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 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 81/80 ^1 D^ di
2 58.54 25/24, 28/27,

33/32

vm2 Ebv ro Hemimiracle
3 87.80 21/20, 22/21 m2 Eb rih 88cET (approx),

octacot

4 117.07 16/15, 15/14 ^m2 Eb^ ra Miracle
5 146.34 12/11 ~2 Evv ru Bohlen-Pierce/bohpier
6 175.61 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 9/8 M2 E re Baldy 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8 234.15 8/7 ^M2 E^ 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 ma Septimin 9-tone MOS: 5 4 5 5 4 5 4 5 4
10 292.68 32/27 m3 F meh Quasitemp
11 321.95 6/5 ^m3 F^ 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^^ mu Hemififths/karadeniz 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13 380.49 5/4 vM3 F#v 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# maa Hocus
15 439.02 9/7 ^M3 F#^ mo 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16 468.29 21/16 v4 Gv fe Barbad
17 497.56 4/3 P4 G fa Schismatic (helmholtz, garibaldi, cassandra)
18 526.83 15/11, 27/20 ^4 G^ fih Trismegistus 9-tone MOS: 5 5 3 5 5 5 5 3 5
19 556.10 11/8 ^^4 G^^ fu
20 585.37 7/5 vA4, d5 G#v,

Ab

fi Pluto
21 614.63 10/7 A4, ^d5 G#,

Ab^

se
22 643.90 16/11 vv5 Avv su
23 673.17 22/15, 40/27 v5 Av sih
24 702.44 3/2 P5 A sol
25 731.71 32/21 ^5 A^ si
26 760.98 14/9, 25/16 vm6 Bbv lo
27 790.24 11/7, 128/81 m6 Bb leh
28 819.51 8/5 ^m6 Bb^ le
29 848.78 18/11, 44/27 ~6 Bvv lu
30 878.05 5/3 vM6 Bv la
31 907.32 27/16 M6 B laa
32 936.59 12/7 ^M6 B^ li
33 965.85 7/4 vm7 Cv ta
34 995.12 16/9 m7 C teh
35 1024.39 9/5, 20/11 ^m7 C^ te
36 1053.66 11/6 ~7 C^^ tu
37 1082.93 15/8 vM7 C#v ti
38 1112.20 40/21, 21/11 M7 C# taa
39 1141.46 48/25, 27/14,

64/33

^M7 C#^ to
40 1170.73 160/81 v8 Dv da
41 1200 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-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.

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


  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