29edo
29 equal divisions of the octave, or 29edo, is the tuning system which divides the 2/1 ratio into 29 equal parts of approximately 41.3 cents each. It is notable for its extremely accurate tuning of prime 3, and for unique melodic properties that proponents of the system consider particularly desirable.
Tuning Theory
JI Approximation
While 29edo excels at prime 3, the rest of the primes up to 31 are relatively lacking. However, primes 5 through 13 have roughly the same amount of error, and in the same direction; difference tones such as 7/5 and 11/7 are thus rather accurate.
| Harmonic | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.5 | -13.9 | -17.1 | -13.4 | -12.9 | +19.2 | -7.9 | -7.6 | +4.9 | +13.6 |
| Relative (%) | +3.6 | -33.6 | -41.3 | -32.4 | -31.3 | +46.4 | -19.0 | -18.3 | +11.9 | +32.8 | |
| Steps
(reduced) |
46
(17) |
67
(9) |
81
(23) |
100
(13) |
107
(20) |
119
(3) |
123
(7) |
131
(15) |
141
(25) |
144
(28) | |
Intervals and Notation
Due to its accurate tuning of prime 3, 29edo can be notated quite cleanly with the familiar circle of fifths; the whole tone is five steps of 29edo, the diatonic semitone is two steps, and the chromatic semitone is three steps. Intervals with double-flats and double-sharps may alternatively be written using ups and downs, smaller accidentals which represent one step of 29edo.
To display how enharmonic equivalences behave, below is a table of the many ways to notate the six notes between D to F, which may be considered to roughly constitute the range of thirds above a tonic C.
| D | ||||
| C𝄪 | ^D | vE♭ | F𝄫 | |
| ^C𝄪 | vD♯ | E♭ | ^F𝄫 | |
| D♯ | ^E♭ | vF♭ | ||
| ^D♯ | vE | F♭ | ||
| vD𝄪 | E | ^F♭ | vG𝄫 | |
| D𝄪 | ^E | vF | G𝄫 | |
| F | ||||
Note that double-sharp / double-flat intervals always differ from a natural ordinal by a single step up or down.
MOS Scales
Diatonic
The Diatonic scale is generated by taking seven adjacent tones from the Circle of Fifths, just as it is in 12edo. Melodies and chords made using this scale will sound nearly identical to those that can be made using 12edo, with the notable exception of the tritone (which comes in two distinct forms depending on which mode it's found in).
| Gens Up | Step Pattern | Notation | Name |
|---|---|---|---|
| 6 | LLLsLLs | C - D - E - F♯ - G - A - B - C | Lydian |
| 5 | LLsLLLs | C - D - E - F - G - A - B - C | Ionian |
| 4 | LLsLLsL | C - D - E - F - G - A - B♭ - C | Mixolydian |
| 3 | LsLLLsL | C - D - E♭ - F - G - A - B♭ - C | Dorian |
| 2 | LsLLsLL | C - D - E♭ - F - G - A♭ - B♭ - C | Aeolian |
| 1 | sLLLsLL | C - D♭ - E♭ - F - G - A♭ - B♭ - C | Phrygian |
| 0 | sLLsLLL | C - D♭ - E♭ - F - G♭ - A♭ - B♭ - C | Locrian |
Chromatic
The Chromatic scale is an extension of the Diatonic scale, which can be found by continuing the sequence along the circle of fifths. Because the circle can be traversed in two possible ways, scales can be extended in an "acute" direction or a "grave" direction.
| Gens Up | Step Pattern | Notation | Name | Notes |
|---|---|---|---|---|
| 11 | LsLsLssLsLss | C - C♯ - D - D♯ - E - E♯ - F♯ - G - G♯ - A - A♯ - B - C | Grave Lydian | Like the seven-note Lydian, lacks a Perfect Fourth over the root. |
| 10 | LsLssLsLsLss | C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - A♯ - B - C | Grave Ionian | |
| 9 | LsLssLsLssLs | C - C♯ - D - D♯ - E - F - F♯ - G - G♯ - A - B♭ - B - C | Grave Mixolydian | |
| 8 | LssLsLsLssLs | C - C♯ - D - E♭ - E - F - F♯ - G - G♯ - A - B♭ - B - C | Grave Dorian | |
| 7 | LssLsLssLsLs | C - C♯ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C | Grave Aeolian | |
| 6 | sLsLsLssLsLs | C - D♭ - D - E♭ - E - F - F♯ - G - A♭ - A - B♭ - B - C | Grave Phrygian | Also accounts for Acute Lydian |
| 5 | sLsLssLsLsLs | C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - B - C | Acute Ionian | Also accounts for Grave Locrian |
| 4 | sLsLssLsLssL | C - D♭ - D - E♭ - E - F - G♭ - G - A♭ - A - B♭ - C♭ - C | Acute Mixolydian | |
| 3 | sLssLsLsLssL | C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - A - B♭ - C♭ - C | Acute Dorian | |
| 2 | sLssLsLssLsL | C - D♭ - D - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C | Acute Aeolian | |
| 1 | ssLsLsLssLsL | C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - G - A♭ - B𝄫 - B♭ - C♭ - C | Acute Phrygian | |
| 0 | ssLsLssLsLsL | C - D♭ - E𝄫 - E♭ - F♭ - F - G♭ - A𝄫 - A♭ - B𝄫 - B♭ - C♭ - C | Acute Locrian | Like the seven-note Locrian, lacks a Perfect Fifth over the root. |
Smitonic
The Smitonic scale can be found as a circle of augmented seconds, or via an "evened out" form of the Harmonic Minor scale. The mode names for this scale are given by Ayceman.
