Well Tempered Tune
“The piano is an untunable instrument.” – Ed Foote
Although funny, this quote from Ed Foote derives from the fact that the strings of a piano or harpsichord are not ideal oscillators and, therefore they cannot be tuned to be perfectly consistent with each other. Even if the strings were ideal, perfect tuning is a mathematical impossibility, And even in the case of unisons, some argue that the best sound is obtained if they are intentionally not tuned to exactly the same frequency.
The phrase “well-tempered” was first used by Andreas Werckmeister in 1691 to describe a method of tuning that removed the “wolves” heard in previous tuning methods. The goal was to provide a “practical organization of the tone system within the twelve steps of an octave so that impeccable performance in all tonalities is enabled.”
Many people are surprised to learn that there is more than one way to tune a piano or harpsichord. There are many parameters that need to be optimized in a tuning, and they cannot all be optimized simultaneously. Sacrifices or compromises must be made, and the final tuning is a subjective decision. As a result, no two tuners will tune a piano in the same way, and occasionally even the same tuner will find it difficult to reproduce what a client previously thought was an ideal tuning.
Ideally an aural tuner or an electronic tuning device (ETD) should take into account the intrinsic contradictions and imperfections of the piano.
There are at least three reasons why piano tuning is not trivial.
First, perfect intervals on the piano (e.g. the perfect fifth and the octave) are mathematically incompatible. The problem is fundamental and has troubled philosophers for thousands of years, as far back as Pythagoras. One solution is to divide each octave into 12 equally spaced tones, i.e. adjust the temperament to equal semitones. Unfortunately, the resulting equal temperament means that all intervals other than the octave are out of tune. The alternative is to allow some intervals to be closer to ideal than others, which was the approach used prior to around 1900. Some people find these “historic temperaments” to be a refreshing alternative to the more modern equal temperament. A nice visual summary of historic temperaments and the time period where they were used can be found here. Playing Beethoven or Bach with a temperament that they most likely used when writing their music can be very revealing, almost mystical.
The second problem arises because of the significant stiffness of metal piano strings, which cause the overtones above the fundamental to be increasingly sharp of the ideal harmonics. This inharmonicity means that not even the octaves can be perfect. The solution is to “stretch” the tuning of the octaves so that the dominant overtones of two or more notes in an interval are congruent (i.e. not beating wildly). The tuner must decide how to achieve the optimal compromise, i.e. tailor the stretch.
The third problem relates to how to tune the unisons, the multiple strings (usually three) for each note. Some tuners believe that tuning each string slightly out of tune relative to the others leads to a richer sound with greater sustain.
Interesting and Useful References
Temperament: How Music Became a Battleground for the Great Minds of Western Civilization, by Suart Isacoff, Vintage Books, NY 2003.
On Pitch, by Rick Baldassin, 2007.
The Craft of Piano Tuning, by Daniel Levitan, The Soundboard Press, NY, 2011.
Tuning, Containing the Perfection of Eighteenth Century Temperament, the Lost Art of Nineteenth Century Temperament, and the Science of Equal Temperament, Complete with Instructions for Aural and Electronic Tuning, by Owen Jorgensen, Michigan State University, 1991.