13edo
| ← 12edo | 13edo | 14edo → |
13 equal divisions of the octave (13edo) is a tuning system which divides the octave into 13 equal parts of approximately 92.3 cents each.
Theory
In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.
As a temperament of 21-odd-limit just intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +36.5 | -17.1 | -45.7 | -19.3 | +2.5 | -9.8 | +19.4 | -12.6 | -20.6 | -9.2 | +17.9 |
| Relative (%) | +39.5 | -18.5 | -49.6 | -20.9 | +2.7 | -10.6 | +21.0 | -13.7 | -22.3 | -10.0 | +19.4 | |
| Steps (reduced) |
21 (8) |
30 (4) |
36 (10) |
41 (2) |
45 (6) |
48 (9) |
51 (12) |
53 (1) |
55 (3) |
57 (5) |
59 (7) | |
Subsets and supersets
13edo is the sixth prime edo, following 11edo and coming before 17edo.
One step of 13edo is very close to 135/128 by direct approximation, in fact one might stress that it is a semiconvergent. The 5-limit aluminium temperament realizes this proximity through a regular temperament perspective, and EDOs supporting it (for example, 494edo or 1547edo), combine the sound of 13edo with relative simplicities and precision of 5-limit JI.
Intervals
| # | Cents | Approximated 21-odd-limit Ratios[1] | Erv Wilson | Archaeotonic | Oneirotonic | 26edo names | Fox-Raven Notation (J = 360Hz) |
Pseudo-Diatonic Category |
Audio |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 1/1 | H | C | C | C | J | Unison | |
| 1 | 92.31 | 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 | β | C#/Db | C#/Db | Cx/Dbb | J#/Kb | Minor second | |
| 2 | 184.62 | 9/8, 10/9, 11/10, 19/17, 21/19 | A | D | D | D | K | Major second | |
| 3 | 276.92 | 7/6, 13/11, 20/17, 19/16, 22/19 | δ | D#/Eb | D#/Eb | Dx/Ebb | L | Minor third | |
| 4 | 369.23 | 5/4, 11/9, 16/13, 26/21 | C | E | E | E | L#/Mb | Major third | |
| 5 | 461.54 | 13/10, 17/13, 21/16, 22/17 | B | E#/Fb | F | Ex/Fb | M | Minor fourth | |
| 6 | 553.85 | 11/8, 18/13, 26/19 | ε | F | F#/Gb | F# | M#/Nb | Major fourth/Minor tritone | |
| 7 | 646.15 | 16/11, 13/9, 19/13 | D | F#/Gb | G | Gb | N | Minor fifth/Major tritone | |
| 8 | 738.46 | 17/11, 20/13, 26/17, 32/21 | γ | G | G#/Hb | G# | O | Major fifth | |
| 9 | 830.77 | 8/5, 13/8, 18/11, 21/13 | F | G#/Ab | H | Ab | O#/Pb | Minor sixth | |
| 10 | 923.08 | 17/10, 12/7, 22/13, 19/11 | E | A | A | A# | P | Major sixth | |
| 11 | 1015.38 | 9/5, 16/9, 20/11, 34/19, 38/21 | α | A#/Bb | A#/Bb | Bb | Q | Minor seventh | |
| 12 | 1107.69 | 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 | G | B/Cb | B | B#/Cbb | Q#/Jb | Major seventh | |
| 13 | 1200.00 | 2/1 | H | C/B# | C | C | J | Octave |
- ↑ Ratios are based on treating 13edo as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.
Notation
13edo can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".
| # | Cents | Up/down notation using the narrow 5th of 7\13, with major wider than minor |
Up/down notation using the narrow 5th of 7\13, with major narrower than minor | ||||
|---|---|---|---|---|---|---|---|
| 0 | 0 | perfect unison | P1 | D | perfect unison | P1 | D |
| 1 | 92 | up unison, minor 2nd | ^1, m2 | ^D, E | up unison, major 2nd | ^1, M2 | ^D, E |
| 2 | 185 | upminor 2nd, minor 3rd | ^m2, m3 | ^E, Fb | upmajor 2nd, major 3rd | ^M2, M3 | ^E, F# |
| 3 | 277 | downmajor 2nd, upminor 3rd | vM2, ^m3 | vE#, ^Fb | downminor 2nd, upmajor 3rd | vm2, ^M3 | vEb, ^F# |
| 4 | 369 | major 2nd, downmajor 3rd | M2, vM3 | E#, vF | minor 2nd, downminor 3rd | m2, vm3 | Eb, vF |
| 5 | 462 | major 3rd, down 4th | M3, v4 | F, vG | minor 3rd, down 4th | m3, v4 | F, vG |
| 6 | 554 | perfect 4th, down 5th | P4, v5 | G, vA | perfect 4th, down 5th | P4, v5 | G, vA |
| 7 | 646 | up 4th, perfect 5th | ^4, P5 | ^G, A | up 4th, perfect 5th | ^4, P5 | ^G, A |
| 8 | 738 | up 5th, minor 6th | ^5, m6 | ^A, B | up 5th, major 6th | ^5, M6 | ^A, B |
| 9 | 831 | upminor 6th, minor 7th | ^m6, m7 | ^B, Cb | upmajor 6th, major 7th | ^M6, M7 | ^B, C# |
| 10 | 923 | downmajor 6th, upminor 7th | vM6, ^m7 | vB#, ^Cb | downminor 6th, upmajor 7th | vm6, ^M7 | vBb, ^C# |
| 11 | 1015 | major 6th, downmajor 7th | M6, vM7 | B#, vC | minor 6th, downminor 7th | m6, vm7 | Bb, vC |
| 12 | 1108 | major 7th, down 8ve | M7, v8 | C, vD | minor 7th, down 8ve | m7, v8 | C, vD |
| 13 | 1200 | perfect 8ve | P8 | D | perfect 8ve | P8 | D |
This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, and heptatonic 2nd-generated.
