Basic Tuning Theory
Most contemporary pianos are tuned based on a system known as equal temperament, which divides octaves into 12 equal intervals known as “semitones”. The frequency ratio between any adjacent semitones (e.g. A-A#, A#-B, B-C, etc.) is always the same – and equal to 2^(1/12) ≈ 1.0595 – with note A4 used as a reference frequency. While only octaves are “perfect” in this system (with a ratio between octaves of exactly 2), equal temperament has become a common tuning standard because it maintains consistent intervals across all keys.
When you pluck or strike a string, you cause it to vibrate and produce a sound wave. The specific shape of the string as it oscillates is a combination of multiple frequencies, called “harmonics”. The fundamental frequency (also called the first harmonic) is the lowest frequency that a string vibrates at, and is what we perceive as the sound’s pitch. For example, note A4 on the piano has a fundamental frequency of f1=440Hz. For ideal/theoretical strings, the subsequent harmonics occur at exact integer multiples of the fundamental frequency; i.e. the second harmonic of A4 has a frequency of f2=880Hz, the third harmonic has a frequency of f3=1320Hz, etc. The overall sound of a note is a complex mixture of the fundamental frequency and its higher harmonics. These harmonics contribute to the sound's unique quality or “timbre”, which is what allows us to distinguish between pianos and other instruments playing the same note.
When the harmonics of a string occur at exact integer multiples of the fundamental frequency, the string is said to be “perfectly harmonic”. In reality, piano strings are not perfectly harmonic, and the frequencies of the harmonics are dependent on factors such as string stiffness, length, and tension. The resulting inharmonicity of the strings is a unique characteristic of a given piano. If octaves were to be tuned “mathematically pure”, the harmonics would not line up, producing an undesirable sound. Instead, using A4 as a reference point, the other strings are “stretched” so that the harmonics line up more closely. The amount of stretch required was first characterized by O.L. Railsback in the 1930s, who quantified how much the stretched tuning should deviate from true equal temperament across the keyboard. The result is the “Railsback curve”, which is essentially a piano’s unique fingerprint. Deviation from equal temperament is typically measured in terms of “cents”, where one cent represents 1/100th of a semitone. My job is to ensure that the strings on your piano are tuned as closely as possible to its fingerprint.