Revision as of 12:25, 13 October 2019

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.C2_16,)	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:

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

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

41edo: Difference between revisions

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Revision as of 12:25, 13 October 2019

Contents

Introduction

Commas

Temperaments

Intervals

Selected just intervals by error

Instruments

Scales and modes

Harmonic Scale

Nonoctave Temperaments

Notation

Music

Links

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 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 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&amp;26) temperament. It is the second smallest equal temperament (after [[29edo|29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|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.
-edo 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|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).
+edo 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|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).
 -ET forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41-ET as the basic [[13-limit|13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41-ET circle in [[205edo|205edo]].
@@ Line 19: / Line 19: @@
 |-
 ! | Name
+! colspan="2" |[[Color notation/Temperament Names|color name]]
 ! | Monzo
 ! | Ratio
@@ Line 24: / Line 25: @@
 |-
 | | odiheim
+|Luluquinzo-aquadgu
+|1uuz<sup>5</sup>g<sup>4</sup>
 | |<nowiki> | -1 2 -4 5 -2 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 29: / Line 32: @@
 |-
 | | harmonisma
+|Thuthutrilo-aru
+|3uu1o<sup>3</sup>r
 | |<nowiki> | 3 -2 0 -1 3 -2 </nowiki>&gt;
 | style="text-align:center;" | 10648/10647
@@ Line 34: / Line 39: @@
 |-
 | | tridecimal schisma, Sagittal schismina
-| |<nowiki> | 12 -2 -1 -1 0 -1/1 </nowiki>&gt;
+|Sathurugu
+|s3urg
+| |<nowiki> | 12 -2 -1 -1 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 4096/4095
 | | 0.42
 |-
 | | Lehmerisma
+|Loloruyoyo
+|1ooryy
 | |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
 | style="text-align:center;" | 3025/3024
@@ Line 44: / Line 53: @@
 |-
 | | Breedsma
+|Bizozogu
+|z<sup>4</sup>gg
 | |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
 | style="text-align:center;" | 2401/2400
@@ Line 49: / Line 60: @@
 |-
 | | Eratosthenes' comma
+|Sanogu
+|s19og
 | |<nowiki> | 6 -5 -1 0 0 0 0 1 </nowiki>&gt;
 | style="text-align:center;" | 1216/1215
@@ Line 54: / Line 67: @@
 |-
 | | schisma
+|Layo
+|Ly
 | |<nowiki> | -15 8 1 </nowiki>&gt;
 | style="text-align:center;" | 32805/32768
@@ Line 59: / Line 74: @@
 |-
 | | squbema
+|Lathuru
+|L3ur
 | |<nowiki> | -3 6 0 -1 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 729/728
@@ Line 64: / Line 81: @@
 |-
 | | septendecimal bridge comma
+|Sutholoruyo
+|17u3o1ory
 | |<nowiki> | -1 -1 1 -1 1 1 -1 </nowiki>&gt;
 | style="text-align:center;" | 715/714
@@ Line 69: / Line 88: @@
 |-
 | | Swets' comma, swetisma
+|Lururuyo
+|1urry
 | |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
 | style="text-align:center;" | 540/539
@@ Line 74: / Line 95: @@
 |-
 | | undevicesimal comma, Boethius' comma
+|Lano
+|L19o
 | |<nowiki> | -9 3 0 0 0 0 0 1 </nowiki>&gt;
 | style="text-align:center;" | 513/512
@@ Line 79: / Line 102: @@
 |-
 | | moctdel
+|Lotriruyo
+|1or<sup>3</sup>y
 | |<nowiki> | -2 0 3 -3 1 </nowiki>&gt;
 | style="text-align:center;" | 1375/1372
@@ Line 84: / Line 109: @@
 |-
 | | Beta 2, septimal schisma, garischisma
+|Sasaru
+|ssr
 | |<nowiki> | 25 -14 0 -1 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 89: / Line 116: @@
 |-
 | | Werckmeister's undecimal septenarian schisma, werckisma
+|Luzozogu
+|1uzzg
