31edo
← 30edo | 31edo | 32edo → |
31 equal divisions of the octave (abbreviated 31edo or 31ed2), also called 31-tone equal temperament (31tet) or 31 equal temperament (31et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 31 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 21/31, or the 31st root of 2.
31edo is also referred to as the tricesimoprimal meantone temperament. The term tricesimoprimal was first used by Adriaan Fokker.
Theory
31edo'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 (and specifically the 11-odd-limit), although it conflates 9/7 with 14/11 and 11/8 with 15/11. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also maps all 15-odd-limit intervals consistently, with the sole exceptions of 13/9, 13/11, 18/13, and 22/13.
Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo 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, meaning it is a strict zeta edo. Other ways in which 31edo is especially accurate is that it represents a record in Pepper ambiguity in the 7-, 9-, and 11-odd-limit, which it is consistent through, and that it is the first non-trivial edo to be consistent in the 11-odd-prime-sum-limit.
One step of 31edo, measuring about 38.7¢, is called a diesis because it stands in for several intervals called "dieses" (most notably, 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 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%). Remarkably, 31edo tempers out 83521/83486, the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's oneirotonic (5L 3s) mos accurate 13:17:19 chords.
Prime harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -5.2 | +0.8 | -1.1 | -9.4 | +11.1 | +11.2 | +12.2 | -8.9 | +15.6 | +16.3 | -19.1 |
Relative (%) | +0.0 | -13.4 | +2.0 | -2.8 | -24.2 | +28.6 | +28.9 | +31.4 | -23.0 | +40.3 | +42.0 | -49.3 | |
Steps (reduced) |
31 (0) |
49 (18) |
72 (10) |
87 (25) |
107 (14) |
115 (22) |
127 (3) |
132 (8) |
140 (16) |
151 (27) |
154 (30) |
161 (6) |
Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -3.3 | -8.3 | -7.4 | +16.8 | -14.0 | +5.7 | -1.9 | +13.9 | +4.5 | -16.1 | +14.5 | +9.8 |
Relative (%) | -8.4 | -21.4 | -19.2 | +43.4 | -36.2 | +14.7 | -4.9 | +35.8 | +11.5 | -41.7 | +37.4 | +25.2 | |
Steps (reduced) |
166 (11) |
168 (13) |
172 (17) |
178 (23) |
182 (27) |
184 (29) |
188 (2) |
191 (5) |
192 (6) |
195 (9) |
198 (12) |
201 (15) |
Subsets and supersets
31edo is the 11th prime edo, following 29edo and coming before 37edo.
Intervals
Degree | Cents | Approximate Ratios[note 1] | Ups and Downs Notation | Extended Pythagorean notation | SKULO notation (S or U = 1) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.00 | 1/1 | P1 | perfect unison | D | 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 | S1/U1 | super/uber unison | SD/UD |
2 | 77.42 | 25/24, 21/20, 22/21, 23/22 | A1, vm2 | aug 1sn, downminor 2nd | D#, vEb | A1 | aug 1sn | D# | sm2 | subminor 2nd | sEb |
3 | 116.13 | 15/14, 16/15 | m2 | minor 2nd | Eb | 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 | N2 | neutral 2nd | UEb/uE |
5 | 193.55 | 9/8, 10/9, 19/17, 28/25 | M2 | major 2nd | E | M2 | major 2nd | E | M2 | major 2nd | E |
6 | 232.26 | 8/7, 144/125 | ^M2 | upmajor 2nd | ^E | d3 | dim 3rd | Fb | SM2 | supermajor 2nd | SE |
7 | 270.97 | 7/6, 75/64 | vm3 | downminor 3rd | vF | A2 | aug 2nd | E# | sm3 | subminor 3rd | sF |
8 | 309.68 | 6/5, 25/21 | m3 | minor 3rd | F | 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 | N3 | neutral 3rd | UF/uF# |
10 | 387.10 | 5/4 | M3 | major 3rd | F# | 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 | SM3 | supermajor 3rd | SF# |
12 | 464.52 | 21/16, 64/49, 13/10, 17/13, 125/96 | v4 | down-4th | vG | A3 | aug 3rd | Fx | s4 | sub 4th | sG |
13 | 503.23 | 4/3 | P4 | perfect 4th | G | P4 | perfect 4th | G | P4 | perfect 4th | G |
14 | 541.94 | 175/128, 11/8, 15/11, 26/19 | ^4, ~4 | up-4th, mid 4th | ^G | AA3, dd5 | double-aug 3rd, double-dim 5th | Fx#, Abb | U4/N4 | uber/neutral 4th | UG |
15 | 580.65 | 7/5, 45/32, 25/18 | A4, vd5 | aug 4th, downdim 5th | G#, vAb | A4 | aug 4th | G# | 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 | d5 | dim 5th | Ab |
17 | 658.06 | 256/175, 16/11, 22/15, 19/13 | v5, ~5 | down-5th, mid 5th | vA | AA4, dd6 | double-aug 4th, double-dim 6th | Gx, Bbbb | u5/N5 | unter/neutral 5th | uA |
18 | 696.77 | 3/2 | P5 | perfect 5th | A | P5 | perfect 5th | A | P5 | perfect 5th | A |
19 | 735.48 | 32/21, 49/32, 20/13, 26/17, 192/125 | ^5 | up-5th | ^A | d6 | dim 6th | Bbb | S5 | super 5th | SA |
20 | 774.19 | 14/9, 11/7, 25/16 | vm6 | downminor 6th | vBb | A5 | aug 5th | A# | sm6 | subminor 6th | sBb |
21 | 812.90 | 8/5 | m6 | minor 6th | Bb | 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 | N6 | neutral 6th | UBb/uB |
23 | 890.32 | 5/3, 42/25 | M6 | major 6th | B | M6 | major 6th | B | M6 | major 6th | B |
24 | 929.03 | 12/7, 128/75 | ^M6 | upmajor 6th | ^B | d7 | dim 7th | Cb | SM6 | supermajor 6th | SB |
25 | 967.74 | 7/4, 125/72 | vm7 | downminor 7th | vC | A6 | aug 6th | B# | sm7 | subminor 7th | sC |
26 | 1006.45 | 16/9, 9/5, 34/19, 25/14 | m7 | minor 7th | C | 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 | N7 | neutral 7th | UC/uC# |
28 | 1083.87 | 28/15, 15/8 | M7 | major 7th | C# | 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 | SM7 | supermajor 7th | SC# |
30 | 1161.29 | 88/45, 96/49, 45/23, 125/64, 35/18 | v8 | down-8ve | vD | A7 | aug 7th | Cx | s8/u8 | sub 8th, unter 8ve | sD/uD |
31 | 1200.00 | 2/1 | P8 | perfect 8ve | D | P8 | perfect 8ve | D | P8 | perfect 8ve | D |
Interval quality and chord names in color notation
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⟩ where 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⟩ where 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 – E – G | Cvm | C subminor |
gu | 10:12:15 | 0 – 8 – 18 | C – E♭ – G | Cm | C minor |
ilo | 18:22:27 | 0 – 9 – 18 | C – E – G | C~ | C neutral |
yo | 4:5:6 | 0 – 10 – 18 | C – E – G | C, Cmaj | C, C major |
ru | 14:18:21 | 0 – 11 – 18 | C – E – G | C^ | C supermajor |
For a more complete list of chords, see 31edo Chord Names and Ups and Downs Notation #Chords and Chord Progressions.
Notation
Step Offset | −4 | −3 | −2 | −1 | 0 | +1 | +2 | +3 | +4 |
---|---|---|---|---|---|---|---|---|---|
Symbol |
Neutral circle-of-fifths notation
Neutral circle-of-fifths notation is much like ups and downs notation. Between C and D (do and re) for example, we have the following notes:
Degree | Letter | Name | English full name |
---|---|---|---|
0 | C | do | C |
1 | C | do | C half-sharp |
2 | C♯ | do ♯ | C sharp |
3 | D♭ | re ♭ | D flat |
4 | D | re | D half-flat |
5 | D | re | D |
Circle-of-fifths notation
Circle-of-fifths notation uses double sharps and double flats:
Degree | Letter | Name | English full name |
---|---|---|---|
0 | C | do | C |
1 | D𝄫 | re 𝄫 | D double flat |
2 | C♯ | do ♯ | C sharp |
3 | D♭ | re ♭ | D flat |
4 | C𝄪 | do 𝄪 | C double sharp |
5 | D | re | D |
While using double sharp and double flat may seem confusing because it alternates between C and D, it provides a way of writing chords that is consistent with traditional notation. For example, the subminor7 chord 12:14:18:21 is written like so:
- C / D♯ / G / A♯
- C♯ / D𝄪 / G♯ / A𝄪
- D♭ / E / A♭ / B
- D / E♯ / A / B♯
In 12edo, the enharmonic equivalences include C♯ = D♭, E♯ = F, and E = F♭. But in 31edo we have:
31edo can be notated with a seperate semi/sesqui sharp/flat chain (like 17edo), with its own enharmonic circle of fifths.
Sagittal notation
The Revo flavor of Sagittal notation from the appendix to The Sagittal Songbook by Jacob A. Barton:
MisterShafXen's notation
31edo can be notated as follows: A A# Bb B B# Cb C C# Db D D# Eb E E# Fb F F# Gb G G# Hb H H# Ib I I# Jb J J#/Kb K K#/Ab A.
Relationship to 12edo
31edo’s circle of 31 fifths can be bent into a 12-spoked "spiral of fifths". This is possible because 18\31 is on the 7\12 kite in the scale tree. Stated another way, it is possible because the absolute value of 31edo’s dodeca-sharpness (edosteps per Pythagorean comma) is 1.
This "spiral of fifths" can be a useful construct for introducing 31edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.
The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.
Scales
MOS scales
The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other MOSes and MOS chains [clarification needed ] are also useful:
- 9\31, the neutral third, generates ultrasoft mosh and superhard dicotonic MOSes.
- 11\31, the supermajor third or diminished fourth, generates a parahard sensoid scale with resolution from neutral thirds, sixths, and sevenths to perfect fourths, fifths, and octaves, and a semihard 3L 8s scale with a jagged-but-chromatic feel.
- 12\31 generator generates a semihard oneirotonic scale, similar to the 5L 3s scale in 13edo but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to 19/13.
- A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
- If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The ultrasoft 9-tone orwelloid (4L 5s) MOS could be treated as a 9-tone well temperament.
- It has close approximations to 6edf (→ miracle) and 9edf (→ Carlos Alpha), fifth-equivalent equal divisions that hit many good JI approximations.
See #Rank-2 temperaments for a table of MOSes and their temperament interpretations.
Harmonic scales
31edo approximates Mode 8 of the harmonic series okay, 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 okay, 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.
The steps are: 5 5 4 4 4 3 3 3.
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 (see mercy temperament).
- 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 subsets
A large open list of subsets from 31edo that people have named:
- 31edo modes
- Strictly proper 7-tone 31edo scales
- Interesting (to somebody) 9-tone 31edo scales
- the Erose-McClain double modes, which are nonoctave
- the Euler-Fokker genus (technically JI but representable in 31)
- the altered pentad
- diasem (2.3.7 subgroup scale; 5 2 5 1 5 2 5 1 5 or 5 1 5 2 5 1 5 2 5 in 31edo)
Approximation to JI
Interval mappings
The following tables show how 15-odd-limit intervals are represented in 31edo. Prime harmonics are in bold; inconsistent intervals are in italics.
Interval and complement | Error (abs, ¢) | Error (rel, %) |
---|---|---|
1/1, 2/1 | 0.000 | 0.0 |
