Monday, March 25, 2024

Shifting along the Chain of Fifths

Musical tuning systems continue to fascinate me. One path of exploration is innovation, to explore fresh territory. Another path is to look at history and to explore the foundations that the present conventions are built on. Meantone is a historically important tuning system that can still offer innovative possibilities.

The conventional 12 tone equal tempered system of today is built on the circle of fifths. The more fundamental meantone system is build on a chain of fifths. Meantone is actually a family of tuning systems. The conventional 12 tone system is just one of these. Some members of the family will close the chain of fifths into circles of different sizes; others will leave the chain unclosed.

More fundamental than the circle of fifths is the circle of octaves. One can start a scale at C, move up through D, E, F, G, A, and B, and then return to C again: a higher C that where the scale started, but the name repeats. It's another C.

The higher C has twice the frequency of the lower C. One C might be at 256 Hz, and the next higher C would be at 512 Hz. Generally, the theory of tuning systems is built on the principle that two pitches will sound consonant together if their frequencies have a simple ratio. 2:1 is about as simple as a ratio can get; that's why pitches an octave apart are so intimately related that we often just ignore their difference. That's what I'll be doing here, largely. I'll just look at pitches within a single octave range.

After 2:1, the next simplest ratio is 3:2. That's the frequency ratio for pitches a perfect fifth apart. That's why the chain of fifths is so important: the chain is like the main highway, the path built by moving from one pitch along to the pitch it is most consonantly related to (again, ignoring octaves).

The picture above shows the conventional names for the pitches along a central section of the chain of fifths. The chain can be extended indefinitely in either direction, just by adding more and more sharp symbols in one direction, or flat symbols in the other. A musical composition generally uses a finite number of pitches, and so will use just a finite section of this infinite chain. Pentatonic music will just use a five pitch section of the chain. A diatonic scale encompasses a seven pitch segment. The note naming convention A, B, C, D, E, F, G is based on the diatonic scale.

Musical compositions can be based on different sections of the chain of fifths, or might be broken into parts that use different sections, or might use more than seven pitches even in a single part. To slide a seven pitch subset up the chain, one adds the next pitch in the direction of motion and removes the trailing pitch. With the seven pitch subset, these two pitches will be nearby. If the chain is built of precise 3:2 perfect fifths, the difference between the new and old pitches will be a ratio of 2187:2048 (folding the octaves together as needed). This small ratio is the basis for the sharp and flat notation: sliding one step down the chain of fifths involves just a small tweak to one pitch of the seven in a diatonic scale.

The tuning system built on a chain of precise 3:2 ratios is called Pythagorean tuning. It runs into trouble. The Pythagorean major third is a ratio of 81:64. This is very close to the simple ratio of 5:4. When two pitches are quite close to a simple ratio like this, but still significantly off that simple ratio, they sound harsh or out of tune.

Meantone tuning is a solution to this problem. The perfect fifths are slightly flattened, in order to reduce the error in the major third. Exactly how much to flatten, that is not fixed. A range of possibilities has been used historically. The conventional 12 tone system is only a little flat, and the error in the major thirds remains significant. The graph above shows the practical range. One can see that, for example, 1/5 comma meantone balances the errors in the perfect fifth and the major third. 2/7 comma meantone balances the errors in the major and minor thirds. These and other choices have been advocated and used historically.

The fundamental problem with meantone tuning is that, in general, it is still working with the infinite chain of fifths. Many musical instruments can only provide a limited choice of pitches to be played. If there are enough choices, composers won't be overly constrained by the requirement that they don't run so far along the chain that they exceed the capacity of the instrument. Another practical possibility is that the size of the perfect fifth is chosen so that the chain is closed into a circle. This is the great virtue of the conventional 12 tone system. 12 pitches per octave is a very practical number. One can wander up and down the chain of fifths, and because it has been closed into a circle, one will never run into a wall. One can use 19 and 31 pitches per octave, along with other choices. The major thirds can be significantly better than those in the conventional 12 tone system, but the extra pitches per octave can be a bit unwieldy.

While conventional piano keyboards have 12 keys per octave, 7 white and 5 black, harpsichords and organs have been built with additional black keys, tucked alongside the conventional 5. Sometimes just one or two black keys are split. The graph above shows the pitches in a range of meantone choices for a fully extended keyboard, where each black key has been split for a total of ten black keys. One can see in this graph that the sharps and flats cross for the 12 tone conventional tuning system. In the exact Pythagorean system C# is higher than Db. For most of the usual meantone choices, such as 1/5 comma meantone, C# is lower than Db.

Another way to manage the chain of fifths, other than walls and circles, is to add shift controls to instruments. Harps are a good example of this approach. A harp has just seven strings per octave, but allows further movement along the chain of fifths by way of pedal controls. Modern electronic instruments could easily support unbounded movement along the chain of fifths. For example, a conventional 12 tone keyboard could work with a full range of the chain of meantone fifths. At any one time, the keyboard would present a range of 12 pitch choices per octave. There would be one "wolf" fifth in the tuning. But a pedal or other control could be provided, to shift the location of this wolf. At one time the keyboard might provide a range along the chain from Eb to G#. If the music requires moving a fifth up from G#, the required D# is not immediately available. But the auxiliary control could shift the Eb to D#. Generally a piece of music is not going to need the Eb at any time close to when it needs the D#, so handling the auxiliary control should not be too burdensome.

The next natural such circulating keyboard would have 19 keys per octave. These could be arranged by splitting the five black keys, and then adding single black keys between B and C, and between E and F. This could work as a simple system with 19 equal steps per octave, which is approximately 1/3 comma meantone. But such a keyboard could also work in arbitrary meantone tunings using an auxiliary shift control. So, for example, in one setting of the auxiliary control, the key between B and C would provide Cb, while in another setting it would provide B#.

1 comment:

  1. ah! Here is a 19edo split black key configuration, as I described: