31edo
User:IlL/Template:RTT restriction
| ← 30edo | 31edo | 32edo → |
Thirty-one tone equal temperament, also called 31-tET, 31-EDO, 31-et, or tricesimoprimal meantone temperament, is the scale derived by dividing the octave into 31 equally large steps. The term Tricesimoprimal was first used by Adriaan Fokker.
Basic theory
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0 | -5.18 | +0.8 | -1.1 | -9.4 | +11.1 | +11.2 | +12.2 | -8.9 |
| relative (%) | 0 | -13 | +2 | -3 | -24 | +29 | +29 | +31 | -23 | |
| nearest edomapping | 31 | 18 | 10 | 25 | 14 | 22 | 3 | 8 | 16 | |
| fifthspan | 0 | +1 | +4 | +10 | -13 | +15 | -5 | -3 | -6 | |
Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 cents. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of quarter-comma meantone. 31's approximation of 7/4, a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the 11-limit, although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).
Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo and a zeta peak integer edo.
31edo's 5\31 neutral third generator generates mosh and dicoid MOSes. Its 12\31 generator generates an oneirotonic scale, similar to the 5L 3s scale in 13edo but with the 9/8 and 5/4 better in tune.
One step of 31edo, measuring about 38.7¢, is called a diesis because it stands in for several intervals called "dieses" (such as 128/125 and 648/625) which are tempered out in 12edo. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in SpiralProgressions. Zhea Erose's 31edo music uses the interval frequently.
31edo is the 11th prime edo, following 29edo and coming before 37edo.
Intervals
| Degree | Cents | Approximate Ratios | Ups and Downs Notation | Extended pythagorean notation | ||||
|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | perfect unison | D | P1 | perfect unison | D |
| 1 | 38.71 | 45/44, 49/48, 46/45, 128/125, 36/35 | ^1, d2 | up-unison, dim 2nd | ^D, Ebb | d2 | dim 2nd | Ebb |
| 2 | 77.42 | 25/24, 21/20, 22/21, 23/22 | A1, vm2 | aug 1sn, downminor 2nd | D#, vEb | A1 | aug 1sn | D# |
| 3 | 116.13 | 15/14, 16/15 | m2 | minor 2nd | Eb | m2 | minor 2nd | Eb |
| 4 | 154.84 | 12/11, 11/10, 35/32 | ~2 | mid 2nd | vE | AA1, dd3 | double-aug 1sn, double-dim 3rd | Dx, Fbb |
| 5 | 193.55 | 9/8, 10/9, 19/17, 28/25 | M2 | major 2nd | E | M2 | major 2nd | E |
| 6 | 232.26 | 8/7, 144/125 | ^M2 | upmajor 2nd | ^E | d3 | dim 3rd | Fb |
| 7 | 270.97 | 7/6, 75/64 | vm3 | downminor 3rd | vF | A2 | aug 2nd | E# |
| 8 | 309.68 | 6/5, 25/21 | m3 | minor 3rd | F | m3 | minor 3rd | F |
| 9 | 348.39 | 11/9, 27/22, 16/13, 60/49, 49/40 | ~3 | mid 3rd | ^F | AA2, dd4 | double-aug 2nd, double-dim 4th | Ex, Gbb |
| 10 | 387.10 | 5/4 | M3 | major 3rd | F# | M3 | major 3rd | F# |
| 11 | 425.81 | 9/7, 14/11, 23/18, 32/25 | ^M3 | upmajor 3rd | ^F# | d4 | dim 4th | Gb |
