41edo

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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. (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	Monzo	Ratio	Cents
odiheim	| -1 2 -4 5 -2 >		0.15
harmonisma	| 3 -2 0 -1 3 -2 >	10648/10647	0.16
tridecimal schisma, Sagittal schismina	| 12 -2 -1 -1 0 -1/1 >	4096/4095	0.42
Lehmerisma	| -4 -3 2 -1 2 >	3025/3024	0.57
Breedsma	| -5 -1 -2 4 >	2401/2400	0.72
Eratosthenes' comma	| 6 -5 -1 0 0 0 0 1 >	1216/1215	1.42
schisma	| -15 8 1 >	32805/32768	1.95
squbema	| -3 6 0 -1 0 -1 >	729/728	2.38
septendecimal bridge comma	| -1 -1 1 -1 1 1 -1 >	715/714	2.42
Swets' comma, swetisma	| 2 3 1 -2 -1 >	540/539	3.21
undevicesimal comma, Boethius' comma	| -9 3 0 0 0 0 0 1 >	513/512	3.38
moctdel	| -2 0 3 -3 1 >	1375/1372	3.78
Beta 2, septimal schisma, garischisma	| 25 -14 0 -1 >		3.80
Werckmeister's undecimal septenarian schisma, werckisma	| -3 2 -1 2 -1 >	441/440	3.93
cuthbert	| 0 0 -1 1 2 -2 >	847/845	4.09
undecimal kleisma, keenanisma	| -7 -1 1 1 1 >	385/384	4.50
gentle comma	| 2 -1 0 1 -2 1 >	364/363	4.76
minthma	| 5 -3 0 0 1 -1 >	352/351	4.93
marveltwin	| -2 -4 2 0 0 1 >	325/324	5.34
Beta 5, Garibaldi comma, hemifamity	| 10 -6 1 -1 >	5120/5103	5.76
hemimage	| 5 -7 -1 3 >	10976/10935	6.48
septendecimal kleisma	| 8 -1 -1 0 0 0 -1 >	256/255	6.78
small BP diesis, mirkwai	| 0 3 4 -5 >	16875/16807	6.99
neutral third comma, rastma	| -1 5 0 0 -2 >	243/242	7.14
kestrel comma	| 2 3 0 -1 1 -2 >	1188/1183	7.30
septimal kleisma, marvel comma	| -5 2 2 -1 >	225/224	7.71
huntma	| 7 0 1 -2 0 -1 >	640/637	8.13
spleen comma	| 1 1 1 1 -1 0 0 -1 >	210/209	8.26
orgonisma	| 16 0 0 -2 -3 >	65536/65219	8.39
gamelan residue, gamelisma	| -10 1 0 3 >	1029/1024	8.43
septendecimal comma	| -7 7 0 0 0 0 -1 >	2187/2176	8.73
mynucuma	| 2 -1 -1 2 0 -1 >	196/195	8.86
quince	| -15 0 -2 7 >		9.15
undecimal semicomma, pentacircle (minthma * gentle)	| 7 -4 0 1 -1 >	896/891	9.69
29th-partial chroma	| -4 -2 1 0 0 0 0 0 0 1 >	145/144	11.98
grossma	| 4 2 0 0 -1 -1 >	144/143	12.06
gassorma	| 0 -1 2 -1 1 -1 >	275/273	12.64
septimal semicomma, octagar	| 5 -4 3 -2 >	4000/3969	13.47
minor BP diesis, sensamagic	| 0 -5 1 2 >	245/243	14.19
secorian	| 12 -7 0 1 0 -1/1 >	28672/28431	14.61
mirwomo comma	| -15 3 2 2 >	33075/32768	16.14
vicesimotertial comma	| 5 -6 0 0 0 0 0 0 1 >	736/729	16.54
small tridecimal comma, animist	| -3 1 1 1 0 -1 >	105/104	16.57
hemimin	| 6 1 0 1 -3 >	1344/1331	16.83
Ptolemy's comma, ptolemisma	| 2 -2 2 0 -1 >	100/99	17.40
'41-tone' comma	| 65 -41 >		19.84
tolerma	| 10 -11 2 1 >		19.95
major BP diesis, gariboh	| 0 -2 5 -3 >	3125/3087	21.18
cassacot	| -1 0 1 2 -2 >	245/242	21.33
keema	| -5 -3 3 1 >	875/864	21.90
blackjackisma	| -10 7 8 -7 >		22.41
roda	| 20 -17 3 >		25.71
minimal diesis, tetracot comma	| 5 -9 4 >	20000/19683	27.66
small diesis, magic comma	| -10 -1 5 >	3125/3072	29.61
thuja comma	| 15 0 1 0 -5 >		29.72
Ampersand's comma	| -25 7 6 >		31.57
great BP diesis	| 0 -7 6 -1 >	15625/15309	35.37
shibboleth	| -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	0.00			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#^^, Ebv	ro	Hemimiracle
3	87.805	93.07	112.39 (70.6416)	21/20, 22/21	vA1, m2	D#v, 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	Evv	ru	Bohlen-Pierce/bohpier
6	175.61	186.15	224.78 (E0.C816)	10/9, 11/10	vM2	Ev	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	Fv	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	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^^	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	F#v	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#	maa	Hocus
15	439.02	465.37	561.95 (231.F3816)	9/7, 32/25	^M3	F#^	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	v4	Gv	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^^	fu
20	585.37	620.49	749.27 (2ED.4516)	7/5	vA4, d5	G#v, 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	vv5	Avv	su
23	673.17	713.56	862.68 (35D.B8816)	22/15, 40/27	v5	Av	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	^5	A^	si
26	760.98	806.63	994.05 (3CE.0C816)	14/9, 25/16	vm6	Bbv	lo
27	790.24	837.66	1011.51 (3F3.83816)	11/7, 128/81	m6	Bb	leh
28	819.51	868.68.	1048.97 (418.FA16)	8/5	^m6	Bb^	le
29	848.78	899.71	1086.44 (43D.716)	18/11, 44/27	~6	Bvv	lu
30	878.05	930.73	1123.9 (463.F716)	5/3	vM6	Bv	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	Cv	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^^	tu
37	1082.93	1147.9	1386.15 (57A.25816)	15/8	vM7	C#v	ti
38	1112.195	1178.93	1423.61 (58F.9C16)	40/21, 21/11	M7	C#	taa
39	1141.46	1209.95	1461.07 (5B5.1416)	48/25, 27/14, 64/33	^M7	C#^	to
40	1170.73	1240.98	1498.54 (5DA.8916)	160/81, 63/32	v8	Dv	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}, 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}, 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 Ebv G	C.vm	C downminor
gu	10:12:15	0-11-24	C Eb^ G	C.^m	C upminor
ilo	18:22:27	0-12-24	C Evv G	C~	C mid
yo	4:5:6	0-13-24	C Ev G	C.v	C downmajor or C dot down
ru	14:18:27	0-15-24	C E^ G	C.^	C upmajor or C dot 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 Avv = Dm(vv5) = "D minor double-down five", or possibly Ddim(^^5)

0-10-23 = D F Av = 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# A#v = 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.

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.

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