User:Overthink/22edo decatonic theory
- Todo: Citation to Paul Erlich's paper, Add diagram
In classical music, we often use the diatonic scale, which is generated by meantone temperament. This gives us relatively simple access to 5-limit harmony, while avoiding wolf intervals. There is a system many consider to be analogous to diatonic in the 7-limit; the decatonic scale of pajara temperament. Strictly speaking, there are two decatonic scales: the symmetrical scale of the 2L 8s mos pattern, with large and small steps in the pattern ssLssssLss; and the pentachordal near-mos scale, with steps in the pattern ssLsssLsss. Here we will mainly be focusing on the pentachordal scale.
Interval classification
We will classify intervals based on the 2L 8s mos, with names "unison", "second", etc. However, since we have 10 note classes, the interval of equivalence is not an "octave" as in diatonic, but a "hendecave". In this system, 1sns, 2nds, 5ths, 6ths, 7ths, 10ths, and 11ves are perfect, while 3rds, 4ths, 8ths, and 9ths are imperfect. Pajara has a half-octave (600 ¢) period representing both 7/5 and 10/7, and a ~3/2 generator (~709 ¢ in 22edo). The half-octave is a perfect 6th, and the 3/2 is a perfect 7th. (From now on, interval names are decatonic unless specified.) Here is a table of intervals classified by the 2L 8s scale.
| Name | Cents (22edo) | Steps of 22edo | JI ratios |
|---|---|---|---|
| Perfect 1sn | 0.0 | 0 | 1/1 |
| Augmented 1sn | 54.5 | 1 | 25/24, 28/27, 49/48 |
| Diminished 2nd | 54.5 | 36/35, 81/80 | |
| Perfect 2nd | 109.1 | 2 | 16/15, 15/14, 21/20 |
| Augmented 2nd | 163.6 | 3 | 10/9 |
| Minor 3rd | 218.2 | 4 | 8/7, 9/8 |
| Major 3rd | 272.7 | 5 | 7/6 |
| Minor 4th | 327.3 | 6 | 6/5 |
| Major 4th | 381.8 | 7 | 5/4 |
| Diminished 5th | 436.4 | 8 | 9/7, 32/25 |
| Perfect 5th | 490.9 | 9 | 4/3, 21/16 |
| Augmented 5th | 545.5 | 10 | 11/8 |
| Diminished 6th | 545.5 | 27/20, 15/11 | |
| Perfect 6th | 600.0 | 11 | 7/5, 10/7 |
| Augmented 6th | 654.5 | 12 | 40/27, 22/15 |
| Diminished 7th | 654.5 | 16/11 | |
| Perfect 7th | 709.1 | 13 | 3/2, 32/21 |
| Augmented 7th | 763.6 | 14 | 14/9, 25/16 |
| Minor 8th | 818.2 | 15 | 8/5 |
| Major 8th | 872.7 | 16 | 5/3 |
| Minor 9th | 927.3 | 17 | 12/7 |
| Major 9th | 981.8 | 18 | 7/4, 16/9 |
| Diminished 10th | 1036.4 | 19 | 9/5 |
| Perfect 10th | 1090.9 | 20 | 15/8, 28/15, 40/21 |
| Augmented 10th | 1145.5 | 21 | 27/14, 48/25, 96/49 |
| Diminished 11ve | 1145.5 | 35/18, 160/81 | |
| Perfect 11ve | 1200.0 | 22 | 2/1 |
The white keys
In diatonic, the white keys are in a mode such that, on C, there is a major triad on the tonic, dominant, and subdominant. Here, the main chords are tetrads, with the major tetrad being P1−M4−P7−M9, approximating 4:5:6:7, and the minor tetrad being P1−m4−P7−m9, approximating 1/(7:8:10:12). More about this will be discussed later. Unfortunately, there is no mode of the pentachordal scale which places the major tetrad on the tonic, dominant, and subdominant of the same key, but the ssLsssLsss mode gets as close as possible. Hence, we will place the white keys on this mode above C.
| Note | Interval above C | Cents | Steps |
|---|---|---|---|
| C | P1 | 0.0 | 0 |
| R | P2 | 109.1 | 2 |
| D | m3 | 218.2 | 4 |
| E | M4 | 381.8 | 7 |
| F | P5 | 490.9 | 9 |
| T | P6 | 600.0 | 11 |
| G | P7 | 709.1 | 13 |
| A | M8 | 872.7 | 16 |
| S | M9 | 981.8 | 18 |
| B | P10 | 1090.9 | 20 |
| C | P11 | 1200.0 | 22 |
The notes are named so that C, D, E, F, G, A, and B represent the Zarlino diatonic scale, with R, S, and T filling the gaps.
Modes
Complete the table
| Name | Brightness | Step pattern | Intervals | Note names on C | Appears in white keys on |
|---|---|---|---|---|---|
| Scandium | +5 | LsssLsssss | P1, A2, M3, M4, P5, A6, A7, M8, M9, P10, P11 | C, R#, D#, E, F, T#, G#, A, S, B, C | D |
| Titanium | +3 | LsssssLsss | P1, A2, M3, M4, P5, P6, P7, M8, M9, P10, P11 | C, R#, D#, E, F, T, G, A, S, B, C | G |
| Vanadium | +3 | sLsssLssss | P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11 | C, R, D#, E, F, T, G#, A, S, B, C | R |
| Chromium | +1 | sLsssssLss | P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11 | C, R, D#, E, F, T, G, Ab, S, B, C | T |
| Manganese (Major) | +1 | ssLsssLsss | P1, P2, M3, M4, P5, P6, A7, M8, M9, P10, P11 | C, R, D, E, F, T, G, A, S, B, C | C |
Chords
First explain what chords are there
Using the Root−4th−7th−9th construction, we find the tetrads on each note.
| Note | Notes of chord | Intervals of chord | Name |
|---|---|---|---|
| C | C−E−G−S | P1−M4−P7−M9 | Major |
| R | R−F−A−B | P1−M4−A7−M9 | Aug. |
| D | D−T−S−C | P1−M4−A7−M9 | Aug. |
| E | E−G−B−R | P1−m4−P7−m9 | Minor |
| F | F−A−C−D | P1−M4−P7−m9 | Major−minor |
| T | T−S−R−E | P1−M4−P7−M9 | Major |
| G | G−B−D−F | P1−M4−P7−M9 | Major |
| A | A−C−E−T | P1−m4−P7−m9 | Minor |
| S | S−R−F−G | P1−m4−P7−m9 | Minor |
| B | B−D−T−A | P1−m4−P7−M9 | Minor−major |