| 12 | 464.52 | 21/16, 13/10, 17/13, 125/96 | v4 | down-4th | vG | A3 | aug 3rd | Fx |
| 13 | 503.23 | 4/3 | P4 | perfect 4th | G | P4 | perfect 4th | G |
| 14 | 541.94 | 11/8, 15/11, 26/19 | ^4, ~4 | up-4th, mid 4th | ^G | AA3, dd5 | double-aug 3rd, double-dim 5th | Fx#, Abb |
| 15 | 580.65 | 7/5, 45/32, 25/18 | A4, vd5 | aug 4th, downdim 5th | G#, vAb | A4 | aug 4th | G# |
| 16 | 619.35 | 10/7, 64/45, 36/25 | ^A4, d5 | upaug 4th, dim 5th | ^G#, Ab | d5 | dim 5th | Ab |
| 17 | 658.06 | 16/11, 22/15, 19/13 | v5, ~5 | down-5th, mid 5th | vA | AA4, dd6 | double-aug 4th, double-dim 6th | Gx, Bbbb |
| 18 | 696.77 | 3/2 | P5 | perfect 5th | A | P5 | perfect 5th | A |
| 19 | 735.48 | 32/21, 20/13, 26/17, 192/125 | ^5 | up-5th | ^A | d6 | dim 6th | Bbb |
| 20 | 774.19 | 14/9, 11/7, 25/16 | vm6 | downminor 6th | vBb | A5 | aug 5th | A# |
| 21 | 812.90 | 8/5 | m6 | minor 6th | Bb | m6 | minor 6th | Bb |
| 22 | 851.61 | 18/11, 44/27, 13/8, 49/30, 80/49 | ~6 | mid 6th | vB | AA5, dd7 | double-aug 5th, double-dim 7th | Ax, Cbb |
| 23 | 890.32 | 5/3, 42/25 | M6 | major 6th | B | M6 | major 6th | B |
| 24 | 929.03 | 12/7, 128/75 | ^M6 | upmajor 6th | ^B | d7 | dim 7th | Cb |
| 25 | 967.74 | 7/4, 125/72 | vm7 | downminor 7th | vC | A6 | aug 6th | B# |
| 26 | 1006.45 | 16/9, 9/5, 34/19, 25/14 | m7 | minor 7th | C | m7 | minor 7th | C |
| 27 | 1045.16 | 11/6, 20/11, 64/35 | ~7 | mid 7th | ^C | AA6, dd8 | double-aug 6th, double-dim 8ve | Bx, Dbb |
| 28 | 1083.87 | 28/15, 15/8 | M7 | major 7th | C# | M7 | major 7th | C# |
| 29 | 1122.58 | 48/25, 40/21, 21/11, 44/23 | ^M7 | upmajor 7th | ^C# | d8 | dim 8ve | Db |
| 30 | 1161.29 | 88/45, 96/49, 45/23, 125/64, 35/18 | v8 | down-8ve | vD | A7 | aug 7th | Cx |
| 31 | 1200.00 | 2/1 | P8 | perfect 8ve | D | P8 | perfect 8ve | D |
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color name | monzo format | examples |
|---|---|---|---|
| downminor | zo | {a, b, 0, 1} | 7/6, 7/4 |
| minor | fourthward wa | {a, b}, b < -1 | 32/27, 16/9 |
| " | 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 |
| major | yo | {a, b, 1} | 5/4, 5/3 |
| " | fifthward wa | {a, b}, b > 1 | 9/8, 27/16 |
| upmajor | ru | {a, b, 0, -1} | 9/7, 12/7 |
All 31edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. 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). Here are the zo, gu, ilo, yo and ru triads:
| color of the 3rd | JI chord | edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| zo | 6:7:9 | 0-7-18 | C vEb G | Cvm | C downminor |
| gu | 10:12:15 | 0-8-18 | C Eb G | Cm | C minor |
| ilo | 18:22:27 | 0-9-18 | C vE G | C~ | C mid |
| yo | 4:5:6 | 0-10-18 | C E G | C | C major or C |
| ru | 14:18:21 | 0-11-18 | C ^E G | C^ | C upmajor or C up |
For a more complete list of chords, see Ups and Downs Notation - Chords and Chord Progressions.
