17edo neutral scale
17edo neutral scale
A lovely system of Middle-Eastern flavored scales!
We can call the Moment of Symmetry scale derived from a 5/17 generator & an octave repeat the 17edo Neutral Scale. It is an example of a neutral thirds scale. We build it by stacking neutral thirds, the generator of the neutrominant temperament. In 17edo that means the interval of five degrees of 17.
Begin anywhere. Let's call our first pitch (& its octave transposition) 0:
0 (0)
Add a note a neutral third (five degrees) up from 0:
0 5 (0)
Add a note a neutral third down from 0 (remember, in 17edo, 0=17):
0 5 12 (0)
Between these notes we have intervals of:
5 7 5
Since we have two different step sizes, we have arrived at a three-note MOS scale. But let's continue; three-note scales don't give us much to work with.
Add an N3 up from 5:
0 5 10 12 (0)
Add an N3 down from 12:
0 5 7 10 12 (0)
Add an N3 up from 10:
0 5 7 10 12 15 (0)
Add an N3 down from 7:
0 2 5 7 10 12 15 (0)
We have arrived again at a MOS scale, of type 3L+4s ("mosh" according to the MOSNamingScheme).
Interval chain
Viewing 17edo as a temperament on the 2.3.7.11.13 subgroup, we get the following interpretation for the 2122122212 mode of the 10-note MOS scale:
| Step# of scale[1] | Steps of 17edo[2] | Note name on C | Harmonics approximated | #Gens up |
|---|---|---|---|---|
| 9 | 14 | Bb | 7/4 | -4 |
| 2 | 2 | Dd | -3 | |
| 5 | 7 | F | -2 | |
| 8 | 12 | Ad | 13/8 | -1 |
| 11 | 17 | C | 2/1 | 0 |
| 4 | 5 | Ed | +1 | |
| 7 | 10 | G | 3/2 | +2 |
| 10 | 15 | Bd | +3 | |
| 3 | 3 | D | 9/8 | +4 |
| 6 | 8 | F+ | 11/8 | +5 |
The 6th degree can be raised by a chroma to a 23/16 (-5 generators). Some may prefer using the sharper 6th degree because it makes a 7/4 with the 8th degree.
7-note neutral scale
degrees from 0: 0 2 5 7 10 12 15 (0)
cents from 0: 0 141 353 494 706 847 1059 (1200)
interval classes from P1: P1 N2 N3 P4 P5 N6 N7 (P8)
degrees between: 2 3 2 3 2 3 2
cents between: 141 212 141 212 141 212 141
interval classes between: N2 M2 N2 M2 N2 M2 N2
modes of 7-note neutral scale
This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.
Terms: The 7 proposed mode names
Naturally, with seven notes we have seven modes, depending on which note we make the starting pitch (tonic) of the scale. I (Andrew Heathwaite) have given these modes a one-syllable name for my own use. Feel free to name (or not name) these modes as you see fit:
| mode 1 : bish | from bottom | in between |
|---|---|---|
| degrees | 0 2 5 7 10 12 15 (0) | 2 3 2 3 2 3 2 |
| cents | 0 141 353 494 706 847 1059 (1200) | 141 212 141 212 141 212 141 |
| interval classes | P1 N2 N3 P4 P5 N6 N7 (P8) | N2 M2 N2 M2 N2 M2 N2 |
| solfege | do ru mu fa sol lu tu (do) | ru re ru re ru re ru |
| mode 2 : dril | from bottom | in between |
|---|---|---|
| degrees | 0 3 5 8 10 13 15 (0) | 3 2 3 2 3 2 2 |
| cents | 0 212 353 565 706 918 1059 (1200) | 212 141 212 141 212 141 141 |
| interval classes | P1 M2 N3 A4 P5 M6 N7 (P8) | M2 N2 M2 N2 M2 N2 N2 |
| solfege | do re mu fu sol la tu (do) | re ru re ru re ru ru |
| mode 3 : fish | from bottom | in between |
|---|---|---|
| degrees | 0 2 5 7 10 12 14 (0) | 2 3 2 3 2 2 3 |
| cents | 0 141 353 494 706 847 988 (1200) | 141 212 141 212 141 141 212 |
| interval classes | P1 N2 N3 P4 P5 N6 m7 (P8) | N2 M2 N2 M2 N2 N2 M2 |
| solfege | do ru mu fa sol lu te (do) | ru re ru re ru ru re |
| mode 4 : gil | from bottom | in between |
|---|---|---|
| degrees | 0 3 5 8 10 12 15 (0) | 3 2 3 2 2 3 2 |
| cents | 0 212 353 565 706 847 1059 (1200) | 212 131 212 141 141 212 141 |
| interval classes | P1 M2 N3 A4 P5 N6 N7 (P8) | M2 N2 M2 N2 N2 M2 N2 |
| solfege | do re mu fu sol lu tu (do) | re ru re ru ru re ru |