5/4, 8/5 | 0.783 | 2.0 |
11/9, 18/11 | 0.979 | 2.5 |
7/4, 8/7 | 1.084 | 2.8 |
7/5, 10/7 | 1.867 | 4.8 |
15/14, 28/15 | 3.314 | 8.6 |
7/6, 12/7 | 4.097 | 10.6 |
11/6, 12/11 | 4.202 | 10.9 |
15/8, 16/15 | 4.398 | 11.4 |
15/11, 22/15 | 4.985 | 12.9 |
3/2, 4/3 | 5.181 | 13.4 |
5/3, 6/5 | 5.964 | 15.4 |
11/7, 14/11 | 8.298 | 21.4 |
9/7, 14/9 | 9.278 | 24.0 |
11/8, 16/11 | 9.382 | 24.2 |
11/10, 20/11 | 10.166 | 26.3 |
13/10, 20/13 | 10.302 | 26.6 |
9/8, 16/9 | 10.362 | 26.8 |
13/8, 16/13 | 11.085 | 28.6 |
9/5, 10/9 | 11.145 | 28.8 |
13/7, 14/13 | 12.169 | 31.4 |
15/13, 26/15 | 15.483 | 40.0 |
13/12, 24/13 | 16.266 | 42.0 |
13/9, 18/13 | 17.263 | 44.6 |
13/11, 22/13 | 18.242 | 47.1 |
Interval and complement | Error (abs, ¢) | Error (rel, %) |
---|---|---|
1/1, 2/1 | 0.000 | 0.0 |
5/4, 8/5 | 0.783 | 2.0 |
11/9, 18/11 | 0.979 | 2.5 |
7/4, 8/7 | 1.084 | 2.8 |
7/5, 10/7 | 1.867 | 4.8 |
15/14, 28/15 | 3.314 | 8.6 |
7/6, 12/7 | 4.097 | 10.6 |
11/6, 12/11 | 4.202 | 10.9 |
15/8, 16/15 | 4.398 | 11.4 |
15/11, 22/15 | 4.985 | 12.9 |
3/2, 4/3 | 5.181 | 13.4 |
5/3, 6/5 | 5.964 | 15.4 |
11/7, 14/11 | 8.298 | 21.4 |
9/7, 14/9 | 9.278 | 24.0 |
11/8, 16/11 | 9.382 | 24.2 |
11/10, 20/11 | 10.166 | 26.3 |
13/10, 20/13 | 10.302 | 26.6 |
9/8, 16/9 | 10.362 | 26.8 |
13/8, 16/13 | 11.085 | 28.6 |
9/5, 10/9 | 11.145 | 28.8 |
13/7, 14/13 | 12.169 | 31.4 |
15/13, 26/15 | 15.483 | 40.0 |
13/12, 24/13 | 16.266 | 42.0 |
13/11, 22/13 | 20.468 | 52.9 |
13/9, 18/13 | 21.447 | 55.4 |
Regular temperament properties
Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.3 | [-49 31⟩ | [⟨31 49]] | +1.63 | 1.64 | 4.22 |
2.3.5 | 81/80, 393216/390625 | [⟨31 49 72]] | +0.98 | 1.63 | 4.20 |
2.3.5.7 | 81/80, 126/125, 1029/1024 | [⟨31 49 72 87]] | +0.83 | 1.43 | 3.70 |
2.3.5.7.11 | 81/80, 99/98, 121/120, 126/125 | [⟨31 49 72 87 107]] | +1.21 | 1.49 | 3.84 |
2.3.5.7.11.13 | 66/65, 81/80, 99/98, 105/104, 121/120 | [⟨31 49 72 87 107 115]] | +0.50 | 2.07 | 5.35 |
31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are 72, 72, 41, and 46, respectively.
31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly consistently: to distance 10.36). In 2.5.7 it tempers out the didacus comma 3136/3125 and the quince comma 823543/819200, thus also tempering out the very small rainy comma, the simplest 2.5.7 comma tempered out by the 7-limit microtemperament 171edo. In the 11-limit, 31edo can be defined as the unique temperament that tempers out 81/80, 99/98, 121/120, and 126/125, and it supports orwell, mohajira, and the relatively high-accuracy temperament miracle. In the 13-limit 31edo 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. In the 17-limit it tempers out 120/119, equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad.
Commas
31edo tempers out the following commas. This assumes the val ⟨31 49 72 87 107 115], comma values rounded to 5 significant digits.