Notations
From the appendix to The Sagittal Songbook by Jacob A. Barton, a diagram of how to notate 31-EDO in the Revo flavor of Sagittal:
Just approximation
Selected just intervals
15-odd-limit interval mappings
The following table shows how 15-odd-limit intervals are represented in 31edo. Prime harmonics are in bold; inconsistent intervals are in italic.
| Interval, complement | Error (abs, ¢) |
|---|---|
| 5/4, 8/5 | 0.783 |
| 11/9, 18/11 | 0.979 |
| 8/7, 7/4 | 1.084 |
| 7/5, 10/7 | 1.867 |
| 15/14, 28/15 | 3.314 |
| 7/6, 12/7 | 4.097 |
| 12/11, 11/6 | 4.202 |
| 16/15, 15/8 | 4.398 |
| 15/11, 22/15 | 4.985 |
| 4/3, 3/2 | 5.181 |
| 6/5, 5/3 | 5.964 |
| 14/11, 11/7 | 8.298 |
| 9/7, 14/9 | 9.278 |
| 11/8, 16/11 | 9.382 |
| 11/10, 20/11 | 10.166 |
| 13/10, 20/13 | 10.302 |
| 9/8, 16/9 | 10.362 |
| 16/13, 13/8 | 11.085 |
| 10/9, 9/5 | 11.145 |
| 14/13, 13/7 | 12.169 |
| 15/13, 26/15 | 15.483 |
| 13/12, 24/13 | 16.266 |
| 18/13, 13/9 | 17.263 |
| 13/11, 22/13 | 18.242 |
| Interval, complement | Error (abs, ¢) |
|---|---|
| 5/4, 8/5 | 0.783 |
| 11/9, 18/11 | 0.979 |
| 8/7, 7/4 | 1.084 |
| 7/5, 10/7 | 1.867 |
| 15/14, 28/15 | 3.314 |
| 7/6, 12/7 | 4.097 |
| 12/11, 11/6 | 4.202 |
| 16/15, 15/8 | 4.398 |
| 15/11, 22/15 | 4.985 |
| 4/3, 3/2 | 5.181 |
| 6/5, 5/3 | 5.964 |
| 14/11, 11/7 | 8.298 |
| 9/7, 14/9 | 9.278 |
| 11/8, 16/11 | 9.382 |
| 11/10, 20/11 | 10.166 |
| 13/10, 20/13 | 10.302 |
| 9/8, 16/9 | 10.362 |
| 16/13, 13/8 | 11.085 |
| 10/9, 9/5 | 11.145 |
| 14/13, 13/7 | 12.169 |
| 15/13, 26/15 | 15.483 |
| 13/12, 24/13 | 16.266 |
| 13/11, 22/13 | 20.468 |
| 18/13, 13/9 | 21.447 |
Selected 19-limit intervals
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) of 31et.
| 3-limit | 5-limit | 7-limit | 11-limit | 13-limit | ||
|---|---|---|---|---|---|---|
| Octave stretch (¢) | +1.63 | +0.98 | +0.83 | +1.21 | +0.50 | |
| Error | absolute (¢) | 1.64 | 1.63 | 1.43 | 1.49 | 2.07 |
| relative (%) | 4.22 | 4.20 | 3.70 | 3.84 | 5.35 | |
- 31et has a lower relative error than any previous ETs in all harmonic limits from 7 to 31.
Relationship to 12-edo
Whereas 12-edo has a circle of twelve 5ths, 31-edo has a spiral of twelve 5ths (since 18\31 is on the 7\12 kite in the scale tree). This spiral of 5th shows 31-edo in a 12-edo-friendly format. Excellent for introducing 31-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
31edo can be notated with a seperate semi/sesqui sharp/flat chain (like 17edo), with its own enharmonic circle of fifths.
