User:VectorGraphics/16edo theory
Vector's 16edo theory was created for Earth#16. 16edo is an extremely easy equal scale to tune, since it ultimately just requires taking lots of square roots, which can be done by geometric methods.
Introduction: note names and intervals
Notes will be named according to VQDMN for armotonic, the larger of the two scales used in Earth#16 theory.
To avoid ambiguity, the VQDMN interval names will be used throughout the entire page. For example, the interval closest to 5/4 will be a "minor fourth", and the interval closest to 3/2 will be a "perfect sixth". However, chords will be named after the mediant's functional name, for familiarity with major and minor chords.
Additionally, to avoid confusion regarding the term "semitone", the 75-cent interval will be called the "eka", as in Armodue theory.
| Cents | Edostep | Note name (from C) | Note name (German) | Note name (fixed do) | Interval name (functional) | Interval name (VQDMN) | Other interval names | Type |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | C | C | do | Unison | Unison | Trivial | |
| 75 | 1 | Minor second | Minor second | Eka, third-tone | Dissonance | |||
| 150 | 2 | D | D | re | Major second | Major second | 2/3-tone | Dissonance |
| 225 | 3 | Arto third | Minor third | Secondary consonance | ||||
| 300 | 4 | E | E | mi | Minor third | Major third | Primary consonance | |
| 375 | 5 | Major third | Minor fourth | Primary consonance | ||||
| 450 | 6 | F | F | fa | Tendo third | Major fourth | Secondary consonance | |
| 525 | 7 | G | G | ku | Perfect fourth | Perfect fifth | Primary consonance | |
| 600 | 8 | Tritone | Semidecim / augmented fifth / diminished sixth | Dissonance | ||||
| 675 | 9 | H | J | bo | Perfect fifth | Perfect sixth | Primary consonance | |
| 750 | 10 | Arto sixth | Minor seventh | Secondary consonance | ||||
| 825 | 11 | J | K | sol | Minor sixth | Major seventh | Primary consonance | |
| 900 | 12 | Major sixth | Minor eighth | Primary consonance | ||||
| 975 | 13 | A | A | la | Tendo sixth | Major eighth | Secondary consonance | |
| 1050 | 14 | B | Minor seventh | Minor ninth | Dissonance | |||
| 1125 | 15 | B | H | si | Major seventh | Major ninth | Dissonance | |
| 1200 | 16 | C | C | do | Octave | Decim | Trivial |
While not shown here, there are 2 accidentals used, as in VQDMN normally: # raises a note by a single eka, and b lowers a note by a single eka.
There are two scales in use in 16edo: antidiatonic ("heptatonic") and armotonic ("nonatonic").
A decim of a 16edo keyboard looks like the following
This contains 7 white keys corresponding to a heptatonic scale starting on C, 7 black keys representing the same mode starting on G#, as well as two "blue notes" covered by neither, which form a a semidecim with each other and which can be added to either to produce a nonatonic scale. Any nonatonic can be thought of as a parent heptatonic plus these two "blue notes" (on either side of the heptatonic on the chain of sixths), which are used significantly more in harmony than in melody, as they make available more possible chord types.
While the modes of a normal diatonic system can be categorized into major and minor based on the size of the third (since the third is the only mediant), in the case of armotonic, there are two mediants to keep track of (thirds and fourths), so the nonatonic modes are categorized based on what pair of third and fourth is available (of which there are 3 possibilities).
A heptatonic can be expanded by adding a "blue note" into each of the 3-eka gaps in the scale. There are 4 possible ways to do this for each heptatonic, 3 of which lead to nonatonic modes, and the fourth leads to a "modmos" similar to melodic minor. Scales that belong to the same parent heptatonic can be used in the same song, much like how you can switch between natural, harmonic, and melodic minor in a song in 12edo. Each nonatonic scale can be reached from three heptatonics.
Here are the scales arranged in a way that makes it clear what corresponds to what: by adding blue notes to a heptatonic, the nonatonic to its left as well as the nonatonics above and below that can be reached, and by removing notes from a nonatonic, the heptatonic to its right as well as the scales above and below that can be reached. A heptatonic can be transformed to its corresponding modmos mode, or the modmos mode to the heptatonic.
