Now that I have all these ratios at my fingertips, having set up my Partch overlay for the Sensel Morph, I thought it might be interesting to take a look at the humble pentatonic scale and the various ways in which it might be tuned.
We can set up the scale by stacking up a series of fifths:
4/3 2/1 3/2 9/8 27/16. Or if I mirror that:
1/1 32/27 4/3 3/2 16/9.
So what we have here is Ben Johnston’s Pythagorean tuning of the minor pentatonic, and you can see that it starts with quite a complex ratio:
32/27 for the minor third. And that got me thinking of Clarence Barlow’s notion of harmonicity and wondering what the scale might sound like when tuned with the simplest ratios possible:
1/1 6/5 4/3 3/2 9/5.
It’s slightly different.
Let me add an arpeggiator and we can compare.
So the Pythagorean version comes quite close to an equally tempered tuning. But the ‘pure’ version has a slightly different character.
The main difference between the two scales is that the Pythagorean version has same size major second throughout: the so-called major whole tone, which corresponds to the ratio of
9/8, whereas the ‘pure’ version starts off with a minor whole tone, corresponding to the ratio of
10/9, followed by a major whole-tone, followed by a pure minor third
6/5, and then a minor whole-tone at the top.
I could also mirror that:
2/1 5/3 3/2 4/3 10/9.
Or in a different transposition:
1/1 10/9 5/4 3/2 5/3 – the ‘major pentatonic´.
I could also mirror that:
2/1 9/5 8/5 4/3 6/5.
And so on…
If I wanted to create a pentatonic scale with 5 equally spaced steps, the closest that I can get with this set of ratios is something like this:
1/1 8/7 21/16 32/21 7/8 2/1.
Since the middle two notes are quite close to the purely tuned fourth and fifth I could also choose to embrace them. Or alternatively choose to obfuscate those pure intervals by adding the adjacent notes:
21/16:4/3 32/21:3/2 – taking a little inspiration from Javanese Gamelan music in which some of the instruments are tuned in pairs quite close to one another, but not exactly, so that beating arises.
A pentatonic scale with five equally spaced steps would have the property of being perfectly balanced – i.e. without any particular point of gravity. If one represented the scale as weights distributed around a suspended wheel, the wheel wouldn’t turn in any particular direction, no matter which position we placed it in. One might say that this corresponds to a spatial mode of hearing, rather than proportional one, as is the case with ratios.
In practice much of the music based around (somewhat) equal distance scales, Gamelan music using the slendro tuning, Mbira music or Chopi Timbila music for example, the individual steps change slightly depending on the player or region. They’re never exactly equal. This probably also has to do with the relation between tuning and timbre, which might differ from note to note given the hand-made nature of the instruments. And that’s something we might take as a point of inspiration when experimenting with our own (equal distance) scales.