MOS scales
| Generator | Cents | MOSes |
|---|---|---|
| 2\31 | 77.42 | 1L 14s 15L 1s |
| 3\31 | 116.13 | 1L 8s 1L 9s 10L 1s 10L 11s |
| 4\31 | 154.84 | 1L 6s 7L 1s 8L 7s 8L 15s |
| 5\31 | 193.55 | 1L 5s 6L 7s 6L 13s 6L 19s |
| 6\31 | 232.26 | 1L 4s 5L 6s 5L 11s 5L 16s 5L 21s |
| 7\31 | 270.97 | 4L 1s 4L 5s 9L 4s 9L 13s |
| 8\31 | 309.68 | 4L 3s 4L 7s 4L 11s 4L 15s |
| 9\31 | 348.39 | 3L 4s 7L 3s 7L 10s 7L 17s |
| 10\31 | 387.10 | 3L 7s 3L 10s 3L 13s 3L 16s 3L 19s 3L 22s 3L 25s |
| 11\31 | 425.81 | 3L 2s 3L 5s 3L 8s 3L 11s 14L 3s |
| 12\31 | 464.52 | 3L 2s 5L 3s 5L 8s 13L 5s |
| 13\31 | 503.23 | 2L 3s 5L 2s 7L 5s 12L 7s |
| 14\31 | 541.94 | 2L 5s 2L 7s 9L 2s 11L 9s |
| 15\31 | 580.65 | 2L 3s 2L 5s 2L 7s 2L 9s 2L 11s 2L 13s 2L 15s 2L 17s 2L 19s |
As a regular temperament
31edo represents a record in Pepper ambiguity in the 7-, 9- and 11-odd-limit, which it is consistent through. In the 13-limit it doesn't do as well, but is the optimal patent val for the rank five temperament tempering out the 13-limit comma 66/65, which equates 6/5 and 13/11. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
Commas
31 EDO tempers out the following commas. (Note: This assumes the val ⟨31 49 72 87 107 115], comma values rounded to 5 significant digits.)
| Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 5 | (16 digits) | [-25 7 6⟩ | 31.567 | Lala-tribiyo | Ampersand |
| 5 | 81/80 | [-4 4 -1⟩ | 21.506 | Gu | Syntonic comma, Didymos comma, meantone comma |
| 5 | (12 digits) | [17 1 -8⟩ | 11.445 | Saquadbigu | Würschmidt comma |
| 5 | (14 digits) | [-21 3 7⟩ | 10.061 | Lasepyo | Semicomma, Fokker comma |
| 5 | (24 digits) | [38 -2 -15⟩ | 1.3843 | Sasa-quintrigu | Hemithirds comma |
| 7 | (18 digits) | [-10 7 8 -7⟩ | 22.413 | Lasepru-aquadbiyo | Blackjackisma |
| 7 | 64827/64000 | [-9 3 -3 4⟩ | 22.227 | Laquadzo-atrigu | Squalentine |
| 7 | 2430/2401 | [1 5 1 -4⟩ | 20.785 | Quadru-ayo | Nuwell |
| 7 | 50421/50000 | [-4 1 -5 5⟩ | 14.516 | Quinzogu | Trimyna |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.795 | Zotrigu | Septimal semicomma, Starling comma |
| 7 | 1728/1715 | [6 3 -1 -3⟩ | 13.074 | Trizo-agu | Orwellisma, Orwell comma |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.4327 | Latrizo | Gamelisma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.7115 | Ruyoyo | Septimal kleisma, Marvel comma |
| 7 | 16875/16807 | [0 3 4 -5⟩ | 6.9903 | Quinru-aquadyo | Mirkwai |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.0832 | Zozoquingu | Hemimean |
| 7 | 6144/6125 | [11 1 -3 -2⟩ | 5.3621 | Sarurutrigu | Porwell |
| 7 | (18 digits) | [-26 -1 1 9⟩ | 3.7919 | Latritrizo-ayo | Wadisma |
| 7 | 65625/65536 | [-16 1 5 1⟩ | 2.3495 | Lazoquinyo | Horwell |
| 7 | (12 digits) | [-11 2 7 -3⟩ | 1.6283 | Latriru-asepyo | Meter |
| 7 | 2401/2400 | [-5 -1 -2 4⟩ | 0.72120 | Bizozogu | Breedsma |
| 11 | 99/98 | [-1 2 0 -2 1⟩ | 17.576 | Loruru | Mothwellsma |
| 11 | 121/120 | [-3 -1 -1 0 2⟩ | 14.367 | Lologu | Biyatisma |
| 11 | 176/175 | [4 0 -2 -1 1⟩ | 9.8646 | Lorugugu | Valinorsma |
