Wednesday, August 18, 2010

Dynamically Tuned Piano

Tuning a keyboard instrument such as a piano so that each half step has exactly the same frequency ratio is the tuning system known as equal temperament. In equal temperament, a sequence of notes can be shifted up or down the keyboard by any number of steps, and the harmonic relationships between the notes will not be changed. This symmetry is a big advantage for equal temperament.

As discussed by Kyle Gann, there are subtly different tunings that have some other advantages. The fundamental problem with equal temperament, or really with any tuning system, is that the mathematical ratios underlying harmony can be combined in an infinite variety which would require densely packed notes infinitesimally spaced, if the mathematical ratios were to be represented precisely by the tuning system. For a tuning system to provide only a manageable set of notes to a performer, some or all of the intervals will have to be adjusted, or tempered.

Any tuning system that is not equal tempered will not be symmetrical, by that very inequality. A sequence of notes will sound somewhat different if it is shifted up or down the keyboard. For example, the fifth from C to G might be slightly different than the fifth from E to B. A musical piece will have a slightly different character when transposed to a different key.

This variety of character opens up an interesting musical possibility. Given modern keyboard technology, the pitch of any note on a keyboard can be shifted slightly in an instant. To change the character of a piece of music, one could leave the music in the same key, but just tweak the tuning of the instrument on the fly, during a performance.

This is analogous to an orchestral harp. A harp has only seven strings. One can use the pedals of the harp to sharpen or flatten strings, one pitch class at a time, so the harp can be played in the different keys. My proposal here is to take a keyboard that can has twelve pitch classes available, and to use a pedal, or perhaps hand operated controls, to shift one or more of those twelve pitch classes slightly sharper or flatter, to make available multiple non-equal temperaments during a performance.

Here is one concrete proposal for a set of operations to shift tunings. Studying Kyle Gann's presentation of Young's Well Temperament from 1799, one can see that the deviation of the pitches of the various notes from equal temperament follows a simple pattern when plotted along the sequence of fifths:

Typically one will want to shift the meanings of the notes, the harmonic structure, by a fifth. One could just shift the tuning pattern directly:

This shift leaves four pitch classes unaltered, sharpens four, and flattens four. One problem with this shift is that the four unaltered pitch classes are in two separate pairs. Musically, probably only two are significant at any time.

One can also provide two alternatives: shifting up

and shifting down

These two shifts also leave four pitch classes unaltered, but the four pitch classes are all together in the circle of fifths, and so would generally work better as a tonal center during a shift.

A complete system could provide these three alternative shifts when moving up a fifth, and the corresponding three when moving down a fifth, for a total of six single step possibilities. Each step could be repeated indefinitely. Shifting over repeatedly would bring one back to the same tuning after twelve shifts. Shifting up or down repeatedly would keep sharpening or flattening notes, so the whole tuning would be drifting up or down in pitch. Of course, a more typical usage would likely be to shift up when moving a fifth in one direction, then to shift down when moving a fifth in the other direction, which would return the keyboard to the starting tuning.


  1. I'm looking at page 57 of the user manual for the Yamaha CP1:

    This piano allows one to shift tunings, but doesn't appear to have the flexibility of control to be allow the range of possibilities I have outlined here.

  2. Ah, but the Yamaha Motif XS synthesizer does allow the flexibility my proposal requires - see page 265 of:

  3. At least one of the latter scaletrons has two keyboards one which could be used to modulate the whole tuning as you suggest. The idea there was to be able to modulate diamonds around an eikosany or vice versa. This idea would also fit well with Boomliter and creels ideas of extended reference.

  4. Another use of dynamic tuning could be with just intonation, to switch on the fly the choice of which intervals are true and which are given up. Maybe a better way to say it is, one could switch the just interval between various pairs of keys. E.g. some major thirds could be 5/4, while others would be 32/25. One could dynamically switch how these choices are made on the keyboard.

    See Easley Blackwood's _The Structure of Recognizable Diatonic Tunings_ for an argument that conventional Euopean music from Bach to Mahler doesn't really work in just intonation. My proposal here is a way to work within that conventional framework, but to start expanding it in a very gradual way.

  5. Some further relevant discussion: Bach's notations on tuning...

  6. This software does all that you're talking about, and more:

    It works with almost any keyboard or softsynth.