41edo

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41edo
Prime factorization 41 (prime)
Step size 29.2683¢
Fifth 24\41 (702.4¢)
Major 2nd 7\41 (204.9¢)
Semitones (A1:m2) 4:3 (117.1¢ : 87.8¢)
Consistency limit 15
Monotonicity limit 21
English Wikipedia has an article on:

The 41 equal divisions of the octave (41edo), or 41(-tone) equal temperament (41tet, 41et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 41 equally-sized steps. Each step is about 29.3 cents, an interval close in size to 64/63, the septimal comma.

Theory

41edo is the second smallest equal division (after 29edo) whose perfect fifth is closer to just intonation than that of 12edo, 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. Anyway, it 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-odd-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent up to 21-odd-limit. Among these, it is more prominent as a 2.3.5.7.11.19 subgroup temperament for its size, and perhaps the smallest system with a satisfactory model of the 9-odd-limit because it is the smallest edo to have the 9-odd-limit be distinctly consistent.

41edo can be seen as a tuning of the garibaldi temperament[1][2], the magic temperament, the superkleismic temperament and multiple temperaments in the tetracot family. 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 also a great tetracot tuning, and works as an alternative to 34edo due to a much better approximation to the 7th harmonic, and supporting monkey, bunya and octacot simultaneously. All three of these extend to the 11-limit by way of interpreting the flat 10/9 as an 11/10 by tempering 100/99. Note that this equivalence is especially nice in 41edo due to also giving a more accurate interpretation of this comma-flat whole tone as a 21/19.

41et forms the foundation of the H-System, which uses the scale degrees of 41et as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41et circle in 205edo. 41et is also used by the Kite Guitar, see below in #Instruments.

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

Prime harmonics

Approximation of prime harmonics in 41edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.0 +0.5 -5.8 -3.0 +4.8 +8.3 +12.1 -4.8 -13.6 -5.2
relative (%) +0 +2 -20 -10 +16 +28 +41 -17 -47 -18
Steps
(reduced) 		41
(0) 		65
(24) 		95
(13) 		115
(33) 		142
(19) 		152
(29) 		168
(4) 		174
(10) 		185
(21) 		199
(35)
  1. ↑ Schismic Temperaments at x31eq.com, the website of Graham Breed
  2. ↑ Lattices with Decimal Notation at x31eq.com