 | |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
 | style="text-align:center;" | 441/440
@@ Line 94: / Line 123: @@
 |-
 | | cuthbert
+|Thuthulolozogu
+|3uu1oozg
 | |<nowiki> | 0 0 -1 1 2 -2 </nowiki>&gt;
 | style="text-align:center;" | 847/845
@@ Line 99: / Line 130: @@
 |-
 | | undecimal kleisma, keenanisma
+|Lozoyo
+|1ozg
 | |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
 | style="text-align:center;" | 385/384
@@ Line 104: / Line 137: @@
 |-
 | | gentle comma
+|Tholuluzo
+|3o1uuz
 | |<nowiki> | 2 -1 0 1 -2 1 </nowiki>&gt;
 | style="text-align:center;" | 364/363
@@ Line 109: / Line 144: @@
 |-
 | | minthma
+|Thulo
+|3u1o
 | |<nowiki> | 5 -3 0 0 1 -1 </nowiki>&gt;
 | style="text-align:center;" | 352/351
@@ Line 114: / Line 151: @@
 |-
 | | marveltwin
+|Thoyoyo
+|3oyy
 | |<nowiki> | -2 -4 2 0 0 1 </nowiki>&gt;
 | style="text-align:center;" | 325/324
@@ Line 119: / Line 158: @@
 |-
 | | Beta 5, Garibaldi comma, hemifamity
+|Saruyo
+|sry
 | |<nowiki> | 10 -6 1 -1 </nowiki>&gt;
 | style="text-align:center;" | 5120/5103
@@ Line 124: / Line 165: @@
 |-
 | | hemimage
+|Satrizo-agu
+|sz<sup>3</sup>g
 | |<nowiki> | 5 -7 -1 3 </nowiki>&gt;
 | style="text-align:center;" | 10976/10935
@@ Line 129: / Line 172: @@
 |-
 | | septendecimal kleisma
+|Sugu
+|17ug
 | |<nowiki> | 8 -1 -1 0 0 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 256/255
@@ Line 134: / Line 179: @@
 |-
 | | small BP diesis, mirkwai
+|Quinru-aquadyo
+|r<sup>5</sup>y<sup>4</sup>
 | |<nowiki> | 0 3 4 -5 </nowiki>&gt;
 | style="text-align:center;" | 16875/16807
@@ Line 139: / Line 186: @@
 |-
 | | neutral third comma, rastma
+|Lulu
+|1uu
 | |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
 | style="text-align:center;" | 243/242
@@ Line 144: / Line 193: @@
 |-
 | | kestrel comma
+|Thuthuloru
+|3uu1or
 | |<nowiki> | 2 3 0 -1 1 -2 </nowiki>&gt;
 | style="text-align:center;" | 1188/1183
@@ Line 149: / Line 200: @@
 |-
 | | septimal kleisma, marvel comma
+|Ruyoyo
+|ryy
 | |<nowiki> | -5 2 2 -1 </nowiki>&gt;
 | style="text-align:center;" | 225/224
@@ Line 154: / Line 207: @@
 |-
 | | huntma
+|Thururuyo
+|3urry
 | |<nowiki> | 7 0 1 -2 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 640/637
@@ Line 159: / Line 214: @@
 |-
 | | spleen comma
+|Nuluzoyo
+|19u1uzy
 | |<nowiki> | 1 1 1 1 -1 0 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 210/209
@@ Line 164: / Line 221: @@
 |-
 | | orgonisma
+|Satrilu-aruru
+|s1u<sup>3</sup>rr
 | |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
 | style="text-align:center;" | 65536/65219
@@ Line 169: / Line 228: @@
 |-
 | | gamelan residue, gamelisma
+|Latrizo
+|Lz<sup>3</sup>
 | |<nowiki> | -10 1 0 3 </nowiki>&gt;
 | style="text-align:center;" | 1029/1024
@@ Line 174: / Line 235: @@
 |-
 | | septendecimal comma
+|Lasu
+|L17u
 | |<nowiki> | -7 7 0 0 0 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 2187/2176
@@ Line 179: / Line 242: @@
 |-
 | | mynucuma
+|Thuzozogu
+|3uzzg
 | |<nowiki> | 2 -1 -1 2 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 196/195
@@ Line 184: / Line 249: @@
 |-
 | | quince
+|Lasepzo-agugu
+|Lz<sup>7</sup>gg
 | |<nowiki> | -15 0 -2 7 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 189: / Line 256: @@
 |-
 | | undecimal semicomma, pentacircle (minthma * gentle)
+|Saluzo
+|s1uz
 | |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
 | style="text-align:center;" | 896/891
@@ Line 194: / Line 263: @@
 |-
 | | 29th-partial chroma
+|Twenty-noyo
+|29oy
 | |<nowiki> | -4 -2 1 0 0 0 0 0 0 1 </nowiki>&gt;
 | style="text-align:center;" | 145/144
@@ Line 199: / Line 270: @@
 |-
 | | grossma
+|Thulu
+|3u1u
 | |<nowiki> | 4 2 0 0 -1 -1 </nowiki>&gt;
 | style="text-align:center;" | 144/143
@@ Line 204: / Line 277: @@
 |-
 | | gassorma
+|Thuloruyoyo
+|3u1oryy
