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41 Tone Equal Temperament
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Introduction | Commas | Temperaments | Intervals | Instruments | Harmonic Scale | Nonoctave Temperaments | Music | Links

Introduction

The 41-tET, 41-EDO, or 41-ET, 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
Approximate
Ratios in the 11-limit
ups and
downs
notation
Proposed names
Andrew's
solfege
syllable
generator for
some MOS and MODMOS Scales available
0
0.00
1/1
P1
D
Unison
do


1
29.27
81/80
^1
D^
Red unison
di


2
58.54
25/24, 28/27,
33/32
vm2
Ebv
Blue minor second
ro
Hemimiracle

3
87.80
21/20, 22/21
m2
Eb
Gray minor second
rih
88cET (approx),
octacot

4
117.07
16/15, 15/14
^m2
Eb^
Red minor second
ra
Miracle

5
146.34
12/11
~2
Evv
Neutral second
ru
Bohlen-Pierce/bohpier

6
175.61
10/9, 11/10
vM2
Ev
Blue major second
reh
Tetracot/bunya/monkey
13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
7
204.88
9/8
M2
E
Gray major second
re
Baldy
11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8
234.15
8/7
^M2
E^
Red major second
ri
Rodan/guiron
11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
9
263.41
7/6, 32/25
vm3
Fv
Blue minor third
ma
Septimin
9-tone MOS: 5 4 5 5 4 5 4 5 4
10
292.68
32/27
m3
F
Gray minor third
meh
Quasitemp

11
321.95
6/5
^m3
F^
Red minor third
me
Superkleismic
11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
12
351.22
11/9,27/22
~3
F^^
Neutral third
mu
Hemififths/karadeniz
10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13
380.49
5/4
vM3
F#v
Blue major third
mi
Magic/witchcraft
10-tone MOS: 2 9 2 2 9 2 2 9 2 2
14
409.76
14/11, 81/64
M3
F#
Gray major third
maa
Hocus

15
439.02
9/7
^M3
F#^
Red major third
mo

11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16
468.29
21/16
v4
Gv
Blue fourth
fe
Barbad

17
497.56
4/3
P4
G
Perfect fourth
fa
Schismatic (helmholtz, garibaldi, cassandra)

18
526.83
15/11, 27/20
^4
G^
Red fourth
fih
Trismegistus
9-tone MOS: 5 5 3 5 5 5 5 3 5
19
556.10
11/8
^^4
G^^
Blue minor tritone
fu


20
585.37
7/5
vA4, d5
G#v,
Ab
Minor tritone / diminished fifth
fi
Pluto

21
614.63
10/7
A4, ^d5
G#,
Ab^
Major tritone / augmented fourth
se


22
643.90
16/11
vv5
Avv
Red major tritone
su


23
673.17
22/15, 40/27
v5
Av
Blue fifth
sih


24
702.44
3/2
P5
A
Perfect fifth
sol


25
731.71
32/21
^5
A^
Red fifth
si


26
760.98
14/9, 25/16
vm6
Bbv
Blue minor sixth
lo


27
790.24
11/7, 128/81
m6
Bb
Gray minor sixth
leh


28
819.51
8/5
^m6
Bb^
Red minor sixth
le


29
848.78
18/11, 44/27
~6
Bvv
Neutral sixth
lu


30
878.05
5/3
vM6
Bv
Blue major sixth
la


31
907.32
27/16
M6
B
Gray major sixth
laa


32
936.59
12/7
^M6
B^
Red major sixth
li


33
965.85
7/4
vm7
vC
Blue minor seventh
ta


34
995.12
16/9
m7
C
Gray minor seventh
teh


35
1024.39
9/5, 20/11
^m7
C^
Red minor seventh
te


36
1053.66
11/6
~7
C^^
Neutral seventh
tu


37
1082.93
15/8
vM7
C#v
Blue major seventh
ti


38
1112.20
40/21, 21/11
M7
C#
Gray major seventh
taa


39
1141.46
48/25, 27/14,
64/33
^M7
C#^
Red major seventh
to


40
1170.73
160/81
v8
Dv
Blue octave
da


41
1200
2/1
P8
D

do



41edo chord names using ups and downs:
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"
0-11-24 = D F^ A = D.^m = "D upminor"
0-12-24 = D F^^ A = D.~ = "D mid"
0-13-24 = D F#v A = D.v = "D downmajor" or "D dot down"
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-EDD elektrische gitaar.jpg
41-EDO Electric guitar, by Gregory Sanchez.

Ron_Sword_with_a_41ET_Guitar.jpg
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.

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.


Music

EveningHorizon play by Cameron Bobro

Links


  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