41edo

Deutsch

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Introduction

The 41-tET, 41-EDO, 41-ET, or 41-Tone Equal Temperament is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 cents, an interval close in size to 64/63, the septimal comma. 41-ET can be seen as a tuning of the Garibaldi temperament [1] , [2] , [3] the Magic temperament [4] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after 29edo) whose perfect fifth is closer to just intonation than that of 12-ET, 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-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).

41-ET forms the foundation of the H-System, which uses the scale degrees of 41-ET as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41-ET circle in 205edo.

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

Commas

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

Name	color name		Monzo	Ratio	Cents
odiheim	Luluquinzo-aquadgu	1uuz5g4	| -1 2 -4 5 -2 >		0.15
harmonisma	Thuthutrilo-aru	3uu1o3r	| 3 -2 0 -1 3 -2 >	10648/10647	0.16
tridecimal schisma, Sagittal schismina	Sathurugu	s3urg	| 12 -2 -1 -1 0 -1 >	4096/4095	0.42
Lehmerisma	Loloruyoyo	1ooryy	| -4 -3 2 -1 2 >	3025/3024	0.57
Breedsma	Bizozogu	z4gg	| -5 -1 -2 4 >	2401/2400	0.72
Eratosthenes' comma	Sanogu	s19og	| 6 -5 -1 0 0 0 0 1 >	1216/1215	1.42
schisma	Layo	Ly	| -15 8 1 >	32805/32768	1.95
squbema	Lathuru	L3ur	| -3 6 0 -1 0 -1 >	729/728	2.38
septendecimal bridge comma	Sutholoruyo	17u3o1ory	| -1 -1 1 -1 1 1 -1 >	715/714	2.42
Swets' comma, swetisma	Lururuyo	1urry	| 2 3 1 -2 -1 >	540/539	3.21
undevicesimal comma, Boethius' comma	Lano	L19o	| -9 3 0 0 0 0 0 1 >	513/512	3.38
moctdel	Lotriruyo	1or3y	| -2 0 3 -3 1 >	1375/1372	3.78
Beta 2, septimal schisma, garischisma	Sasaru	ssr	| 25 -14 0 -1 >		3.80
Werckmeister's undecimal septenarian schisma, werckisma	Luzozogu	1uzzg	| -3 2 -1 2 -1 >	441/440	3.93
cuthbert	Thuthulolozogu	3uu1oozg	| 0 0 -1 1 2 -2 >	847/845	4.09
undecimal kleisma, keenanisma	Lozoyo	1ozg	| -7 -1 1 1 1 >	385/384	4.50
gentle comma	Tholuluzo	3o1uuz	| 2 -1 0 1 -2 1 >	364/363	4.76
minthma	Thulo	3u1o	| 5 -3 0 0 1 -1 >	352/351	4.93
marveltwin	Thoyoyo	3oyy	| -2 -4 2 0 0 1 >	325/324	5.34
Beta 5, Garibaldi comma, hemifamity	Saruyo	sry	| 10 -6 1 -1 >	5120/5103	5.76
hemimage	Satrizo-agu	sz3g	| 5 -7 -1 3 >	10976/10935	6.48