| 11 | 243/242 | [-1 5 0 0 -2⟩ | 7.1391 | Lulu | Rastma |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.5026 | Lozoyo | Keenanisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.9302 | Luzozogu | Werckisma |
| 11 | 540/539 | [2 3 1 -2 -1⟩ | 3.2090 | Lururuyo | Swetisma |
| 11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.57240 | Loloruyoyo | Lehmerisma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Rank-2 temperaments
List of 31et rank two temperaments by badness
List of edo-distinct 31et rank two temperaments
| Generator | Cents | Temperaments | Pergen |
|---|---|---|---|
| 1\31 | 38.71 | Slender | (P8, P4/13) |
| 2\31 | 77.42 | Valentine/Lupercalia | (P8, P5/9) |
| 3\31 | 116.13 | Miracle | (P8, P5/6) |
| 4\31 | 154.84 | Nusecond | (P8, P11/11) |
| 5\31 | 193.55 | Luna/Hemithirds/Hemiwürschmidt | (P8, WWP4/15) |
| 6\31 | 232.26 | Mothra/Mosura | (P8, P5/3) |
| 7\31 | 270.97 | Orwell/Orson/Winston | (P8, P12/7) |
| 8\31 | 309.68 | Myna | (P8, WWP5/10) |
| 9\31 | 348.39 | Vicentino/Mohajira/Migration | (P8, P5/2) |
| 10\31 | 387.10 | Würschmidt/Worschmidt | (P8, WWP5/8) |
| 11\31 | 425.81 | Squares/Sentinel | (P8, P11/4) |
| 12\31 | 464.52 | A-Team/Semisept | (P8, M9/3) |
| 13\31 | 503.23 | Meantone/Meanpop | (P8, P5) |
| 14\31 | 541.94 | Casablanca/Cypress/Oracle | (P8, W5P4/12) |
| 15\31 | 580.65 | Tritonic/Tritoni | (P8, WWP4/5) |
Scales
Harmonic Scale
31edo approximates Mode 8 of the harmonic series O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the 13-limit.
| 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 31edo: | 0 | 5 | 10 | 14 | 18 | 22 | 25 | 28 | 31 |
| ...in cents: | 0 | 193.5 | 387.1 | 541.9 | 696.8 | 851.6 | 967.7 | 1083.9 | 1200.0 |
In mode 16, the most closely-matched harmonics are the composite ones, 21 and 25. Of the other harmonics:
- 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent.
- 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent.
- 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent.
- 27 is quite flat, as it's 3^3 and the error from the meantone fifths accumulates.
- 29 and 31 are both very sharp, and intervals involving them are unlikely to play any major role.
| Odd overtones in "Mode 16": | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 |
| ...as JI Ratio from 1/1: | 17/16 | 19/16 | 21/16 | 23/16 | 25/16 | 27/16 | 29/16 | 31/16 |
| ...in cents: | 105.0 | 297.5 | 470.8 | 628.3 | 772.6 | 905.9 | 1029.6 | 1145.0 |
| Nearest degree of 31edo: | 3 | 8 | 12 | 16 | 20 | 23 | 27 | 30 |
| ...in cents: | 116.1 | 309.7 | 464.5 | 619.4 | 774.2 | 890.3 | 1045.1 | 1161.3 |
Various Modes
A large open list of modes (subsets) from 31edo that people have named: 31edo modes. Strictly proper 7-note 31edo scales in the sense of David Rothenberg. Interesting (to somebody) 9-note 31edo scales. See also 31edo MOS scales. Some of the popular ones:
- 31-tone major: 5 5 3 5 5 5 3
- Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
- Harmonic scale 8: 5 5 4 4 4 3 3 3
- the Euler-Fokker genera (technically JI but representable in 31)