| Gens Up | Step Pattern | Notation | Name (Ayceman) | Altered Diatonic Mode |
|---|---|---|---|---|
| 6 | LLsLsLs | C - D - E - ^F - ^G - vA - vB - C | Nerevarine | Major Augmented |
| 5 | LsLLsLs | C - D - ^E♭ - ^F - ^G - vA - vB - C | Vivecan | Harmonic Minor |
| 4 | LsLsLLs | C - D - ^E♭ - ^F - vG - vA - vB - C | Lorkhanic | Lydian ♯2 |
| 3 | LsLsLsL | C - D - ^E♭ - ^F - vG - vA - B♭ - C | Sothic | Dorian ♯4 |
| 2 | sLLsLsL | C - ^D♭ - ^E♭ - ^F - vG - vA - B♭ - C | Kagrenacan | Locrian ♯6 |
| 1 | sLsLLsL | C - ^D♭ - ^E♭ - vF - vG - vA - B♭ - C | Almalexian | Ultralocrian |
| 0 | sLsLsLL | C - ^D♭ - ^E♭ - vF - vG - A♭ - B♭ - C | Dagothic | Phrygian Dominant |
Gramitonic
The Gramitonic scale takes the role of a diminished scale in 29edo: since four minor thirds fall short of the octave, the chain of minor thirds can be extended into this nine-note scale. Note how the four bright modes resemble the pattern of the familiar octatonic scale, with one of the small steps duplicated, and the four darkest modes resemble the rotated variant of that scale; additionally, there is a symmetrical mode that is entirely new to 29edo. The mode names for this scale are given by Lilly Flores.
| Gens Up | Step Pattern | Notation | Name (Flores) |
|---|---|---|---|
| 8 | LsLsLsLss | C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - E𝄫𝄫 - D𝄫 - C | Roi |
| 7 | LsLsLssLs | C - F𝄫 - E♭ - A𝄫♭ - G♭ - C𝄫♭ - B𝄫 - A - D𝄫 - C | Steno |
| 6 | LsLssLsLs | C - F𝄫 - E♭ - A𝄫♭ - G♭ - F♯ - B𝄫 - A - D𝄫 - C | Limni |
| 5 | LssLsLsLs | C - F𝄫 - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C | Telma |
| 4 | sLsLsLsLs | C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - D𝄫 - C | Krini |
| 3 | sLsLsLssL | C - B♯ - E♭ - D♯ - G♭ - F♯ - B𝄫 - A - G𝄪 - C | Elos |
| 2 | sLsLssLsL | C - B♯ - E♭ - D♯ - G♭ - F♯ - E𝄪 - A - G𝄪 - C | Mychos |
| 1 | sLssLsLsL | C - B♯ - E♭ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C | Akti |
| 0 | ssLsLsLsL | C - B♯ - A𝄪♯ - D♯ - C𝄪♯ - F♯ - E𝄪 - A - G𝄪 - C | Dini |
Checker
Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an 8-note augmented scale. Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the Tcherepnin scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin. The mode names for this scale are given by R-4981.
| Gens Up | Step Pattern | Notation | Name (R-4981) |
|---|---|---|---|
| 7 | LsLssLss | C - G𝄫♭ - F♭ - C𝄫♭ - B𝄫♭ - A♭ - E𝄫♭ - D𝄫 - C | King |
| 6 | LssLsLss | C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - E𝄫𝄫 - D𝄫 - C | Queen |
| 5 | LssLssLs | C - G𝄫♭ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C | Marshall |
| 4 | sLsLssLs | C - B♯ - F♭ - E - B𝄫♭ - A♭ - G♯ - D𝄫 - C | Cardinal |
| 3 | sLssLsLs | C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - D𝄫 - C | Rook |
| 2 | sLssLssL | C - B♯ - F♭ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C | Bishop |
| 1 | ssLsLssL | C - B♯ - A𝄪♯ - E - D𝄪 - A♭ - G♯ - F𝄪♯ - C | Knight |
| 0 | ssLssLsL | C - B♯ - A𝄪♯ - E - D𝄪 - C𝄪𝄪 - G♯ - F𝄪♯ - C | Pawn |
Machinoid
Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that six whole tones reach an up-octave, not a perfect octave. However, the octave can still be closed by employing one diminished third to act as a "wolf" version of the whole tone; this leads to Machinoid, a whole tone scale that has six distinct modes. The mode names for this scale are given by Lilly Flores.
| Gens Up | Step Pattern | Notation | Name (Flores) |
|---|---|---|---|
| 5 | LLLLLs | C - D - E - F♯ - G♯ - A♯ - C | Erev |
| 4 | LLLLsL | C - D - E - F♯ - G♯ - B♭ -C | Oplen |
| 3 | LLLsLL | C - D - E - F♯ - A♭ - B♭ - C | Layla |
| 2 | LLsLLL | C - D - E - G♭ - A♭ - B♭ - C | Shemesh |
| 1 | LsLLLL | C - D - F♭ - G♭ - A♭ - B♭ - C | Boqer |
| 0 | sLLLLL | C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C | Tsohorayim |