| Mode name | Pattern | Pattern (16edo notes) | Accidentals from C Ionian | Blue notes | Third | Fourth | Heptatonic mode name | Heptatonic pattern | Heptatonic pattern (16edo notes) | Modmos scale |
|---|---|---|---|---|---|---|---|---|---|---|
| Lydian | LLLLsLLLs | #_#_#_#_##_#_#_## | G# | 1, 5 | Major | Major | - | - | - | |
| Ionian | LLLsLLLLs | #_#_#_##_#_#_#_## | 5, 9 | Major | Major | Alsatian | sssLssL | #_#_#_#__#_#_#__# | #_#_#_#_##_#_##_# | |
| Mixolydian | LLLsLLLsL | #_#_#_##_#_#_##_# | Bb | 4, 9 | Major | Major | Provencal | ssLsssL | #_#_#__#_#_#_#__# | #_#_##_#_#_#_#_## |
| Corinthian | LLsLLLLsL | #_#_##_#_#_#_##_# | Bb Fb | 4, 8 | Major | Minor | Norman | ssLssLs | #_#_#__#_#_#__#_# | #_#_#_##_#_##_#_# |
| Olympian | LLsLLLsLL | #_#_##_#_#_##_#_# | Bb Fb Ab | 3, 8 | Major | Minor | Picardian | sLsssLs | #_#__#_#_#_#__#_# | #_##_#_#_#_#_##_# |
| Dorian | LsLLLLsLL | #_##_#_#_#_##_#_# | Bb Fb Ab Eb | 3, 7 | Minor | Minor | Burgundian | sLssLss | #_#__#_#_#__#_#_# | #_#_##_#_##_#_#_# |
| Aeolian | LsLLLsLLL | #_##_#_#_##_#_#_# | Bb Fb Ab Eb Jb | 2, 7 | Minor | Minor | Breton | LsssLss | #__#_#_#_#__#_#_# | ##_#_#_#_#_##_#_# |
| Phrygian | sLLLLsLLL | ##_#_#_#_##_#_#_# | Bb Fb Ab Eb Jb Db | 2, 6 | Minor | Minor | Corsican | LssLsss | #__#_#_#__#_#_#_# | #_##_#_##_#_#_#_# |
| Locrian | sLLLsLLLL | ##_#_#_##_#_#_#_# | Bb Fb Ab Eb Jb Db Hb | 1, 6 | Minor | Minor | - | - |
Chords
A conventional triad may have one of 4 possible qualities: tendo (major fourth), major (minor fourth), minor (major third), and arto (minor third). Additionally, there are two diminished triads available: the arto diminished and major diminished.
Additionally, there is the "slendric tetrad", formed by playing the minor third and major fourth together in the same chord (with a root and a sixth), as in arto and tendo theory.
A fourth note can also be added to the chord; this can be an eighth or ninth. There are then 12 distinct tetrads (four of which with inversions that also follow the pattern of tetrads) available in the armotonic scale, arranged here in rough order of consonance to dissonance:
| Chord name | Notes | Mediant | Fourth note |
|---|---|---|---|
| Dominant tetrad, arto diminished tetrad | 1 m4 P6 M8 | Third or fourth | Eighth |
| Major tetrad | 1 m4 P6 m9 | Fourth | Ninth |
| Minor tetrad, dominant b8 tetrad | 1 M3 P6 M8 | Third or fourth | Eighth |
| Tendo b9 tetrad | 1 M4 P6 m9 | Fourth | Ninth |
| Arto tetrad, tendo b8 tetrad | 1 m3 P6 m8 | Third or fourth | Eighth |
| Tendo tetrad | 1 M4 P6 M9 | Fourth | Ninth |
| Arto b9 tetrad | 1 m3 P6 m9 | Third | Ninth |
| Minor b8 tetrad, major diminished b8 tetrad | 1 M3 P6 m8 | Third or fourth | Eighth |
| Minor b9 tetrad | 1 M3 P6 m9 | Third | Ninth |
| Minor #9 tetrad | 1 M3 P6 M9 | Third | Ninth |
| Arto diminished b9 tetrad | 1 m3 d6 m9 | Third | Ninth |
| Major diminished tetrad | 1 m4 d6 m9 | Fourth | Ninth |
There is also the slendric pentad, created by stacking an additional minor third on top of the slendric tetrad.
Here is a 16edo circle of sixths (remember, a sixth is the same as a diatonic fifth)
Suspended chords do not work as well in 16edo, because the interval produced by stacking two sixths is 150c, way smaller than can be considered consonant in a chord. However, the presence of arto and tendo chords more than makes up for this by introducing more variety in standard triads.
Staff notation
This is a 16edo staff. There are 6 lines per staff, to accommodate the size of the decim in scale steps. The treble clef is the same G clef you're familiar with, but the bass clef is a new shape, because it marks the note H below middle C, as opposed to F (according to VQDMN convention for 9-note scales). The C clef is still the same as well.
Functional harmony
Functional harmony is a system for assigning functions to different triads in a scale. Chords that are separated by a third often have the same function in 12edo; this generalizes to 16edo, where the cycle of mediants is 18 entries long (due to the two options for mediants for each root).
The following table shows the functions for the Ionian nonatonic; more chords are available in Alsatian, however they are not covered here, as the difference between the Alsatian nonatonics is similar to the difference between minor scales in 12edo.
(xxxa = arto, xxx = minor, XXX=major, XXXT = tendo)
| Function | Chords with Alsatian root notes | Other chords | Resolves to |
|---|---|---|---|
| Function 1 | i, III, *VII | Function 2, Function 5 | |
| Function 2 | iiia, VT, *II, v | Function 3 | |
| Function 3 | *vii, VIT | IXMdim, ixadim | Function 4, Function 5 |
| Function 4 | viii, IT | iva | Function 2, Function 5 |
| Function 5 | *ii | IV | Function 6 |
| Function 6 | vi, VIII | Function 1 |
*A resolution can also be done from a minor chord to its parallel major chord.
Functions 6 and 3 are similar to the dominant, functions 5 and 2 are similar to the subdominant, and functions 4 and 1 are similar to the tonic.
The functional names of notes are the same as in diatonic, except that "mediant" and "submediant" cover 2 pitch classes as opposed to 1.