# 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:

1. the unison - 0 (degrees of 17edo) - solfege name 'do'.
2. a second - includes 1 (ra, a minor second), 2 (ru, a neutral second), and 3 (re, a major second).
3. a third - includes 4 (me, a minor third), 5 (mu, a neutral third), and 6 (mi, a major third).
4. the perfect fourth - 7 (fa).

## Contents

### 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 diatonic (phrygian)
1-4-2 do ra mu fa
1-5-1 do ra mi fa balkan, Hijaz (arabic)
2-2-3 do ru me fa Bayyati (arabic) 17edo neutral scale (led)
2-3-2 do ru mu fa Iraq (arabic) 17edo neutral scale (bish, fish, jwl)
2-4-1 do ru mi fa
3-1-3 do re me fa aeolian diatonic (aolian, dorian) ; scorp (mode 3)
3-2-2 do re mu fa Rast (arabic) 17edo neutral scale (dril, gil, kleeth)
3-3-1 do re mi fa ionian diatonic (ionian, mixolydian)

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