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I’ve read about somewhat arcane nature of piano tuning and the various temperaments used through the ages, but Minute Physics succinctly describes what is going with all the math behind the production of sound.

From Wikipedia:

Many factors cause pianos to go out of tune, particularly atmospheric changes. For instance, changes in humidity will affect the pitch of a piano; high humidity causes the sound board to swell, stretching the strings and causing the pitch to go sharp, while low humidity has the opposite effect.^[1] Changes in temperature can also affect the overall pitch of a piano. In newer pianos the strings gradually stretch and wooden parts compress, causing the piano to go flat, while in older pianos the tuning pins (that hold the strings in tune) can become loose and don’t hold the piano in tune as well.^[2] Frequent and hard playing can also cause a piano to go out of tune.^[2] For these reasons, many piano manufacturers recommend that new pianos be tuned four times during the first year and twice a year thereafter.^[3]

An out-of-tune piano can often be identified by the characteristic “honky tonk” wah-wah or beating sound it produces. This fluctuation in the sound intensity is a result of two (or more) tones of similar frequencies being played together. For example, if a piano string tuned to 440 Hz (vibrations per second) is played together with a piano string tuned to 442 Hz, the resulting tone beats at a frequency of 2 Hz, due to the constructive and destructive interference between the two tones. Likewise, if a string tuned to 220 Hz (with a harmonic at 440 Hz) is played together with a string tuned at 442 Hz, the same 2 Hz beat is heard.^[4] Because pianos typically have multiple strings for each piano key, these strings must be tuned to the same frequency to eliminate beats.

The pitch of a note is determined by the frequency of vibrations. For a vibrating string, the frequency is determined by the string’s length, mass, and tension.^[5] Piano strings are wrapped around tuning pins, which are turned to adjust the tension of the strings.

History

Piano tuning became a profession around the beginning of the 1800s, as the “pianoforte” became mainstream.^[6] Previously musicians owned harpsichords, which were much easier to tune, and which the musicians generally tuned themselves. Early piano tuners were trained and employed in piano factories, and often underwent an apprenticeship of about 5–7 years. Early tuners faced challenges related to a large variety of new and changing pianos and non-standardized pitches.

Historically, keyboard instruments were tuned using just intonation, pythagorean tuning and meantone temperament meaning that such instruments could sound “in tune” in one key, or some keys, but would then have more dissonance in other keys.^[7] The development of well temperament allowed fixed-pitch instruments to play reasonably well in all of the keys. The famous “Well-Tempered Clavier” by Johann Sebastian Bach took advantage of this breakthrough, with preludes and fugues written for all 24 major and minor keys.^[8] However, while unpleasant intervals (such as the wolf interval) were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own distinctive character. During the 1700s this variation led to an increase in the use of equal temperament, in which the frequency ratio between each pair of adjacent notes on the keyboard was made equal, allowing music to be transposed between keys without changing the relationship between notes.^[9]

Pianos are generally tuned to an A440 pitch standard that was adopted during the early 1900s in response to widely varying standards.^[10] Previously the pitch standards had gradually risen from about A415 during the late 1700s and early 1800s to A435 during the late 1800s. Though A440 is generally the standard, some orchestras, particularly in Europe, use a higher pitch standard, such as A444.^[11]