41edo

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Theory

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. 41-ET is also used by the Kite Guitar, see below in "Instruments".

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

Notation

Cents Value Approximate Ratios

in the 11-limit

Ups and Downs Notation Andrew's

Solfege Syllables

0 0.00 1/1 perfect unison P1 D do
1 29.27 81/80, 64/63 up-unison ^1 ^D di
2 58.54 25/24, 28/27, 33/32 double-up unison, downminor 2nd ^^1, vm2 ^^D, vEb ro
3 87.805 21/20, 22/21 down-aug 1sn, minor 2nd vA1, m2 vD#, Eb rih
4 117.07 16/15, 15/14 augmented 1sn, upminor 2nd A1, ^m2 D#, ^Eb ra
5 146.34 12/11 mid 2nd ~2 ^D#, vvE ru
6 175.61 10/9, 11/10 downmajor 2nd vM2 vE reh
7 204.88 9/8 major 2nd M2 E re
8 234.15 8/7 upmajor 2nd ^M2 ^E ri
9 263.415 7/6 downminor 3rd vm3 vF ma
10 292.68 32/27, 13/11 minor 3rd m3 F meh
11 321.95 6/5 upminor 3rd ^m3 ^F me
12 351.22 11/9, 27/22 mid 3rd ~3 ^^F, vGb mu
13 380.49 5/4 downmajor 3rd vM3 vF#, Gb mi
14 409.76 14/11, 81/64 major 3rd M3 F#, ^Gb maa
15 439.02 9/7, 32/25 upmajor 3rd ^M3 ^F#, vvG mo
16 468.29 21/16, 13/10 down-4th v4 vG fe
17 497.56 4/3 perfect 4th P4 G fa
18 526.83 15/11, 27/20 up-4th ^4 ^G fih
19 556.1 11/8 mid-4th ~4 ^^G, vAb fu
20 585.37 7/5 downaug 4th, dim 5th vA4, d5 vG#, Ab fi
21 614.63 10/7 aug 4th, updim 5th A4, ^d5 G#, ^Ab se
22 643.90 16/11, 13/9 mid-5th ~5 vvA su
23 673.17 22/15, 40/27 down-5th v5 vA sih
24 702.44 3/2 perfect 5th P5 A sol
25 731.71 32/21, 20/13 up-5th ^5 ^A si
26 760.98 14/9, 25/16 downminor 6th vm6 ^^A, vBb lo
27 790.24 11/7, 128/81 minor 6th m6 vA#, Bb leh
28 819.51 8/5 upminor 6th ^m6 A#, ^Bb le
29 848.78 18/11, 44/27 mid 6th ~6 ^A#, vvB lu
30 878.05 5/3 downmajor 6th vM6 vB la
31 907.32 27/16 major 6th M6 B laa
32 936.59 12/7 upmajor 6th ^M6 ^B li
33 965.85 7/4 downminor 7th vm7 vC ta
34 995.12 16/9 minor 7th m7 C teh
35 1024.39 9/5, 20/11 upminor 7th ^m7 ^C te
36 1053.66 11/6 mid 7th ~7 ^^C, vDb tu
37 1082.93 15/8 downmajor 7th vM7 vC#, Db ti
38 1112.195 40/21, 21/11 major 7th M7 C#, ^Db taa
39 1141.46 48/25, 27/14, 64/33 upmajor 7th ^M7 C#^, vvD to
40 1170.73 160/81, 63/32 dim 8ve v8 vD da
41 1200 2/1 perfect 8ve 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

Chord Names

All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. 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# = Daug = D aug

For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.

Red-Blue 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.

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

Relationship to 12-edo

Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. 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.

