An ingenious method of tuning pianos, based on the concept of entropy, has recently been devised by Haye Hinrichsen of Würzburg University. Entropy, which first appeared in the mid-nineteenth century in thermodynamics and later in statistical mechanics, is a measure of disorder. Around 1948 Claude Shannon developed a mathematical theory of communications and used entropy as an indicator of information content [TM084, or search for “thatsmaths” at irishtimes.com].
Tuning curve showing the stretch for high and low notes (Image: Wikimedia Commons).
In the context of piano tuning, entropy is a measure of disharmony or discord, and the aim is to make it as small as possible. Last week’s blogpost showed how the entropy of twin Gaussian peaks decreases as they move closer together. This is a key idea behind Entropy Piano Tuning.
Sound Waves
Sound is a series of pressure variations in the air. Although all sounds travel at the same speed, they have many different patterns of pressure. They may be analysed into wave-like components called Fourier modes. Each component is a pure sine wave with fixed amplitude and frequency. The amplitude gives the loudness of the wave while the frequency – the number of pressure oscillations per second – gives the pitch or tone.
Western music is based predominantly on a tuning system called equal temperament, where the ratio of frequencies of neighbouring notes is constant. With twelve semi-tones in an octave, this ratio is the twelfth root of 2. Thus, by advancing a full octave the frequency is doubled. The system allows music in all keys, but there is a price to pay: all intervals other than octaves are slightly out of tune. With the exception of purists, we have learned to tolerate the slight imperfections of equal temperament.
Each note on the piano has a fundamental frequency but, in addition, it has overtones or harmonics. For an idealized vibrating string, the harmonics are exact multiples of the fundamental frequency: the second overtone has twice the base frequency, the third is three times larger, and so on. A piano string is more complex, with vibrations between those of an ideal string and a thin rod. As a result, the overtones are not whole-number multiples of the fundamental frequency, but are slightly different from these values.
Image from blog http://www.21harnomy.com.
Inharmonicity
We call this discrepancy inharmonicity, and piano tuners must take it into account. The pattern of inharmonicity differs from one piano to another and the tuning must be tailored to suit each individual instrument. One consequence of inharmonicity is that a piano sounds better when its tuning deviates from a pure harmonic sequence. Higher pitches are tuned up and lower ones down, a process called stretching. Aural tuners are skilled at choosing the stretch to suit the instrument. They also allow deviations for individual notes.
Entropy Piano Tuning
The entropy tuning method also produces a tuning that allows for these effects. The entropy of two notes sounded together decreases as their harmonics or overtones come into alignment. Thus, the method produces a sequence of pitches that minimises the total discordance between the harmonics of all the notes. The method is inherently sensitive to all intervals, not just to octaves. It can produce stretch deviations for individual notes comparable to those found by an aural tuner.
The entropy for a given tuning is given by the sum over all fundamental tones and overtones:
S = – Σj Σm pjm log pjm
where pjm is the sound intensity of the m-th harmonic of the j-th fundamental tone. When two pitches pjm and pkn coalesce, the value of S goes down. The goal is to find a set of fundamental tones p0m that result in a local minimum of the entropy S. This is done by a Monte Carlo method of randomly varying a fundamental tone, keeping the variation if it reduces S and rejecting it otherwise. The method converges to a satisfactory tuning curve.
In a practical implementation of the method, each piano note is recorded and analysed into harmonics. Random variations are tested, and accepted if they reduce the entropy. The process continues until a satisfactory tuning is reached. This gives the target pitch for each note. The method is increasingly being used by professional tuners. Of course, some musicians feel that a skilled piano technician can produce a superior result. They are probably correct; but for how long?
Sources:
- Haye Hinrichsen, 2012: Entropy-based Tuning of Musical Instruments. Rev. Bras. Ens. Fis. 34(2) (2012), 2301. arXiv:1203.5101.
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The "Entropy-Piano-Tuner" software can be downloaded
free of charge from: www.piano-tuner.org
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There is a long discussion about the project on
the World Piano Forum:
http://www.pianoworld.com/forum/ubbthreads.php/topics/2435332/Entropy_Piano_Tuner.html
ThatsMaths 