41edo
Contents
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). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.
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.
Intervals
| # | Cents | Approximate Ratios* | 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, 49/48 | up-unison | ^1 | ^D | di |
| 2 | 58.54 | 25/24, 28/27, 36/35, 33/32 | double-up unison, downminor 2nd | ^^1, vm2 | ^^D, vEb | ro |
| 3 | 87.80 | 21/20, 22/21, 19/18, 20/19 | down-aug 1sn, minor 2nd | vA1, m2 | vD#, Eb | rih |
| 4 | 117.07 | 16/15, 15/14, 14/13 | augmented 1sn, upminor 2nd | A1, ^m2 | D#, ^Eb | ra |
| 5 | 146.34 | 12/11, 13/12 | mid 2nd | ~2 | ^D#, vvE | ru |
| 6 | 175.61 | 10/9, 11/10, 21/19 | downmajor 2nd | vM2 | vE | reh |
| 7 | 204.88 | 9/8 | major 2nd | M2 | E | re |
| 8 | 234.15 | 8/7, 15/13 | upmajor 2nd | ^M2 | ^E | ri |
| 9 | 263.41 | 7/6, 22/19 | downminor 3rd | vm3 | vF | ma |
| 10 | 292.68 | 32/27, 13/11, 19/16 | minor 3rd | m3 | F | meh |
| 11 | 321.95 | 6/5 | upminor 3rd | ^m3 | ^F | me |
| 12 | 351.22 | 11/9, 27/22, 16/13 | mid 3rd | ~3 | ^^F, vGb | mu |
| 13 | 380.49 | 5/4, 26/21 | downmajor 3rd | vM3 | vF#, Gb | mi |
| 14 | 409.76 | 81/64, 14/11, 24/19, 19/15 | 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 | 27/20, 15/11, 19/14 | up-4th | ^4 | ^G | fih |
| 19 | 556.10 | 11/8, 18/13, 26/19 | 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, 19/13 | mid-5th | ~5 | vvA | su |
| 23 | 673.17 | 40/27, 22/15, 28/19 | 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 | 128/81, 11/7, 19/12, 30/19 | minor 6th | m6 | vA#, Bb | leh |
| 28 | 819.51 | 8/5, 21/13 | upminor 6th | ^m6 | A#, ^Bb | le |
| 29 | 848.78 | 18/11, 44/27, 13/8 | mid 6th | ~6 | ^A#, vvB | lu |
| 30 | 878.05 | 5/3 | downmajor 6th | vM6 | vB | la |
| 31 | 907.32 | 27/16, 22/13, 32/19 | major 6th | M6 | B | laa |
| 32 | 936.59 | 12/7, 19/11 | upmajor 6th | ^M6 | ^B | li |
| 33 | 965.85 | 7/4, 26/15 | downminor 7th | vm7 | vC | ta |
| 34 | 995.12 | 16/9 | minor 7th | m7 | C | teh |
| 35 | 1024.39 | 9/5, 20/11, 38/21 | upminor 7th | ^m7 | ^C | te |
| 36 | 1053.66 | 11/6, 24/13 | mid 7th | ~7 | ^^C, vDb | tu |
| 37 | 1082.93 | 15/8, 28/15, 13/7 | downmajor 7th | vM7 | vC#, Db | ti |
| 38 | 1112.20 | 40/21, 21/11, 36/19, 19/10 | major 7th | M7 | C#, ^Db | taa |
| 39 | 1141.46 | 48/25, 27/14, 35/18, 64/33 | upmajor 7th | ^M7 | C#^, vvD | to |
| 40 | 1170.73 | 160/81, 63/32, 96/49 | dim 8ve | v8 | vD | da |
| 41 | 1200.00 | 2/1 | perfect 8ve | P8 | D | do |
* Based on treating 41-edo 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 = 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 half-aug
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.
Notation
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
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | ||
|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.0 | +0.48 | -5.8 | -3.0 | +4.8 | +8.3 | +12.1 | -4.8 |
| relative (%) | 0.0 | +1.7 | -19.9 | -10.2 | +16.3 | +28.2 | +41.4 | -16.5 | |
| fifthspan | 0 | +1 | -8 | -14 | -18 | +20 | +7 | -3 | |
The following table shows how 15-odd-limit intervals are represented in 41edo. Prime harmonics are in bold.
| Interval, complement | Error (abs, ¢) |
|---|---|
| 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.
The same spiral, but with notes not intervals:
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 |
| " | 361/360 | 4.80 | [-3 -2 -1 0 0 0 0 2⟩ | Nonogu | 19oog2 | go comma |
| " | 513/512 | 3.38 | [-9 3 0 0 0 0 0 1⟩ | Lano | L19o | undevicesimal comma, Boethius' comma |
| " | 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⟩ | 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 | Temperament(s) | Pergen | Some MOS Scales Available |
|---|---|---|---|---|
| 1 | 29.27 | Sepla-sezo = [-100 33 0 17⟩ | (P8, P4/17) | |
| 2 | 58.54 | Hemimiracle | (P8, P5/12) | |
| 3 | 87.80 | 88cET (approx), Octacot | (P8, P5/8) | |
| 4 | 117.07 | Miracle | (P8, P5/6) | |
| 5 | 146.34 | Bohlen-Pierce/Bohpier | (P8, P12/13) | |
| 6 | 175.61 | Tetracot/Bunya/Monkey | (P8, P5/4) | 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5 |
| 7 | 204.88 | Baldy | (P8, c3P4/20) | 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1 |
| 8 | 234.15 | 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) | |
| 11 | 321.95 | Superkleismic | (P8, ccP4/9) | 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3 |
| 12 | 351.22 | Hemififths/Karadeniz | (P8, P5/2) | 10-tone MOS: 5 2 5 5 2 5 5 5 2 5 |
| 13 | 380.49 | Magic/Witchcraft | (P8, P12/5) | 10-tone MOS: 2 9 2 2 9 2 2 9 2 2 |
| 14 | 409.76 | Hocus | (P8, c3P4/10) | |
| 15 | 439.02 | Sasa-tritribizo = [5 -35 0 18⟩ | (P8, c6P5/18) | 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4 |
| 16 | 468.29 | Barbad | (P8, c7P4/19) | |
| 17 | 497.56 | Schismatic (Helmholtz, Garibaldi, Cassandra) | (P8, P5) | |
| 18 | 526.83 | Trismegistus | (P8, c6P5/15) | 9-tone MOS: 5 5 3 5 5 5 5 3 5 |
| 19 | 556.10 | Sasa-quadquadlu = [57 -1 0 0 -16⟩ | (P8, c7P4/16) | |
| 20 | 585.37 | Pluto | (P8, c3P4/7) |
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 | ||||
|---|---|---|---|---|---|---|
| 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-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. 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.
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
- 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