| mode 5 : jwl | from bottom | in between |
|---|---|---|
| degrees | 0 2 5 7 9 12 14 (0) | 2 3 2 2 3 2 3 |
| cents | 0 141 353 494 635 847 988 (1200) | 141 212 141 141 212 141 212 |
| interval classes | P1 N2 N3 P4 d5 N6 m7 (P8) | N2 M2 N2 N2 M2 N2 M2 |
| solfege | do ru mu fa su lu te (do) | ru re ru ru re ru re |
| mode 6 : kleeth | from bottom | in between |
|---|---|---|
| degrees | 0 3 5 7 10 12 15 (0) | 3 2 2 3 2 3 2 |
| cents | 0 212 353 494 706 847 1059 (1200) | 212 141 141 212 141 212 141 |
| interval classes | P1 M2 N3 P4 P5 N6 N7 (P8) | M2 N2 N2 M2 N2 M2 N2 |
| solfege | do re mu fa sol lu tu (do) | re ru ru re ru re ru |
| mode 7 : led | from bottom | in between |
|---|---|---|
| degrees | 0 2 4 7 9 12 14 (0) | 2 2 3 2 3 2 3 |
| cents | 0 141 282 494 635 847 988 (1200) | 141 141 212 141 212 141 212 |
| interval classes | P1 N2 m3 P4 d5 N6 m7 (P8) | N2 N2 M2 N2 M2 N2 M2 |
| solfege | do ru me fa su lu te (do) | ru ru re ru re ru re |
As you can see, these modes contain many neutral 2nds & 3rds, making it sound very different from the traditional major-minor Western harmonic & melodic system, while having a coherent structure including ample 4ths & 5ths that help ground the scale. The 17edo neutral sixths, at 847 cents, come very close to the 13th harmonic - JI interval 13/8 - 841 cents. Thus, their inversions, the 17edo neutral thirds come very close to 16/13.
The 17edo neutral 2nds, at 141 cents, fall between 13/12 (139 cents) & 12/11 (151) cents. I've found that they generally function as 13/12, since they fall 3/2 away from 13/8. But you can discover these things for yourself, if you like, & feel free to think of them in different ways entirely.
Interestingly, the 7-note neutral scale does not allow you to build any minor or major triads whatsoever. You have only one minor 3rd, which occurs with a diminished 5th, but no perfect fifth, allowing you to build a diminished triad, but no minor triad. You have no major thirds at all. In JI-terms, you might say that it contains harmonies based on 2, 3, & 13, while skipping 7 & 11.
17-tonists may find these scales helpful for escaping the familiar. Just because you can play diatonic music in 17edo, doesn't mean you have to. These neutral scales give you a more xenharmonic modal system to play with.
If you continue stacking neutral thirds, you will soon come to a rather lovely 10-note neutral scale. I (or someone) will come back to that sooner or later.
Some brief note on the 3, 7 and 10 note MOS
You can also take call the neutral sixth the generator, which I (Andrew Heathwaite) personally favour as it is an (approximate) harmonic rather than a subharmonic. But that's because it's how I use it, you might not. If you see it this way, the 3rd harmonic is harmonically opposite to the 13th harmonic, because, (13/8)^2 ~ 4/3, the perfect fourth being an upside down perfect fifth.
You might also find that the 10-note scale can be formed by two 17-tone pythagoresque pentatonic scales a neutral interval apart, implying something of a different approach. And one of the loveliest things I find about them is the ease with which one can play 8:11:13 chords, so there are some frightening blues licks in this decatonic scale. R'lyeh blues anyone?
(Note that you will come up with similarly structured scales by using other neutral thirds as generators, although some of them will sound quite different. A neutral sixth about sharp of the 13th harmonic leads to 7L+3s like in 17-tone, whereas going flat of the 13th harmonic can lead to 7s+3L. (This boast is possible because 10-edo sits right on it.) Some equal divisions of the octave containing neutral scales: 10edo, 13edo, 16edo, 19edo, 24edo, 31edo....)