 1080 (1742–1749) – rendered by Claudi Meneghin (2024)

Nicolaus Bruhns

Prelude in E Minor "The Great" – rendered by Claudi Meneghin (2023)
Prelude in E Minor "The Little" – rendered by Claudi Meneghin (2024)

Scott Joplin

Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)

20th century

Joseph Monzo

Theme from Invisible Haircut (1990)

21st century

Abnormality

FUZZ (2024)

Beheld

Subsidence vibe (2024)

Cameron Bobro

Evening Horizon ^{[dead link]} play

Flora Canou

Notes of the Generation (2023) – an 8-piece album in 41et

"Chaotic Witch #1" · "Party Cubes" · "Big Dreamer Pavilion" · "Lost Cyclops" · "Sky Tree" · "Long Night Ahead" · "Fractocraft" · "After the Generation"

Francium

Tetracotta (2024) – tetracot[13] in 41edo tuning
harmon (2024)

Jake Freivald

Little Magical Object – magic[19] in 41edo tuning

Ray Perlner

Tapeworm Saga

Preludium, for microtonal video game ensemble (2023)
Spring's Arrival, for synth septet (2024)

Chris Vaisvil (site)

The Magic of Belief (2013) – magic[19] in 41edo tuning

Xeno Ov Eleas

A Treasure Lost and Must Be Found (2022)

Kite Guitar Recordings

Kite Giedraitis

Evening Rondo
Downminor Etude (midi demo)

Igliashon Jones

Pixel Archipelago

Intervallic Prism (2020) – a 7-track album

"Red" · "Orange" · "Yellow" · "Green" · "Blue" · "Indigo" · "Violet"

Aaron Wolf

12-Bar-Blues on Kite Guitar – a simple 12-bar blues
Fourthward Lang Syne – an arrangement of Auld Lang Syne

Kite Guitar Videos

Timmy Barnett

Downminor Etude ^{[dead link]}

Wilckerson Ganda

Vintage Rock

Travis Johnson

Evening Rondo

Kite Guitar Scores

Kite Guitar originals
Kite Guitar translations

External links

Tetracontamonophonic Scales for Guitar by Ron Sword
Intervals, Scales and Chords in 41EDO by Cam Taylor – a work in progress using just intonation concepts and simplified Sagittal notation.

Notes

↑ Schismic Temperaments at x31eq.com, the website of Graham Breed
↑ Lattices with Decimal Notation at x31eq.com
↑ Ratios with more than 8 digits are presented by placeholders with informative hints

[1] Schismic Temperaments at x31eq.com, the website of Graham Breed

[2] Lattices with Decimal Notation at x31eq.com

[3] Ratios with more than 8 digits are presented by placeholders with informative hints

[1]

[2]

[3]