Prime limit |
Ratio[note 2] | Monzo | Cents | Color name | Name |
---|---|---|---|---|---|
3 | (30 digits) | [-49 31⟩ | 160.605 | Quadlawa | 31-comma |
5 | (16 digits) | [-25 7 6⟩ | 31.567 | Lala-tribiyo | Ampersand |
5 | 81/80 | [-4 4 -1⟩ | 21.506 | Gu | Syntonic comma |
5 | (12 digits) | [17 1 -8⟩ | 11.445 | Saquadbigu | Würschmidt comma |
5 | (14 digits) | [-21 3 7⟩ | 10.061 | Lasepyo | Semicomma |
5 | (24 digits) | [38 -2 -15⟩ | 1.3843 | Sasa-quintrigu | Hemithirds comma |
7 | 59049/57344 | [-13 10 0 -1⟩ | 50.72 | Laru | Harrison's comma |
7 | 3645/3584 | [-9 6 1 -1⟩ | 29.22 | Laruyo | Schismean 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 comma |
7 | 2430/2401 | [1 5 1 -4⟩ | 20.785 | Quadru-ayo | Nuwell comma |
7 | 50421/50000 | [-4 1 -5 5⟩ | 14.516 | Quinzogu | Trimyna comma |
7 | 126/125 | [1 2 -3 1⟩ | 13.795 | Zotrigu | Starling comma, septimal semicomma |
7 | 1728/1715 | [6 3 -1 -3⟩ | 13.074 | Trizo-agu | Orwellisma |
7 | 1029/1024 | [-10 1 0 3⟩ | 8.4327 | Latrizo | Gamelisma |
7 | 225/224 | [-5 2 2 -1⟩ | 7.7115 | Ruyoyo | Marvel comma, septimal kleisma |
7 | 16875/16807 | [0 3 4 -5⟩ | 6.9903 | Quinru-aquadyo | Mirkwai comma |
7 | 3136/3125 | [6 0 -5 2⟩ | 6.0832 | Zozoquingu | Hemimean comma |
7 | 6144/6125 | [11 1 -3 -2⟩ | 5.3621 | Sarurutrigu | Porwell comma |
7 | (18 digits) | [-26 -1 1 9⟩ | 3.7919 | Latritrizo-ayo | Wadisma |
7 | 65625/65536 | [-16 1 5 1⟩ | 2.3495 | Lazoquinyo | Horwell comma |
7 | (12 digits) | [-11 2 7 -3⟩ | 1.6283 | Latriru-asepyo | Meter comma |
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 |
13 | 105/104 | [-3 1 1 1 0 -1⟩ | 16.567 | Thuzoyo | Animist comma |
13 | 144/143 | [4 2 0 0 -1 -1⟩ | 12.064 | Thulu | Grossma |
13 | 196/195 | [2 -1 -1 2 0 -1⟩ | 8.8554 | Thuzozogu | Mynucuma |
13 | 625/624 | [-4 -1 4 0 0 -1⟩ | 2.77 | Thuquadyo | Tunbarsma |
13 | 1001/1000 | [-3 0 -3 1 1 1⟩ | 1.73 | Tholozotrigu | Fairytale comma, sinbadma |
13 | 4096/4095 | [12 -2 -1 -1 0 -1⟩ | 0.42 | Sathurugu | Schismina |
Rank-2 temperaments
- List of 31et rank two temperaments by badness
- List of edo-distinct 31et rank two temperaments
- Syntonic–31 equivalence continuum
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Instruments
Keyboard Instruments
String Instruments
Other Instruments
Music
- See also: Category:31edo tracks
See also
- Lumatone mapping for 31edo
- List of 31edo Chords
- Skip fretting system 31 2 9
- Pentachords of 31edo
- Tricesimoprimal Tetrachordal Tesseract
- MicroPedagogyCollective – is at work (as of 2012) producing demonstrative material which will encourage and enable more people to learn this system. There have been two ThirtyOneToneSinginCamps as well.
- CG-31
Notes
Further reading
Books
- Coates, Bill. Diesis: An Introduction to the Temperament of 31 Notes to Each Octave. Self-published, 1992.
- Sword, Ron. Tricesimoprimal Scales for Guitar: Scales for 31-EDO. 2009. (Metatonal Music link) (A comprehensive approach to 31edo and all the families associated for guitar. Features over 300 scale charts/scale examples.)
Articles
- The Development of 31-tone Music Permalink by Anton de Beer
- Equal Temperament and the Thirty-one-keyed organ Permalink by Adriaan Daniël Fokker
- New Music with 31 Notes by Adriaan Daniël Fokker, translated by Leigh Gerdine
- About 31-tone Equal Temperament Permalink by Paul Rapoport
- Toward a Theory of Meantone (and 31-et) Harmony Permalink by Siemen Terpstra
- 31-ed2 / 31-edo / 31-ET / 31-tone equal-temperament Permalink on Tonalsoft Encyclopedia
- Harmonic Resources of 31Et EMT and 31EBMT by Juhan Puhm (2016)
External links
Websites
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.
- Playlist of 31edo music theory videos on YouTube by Zhea Erose
Software
Diagrams
- Keys and Modes of 31Et by Juhan Puhm (2016)
- Keyboard Mapping for 31Et by Juhan Puhm (2017)
- Mapping Range for 31Et by Juhan Puhm (2017)