This video captures Lassus’ Ave Regina Coelorum, performed on the Hi-Pi Tonal Plexus keyboard.

This excerpt by Lassus (16th century composer) presents a puzzle for tuning with pure intervals, where the closing chord should be the same as the opening chord, but if strict theoretical rules are followed the closing chord will fall about a quartertone lower than the opening chord. The passage is played twice here on a 2 octave Tonal Plexus keyboard, showing how this pitch shifting occurs.

Details on the two performances below.

via AaronAndrewHunt:

In the first performance, only pure intervals are used, which lead to the quartertone lower closing chord. The drop in pitch at 2 junctures should be clearly audible. The second time the quartertone distance between the opening and closing chord is distributed throughout the passage at almost every change in harmony, so that the final chord ends up being the same as the opening chord instead of lower. This means that common tones and pitches normally held as suspensions (which should not change) actually do change in the second example, shifting up by 1 JND at almost every change of harmony. This is usually called ‘adaptive’ tuning.

I limited the adaptive shifts to 1 JND, which is the distance between 1 key vertically and the next in the default tuning of 205ET which is used here. The adapting could be done in slightly different ways to end up with the desired outcome. I personally like the small lift in pitch which is felt with every change in harmony in the adaptive version. This lifting of pitch gives a feeling that the piece is constantly rising when in fact it is just compensating for the naturally falling pitch. I like the subtle changes. A different example could show the opposite, where small lowing of pitch over time could compensate for rising pitch, and a longer passage could mix both things.

Just a note here – I have changed some registers of pitches in this example because the bass notes would not all fit on a 2 octave keyboard. Also, I admit that I did not consult a score for this; I only listened to a tuning example made by Kalle Aho and figured out the passage by ear at the keyboard. The basic voice leading is correct (compare with a vocal performance at http://www.youtube.com/watch?v=JSAqBnI7ukw – the opening phrase; note that in some other performances a B-flat is used in the first chord of the final elaborated cadence instead of a B, which also sounds nice).

Thanks to Kalle for posting the adaptive tuning example which gave me the idea to do this. And, thank you for watching and listening (and reading).

Check out the Tonal Plexus at http://www.h-pi.com