| 2 3 3 2 3 2 3 2 3 3 2 3 | Meantone Chromatic (53/220-comma) |
| 5 5 3 5 5 5 3 | Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian |
| 5 3 5 5 3 5 5 | Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian |
| 5 3 5 5 5 5 3 | Thirty-one tone Melodic Minor |
| 5 3 5 5 3 7 3 | Thirty-one tone Harmonic Minor |
| 5 5 3 5 3 7 3 | Thirty-one tone Harmonic Major |
| 5 5 3 5 3 5 5 | Thirty-one tone Major-Minor |
| 5 8 5 13 | Genus primum |
| 10 3 5 5 5 3 | Genus secundum |
| 8 2 8 3 7 3 | Genus tertium |
| 10 10 10 1 | Genus quartum |
| 5 7 6 7 5 1 | Genus quintum |
| 4 6 2 6 4 3 3 3 | Genus sextum |
| 4 6 5 6 4 6 | Genus septimum |
| 6 6 6 1 6 6 | Genus octavum |
| 4 6 9 6 4 2 | Genus nonum |
| 13 6 6 6 | Genus decimum |
| 5 5 3 5 5 3 2 3 | Genus diatonicum |
| 3 5 2 3 5 3 2 5 3 | Genus chromaticum |
| 5 5 2 1 5 5 2 3 3 | Genus diatonicum cum septimis |
| 3 4 3 3 2 1 4 1 4 1 2 3 | Genus enharmonicum vocale |
| 2 2 4 2 2 3 3 3 1 3 3 3 | Genus enharmonicum instrumentale |
| 3 2 3 2 3 2 3 3 2 3 2 3 | Genus diatonico-chromaticum |
| 5 2 1 2 5 3 2 1 4 1 2 3 | Genus bichromaticum |
| 4 4 5 4 4 5 5 | Neutral Diatonic Mixolydian |
| 4 5 4 4 5 5 4 | Neutral Diatonic Lydian |
| 5 4 4 5 5 4 4 | Neutral Diatonic Phrygian |
| 4 4 5 5 4 4 5 | Neutral Diatonic Dorian |
| 4 5 5 4 4 5 4 | Neutral Diatonic Hypolydian |
| 5 5 4 4 5 4 4 | Neutral Diatonic Hypophrygian |
| 5 4 4 5 4 4 5 | Neutral Diatonic Hypodorian |
| 4 5 4 4 5 4 5 | Neutral Mixolydian |
| 5 4 4 5 4 5 4 | Neutral Lydian |
| 4 4 5 4 5 4 5 | Neutral Phrygian |
| 4 5 4 5 4 5 4 | Neutral Dorian |
| 5 4 5 4 5 4 4 | Neutral Hypolydian |
| 4 5 4 5 4 4 5 | Neutral Hypophrygian |
| 5 4 5 4 4 5 4 | Neutral Hypodorian |
| 2 2 9 2 2 9 5 | Hemiolic Chromatic Mixolydian |
| 2 9 2 2 9 5 2 | Hemiolic Chromatic Lydian |
| 9 2 2 9 5 2 2 | Hemiolic Chromatic Phrygian |
| 2 2 9 5 2 2 9 | Hemiolic Chromatic Dorian |
| 2 9 5 2 2 9 2 | Hemiolic Chromatic Hypolydian |
| 9 5 2 2 9 2 2 | Hemiolic Chromatic Hypophrygian |
| 5 2 2 9 2 2 9 | Hemiolic Chromatic Hypodorian |
| 2 3 8 2 3 8 5 | Ratio 2:3 Chromatic Mixolydian |
| 3 8 2 3 8 5 2 | Ratio 2:3 Chromatic Lydian |
| 8 2 3 8 5 2 3 | Ratio 2:3 Chromatic Phrygian |
| 2 3 8 5 2 3 8 | Ratio 2:3 Chromatic Dorian |
| 3 8 5 2 3 8 2 | Ratio 2:3 Chromatic Hypolydian |
| 8 5 2 3 8 2 3 | Ratio 2:3 Chromatic Hypophrygian |
| 5 2 3 8 2 3 8 | Ratio 2:3 Chromatic Hypodorian |
| 3 5 5 3 5 5 5 | Intense Diatonic Mixolydian, M.Locrian |
| 5 3 5 5 5 3 5 | Intense Diatonic Phrygian, M.Dorian |
| 3 5 5 5 3 5 5 | Intense Diatonic Dorian, M.Phrygian |
| 5 5 5 3 5 5 3 | Intense Diatonic Hypolydian, M.Lydian |
| 5 5 3 5 5 3 5 | Intense Diatonic Hypophrygian, M.Mixolydian |
| 2 5 6 2 5 6 5 | Soft Diatonic Mixolydian |
| 5 6 2 5 6 5 2 | Soft Diatonic Lydian |
| 6 2 5 6 5 2 5 | Soft Diatonic Phrygian |
| 2 5 6 5 2 5 6 | Soft Diatonic Dorian |
| 5 6 5 2 5 6 2 | Soft Diatonic Hypolydian |
| 6 5 2 5 6 2 5 | Soft Diatonic Hypophrygian |
| 5 2 5 6 2 5 6 | Soft Diatonic Hypodorian |