 | |<nowiki> | 0 -1 2 -1 1 -1 </nowiki>&gt;
 | style="text-align:center;" | 275/273
@@ Line 209: / Line 284: @@
 |-
 | | septimal semicomma, octagar
+|Rurutriyo
+|rry<sup>3</sup>
 | |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
 | style="text-align:center;" | 4000/3969
@@ Line 214: / Line 291: @@
 |-
 | | minor BP diesis, sensamagic
+|Zozoyo
+|zzy
 | |<nowiki> | 0 -5 1 2 </nowiki>&gt;
 | style="text-align:center;" | 245/243
@@ Line 219: / Line 298: @@
 |-
 | | secorian
-| |<nowiki> | 12 -7 0 1 0 -1/1 </nowiki>&gt;
+|Sathuzo
+|s3uz
+| |<nowiki> | 12 -7 0 1 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 28672/28431
 | | 14.61
 |-
 | | mirwomo comma
+|Labizoyo
+|Lzzyy
 | |<nowiki> | -15 3 2 2 </nowiki>&gt;
 | style="text-align:center;" | 33075/32768
@@ Line 229: / Line 312: @@
 |-
 | | vicesimotertial comma
+|Sa-twenty-tho
+|s23o
 | |<nowiki> | 5 -6 0 0 0 0 0 0 1 </nowiki>&gt;
 | style="text-align:center;" | 736/729
@@ Line 234: / Line 319: @@
 |-
 | | small tridecimal comma, animist
+|Thuzoyo
+|3uzy
 | |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 105/104
@@ Line 239: / Line 326: @@
 |-
 | | hemimin
+|Trilu-azo
+|1u<sup>3</sup>z
 | |<nowiki> | 6 1 0 1 -3 </nowiki>&gt;
 | style="text-align:center;" | 1344/1331
@@ Line 244: / Line 333: @@
 |-
 | | Ptolemy's comma, ptolemisma
+|Luyoyo
+|1uyy
 | |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
 | style="text-align:center;" | 100/99
@@ Line 249: / Line 340: @@
 |-
 | | '41-tone' comma
+|Wa-41
+|41-edo
 | |<nowiki> | 65 -41 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 254: / Line 347: @@
 |-
 | | tolerma
+|Sazoyoyo
+|szyy
 | |<nowiki> | 10 -11 2 1 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 259: / Line 354: @@
 |-
 | | major BP diesis, gariboh
+|Triru-aquinyo
+|r<sup>3</sup>y<sup>5</sup>
 | |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
 | style="text-align:center;" | 3125/3087
@@ Line 264: / Line 361: @@
 |-
 | | cassacot
+|Luluzozoyo
+|1uuzzy
 | |<nowiki> | -1 0 1 2 -2 </nowiki>&gt;
 | style="text-align:center;" | 245/242
@@ Line 269: / Line 368: @@
 |-
 | | keema
+|Zotriyo
+|zy<sup>3</sup>
 | |<nowiki> | -5 -3 3 1 </nowiki>&gt;
 | style="text-align:center;" | 875/864
@@ Line 274: / Line 375: @@
 |-
 | | blackjackisma
+|Lasepru-aquadbiyo
+|Lr<sup>7</sup>y<sup>8</sup>
 | |<nowiki> | -10 7 8 -7 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 279: / Line 382: @@
 |-
 | | roda
+|Sasa-triyo
+|ssy<sup>3</sup>
 | |<nowiki> | 20 -17 3 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 284: / Line 389: @@
 |-
 | | minimal diesis, tetracot comma
+|Saquadyo
+|sy<sup>4</sup>
 | |<nowiki> | 5 -9 4 </nowiki>&gt;
 | style="text-align:center;" | 20000/19683
@@ Line 289: / Line 396: @@
 |-
 | | small diesis, magic comma
+|Laquinyo
+|Ly<sup>5</sup>
 | |<nowiki> | -10 -1 5 </nowiki>&gt;
 | style="text-align:center;" | 3125/3072
@@ Line 294: / Line 403: @@
 |-
 | | thuja comma
+|Saquinlu-ayo
+|s1u<sup>5</sup>y
 | |<nowiki> | 15 0 1 0 -5 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 299: / Line 410: @@
 |-
 | | Ampersand's comma
+|Lala-tribiyo
+|LLy<sup>3</sup>
 | |<nowiki> | -25 7 6 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 304: / Line 417: @@
 |-
 | | great BP diesis
+|Rutribiyo
+|ry<sup>6</sup>
 | |<nowiki> | 0 -7 6 -1 </nowiki>&gt;
 | style="text-align:center;" | 15625/15309
@@ Line 309: / Line 424: @@
 |-
 | | shibboleth
+|Tritriyo
+|y<sup>9</sup>
 | |<nowiki> | -5 -10 9 </nowiki>&gt;
 | style="text-align:center;" |
@@ Line 446: / Line 563: @@
 |310.24
 |374.63 (176.A2<sub>16</sub>)
-| | [[32/27|32/27]]
+| | [[32/27|32/27]], [[13/11]]
 | style="text-align:center;" | m3
 | style="text-align:center;" | F
@@ Line 512: / Line 629: @@
 |496.39
 |599.415 (257.6A<sub>16</sub>)
-| | [[21/16|21/16]]
