Thursday, February 29, 2024

A Tale of Two Unconventional Tunings

Nowadays the notes available on a conventional piano, seven white keys and five black keys per octave, form the building blocks for almost all the music in circulation. And of course these building blocks have been very effective at enabling the crafting of a vast treasure chest of music, diverse, profound, and beautiful. And yet, there is value in exploring other tuning systems.
  • Around the world, there are still many different traditional tuning systems in use.
  • Tuning systems evolved over the centuries in Europe, only settling on the present convention some two centuries ago.
  • Different tuning systems enable different musical structures; they are a rich compositional resource.
  • Conventional tuning can be better understood in perspective, as being one alternative among many.

One could spend a lifetime learning about different tuning systems, their histories and features etc. But sometimes when encountering a large building it can be difficult to find an entrance! Recently I have been exploring the tuning system that divides octaves into fifty equal intervals, rather than the conventional twelve. Dividing octaves into fifty three equal intervals is another useful tuning. The sizes of the intervals in these two tunings is not very different, yet the tunings have quite different strengths. Comparing these two systems could work as a doorway into the world of alternate tunings.

A general foundation for tuning theory is the observation that two pitches sound consonant when the ratio between their frequencies is a simple rational fraction. For example, the A above middle C is conventionally tuned to 440 Hertz. The next higher A, an octave higher, is at 880 Hz, a ratio of 2:1. If one tunes the E in between to 660 Hz, it will sound very nicely consonant with either of the As, with ratios of a perfect fifth, 3:2, or a perfect fourth, 4:3. Tuning the C# to 550 Hz will complete a consonant major triad. The interval from the A of 440 Hz and the C# of 550 Hz is a major third, with a frequency ratio of 5:4. The interval from the C# of 550 Hz to the E of 660 Hz is a minor third, with frequency ratio 6:5.

The pitches involved in a piece of music form a network. Each pitch is related to several other pitches, and these related pitchs then relate to yet other pitches. Pitches are thus related by chains of simple intervals. The whole pitch space forms a kind of network. If the simple relationships are built from the consonant relationships of fifths and thirds described above, the network of pitches will look something like this:

These pitches are all inside a single octave range - the network could be replicated in as many octaves as needed. The network can also be extended arbitrarily in any and all directions.

While one can make perfectly good music with a tuning system like this, with very precisely consonant frequency ratios, it does run into difficulties. As the network is extended, each octave gets broken up more and more finely, without limit. It's hard to build instruments that can play so many notes, hard for players to hit the right notes, and hard for listeners to distinguish among so many notes. Over the centuries, musicians, composers, and instrument builders have developed simpler tuning systems that approximate these ideal intervals while avoiding the infinite division problem. And then music has evolved to take advantage of opportunities these simpler tuning systems provide for harmonic movement. A tuning system is a network of pitches with a particular shape. Music is then a kind of dance that moves around through that shape.

The fine divisions brought about by precise consonance first arise with the 81:80 pitch on the right column of the tuning network above. It is very difficult to distinguish that from the 1:1 in the center. So the first tuning simplification is to adjust the pitches in the network somehow so that 81:80 is changed back to 1:1. This changes the shape of the tuning network from a flat plane to a cylinder. If one travels in a suitable constant direction on the surface of a cylinder, one can end up back where one started.

There are many ways to adjust the pitches in the network so the 81:80 is flattened slightly to become 1:1, but in general this tuning system is known as meantone. The way pitches are named in European music is a reflection of the meantone system:

While this system does allow unbounded movement, that movement needs to flow around the cylinder, along the diagonal strip where the sharps and flats don't get too wild. Old keyboard instruments sometimes have extra black keys to accommodate a wider range of movement, but still, it can be challenging to dance freely when there is an abrupt edge that one must steer away from. So the next step of evolution is to wrap the cylinder into a torus:

If one moves a perfect fifth from G#, one arrives at Eb. The network of pitches has been tweaked somehow so that D# and Eb are the same pitch. There are various ways to do this, but the simplest way is to adjust all the fiths and thirds in the same way, so the system is totally uniform. This is our conventional tuning of today.

To review the development so far: A network of precise consonances splinters the pitch space to an impractical unbounded extent. Adjusting, or tempering, the intervals allow the network to wrap back on itself, so the number of pitches required can be limited.

Fundamentally, a tuning system is a compromise between simplicity and precision. But tuning must serve music. The shape of the tuning network enables some kinds of harmonic movement but prevents other sorts. Music and tuning evolve in response to each other, meeting each other's demands and taking advantage of each other's opportunities.

One can build a tuning system by dividing octaves into equal intervals of any number. A good tuning system will provide intervals that are close approximations of the precise consonances of 3:2 and 5:4. Dividing octaves into 50 or 53 equal parts will provide reasonable approximations:

This table gives the error, in cents, for each tuning system for each consonant interval. One can see that the conventional tuning system has somewhat large errors for several intervals, though it comes quite close for 3:2. The 53 steps per octave system is quite accurate for all the intervals. The 50 step system is not so good for 3:2, but it is at least better than conventional tuning for the thirds 5:4 and 6:5.

It might seem that, since 53 steps per octave is only slightly more than 50 steps per octave, and provides a significant improvement in precision, that the 50 step per octave system is not very useful. But beyond simplicity and precision, one must look at the shape of the tuning network:

The bright blue highlighted cells marked "0" show the way the torus is wrapped back on itself. Those closely spaced "0" cells along a line sloping slightly down to the right, those cells are wrapped in exactly the way that the meantone tuning system is wrapped. What this means is that most any music written for the meantone system will be playable in the 50 step per octave system. The 50 step system will support even triple sharps and triple flats. It would be a rare piece of music that requires more sharps and flats than that!

The 53 step per octave system has a very different shape:

The pattern of repeated cells, the way the tuning torus is wrapped, does not match the meantone system at all. Music written in the meantone system will fail to return or connect back properly if one tries to play it in the 53 step system.

I have been exploring some of the unconventional musical possibilities of these two tuning systems. For each, I picked a subset of the available pitches to work as a scale. In both systems I built the scale to form a path from lower left to upper right, which is, roughly speaking, a chromatic scale. I then used my algorithmic composition software to generate some music that would flow with the shapes of the scales: