17edo tetrachords
Let a "17edo primary tetrachord" mean a set of four pitches in 17edo that span a perfect fourth (seven degrees) and include one of each of the following:
- the unison - 0 (degrees of 17edo) - solfege name 'do'.
- a second - includes 1 (ra, a minor second), 2 (ru, a neutral second), and 3 (re, a major second).
- a third - includes 4 (me, a minor third), 5 (mu, a neutral third), and 6 (mi, a major third).
- the perfect fourth - 7 (fa).
correspondance:
| degrees | cents | name | solfege |
|---|---|---|---|
| 0 | 0 | unison | do |
| 1 | 71 | minor second (a.k.a third-tone) | ra |
| 2 | 141 | neutral second (a.k.a. two-thirds-tone) | ru |
| 3 | 212 | major second (a.k.a. tone) | re |
| 4 | 282 | minor third (a.k.a. subminor third) | me |
| 5 | 353 | neutral third | mu |
| 6 | 424 | major third (a.k.a. supermajor third) | mi |
| 7 | 494 | perfect fourth | fa |
tetrachord notation
Tetrachord notation will show three scalar steps (as degrees of 17edo) separated by hyphens.
For instance, tetrachord 3-3-1 consists of
0 (do), the unison;
3 (re), a major second, 3 degrees up from 0;
6 (mi), a major third, 3 degrees up from 3; and
7 (fa), the perfect fourth, 1 degree up from 6.
The numbers in a tetrachord name will always add to 7.
17edo primary tetrachords
We have 9 primary tetrachords in 17edo.
| tetrachord notation | solfege | name (suggestions?) | used in |
|---|---|---|---|
| 1-3-3 | do ra me fa | phrygian (jins Kurd) | diatonic (phrygian) |
| 1-4-2 | do ra mu fa | ||
| 1-5-1 | do ra mi fa | balkan, jins Hijaz | |
| 2-2-3 | do ru me fa | jins Bayyati | 17edo neutral scale (led) |
| 2-3-2 | do ru mu fa | "ʻIraq" tetrachord | 17edo neutral scale (bish, fish, jwl) |
| 2-4-1 | do ru mi fa | ||
| 3-1-3 | do re me fa | aeolian (jins Nahawand) | diatonic (aolian, dorian) ; scorp (mode 3) |
| 3-2-2 | do re mu fa | jins Rast | 17edo neutral scale (dril, gil, kleeth) |
| 3-3-1 | do re mi fa | ionian (jins ʻAjam) | diatonic (ionian, mixolydian) |
Notes on the above:
- Many references say that the semitones in Hijaz should be greater than 100 cents, but in 17edo they are significantly smaller (about 70 cents).
- The "ʻIraq" tetrachord is similar to the first four notes of Maqam ʻIraq, but those are not usually considered a jins because the fourth note is not a place of rest (in particular it is not the ghammaz). This tetrachord is jins Sikah with an extra note on top making a perfect fourth with the tonic.
Another way of showing them:
| ra | ru | re | |
|---|---|---|---|
| me | 1-3-3 | 2-2-3 | 3-1-3 |
| mu | 1-4-2 | 2-3-2 | 3-2-2 |
| mi | 1-5-1 | 2-4-1 | 3-3-1 |
17edo tetrachords complete
A more generalized tetrachord system would allow multiple seconds or multiple thirds: for instance, 1-1-5 or 5-1-1. Thus, a complete chart of 17edo tetrachords looks like this (with primary tetrachords in bold):
| 1-1-5 | 2-1-4 | 3-1-3 | 4-1-2 | 5-1-1 |
| 1-2-4 | 2-2-3 | 3-2-2 | 4-2-1 | |
| 1-3-3 | 2-3-2 | 3-3-1 | ||
| 1-4-2 | 2-4-1 | |||
| 1-5-1 |
Thus, by allowing multiples seconds or multiple thirds, we add 6 new tetrachords to our 9 primary tetrachords, for a total of 15. Our new ones:
| tetrachord notation | solfege | name (suggestions?) | used in |
|---|---|---|---|
| 1-1-5 | do ra ru fa | ||
| 1-2-4 | do ra re fa | ||
| 2-1-4 | do ru re fa | ||
| 4-1-2 | do me mu fa | ||
| 4-2-1 | do me mi fa | ||
| 5-1-1 | do mu mi fa |
See also: tetrachord, 22edo tetrachords, Tricesimoprimal Tetrachordal Tesseract.