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Entropy-based tuning

The idea behind entropy-based tuning is very simple: Since entropy is a measure of disorder, we expect the entropy of a well-tuned piano to be lower than the entropy of an instrument out of tune.

articleA first test with real data led to the surprising result that a simple entropy-based iteration algorithm produces a resonably-looking tuning curve. In spring 2012 we published these findings in a Brazilian physics journal (see right). This article triggered a huge but misleading echo in the media (e.g. in Daily Mail and the Wall Street Journal), claiming that the new method puts aural tuners out of job. However, at that time it was not even clear whether the method would work at all, and not a single piano had been tuned.

Now we present a software which allows everyone to evaluate this new tuning method. With this software we do not aim to compete with high-end tuning devices and professional aural tuners, rather we would like to demonstrate that a single extremely simple formula can generate an acceptable tuning.

The problem: A piano cannot be tuned with a conventional tuning device because of the so-called inharmonicity, a physical effect caused by the stiffness of the steel strings. In fact, the art of piano construction and tuning is to compensate these effects by putting the instrument slightly out of the mathematical tune. This creates the typical texture of the sound which may have played an important role for the world-wide dissemination of the piano.

How the EPT works: The entropy piano tuner (EPT) records all keys of the piano and determines the spectra of their higher partials (overtones). These power spectra are filtered, added up and used to compute the entropyformula

This is like pressing all keys of the piano simultaneously and computing the disorder of the resulting sound. By shifting the spectra of individual keys, which correspond to the process of tuning, it is possible to minimize the entropy. In the EPT we use a simple Monte-Carlo algorithm: A key is selected at random and the pitch of the key is altered by a small random amount. If this leads to a decreasing entropy, the change in the pitch is accepted, otherwise it is discarded. In this way it is possible to reduce the entropy successively, producing a certain tuning curve.

The EPT realizes the three steps – recording of the keys, minimization of entropy, and tuning according to the computed pitches – in a single application.