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    Tuning the piano


    The relationship between two pitches, called an interval, is the ratio of their absolute frequencies. Two different intervals are perceived as the same when the pairs of pitches involved share the same frequency ratio. The easiest intervals to identify, and the easiest intervals to tune, are those that are just, meaning they have a simple whole-number ratio. The term temperament refers to a tuning system that tempers the just intervals (usually the perfect fifth, which has the ratio 3:2) to satisfy another mathematical property; in equal temperament, a fifth is tempered by narrowing it slightly, achieved by flattening its upper pitch slightly, or raising its lower pitch slightly. A temperament system is also known as a set of bearings.
    Tempering an interval causes it to beat, which is a fluctuation in perceived sound intensity due to interference between close (but unequal) pitches. The rate of beating is equal to the frequency differences of any harmonics that are present for both pitches and that coincide or nearly coincide.
    Piano tuning is the act of adjusting the tensions of the piano's strings, thereby aligning the intervals among their tones so that the instrument is in tune. The meaning of the term in tune in the context of piano tuning is not simply a particular fixed set of pitches. Fine piano tuning carefully assesses the interaction among all notes of the chromatic scale, different for every piano, and thus requires slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament (see Piano key frequencies for the theoretical piano tuning). In all systems of tuning, each pitch is derived from its relationship to a chosen fixed pitch, usually A440.[12]
    ~{reewikipediapiano}