22edo/Vector's approach
This page documents Vector's info on 22EDO, for the Tuning of the Year project.
Notes and intervals
Intro
22edo contains 22 notes per octave, and thus 22 distinct intervals within the octave to use in your music.
Zarlino (intense diatonic) is the basic diatonic for 22edo, as with 15edo, so that the same porcupine-based notation may be used for both. The scale has two different types of whole tones, and was chosen so that the basic major and minor triads would represent 4:5:6 and its minor counterpart.
However, the main scale that will be used is jaric (pajara[10]), which has 10 interval names taken from Leriendil's decatonic system mixed with standard interval names.
Both scales have the same chroma size, making interval logic pretty easy to apply to both.
| Name | Degree | Cents | Approximate Ratios | Note (Diatonic) | Note (Pajara) |
|---|---|---|---|---|---|
| perfect unison | 0 | 0 | 1/1 | C | 0 |
| augmented unison | 1 | 54.5 | 25/24, 81/80, 36/35, 33/32 | C# | 0# |
| minor second | 2 | 109.1 | 16/15, 15/14 | Db | 1 |
| major second | 3 | 163.6 | 12/11, 11/10, 10/9 | D | 1# |
| minor unilatus | 4 | 218.2 | 9/8, 8/7 | D# | 2 |
| major unilatus | 5 | 272.7 | 7/6 | Ebb | 2# |
| minor third | 6 | 327.3 | 6/5, 11/9 | Eb | 3b |
| major third | 7 | 381.8 | 5/4 | E | 3 |
| diminished fourth | 8 | 436.4 | 9/7, 14/11 | E# | 4b |
| perfect fourth | 9 | 490.9 | 4/3 | F | 4 |
| augmented fourth / diminished median | 10 | 545.5 | 15/11, 11/8 | F# | 4# |
| perfect median | 11 | 600.0 | 10/7, 7/5 | Gbb | 5 |
| diminished fifth / augmented median | 12 | 654.5 | 16/11, 22/15 | Gb | 6b |
| perfect fifth | 13 | 709.1 | 3/2 | G | 6 |
| augmented fifth | 14 | 763.6 | 11/7, 14/9 | G# | 7b |
| minor sixth | 15 | 818.2 | 8/5 | Ab | 7 |
| major sixth | 16 | 872.7 | 18/11, 5/3 | A | 7# |
| minor antilatus | 17 | 927.3 | 12/7 | A# | 8b |
| major antilatus | 18 | 981.8 | 7/4, 16/9 | Bbb | 8 |
| minor seventh | 19 | 1036.4 | 9/5, 20/11, 11/6 | Bb | 8# |
| major seventh | 20 | 1090.9 | 28/15, 15/8 | B | 9 |
| diminished octave / augmented seventh | 21 | 1145.5 | 48/25, 64/33, 35/18 | B# | 9# |
| perfect octave | 22 | 1200.0 | 2/1 | C | 0 |
Just intonation
Microtonal theorists and composers like to use just intonation as a basis for their theory. Meaning that they think of harmony in terms of either just intonation or some abstraction away from it. Just intonation is the tuning where all intervals are pure integer ratios, which is easy to tune, and provides a distinct harmonious sound. The most important ratios of just intonation are 2/1, 3/2, and 4/3, along with to a lesser extent 5/3 and 5/4. These are the ratios that are most likely to be picked up by historical musical cultures, and explains the convergence on pentatonic and heptatonic scale forms.
However, just intonation has caught modern microtonal theorists' attention for a different reason: the idea of stacking (multiplying or dividing) "prime" intervals to obtain more complex intervals. Stacking is multiplying, but it looks like adding on paper because pitch is logarithmic.
The decatonic scale
The 10-form (represented as pajara[10] here) is arguably a more intuitive categorization scheme for intervals than the 7-form we're used to. While most microtonal systems simply add new accidentals and interval qualities, it's best to think of 22edo as adding three distinct interval ordinal categories "in between the gaps" of the standard 7.
Unilatus
From the Latin for "carried once", this refers to the separation of roughly one 5edo-step, or two decatonic steps, and sits between the standard seconds and thirds. The minor unilatus in 22edo is the 4-step interval representing 8/7 and 9/8. Meanwhile, the major unilatus represents 7/6. Two unilati make a fourth, just as two thirds make a fifth.
Antilatus
The antilatus is the separation of roughly four 5edo-steps, or eight decatonic steps, and sits roughly between the standard sixths and sevenths. The major antilatus represents 7/4, and the minor antilatus represents 12/7. The antilatus essentially gives the simplest 7-limit intervals their own interval category, much as the simplest 5-limit intervals do, properly cementing 22edo as a 7-limit system.
Median
The perfect median is the interval of exactly 600 cents - that is, the perfect semioctave. The median splits the octave in two, reflecting the tritone as an intuitive interval category even in heptatonic systems, where the fourth and the fifth intersect to create it. But here, it is its own degree. Antilati are separated from thirds by the perfect median, and same with sixths from unilati. The median also solves the pesky 14/11 problem in heptatonic schemes: either 11/8 is a median, in which case the separation between it and a 7/4 antilatus (that is, 14/11) is a third, or it is a fourth, in which case 14/11 is also a fourth - however, the diminished fourth is the basic decatonic "imperfect" fourth, rather than the augmented fourth, so it is less of a problem.