Pythagorean Shifts

I see that the Yamaha CP1 electronic piano can be switched to play in Pythagorean tuning. This means that fifths are true, i.e. a frequency ratio of 3/2. This is about 2 cents sharper than an equal tempered fifth - a cent is 1% of a half step. Of course, if you follow the circle of fifths and every fifth is 2 cents sharper than equal tempered, the circle will not actually close. The last fifth must be 22 cents flat! Thus, the Pythagorean tuning can be diagrammed with:



I don't see in the CP1 manual which fifth they chose to be flat, so I just took a guess.

This choice of the flat fifth might be something a performer would like to vary, even during a performance. How should the tuning of the piano best change when the performer shifts the flat fifth up or down a fifth?

One approach would just be to shift the whole diagram over a fifth - to keep the same set of differences from equal temperament, but just to shift the assignment of those differences a fifth. This can be diagrammed with:


This is clearly not very satisfactory. Every note on the piano gets adjusted - most notes a flattened two cents, but then one note is sharpened by twenty two cents. This global shuffling could be disruptive in the middle of a performance.

Another approach is simply to sharpen a single note by twenty four cents:


The great advantage of this approach is that, when shifting the flattened fifth one step around the circle of fifths, only one note is changed: the eleven others are left unaltered. This should eliminate any sense of disruption during a performance.

However, with this approach, the whole set of differences from equal temperament drifts sharp by two cents. Shifting again and again in the same direction would just keep sharpening the tuning of the piano. But that is the nature of real Pythagorean tuning: the circle of fifths just doesn't close!

Of course the performer would also have the inverse option, to move the flattened fifth back in the circle of fifths. Again eleven note are unaltered, and then one note would be flattened by twenty four cents.

It looks like this approach to dynamically shifts for Pythagorean tuning could be implemented naturally on the Yamaha Motif XS synthesizer, using the stock tuning controls.

Step one is just to work with Pythagorean tuning, to explore the musical value of the just tuned perfect fifths, and then to see how that awkward very flat fifth gets in the way. Then one can start to explore how dynamically shifted tuning allows one to push the piano past its usual limits - it's as if the piano actually has more keys than twelve per octave!