+| | [[21/16|21/16]], [[13/10]]
 | style="text-align:center;" | v4
 | style="text-align:center;" | vG
@@ Line 611: / Line 728: @@
 |775.61
 |936.585 (3A8.96<sub>16</sub>)
-| | [[32/21|32/21]]
+| | [[32/21|32/21]], [[20/13]]
 | style="text-align:center;" | ^5
 | style="text-align:center;" | ^A
@@ Line 890: / Line 1,007: @@
 | style="text-align:center;" | C upmajor or C up
 |}
--10-20 = D F Ab = Ddim = "D dim"
+-10-20 = D F Ab = Ddim = D dim
--10-21 = D F ^Ab = Ddim(^5) = "D dim up-five"
+-10-21 = D F ^Ab = Ddim(^5) = D dim up-five
--10-22 = D F vvA = Dm(~5) = "D minor mid-five"
+-10-22 = D F vvA = Dm(~5) = D minor mid-five
--10-23 = D F vA = Dm(v5) = "D minor down-five"
+-10-23 = D F vA = Dm(v5) = D minor down-five
--10-24 = D F A = Dm = "D minor"
+-10-24 = D F A = Dm = D minor
--14-24 = D F# A = D = "D" or "D major"
+-14-24 = D F# A = D = D or D major
--14-25 = D F# ^A = D(^5) = "D up-five"
+-14-25 = D F# ^A = D(^5) = D up-five
--14-26 = D F# ^^A = D(^^5) = "D double-up-five", or possibly Daug(vv5)
+-14-26 = D F# ^^A = D(^^5) = D double-up-five, or possibly Daug(vv5)
--14-27 = D F# vA# = Daug(v5) = "D aug down-five"
+-14-27 = D F# vA# = Daug(v5) = D aug down-five
--14-28 = D F# A# is Daug = "D aug"
+-14-28 = D F# A# is Daug = D aug
 etc.
@@ Line 914: / Line 1,031: @@
 For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|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. The 6 mid intervals are uncategorized, since they are all so far from 12edo.
+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.
 [[File:41-edo spiral.png|673x673px]]
@@ Line 926: / Line 1,043: @@
 | | '''Error (abs., in [[cent|cents]])'''
 |-
-| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
+| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
-| style="text-align:center;" | 0.484
+| style="text-align:center;" | '''0.484'''
 |-
 | style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
@@ Line 938: / Line 1,055: @@
 | style="text-align:center;" | 2.854
 |-
-| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
+| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
-| style="text-align:center;" | 2.972
+| style="text-align:center;" | '''2.972'''
 |-
 | style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
@@ Line 956: / Line 1,073: @@
 | style="text-align:center;" | 4.296
 |-
-| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
+| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
-| style="text-align:center;" | 4.780
+| style="text-align:center;" | '''4.780'''
 |-
 | style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
 | style="text-align:center;" | 5.342
 |-
-| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
+| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
-| style="text-align:center;" | 5.826
+| style="text-align:center;" | '''5.826'''
 |-
 | style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
@@ Line 980: / Line 1,097: @@
 | style="text-align:center;" | 7.769
 |-
-| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
+| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
-| style="text-align:center;" | 8.253
+| style="text-align:center;" | '''8.253'''
 |-
 | style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
@@ Line 1,008: / Line 1,125: @@
 ''41-EDO Classical guitar, by Ron Sword.''
-The [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf 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.
+The [[The Kite Guitar|Kite Guitar]] (see also [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf 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.
 [[File:The Kite Tuning smaller.jpg|1122x1122px]]

41edo: Difference between revisions

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Revision as of 12:25, 13 October 2019

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