41-edo spiral.png

The same spiral, but with notes not intervals:

41-edo spiral with notes.png

Commas

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

Prime

Limit

Ratio Cents Monzo color name Name
3 19.84 | 65 -41 > Wa-41 41-edo '41-tone' comma
5 57.27 | -5 -10 9 > Tritriyo y9 shibboleth
" 31.57 | -25 7 6 > Lala-tribiyo LLy3 Ampersand's comma
" 3125/3072 29.61 | -10 -1 5 > Laquinyo Ly5 small diesis, magic comma
" 20000/19683 27.66 | 5 -9 4 > Saquadyo sy4 minimal diesis, tetracot comma
" 25.71 | 20 -17 3 > Sasa-triyo ssy3 roda
" 32805/32768 1.95 | -15 8 1 > Layo Ly schisma
7 15625/15309 35.37 | 0 -7 6 -1 > Rutribiyo ry6 great BP diesis
" 22.41 | -10 7 8 -7 > Lasepru-aquadbiyo Lr7y8 blackjackisma
" 875/864 21.90 | -5 -3 3 1 > Zotriyo zy3 keema
" 3125/3087 21.18 | 0 -2 5 -3 > Triru-aquinyo r3y5 major BP diesis, gariboh
" 19.95 | 10 -11 2 1 > Sazoyoyo szyy tolerma
" 33075/32768 16.14 | -15 3 2 2 > Labizoyo Lzzyy mirwomo comma
" 245/243 14.19 | 0 -5 1 2 > Zozoyo zzy minor BP diesis, sensamagic
" 4000/3969 13.47 | 5 -4 3 -2 > Rurutriyo rry3 septimal semicomma, octagar
" 9.15 | -15 0 -2 7 > Lasepzo-agugu Lz7gg quince
" 1029/1024 8.43 | -10 1 0 3 > Latrizo Lz3 gamelan residue, gamelisma
" 225/224 7.71 | -5 2 2 -1 > Ruyoyo ryy septimal kleisma, marvel comma
" 16875/16807 6.99 | 0 3 4 -5 > Quinru-aquadyo r5y4 small BP diesis, mirkwai
" 10976/10935 6.48 | 5 -7 -1 3 > Satrizo-agu sz3g hemimage
" 5120/5103 5.76 | 10 -6 1 -1 > Saruyo sry Beta 5, Garibaldi comma, hemifamity
" 3.80 | 25 -14 0 -1 > Sasaru ssr Beta 2, septimal schisma, garischisma
" 2401/2400 0.72 | -5 -1 -2 4 > Bizozogu z4gg Breedsma
11 29.72 | 15 0 1 0 -5 > Saquinlu-ayo s1u5y thuja comma
" 245/242 21.33 | -1 0 1 2 -2 > Luluzozoyo 1uuzzy cassacot
" 100/99 17.40 | 2 -2 2 0 -1 > Luyoyo 1uyy Ptolemy's comma, ptolemisma
" 1344/1331 16.83 | 6 1 0 1 -3 > Trilu-azo 1u3z hemimin
" 896/891 9.69 | 7 -4 0 1 -1 > Saluzo s1uz undecimal semicomma, pentacircle (minthma * gentle)
" 65536/65219 8.39 | 16 0 0 -2 -3 > Satrilu-aruru s1u3rr orgonisma
" 243/242 7.14 | -1 5 0 0 -2 > Lulu 1uu neutral third comma, rastma
" 385/384 4.50 | -7 -1 1 1 1 > Lozoyo 1ozg undecimal kleisma, keenanisma
" 441/440 3.93 | -3 2 -1 2 -1 > Luzozogu 1uzzg Werckmeister's undecimal septenarian schisma, werckisma
" 1375/1372 3.78 | -2 0 3 -3 1 > Lotriruyo 1or3y moctdel
" 540/539 3.21 | 2 3 1 -2 -1 > Lururuyo 1urry Swets' comma, swetisma
" 3025/3024 0.57 | -4 -3 2 -1 2 > Loloruyoyo 1ooryy Lehmerisma
" 0.15 | -1 2 -4 5 -2 > Luluquinzo-aquadgu 1uuz5g4 odiheim
13 105/104 16.57 | -3 1 1 1 0 -1 > Thuzoyo 3uzy small tridecimal comma, animist
" 28672/28431 14.61 | 12 -7 0 1 0 -1 > Sathuzo s3uz secorian
" 275/273 12.64 | 0 -1 2 -1 1 -1 > Thuloruyoyo 3u1oryy gassorma
" 144/143 12.06 | 4 2 0 0 -1 -1 > Thulu 3u1u grossma
" 196/195 8.86 | 2 -1 -1 2 0 -1 > Thuzozogu 3uzzg mynucuma
" 640/637 8.13 | 7 0 1 -2 0 -1 > Thururuyo 3urry huntma
" 1188/1183 7.30 | 2 3 0 -1 1 -2 > Thuthuloru 3uu1or kestrel comma
" 325/324 5.34 | -2 -4 2 0 0 1 > Thoyoyo 3oyy marveltwin
" 352/351 4.93 | 5 -3 0 0 1 -1 > Thulo 3u1o minthma
" 364/363 4.76 | 2 -1 0 1 -2 1 > Tholuluzo 3o1uuz gentle comma
" 847/845 4.09 | 0 0 -1 1 2 -2 > Thuthulolozogu 3uu1oozg cuthbert
" 729/728 2.38 | -3 6 0 -1 0 -1 > Lathuru L3ur squbema
" 4096/4095 0.42 | 12 -2 -1 -1 0 -1 > Sathurugu s3urg tridecimal schisma, Sagittal schismina
" 10648/10647 0.16 | 3 -2 0 -1 3 -2 > Thuthutrilo-aru 3uu1o3r harmonisma
17 2187/2176 8.73 | -7 7 0 0 0 0 -1 > Lasu L17u septendecimal comma
" 256/255 6.78 | 8 -1 -1 0 0 0 -1 > Sugu 17ug septendecimal kleisma
" 715/714 2.42 | -1 -1 1 -1 1 1 -1 > Sutholoruyo 17u3o1ory septendecimal bridge comma
19 210/209 8.26 | 1 1 1 1 -1 0 0 -1 > Nuluzoyo 19u1uzy spleen comma
" 1216/1215 1.42 | 6 -5 -1 0 0 0 0 1 > Sanogu s19og Eratosthenes' comma
" 513/512 3.38 | -9 3 0 0 0 0 0 1 > Lano L19o undevicesimal comma, Boethius' comma
23 736/729 16.54 | 5 -6 0 0 0 0 0 0 1 > Sa-twenty-tho s23o vicesimotertial comma
29 145/144 11.98 | -4 -2 1 0 0 0 0 0 0 1 > Twenty-noyo 29oy 29th-partial chroma

