Disclosed herein are exemplary embodiments of methods, systems, and apparatus for improving pianos and like musical instruments. For example, in certain embodiments, the improvements are in the structures of the piano (or like musical instruments) that affect the vibratory relationships between the struck string and other segments of said string. The improvements affect how these vibratory relationships of the string segments interact with the bridge, soundboard, the plate of a piano, the string rests, and/or the string terminations. For example, in embodiments of the disclosed technology, the vibratory relationships of the string segments involve transient transverse (T) and longitudinal (L) frequencies as well as sustained transverse, longitudinal, and horizontal (H) modes carried by said string segments and their support structures. Embodiments of the disclosed technology also affect how these relationships are defined across the compass of the standard piano keyboard.