septendecimal kleisma	Sugu	17ug	| 8 -1 -1 0 0 0 -1 >	256/255	6.78
small BP diesis, mirkwai	Quinru-aquadyo	r5y4	| 0 3 4 -5 >	16875/16807	6.99
neutral third comma, rastma	Lulu	1uu	| -1 5 0 0 -2 >	243/242	7.14
kestrel comma	Thuthuloru	3uu1or	| 2 3 0 -1 1 -2 >	1188/1183	7.30
septimal kleisma, marvel comma	Ruyoyo	ryy	| -5 2 2 -1 >	225/224	7.71
huntma	Thururuyo	3urry	| 7 0 1 -2 0 -1 >	640/637	8.13
spleen comma	Nuluzoyo	19u1uzy	| 1 1 1 1 -1 0 0 -1 >	210/209	8.26
orgonisma	Satrilu-aruru	s1u3rr	| 16 0 0 -2 -3 >	65536/65219	8.39
gamelan residue, gamelisma	Latrizo	Lz3	| -10 1 0 3 >	1029/1024	8.43
septendecimal comma	Lasu	L17u	| -7 7 0 0 0 0 -1 >	2187/2176	8.73
mynucuma	Thuzozogu	3uzzg	| 2 -1 -1 2 0 -1 >	196/195	8.86
quince	Lasepzo-agugu	Lz7gg	| -15 0 -2 7 >		9.15
undecimal semicomma, pentacircle (minthma * gentle)	Saluzo	s1uz	| 7 -4 0 1 -1 >	896/891	9.69
29th-partial chroma	Twenty-noyo	29oy	| -4 -2 1 0 0 0 0 0 0 1 >	145/144	11.98
grossma	Thulu	3u1u	| 4 2 0 0 -1 -1 >	144/143	12.06
gassorma	Thuloruyoyo	3u1oryy	| 0 -1 2 -1 1 -1 >	275/273	12.64
septimal semicomma, octagar	Rurutriyo	rry3	| 5 -4 3 -2 >	4000/3969	13.47
minor BP diesis, sensamagic	Zozoyo	zzy	| 0 -5 1 2 >	245/243	14.19
secorian	Sathuzo	s3uz	| 12 -7 0 1 0 -1 >	28672/28431	14.61
mirwomo comma	Labizoyo	Lzzyy	| -15 3 2 2 >	33075/32768	16.14
vicesimotertial comma	Sa-twenty-tho	s23o	| 5 -6 0 0 0 0 0 0 1 >	736/729	16.54
small tridecimal comma, animist	Thuzoyo	3uzy	| -3 1 1 1 0 -1 >	105/104	16.57
hemimin	Trilu-azo	1u3z	| 6 1 0 1 -3 >	1344/1331	16.83
Ptolemy's comma, ptolemisma	Luyoyo	1uyy	| 2 -2 2 0 -1 >	100/99	17.40
'41-tone' comma	Wa-41	41-edo	| 65 -41 >		19.84
tolerma	Sazoyoyo	szyy	| 10 -11 2 1 >		19.95
major BP diesis, gariboh	Triru-aquinyo	r3y5	| 0 -2 5 -3 >	3125/3087	21.18
cassacot	Luluzozoyo	1uuzzy	| -1 0 1 2 -2 >	245/242	21.33
keema	Zotriyo	zy3	| -5 -3 3 1 >	875/864	21.90
blackjackisma	Lasepru-aquadbiyo	Lr7y8	| -10 7 8 -7 >		22.41
roda	Sasa-triyo	ssy3	| 20 -17 3 >		25.71
minimal diesis, tetracot comma	Saquadyo	sy4	| 5 -9 4 >	20000/19683	27.66
small diesis, magic comma	Laquinyo	Ly5	| -10 -1 5 >	3125/3072	29.61
thuja comma	Saquinlu-ayo	s1u5y	| 15 0 1 0 -5 >		29.72
Ampersand's comma	Lala-tribiyo	LLy3	| -25 7 6 >		31.57
great BP diesis	Rutribiyo	ry6	| 0 -7 6 -1 >	15625/15309	35.37
shibboleth	Tritriyo	y9	| -5 -10 9 >		57.27