Intervals

# Cents Approximate Ratios* Ups and Downs Notation Andrew's
Solfege 		Kite's
Solfege
0 0.00 1/1 perfect unison P1 D do do
1 29.27 81/80, 64/63, 49/48 up-unison ^1 ^D di da
2 58.54 25/24, 28/27, 36/35, 33/32 double-up 1sn, downminor 2nd ^^1, vm2 ^^D, vEb ro ru
3 87.80 21/20, 22/21, 19/18, 20/19 down-aug 1sn, minor 2nd vA1, m2 vD#, Eb rih ro
4 117.07 16/15, 15/14, 14/13 augmented 1sn, upminor 2nd A1, ^m2 D#, ^Eb ra ra
5 146.34 12/11, 13/12 mid 2nd ~2 ^D#, vvE ru ruh
6 175.61 10/9, 11/10, 21/19 downmajor 2nd vM2 vE reh reh
7 204.88 9/8 major 2nd M2 E re rih
8 234.15 8/7, 15/13 upmajor 2nd ^M2 ^E ri ri
9 263.41 7/6, 22/19 downminor 3rd vm3 vF ma mu
10 292.68 32/27, 13/11, 19/16 minor 3rd m3 F meh mo
11 321.95 6/5 upminor 3rd ^m3 ^F me ma
12 351.22 11/9, 27/22, 16/13 mid 3rd ~3 ^^F, vGb mu muh
13 380.49 5/4, 26/21 downmajor 3rd vM3 vF#, Gb mi meh
14 409.76 81/64, 14/11, 24/19, 19/15 major 3rd M3 F#, ^Gb maa mih
15 439.02 9/7, 32/25 upmajor 3rd ^M3 ^F#, vvG mo mi
16 468.29 21/16, 13/10 down-4th v4 vG fe fu
17 497.56 4/3 perfect 4th P4 G fa fo
18 526.83 27/20, 15/11, 19/14 up-4th ^4 ^G fih fa
19 556.10 11/8, 18/13, 26/19 mid-4th ~4 ^^G, vAb fu fuh
20 585.37 7/5 downaug 4th, dim 5th vA4, d5 vG#, Ab fi feh / so
21 614.63 10/7 aug 4th, updim 5th A4, ^d5 G#, ^Ab se fih / sa
22 643.90 16/11, 13/9, 19/13 mid-5th ~5 vvA su suh
23 673.17 40/27, 22/15, 28/19 down-5th v5 vA sih seh
24 702.44 3/2 perfect 5th P5 A sol sih
25 731.71 32/21, 20/13 up-5th ^5 ^A si si
26 760.98 14/9, 25/16 downminor 6th vm6 ^^A, vBb lo lu
27 790.24 128/81, 11/7, 19/12, 30/19 minor 6th m6 vA#, Bb leh lo
28 819.51 8/5, 21/13 upminor 6th ^m6 A#, ^Bb le la
29 848.78 18/11, 44/27, 13/8 mid 6th ~6 ^A#, vvB lu luh
30 878.05 5/3 downmajor 6th vM6 vB la leh
31 907.32 27/16, 22/13, 32/19 major 6th M6 B laa lih
32 936.59 12/7, 19/11 upmajor 6th ^M6 ^B li li
33 965.85 7/4, 26/15 downminor 7th vm7 vC ta tu
34 995.12 16/9 minor 7th m7 C teh to
35 1024.39 9/5, 20/11, 38/21 upminor 7th ^m7 ^C te ta
36 1053.66 11/6, 24/13 mid 7th ~7 ^^C, vDb tu tuh
37 1082.93 15/8, 28/15, 13/7 downmajor 7th vM7 vC#, Db ti teh
38 1112.20 40/21, 21/11, 36/19, 19/10 major 7th M7 C#, ^Db taa tih
39 1141.46 48/25, 27/14, 35/18, 64/33 upmajor 7th ^M7 C#^, vvD to ti
40 1170.73 160/81, 63/32, 96/49 dim 8ve v8 vD da du
41 1200.00 2/1 perfect 8ve P8 D do do

* Based on treating 41edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.

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 (7-over) 6:7:9 0-9-24 C vEb G Cvm C downminor
gu (5-under) 10:12:15 0-11-24 C ^Eb G C^m C upminor
ilo (11-over) 18:22:27 0-12-24 C vvE G C~ C mid
yo (5-over) 4:5:6 0-13-24 C vE G Cv C downmajor or C down
ru (7-under) 14:18:21 0-15-24 C ^E G C^ C upmajor or C up

0-10-20 = D F Ab = Dd = D dim

0-10-21 = D F ^Ab = Dd(^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 half-aug

0-14-27 = D F# vA# = Da(v5) = D aug down-five or perhaps D(v#5) = D downsharp-five

0-14-28 = D F# A# = Da = D aug

For a more complete list, see 41edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.

Notations

Red-Blue Notation

A red-note/blue-note system, similar to the one proposed for 36edo, is one option for notating 41edo. This is a special case of Kite's color notation, treating 41edo as a temperament of the 2,3,7 subgroup. 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.

Sagittal

From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 41-EDO in the Revo flavor of Sagittal:

41edo Sagittal.png

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 41edo. Prime harmonics are in bold. As 41edo is consistent in the 15-odd-limit, the results by direct approximation and patent val mapping are the same.