Temperaments

List of edo-distinct 41et rank two temperaments

Table of Temperaments by generator

Degree Cents Generator Some MOS and MODMOS Scales Available
0 0.00
1 29.27
2 58.54 Hemimiracle
3 87.805 88cET (approx),

octacot

4 117.07 Miracle
5 146.34 Bohlen-Pierce/bohpier
6 175.61 Tetracot/bunya/monkey 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
7 204.88 Baldy 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8 234.15 Rodan/guiron 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
9 263.415 Septimin 9-tone MOS: 5 4 5 5 4 5 4 5 4
10 292.68 Quasitemp
11 321.95 Superkleismic 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
12 351.22 Hemififths/karadeniz 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13 380.49 Magic/witchcraft 10-tone MOS: 2 9 2 2 9 2 2 9 2 2
14 409.76 Hocus
15 439.02 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16 468.29 Barbad
17 497.56 Schismatic (helmholtz, garibaldi, cassandra)
18 526.83 Trismegistus 9-tone MOS: 5 5 3 5 5 5 5 3 5
19 556.1
20 585.37 Pluto

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

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.

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. If the interval between strings is 13\41, 25 of the 41 intervals are in easy reach: vm2, ^m2, vM2, M2, ^M2, vm3, ^m3, vM3, ^M3, P4, ~4, d5, A4, ~5, P5, vm6, ^m6, vM6, ^M6, vm7, m7, ^m7, vM7, ^M7, P8.

Caleb's Kite guitar.jpg

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

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