Temperaments

List of edo-distinct 41et rank two temperaments

Intervals

	Cents Value	pions	7mus	Approximate Ratios in the 11-limit	Ups and Downs Notation		Andrew's Solfege Syllables	Generator	Some MOS and MODMOS Scales Available
0				1/1	P1	D	do
1	29.27	31.02	37.46 (25.7716)	81/80, 64/63	^1	^D	di
2	58.54	62.05	74.93 (4A.EB16)	25/24, 28/27, 33/32	^^1, vm2	^^D, vEb	ro	Hemimiracle
3	87.805	93.07	112.39 (70.6416)	21/20, 22/21	vA1, m2	vD#, Eb	rih	88cET (approx), octacot
4	117.07	124.1	149.85 (95.DA816)	16/15, 15/14	A1, m2	D#, ^Eb	ra	Miracle
5	146.34	155.12	187.32 (BB.5116)	12/11	~2	^D#, vvE	ru	Bohlen-Pierce/bohpier
6	175.61	186.15	224.78 (E0.C816)	10/9, 11/10	vM2	vE	reh	Tetracot/bunya/monkey	13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
7	204.88	217.17	262.24 (106.3E16)	9/8	M2	E	re	Baldy	11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8	234.15	248.195	299.71 (12B.B516)	8/7	^M2	^E	ri	Rodan/guiron	11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
9	263.415	279.22	337.17 (151.2C16)	7/6	vm3	vF	ma	Septimin	9-tone MOS: 5 4 5 5 4 5 4 5 4
10	292.68	310.24	374.63 (176.A216)	32/27, 13/11	m3	F	meh	Quasitemp
11	321.95	341.27	412.1 (1A0.1916)	6/5	^m3	^F	me	Superkleismic	11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
12	351.22	372.29	449.56 (1C5.916)	11/9, 27/22	~3	^^F, vGb	mu	Hemififths/karadeniz	10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13	380.49	403.32	487.02 (1EB.0616)	5/4	vM3	vF#, Gb	mi	Magic/witchcraft	10-tone MOS: 2 9 2 2 9 2 2 9 2 2
14	409.76	434.34	524.49 (20C.7C816)	14/11, 81/64	M3	F#, ^Gb	maa	Hocus
15	439.02	465.37	561.95 (231.F3816)	9/7, 32/25	^M3	^F#, vvG	mo		11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16	468.29	496.39	599.415 (257.6A16)	21/16, 13/10	v4	vG	fe	Barbad
17	497.56	527.415	636.88 (27C.E116)	4/3	P4	G	fa	Schismatic (helmholtz, garibaldi, cassandra)
18	526.83	558.44	674.34 (2A2.57816)	15/11, 27/20	^4	^G	fih	Trismegistus	9-tone MOS: 5 5 3 5 5 5 5 3 5
19	556.1	589.46	711.805 (2C7.CE16)	11/8	~4	^^G, vAb	fu
20	585.37	620.49	749.27 (2ED.4516)	7/5	vA4, d5	vG#, Ab	fi	Pluto
21	614.63	651.51	786.73 (312.BB16)	10/7	A4, ^d5	G#, ^Ab	se
22	643.90	683.54	825.195 (347.3216)	16/11, 13/9	~5	vvA	su
23	673.17	713.56	862.68 (35D.B8816)	22/15, 40/27	v5	vA	sih
24	702.44	744.585	899.12 (383.1F16)	3/2	P5	A	sol
25	731.71	775.61	936.585 (3A8.9616)	32/21, 20/13	^5	^A	si
26	760.98	806.63	994.05 (3CE.0C816)	14/9, 25/16	vm6	^^A, vBb	lo
27	790.24	837.66	1011.51 (3F3.83816)	11/7, 128/81	m6	vA#, Bb	leh
28	819.51	868.68.	1048.97 (418.FA16)	8/5	^m6	A#, ^Bb	le
29	848.78	899.71	1086.44 (43D.716)	18/11, 44/27	~6	^A#, vvB	lu
30	878.05	930.73	1123.9 (463.F716)	5/3	vM6	vB	la
31	907.32	961.76	1161.37 (489.5E16)	27/16	M6	B	laa
32	936.59	992.78	1198.83 (4AE.D416)	12/7	^M6	^B	li
33	965.85	1023.805	1236.29 (4D4.4B16)	7/4	vm7	vC	ta
34	995.12	1064.83	1273.76 (4F9.C216,)	16/9	m7	C	teh
35	1024.39	1085.85	1311.22 (51F.3816)	9/5, 20/11	^m7	^C	te
36	1053.66	1116.88	1348.32 (544.AF16)	11/6	~7	^^C, vDb	tu
37	1082.93	1147.9	1386.15 (57A.25816)	15/8	vM7	vC#, Db	ti
38	1112.195	1178.93	1423.61 (58F.9C16)	40/21, 21/11	M7	C#, ^Db	taa
39	1141.46	1209.95	1461.07 (5B5.1416)	48/25, 27/14, 64/33	^M7	C#^, vvD	to
40	1170.73	1240.98	1498.54 (5DA.8916)	160/81, 63/32	v8	vD	da
41	1200	1272	1536 (60016)	2/1	P8	D	do

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. 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	6:7:9	0-9-24	C vEb G	Cvm	C downminor
gu	10:12:15	0-11-24	C ^Eb G	C^m	C upminor
ilo	18:22:27	0-12-24	C vvE G	C~	C mid
yo	4:5:6	0-13-24	C vE G	Cv	C downmajor or C down
ru	14:18:27	0-15-24	C ^E G	C^	C upmajor or C up

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

0-10-21 = D F ^Ab = Ddim(^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 double-up-five, or possibly Daug(vv5)

0-14-27 = D F# vA# = Daug(v5) = D aug down-five

0-14-28 = D F# A# is Daug = D aug

etc.

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

This chart shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with xen. 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.

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 41edo (ordered by absolute error).

Interval, complement	Error (abs., in cents)
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

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.

A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.

Scales and modes

A list of 41edo modes (MOS and others).

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 chart:

3 degrees of 41edo (near 88cET)			overlap	5 degrees of 41edo (near BP)
deg of 41edo	deg of 88cET	cents	cents	cents	deg of BP	deg of 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

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.

Music

EveningHorizon play by Cameron Bobro

Links

• Wikipedia article on 41edo
• Magic22 as srutis describes a possible use of 41edo for indian music.
• see also Magic family
• Sword, Ron. "Tetracontamonophonic Scales for Guitar"
• Taylor, Cam. Intervals, Scales and Chords in 41EDO, a work in progress using just intonation concepts and simplified Sagittal notation.

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1. ^ "Schismic Temperaments" at x31eq.com the website of Graham Breed
2. ^ "Lattices with Decimal Notation" at x31eq.com
3. ^ Schismatic temperament
4. ^ Magic temperament