We’re going a little far afield from the usual fare this week, but there’s been a breakthrough in piano tuning you should know about. It’s called an algorithm.
According to the physics blog of the MIT Technology Review, Haye Hinrichsen at the University of Wurzburg in Germany has come up with a computer algorithm that now makes it possible for electronic tuning software to match the performance of the best human tuners.
Scientists have been experimenting in this area for some time now. Pluck a string on a guitar or hit a key on a piano and the sound it produces is the result of its fundamental frequency and its harmonics at frequencies that are whole number multiples of the fundamental. The frequency is easy enough to find, but music is organized into octaves, and the frequency of a note doubles from octave to octave–– so it grows exponentially as the octaves increase.
Notes that are an octave apart can all be in tune but at other intervals––like perfect fourths or fifths––they will always be slightly out of tune. Skilled professional tuners get around this problem by “stretching” the interval between some notes using their intuitive sense of what it takes to bring the musical notes into balance. Up until now, electronic tuning programs have been able to calculate the “average stretch” but nowhere near as effectively as the human ear.
Enter Hinrichsen and his theory of “entropy minimization.”
“When humans listen to two notes an octave apart, say A2 and A3, they compare not just the fundamental frequencies buy also the harmonics,” according to the Technology Review blog. “In theory, the second, third and fourth harmonics of A2 should correspond exactly with the first, second and third harmonics of A3 (and so on). The notes are in tune when the harmonics lock exactly. (But) the higher harmonics do not match exactly and the slight mismatch produces a beat frequency that a professional tuner tries to minimize.”
What the professional tuner does with his ears, Hinrichsen attempts to replicate by using the conventional “equal temperament method” of tuning, then dividing up the audio spectrum with a resolution that matches the human ear’s.
“Since the entropy of two spectral lines decreases as they begin to overlap, the problem of finding the best possible compromise when matching harmonics is equivalent to minimizing the entropy of the system,” the Technology Review reports. “So the new method is to measure the entropy of the system, apply a small random change to the frequency of a note and measure the entropy again.” If it drops, the system is considered more in tune. Repeating this with another random change allows you to hone the process even more.
There is no perfect pitch in the “entropy minimization” system––nor, for that matter, is there perfect pitch in the human ear––but the results come close enough. Professional tuners will now have to go to bed wondering if their days are numbered. “The implementation of the method is very easy,” Hinrichsen says. It is just an algorithm that can be inserted into any number of electronic tuners, so watch for it at a Radio Shack near you.