15-odd-limit intervals by patent val mapping
Interval, complement Error (abs, ¢) Error (rel, %)
3/2, 4/3 0.484 1.7
9/8, 16/9 0.968 3.3
15/14, 28/15 2.370 8.1
7/5, 10/7 2.854 9.8
7/4, 8/7 2.972 10.2
7/6, 12/7 3.456 11.8
13/11, 22/13 3.473 11.9
11/9, 18/11 3.812 13.0
9/7, 14/9 3.940 13.5
11/6, 12/11 4.296 14.7
11/8, 16/11 4.780 16.3
15/8, 16/15 5.342 18.3
5/4, 8/5 5.826 19.9
5/3, 6/5 6.310 21.6
9/5, 10/9 6.794 23.2
13/9, 18/13 7.285 24.9
11/7, 14/11 7.752 26.5
13/12, 24/13 7.769 26.5
13/8, 16/13 8.253 28.2
15/11, 22/15 10.122 34.6
11/10, 20/11 10.606 36.2
13/7, 14/13 11.225 38.4
15/13, 26/15 13.595 46.4
13/10, 20/13 14.079 48.1

Relationship to 12-edo

Whereas 12edo has a circle of twelve 5ths, 41edo has a spiral of twelve 5ths, since 24\41 is on the 7\12 kite in the scale tree. (See Chapter 5.7 of Kite's book for an explanation of kites.) This spiral of 5th shows 41edo in a 12edo-friendly format. Excellent for introducing 41edo 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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [65 -41⟩ [⟨41 65]] -0.153 0.15 0.52
2.3.5 3125/3072, 20000/19683 [⟨41 65 95]] +0.734 1.26 4.31
2.3.5.7 225/224, 245/243, 1029/1024 [⟨41 65 95 115]] +0.815 1.10 3.76
2.3.5.7.11 100/99, 225/224, 243/242, 245/242 [⟨41 65 95 115 142]] +0.375 1.32 4.51
2.3.5.7.11.13 100/99, 105/104, 144/143, 196/195, 243/242 [⟨41 65 95 115 142 152]] -0.060 1.55 5.29
2.3.5.7.11.13.19 100/99, 105/104, 133/132, 144/143, 171/169, 196/195 [⟨41 65 95 115 142 152 174]] +0.111 1.49 5.10

41et is lower in relative error than any previous equal temperaments in the 3-, 13- and 19-limit. The next ETs doing better in these subgroups are 53, 53, and 46, respectively. It is even more prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next ETs doing better in these subgroups are 72 and 53, respectively.

Commas

41edo tempers out the following commas using its patent val, ⟨41 65 95 115 142 152 168 174 185 199 203].

Prime
limit 		Ratio[1] Cents Monzo Color name Name(s)
3 (40 digits) 19.84 [65 -41⟩ Wa-41 41-edo 41-comma
5 (14 digits) 57.27 [-5 -10 9⟩ Tritriyo y9 Shibboleth comma
5 (16 digits) 31.57 [-25 7 6⟩ Lala-tribiyo LLy3 Ampersand
5 3125/3072 29.61 [-10 -1 5⟩ Laquinyo Ly5 Magic comma
5 (10 digits) 27.66 [5 -9 4⟩ Saquadyo sy4 Tetracot comma
5 (18 digits) 25.71 [20 -17 3⟩ Sasa-triyo ssy3 Roda
5 (10 digits) 1.95 [-15 8 1⟩ Layo Ly Schisma
7 (10 digits) 35.37 [0 -7 6 -1⟩ Rutribiyo ry6 Great BP diesis
7 (18 digits) 22.41 [-10 7 8 -7⟩ Lasepru-aquadbiyo Lr7y8 Blackjackisma
7 875/864 21.90 [-5 -3 3 1⟩ Zotriyo zy3 Keema
7 3125/3087 21.18 [0 -2 5 -3⟩ Triru-aquinyo r3y5 Gariboh comma
7 (12 digits) 19.95 [10 -11 2 1⟩ Sazoyoyo szyy Tolerma
7 (10 digits) 16.14 [-15 3 2 2⟩ Labizoyo Lzzyy Mirwomo comma
7 245/243 14.19 [0 -5 1 2⟩ Zozoyo zzy Sensamagic comma
7 4000/3969 13.47 [5 -4 3 -2⟩ Rurutriyo rry3 Octagar comma
7 (12 digits) 9.15 [-15 0 -2 7⟩ Lasepzo-agugu Lz7gg Quince comma
7 1029/1024 8.43 [-10 1 0 3⟩ Latrizo Lz3 Gamelisma
7 225/224 7.71 [-5 2 2 -1⟩ Ruyoyo ryy Marvel comma
7 (10 digits) 6.99 [0 3 4 -5⟩ Quinru-aquadyo r5y4 Mirkwai comma
7 (10 digits) 6.48 [5 -7 -1 3⟩ Satrizo-agu sz3g Hemimage comma
7 5120/5103 5.76 [10 -6 1 -1⟩ Saruyo sry Hemifamity comma
7 (16 digits) 3.80 [25 -14 0 -1⟩ Sasaru ssr Garischisma
7 2401/2400 0.72 [-5 -1 -2 4⟩ Bizozogu z4gg Breedsma
11 (12 digits) 29.72 [15 0 1 0 -5⟩ Saquinlu-ayo s1u5y Thuja comma
11 245/242 21.33 [-1 0 1 2 -2⟩ Luluzozoyo 1uuzzy Cassacot
11 100/99 17.40 [2 -2 2 0 -1⟩ Luyoyo 1uyy Ptolemisma
11 1344/1331 16.83 [6 1 0 1 -3⟩ Trilu-azo 1u3z Hemimin
11 896/891 9.69 [7 -4 0 1 -1⟩ Saluzo s1uz Pentacircle
11 (10 digits) 8.39 [16 0 0 -2 -3⟩ Satrilu-aruru s1u3rr Orgonisma
11 243/242 7.14 [-1 5 0 0 -2⟩ Lulu 1uu Rastma
11 385/384 4.50 [-7 -1 1 1 1⟩ Lozoyo 1ozg Keenanisma
11 441/440 3.93 [-3 2 -1 2 -1⟩ Luzozogu 1uzzg Werckisma
11 1375/1372 3.78 [-2 0 3 -3 1⟩ Lotriruyo 1or3y Moctdel
11 540/539 3.21 [2 3 1 -2 -1⟩ Lururuyo 1urry Swetisma
11 3025/3024 0.57 [-4 -3 2 -1 2⟩ Loloruyoyo 1ooryy Lehmerisma
11 (12 digits) 0.15 [-1 2 -4 5 -2⟩ Luluquinzo-aquadgu 1uuz5g4 Odiheim
13 343/338 25.42 [-1 0 0 3 0 -2⟩ Thuthutrizo 3uuz3
13 105/104 16.57 [-3 1 1 1 0 -1⟩ Thuzoyo 3uzy Animist
13 (10 digits) 14.61 [12 -7 0 1 0 -1⟩ Sathuzo s3uz Secorian
13 275/273 12.64 [0 -1 2 -1 1 -1⟩ Thuloruyoyo 3u1oryy Gassorma
13 144/143 12.06 [4 2 0 0 -1 -1⟩ Thulu 3u1u Grossma
13 196/195 8.86 [2 -1 -1 2 0 -1⟩ Thuzozogu 3uzzg Mynucuma
13 640/637 8.13 [7 0 1 -2 0 -1⟩ Thururuyo 3urry Huntma
13 1188/1183 7.30 [2 3 0 -1 1 -2⟩ Thuthuloru 3uu1or Kestrel comma
13 325/324 5.34 [-2 -4 2 0 0 1⟩ Thoyoyo 3oyy Marveltwin comma
13 352/351 4.93 [5 -3 0 0 1 -1⟩ Thulo 3u1o Minthma
13 364/363 4.76 [2 -1 0 1 -2 1⟩ Tholuluzo 3o1uuz Gentle comma
13 847/845 4.09 [0 0 -1 1 2 -2⟩ Thuthulolozogu 3uu1oozg Cuthbert
13 729/728 2.38 [-3 6 0 -1 0 -1⟩ Lathuru L3ur Squbema
13 2080/2079 0.83 [5 -3 1 -1 -1 1⟩ Tholuruyo 3o1ury Ibnsinma
13 4096/4095 0.42 [12 -2 -1 -1 0 -1⟩ Sathurugu s3urg Schismina
13 6656/6655 0.26 [9 0 -1 0 -3 1⟩ Thotrilo-agu 3u1o3g2 Jacobin comma
13 (10 digits) 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 schisma
17 256/255 6.78 [8 -1 -1 0 0 0 -1⟩ Sugu 17ug Septendecimal kleisma
17 715/714 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
19 361/360 4.80 [-3 -2 -1 0 0 0 0 2⟩ Nonogu 19oog2 Go comma
19 513/512 3.38 [-9 3 0 0 0 0 0 1⟩ Lano L19o Boethius' comma
19 1216/1215 1.42 [6 -5 -1 0 0 0 0 1⟩ Sanogu s19og Eratosthenes' comma
23 736/729 16.54 [5 -6 0 0 0 0 0 0 1⟩ Satwetho s23o Vicesimotertial comma
29 145/144 11.98 [-4 -2 1 0 0 0 0 0 0 1⟩ Twenoyo 29oy 29th-partial chroma
  1. ↑ Ratios with more than 8 digits are presented by placeholders with informative hints

Rank-2 temperaments

Table of temperaments by generator
Degree Cents Temperament(s) Pergen Some MOS Scales Available
1 29.27 Slendi (P8, P4/17) Pathological 38-tone MOS
2 58.54 Hemimiracle
Dodecacot 		(P8, P5/12) 21-tone MOS
3 87.80 Octacot (P8, P5/8) 14-tone MOS: 3 3 3 3 3 3 3 3 3 3 3 3 3 2
4 117.07 Miracle (P8, P5/6) 11-tone MOS: 4 4 4 4 4 4 4 4 4 4 1
5 146.34 BPS / bohpier (P8, P12/13) 20-tone MOS
6 175.61 Tetracot / bunya / monkey
Sesquiquartififths / sesquart 		(P8, P5/4) 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
7 204.88 Baldy
Quadrimage 		(P8, c3P4/20) 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
8 234.15 Slendric / rodan / guiron (P8, P5/3) 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
9 263.41 Septimin (P8, ccP4/11) 9-tone MOS: 5 4 5 5 4 5 4 5 4
10 292.68 Quasitemp (P8, c3P4/14) 29-tone MOS
11 321.95 Superkleismic (P8, ccP4/9) 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
12 351.22 Hemif / hemififths / salsa
Karadeniz 		(P8, P5/2) 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
13 380.49 Magic / witchcraft
Quanharuk 		(P8, P12/5) 10-tone MOS: 2 9 2 2 9 2 2 9 2 2
14 409.76 Hocum
Hocus 		(P8, c3P4/10) 32-tone MOS
15 439.02 Superthird (P8, c6P5/18) 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
16 468.29 Barbad (P8, c7P4/19) 8-tone MOS: 7 2 7 7 2 7 7 2
17 497.56 Helmholtz / garibaldi / cassandra / andromeda
Kwai 		(P8, P5) 12-tone MOS: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
18 526.83 Trismegistus (P8, c6P5/15) 9-tone MOS: 5 5 3 5 5 5 5 3 5
19 556.10 Alphorn (P8, c7P4/16) 9-tone MOS: 3 3 3 10 3 3 3 3 10
20 585.37 Pluto
Merman 		(P8, c3P4/7) Pathological 35-tone MOS

Scales and modes

A list of 41edo modes (MOS and others). See also Kite Guitar Scales and Kite Giedraitis's Categorizations of 41edo Scales.

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 Relationship between Bohlen-Pierce and octave-ful temperaments, and see this chart:

3 degrees of 41edo near 88cET overlap 5 degrees of 41edo near BP
41edo 88cET cents cents cents BP 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

41edo Electric guitar, by Gregory Sanchez.

Ron_Sword_with_a_41ET_Guitar.jpg

41edo Classical guitar, by Ron Sword.

The Kite Guitar (see also Kite Tuning) is a guitar fretting using every other step of 41edo, i.e. 41ed4 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 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, 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 41edo keyboard design:

41edo keyboard layout.png

See also 41-edo Keyboards for Lumatone, Linnstrument and Harpejji options, as well as DIY options.

41edo as a Universal Tuning

41's claim to fame as a "universal tuning" is the fact that it approximates scales present in many important world music traditions, and thus is good for both combining and exploring cultural playstyles.

Western

Due to 41edo's extremely accurate perfect fifth, it makes a good tuning for schismatic temperament, which in turn is a good approximation of the standard 12edo scale, and when arranged as a Bbb-D gamut, approximates the 12 note Pythagorean tuning known as Kirnberger I. This extends the Ptolemy Diatonic Scale (7 6 4 7 6 7 4), which 41 approximates excellently, by completing the circle of fifths with pure 3/2s. By using this system and occasionally substituting in alternate major seconds and sixths when necessary, it becomes quite reminiscent of (and improves on) 12edo harmony. Additionally, the Pythagorean Pentatonic scale can be used for melodies overtop due to the strong quartal nature of the scale. The Pythagorean diatonic scale exists as an option as well, but use may be limited unless Gentle triads are ideal.

Middle Eastern

See also: Arabic, Turkish, Persian

While the Hemif[7] scale itself and MODMOSes related to it give the middle eastern sound well, 41 has other interesting properties that make it an ideal system for Arabic and Turkish music. It is considered a "Level 2 EDO" due to the fact that it has neutral seconds and thirds as well as submajor and supraminor ones added to a Pythagorean skeleton, with small semitones as minor seconds and major whole tones as major seconds. The submajor third is great for Turkish Rast, around Ozan Yarman’s ideal size, and is sharp enough to sound close to a 5/4, while the neutral third exists as half of a 3/2 and works well for Arabic Rast and some Persian scales. Additionally, a large apotome exists for the Hijaz maqam.

Indonesian

Gamelan music is mainly based on two scales, the older Slendro and newer Pelog, though these scales are expanded on extensively through octave stretching, extensions and combination of the scales, and more. Slendro is excellently approximated by the 8\41 generator. Pelog is approximated quite well also, this time by mavila temperament, using the "grave" fifth of 41 as the generator (23\41).

Thai

Classical Thai music, which often is played in 7edo or a system resembling it, is well represented in 41, due to the tetracot temperament that splits the perfect fifth into 4 equal parts, each about the size of a 7edo second.

Indian

See also: Magic22 as srutis

Carnatic music, which is normally based on a 22-note unequal scale, has found some use from 22edo as a good approximation, but 41 offers another option with Magic[22], which not only represents 22edo closely, but preserves accurate perfects fifths and the unequal quality of a more typical carnatic scale.

Japanese

Japanese classical music known as Gagaku is largely built around winds, strings, and percussion, and the melodies, like many Asian cultures, are built around Pythagorean pentatonic scales, alongside chromaticism with narrow semitones, which are well represented by Pythagorean limmas.

Blues

Due to its pure sounding major thirds and approximations of standard western harmony, 41 naturally is good for jazz and blues music, though a great strength of this system as opposed to many others is its excellent harmonic seventh, alongside MOS scales that supply them, Miracle in particular.

Coltrane changes can be represented with two Pythagorean major thirds and a pental one, or an MOS like Magic, whose circles of major thirds give options for rotating major and minor triads within one scale.

Blue notes, rather than being considered inflections, can be notated as accidentals instead, such as the "blue third" which is represented by a neutral third, or any number of septimal intervals that sound great in a blues context.

Other

Georgian Polyphonic singing can be done in a 41edo context due to its excellent approximations of prime harmonics and neutral third, as well as Pythagorean seconds and sevenths. Asian musical traditions built around pentatonic scales can use both Pythagorean and Barbad[5].

Music

See also

External links

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