Pentatonic 5th-generated: D * * E * G * * A * C * * D (generator = wide 3/2 = 8\13 = perfect 5thoid)
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d (s = sub-, d = -oid)
pentatonic genchain of fifths: ...Ebb - Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# - Cx...
pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)
Octatonic 5th-generated: A * B C * D * E F * G H * A (generator = wide 3/2 = 8\13 = perfect 6th)
A - A#/Bb - B - C - C#/Db - D - D#/Eb - E - F - F#/Gb - G - H - H#/Ab - A
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
octotonic genchain of sixths: ..D# - A# - F# - C# - H# - E - B - G - D - A - F - C - H - Eb - Bb - Gb - Db - Ab...
octotonic genchain of sixths: ...M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7...
Heptatonic 2nd-generated: D * E * F * G A * B * C * D (generator = 2\13 = perfect 2nd)
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
genchain of seconds: ...Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#...
genchain of seconds: ...d6 - d7 - d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 - A2 - A3...
13edo chromatic ascending and descending scale on C (MIDI)
JI approximation
Tuning by ear
13edo can be approximated by a circle of 64/49 subminor fourths (which can be tuned by tuning two 7/4 subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.
Phi vibes
Acoustic phi
13edo has a very close approximation of acoustic phi (9\13), with only -2.3 cents of error. 23edo and 36edo are even closer, but unlike all closer EDOs, 13-EDO has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13-EDO is the only EDO that tempers the ratios of the Fibonacci sequence into a single interval.
See also: 9edϕ
Logarithmic phi
As a coïncidence, 13edo also has a very close appoximation of logarithmic phi (21\13), with only -3.2 cents of error.
Not until 144 do we find a better EDO in terms of relative error on these two intervals.
However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2**(phi) ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant phi ≃ 1.6180.
That being said, logarithmic phi has interesting applications as Metallic MOS, and in particular the fractal-like possibilities of self-similar subdivision of musical scales.
| Interval | Error (abs, ¢) |
|---|---|
| 2**(ϕ) / ϕ | 0.858 |
| ϕ | 2.321 |
| 2**(ϕ) | 3.179 |
Local zeta peak
At the local zeta peak of 13edo, there is an improvement in both acoustic phi and logarithmic phi.
Scales
Important mosses (values in parentheses are (period, generator)):
- oneirotonic 5L 3s 22122121 (5\13, 1\1)
- archeotonic 6L 1s 2222221 (2\13, 1\1)
- lovecraft 4L 5s 212121211 (3\13, 1\1)
- Sephiroth 3L 4s 3131311 (4\13, 1\1)
Due to the prime character of the number 13, 13edo can form several xenharmonic moment of symmetry scales. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.
~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson
Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.
Pathological Modes
2 1 1 1 1 2 1 1 1 1 1 2L 9s MOS
3 1 1 1 1 1 1 1 1 1 1 1L 10s MOS
2 1 1 1 1 1 1 1 1 1 1 1 1L 11s MOS
Harmony in 13edo
Contrary to popular belief, consonant harmony is possible in 13edo, but it requires a radically different approach than that used in 12edo (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12edo within 13edo is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13edo. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12edo, since the strongest dissonances in 13edo are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13edo is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the 2*13 subgroup 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26edo.
By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.
Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.
Play the 4:5:9 chord:
Play the 4:5:9:11 chord:
Play the 4:5:9:13 chord:
Play the 4:5:9:21 chord:
Notational and compositional approaches
13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.
The Cryptic Ruse Methods
13edo offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are Rothenberg proper, and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L 1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.
To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.
Modes and Harmony in the Archeotonic Scale
The 2\13-based heptatonic has been named archeotonic after the "Old Ones" that rule the Dreamlands. Modes of the archeotonic are named after the individual Old Ones.
A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.
Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.
There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.
Modes and Harmony in the Oneirotonic Scale
The 5\13-based octatonic has been named oneirotonic after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.
Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.
There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.
Fox, Inthar and Jaimbee's approach
Our approach is based on 5L 3s. In fact, we have an absolute pitch notation for 5L 3s edos called the Fox-Raven notation.
Mapping to Standard Keyboards
The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
| 1 | 6 | 11 | 3 | 8 | (13) | 5 | 10 | 2 | 7 | 12 | 4 | 9 | 1 | Place in Chain of 738.5 cent intervals |
| X | * | * | * | * | * | * | * | X | Marked are the octatonic scales (X=Sarnathian) | |||||
| * | * | * | * | X | * | * | * | |||||||
| * | X | * | * | * | * | * | * | |||||||
| * | * | * | * | * | * | X | * | |||||||
| * | * | * | X | * | * | * | * | |||||||
| D | Eb | E | F | Gb | G | Ab | A | Bb | B | C | Db | D | Keeps the pentatonic scale on the white keys | |
| A | Bb | B | C | Db | D | Eb | E | F | Gb | G | Ab | A | ||
| E | F | Gb | G | Ab | A | Bb | B | C | Db | D | Eb | E | ||
| B | C | Db | D | Eb | E | F | Gb | G | Ab | A | Bb | B | ||
| C | Db | D | Eb | E | F | Gb | G | Ab | A | Bb | B | C | Puts the missing key between a semitone | |
| G | Ab | A | Bb | B | C | Db | D | Eb | E | F | Gb | G | (if that were to be valuable in any way) |
The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.
Regular temperament properties
Uniform maps
Lua error in Module:Uniform_map at line 135: Must provide edo if not min or max given..
Commas
13edo tempers out the following commas. (Note: This assumes the val ⟨13 21 30 36 45 48].)
| Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 5 | (14 digits) | [-21 3 7⟩ | 10.06 | Lasepyo | Semicomma, Fokker comma |
| 7 | 1029/1000 | [-3 1 -3 3⟩ | 49.49 | Trizogu | Keega |
| 7 | 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicennma, Avicenna's Enharmonic Diesis |
| 7 | 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal Comma, Archytas' Comma, Leipziger Komma |
| 7 | 64827/64000 | [-9 3 -3 4⟩ | 22.23 | Laquadzo-atrigu | Squalentine |
| 7 | 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyo | Gariboh |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquingu | Hemimean comma |
| 11 | 56/55 | [3 0 -1 1 -1⟩ | 31.19 | Luzogu | Undecimal diesis |
| 11 | 121/120 | [-3 -1 -1 0 2⟩ | 14.37 | Lologu | Biyatisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozogu | Werckisma |
| 13 | 40/39 | [3 -1 1 0 0 -1⟩ | 43.83 | Thuyo | Tridecimal minor diesis |
| 13 | 105/104 | [-3 1 1 1 0 -1⟩ | 16.57 | Thuzoyo | Animist |
| 13 | 169/168 | [-3 -1 0 -1 0 2⟩ | 10.27 | Thothoru | Buzurgisma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Animism
The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:
0 4 5 8 9 13 pentatonic
and
0 1 3 4 5 8 9 10 12 13 nonatonic
Introductory Materials
By Inthar
- 13edo example composition (File:13edo 1MC score.pdf)
Oneirotonic Modal Studies
- : Tonal Study in Dylathian
- : Tonal Study in Ultharian
- : Tonal Study in Hlanithian
- : Tonal Study in Illarnekian
- : Tonal Study in Mnarian
- : Tonal Study in Sarnathian
- : Tonal Study in Celephaïsian
- : Tonal Study in Kadathian
Music
- See also: Category:13edo tracks
- Fuzz Reef in Glacial[7]
- Lunar Approach in Father[8]
- Tocada (2016)
- Ménuet in 13edo for Organ
- Micropiece in 13edo
- Canon at the Semitone on The Mother's Malison Theme, for Organ
- 13 Miles (blog entry)
- Fast And Furious 13 (blog entry)
- SazDul 13 (blog entry)
- Thirteenstan (blog entry)
- Concertina 13 (blog entry)
- Hocus-Bogus 13 (blog entry)
- MP13 (blog entry)
- Berimbau 13 (blog entry)
- Tenori Blofeld (video) (blog entry)
- From The Diary Of The 13th Fly (blog entry)
- Lotus 13 (blog entry)
- "Sean's Bits", from Golden Hour (2010)
- (iNeed) SyNthetikLove[dead link] play[dead link]
- Strange, Strange Day[dead link] play[dead link]
- Stare (into The Sun)[dead link] play[dead link]
- Solo Viola Sonata No. 1 "Dark Elven Folk Tunes I"
- Duet Miniature for Violin & Viola "Dark Elven Folk Tunes II"
- 13edo Bebop (2006)
From the (defunct) WT13C project
- WT13C: Prelude XII in F# (Score)
- WT13C: Fugue XII in F# (Form chart)