| 1 2 10 1 2 10 5 | Enharmonic Mixolydian |
| 2 10 1 2 10 5 1 | Enharmonic Lydian |
| 10 1 2 10 5 1 2 | Enharmonic Phrygian |
| 1 2 10 5 1 2 10 | Enharmonic Dorian |
| 2 10 5 1 2 10 1 | Enharmonic Hypolydian |
| 10 5 1 2 10 1 2 | Enharmonic Hypophrygian |
| 5 1 2 10 1 2 10 | Enharmonic Hypodorian |
| 6 6 7 6 6 | Quasi-equal Pentatonic |
| 3 2 2 3 3 2 3 3 2 2 3 3 | Fokker 12-tone |
| 5 3 5 3 5 2 5 3 | Modus conjunctus |
| 3 5 2 5 3 5 3 5 | Octatonic |
| 3 3 4 3 5 3 4 3 3 | Hahn symmetric pentachordal |
| 3 4 3 3 5 3 4 3 3 | Hahn pentachordal |
| 4 4 2 5 3 3 4 3 3 | Hahn Nonatonic |
| 5 1 5 1 5 1 5 1 5 1 1 | de Vries 11-tone |
| 4 1 4 4 4 1 4 4 1 4 | Breed 10-tone |
| 4 2 4 2 4 2 4 3 3 3 | Lumma Decatonic |
| 5 3 3 3 3 5 3 3 3 | Rothenberg Generalized Diatonic |
| 5 2 6 5 2 5 6 | "Septimal" Natural Minor |
| 4 3 4 3 4 3 4 3 3 | Thirty-one tone Orwell |
| 2 5 2 2 5 2 2 2 5 2 2 | Secor Sentinel |
Trivia
31edo is close to a circle made by stacking 31 pure 17/13 subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents (relative error 7.1%).
Music
- An alphabetical list of Tricesimoprimal Compositions.
- Aire #2 in 31-equal temperament by Jon Lyle Smith [dead link]
- I Stand Hopeless Before the Gray Sea by Chuckles McGee
- Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’) by Claudi Meneghin
- earwig by Andrew Heathwaite
- Fanfare and Toccata by Juhani Nuorvala
- Orphanage of the Dutch Music IX: Sonate no. II in the 31-tone temperament - YouTube
- "Lively Up Yourself" cover featuring Paul Erlich on 31edo guitar
- Q: IS ART REVOLUTIONARY? A: LOL NO by Diamond Doll (xen-pop)
- Cool Guitar Girl by Diamond Doll (xen-pop)
- Some songs by Benyamin Linus
By Cam Taylor
By Johann alias Circular17
- Mystère et tolérance
- Wave from the past
- Curieuse planète
- Deep but not too much
- Heal
- Désir mimétique (virtually 31edo)
By Zhea Erose
See Also
Pedagogy
The MicroPedagogyCollective is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two ThirtyOneToneSinginCamps as well.
See also: 31edo solfege, Tricesimoprimal Tetrachordal Tesseract, Pentachords of 31edo.
Books
External image: http://ronsword.com/images/TSG_sm.jpg [dead link]
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- Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.
Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009. [dead link] - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.
Articles
- Wikipedia: 31 equal temperament
- de Beer, Anton, The Development of 31-tone Music Permalink
- Fokker, Adriaan Daniël, Equal Temperament and the Thirty-one-keyed organ Permalink
- Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
- Rapoport, Paul, About 31-tone Equal Temperament Permalink
- Terpstra, Siemen, Toward a Theory of Meantone (and 31-et) Harmony Permalink
- Tonalsoft Encyclopedia article Permalink
Videos
- Quarter sharps and flats in the same diatonic key signature – Youtube by Stephen Weigel – a list of diatonic key signatures and major scales in 31edo (including semi- and sesqui-sharps); and docs in its description.