One thing is having all the ratios of the Partch 43-tone scale nicely presented with this keyboard layout but how does one get a handle on the ratios between the ratios?

In his book Genesis of a Music Partch himself points out that it’s easy to calculate the ratio between two ratios by simply inverting the smaller of the two and using that to multiply the other. So in the case of 4/3 and 3/2 we would have 3×3=9 and 4×2=8, which gives us the ratio of 9/8, which we know is the major whole tone that we have between a perfectly tuned fourth and fifth.

With some of the more complex ratios, e.g. 81/80:32/31, things can get a little more complicated and so I’ve set up a spreadsheet to do the work for me, also including a representation of the interval in terms of cents using the following formula: 1200*log(n/d)/log(2) where n/d is your ratio.

And to make things a little more fun I’ve set up a small device using the Audulus 4 beta that uses the same principles to display both the ratio and the cents of any two notes played on the keyboard in real time.

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.

The ID700 is a modern interpretation of the Buchla 700, and it comes with 4000 tunings pre-installed, one of which is the Partch 43-tone scale. The overlay I created for the SenselMorph has C as its root note but the scales in the 700 have A as their starting point ¹, which means that I have to shift the incoming MIDI notes either up by 9 steps, or down by 34 in order to have them work with the overlay.

I’ve done that using Bram Bos’ Mozaic MIDI filter, and I’ve set up quite a few instances so that I can quickly shift between octaves or play them simultaneously.

One interesting thing about working with a 43-note scale and only having 128 MIDI notes to work with, is that one is just one MIDI note short of having a full three octave range.

So in this case this (21/16) is the lowest note, and if I go to the top register, that’s the highest (27/20), and after that, I run out of MIDI notes.

If I come back to a middle register, one of the nice things about the ID700 is that it offers MPE support, which means that I can shape each note independently of the others. And in this case I’ve mapped pressure to index number 2, so what sounds a little bit like opening up a filter is actually frequency modulation.

One of the joys of just intonation is rediscovering the beauty of these pure, simple triads, and then with the synth one can add a little bit of character.

As I mentioned in a previous video the symmetry of both the scale and the layout are useful to keep in mind when navigating it. So for example a major triad, when mirrored, gives a minor. So there are these music-theoretical aspects that are mirrored in the visual layout.

If one wants a little more fun one could add an arpeggiator… and so on…

I’ve been exploring with the Partch 43-tone scale and one of the things I’ve been curious about is how one might go about navigating a scale that has so many degrees.

My first step has been to add a colour scheme to the overlay that I created for the Sensel Morph, based on the colours that Partch himself used on his Chromolodeons.

With that the symmetry of the scale becomes immediately apparent. The coloured hexagons are the ratios from the 11-limit tonality diamond that Partch has at the core of his scale, and the grey hexagons are the additional tones that he added in-between those core ratios as a way of filling in some of the wider gaps, and making a wider range of harmonies possible.

So the ratios in the top right hand quadrant will be mirrored in the bottom left, and vice versa. To demonstrate: a pure fifth 1/1:3/2 will correspondingly give me a pure fifth here 2/1:4/3. I can also go to another degree of the scale, 16/15:8/5 will corresponding give me a pure fifth here 15/8:5/4. You can also do that on other degrees of the scale 81/80:32/21, but here you won’t get a pure fifth, you can hear some beating, and the same thing here 21/20:11/7, 33/32:14/9 slightly faster beating. But if I go back to one of the pure fifths 1/1:3/2 and add a third 5/4 you get a nice pure major triad, or minor 1/1:6/5:3/2. There are also two ‘alternative’ degrees of the scale, e.g. here 10/9:4/3 I get a minor third, but if I take this one 11/10:4/3 it’s no longer in tune, but what it does give me is a purely tuned major second (10/9), and correspondingly 20/11:18/11, I get the pure major second over there.

So that’s a little start in how one might go about navigating the 43 tones of the scale. I find the the symmetries are really useful in reducing the amount of information that one has to deal with.

I’ve recently experimented with the Partch 43-tone scale, here in the Wilsonic app, sending it over to the Audiokit SynthOne in order to play it with those sounds using the overlay I created for the Sensel Morph, and I was curious to try something similar with the Audio Damage Continua synth. Unfortunately there’s no easy way to send the tuning directly from Wilsonic to Continua but it is possible to go over to sevish.com, to the scale workshop there, where the Partch scale is available as one of the presets that can be loaded.

I’ve set the base MIDI note to 60 and the base frequency to 261.626 Hz, which corresponds to an equally tempered Middle C. That can then be exported as a .tun file which can in turn be imported into Continua. And voila, the Partch scale in Continua.

The nice thing about Continua is that it offers MIDI MPE which means that you can alter the pitch the timbre of each note independently of the others. So, for example, I can alter the waveform of the 5th, or its pitch, or the root.

And so suddenly, in addition to the scale, you have a whole range of expressive possibilities at your fingertips.