Where does harmonic collide?

A component of a periodic sound or signal whose frequency is a whole-number multiple of the lowest, or fundamental, frequency: a plucked string or a blown pipe vibrates not only at its fundamental but at twice, three times, four times that rate, and each of these components is a harmonic, together forming the harmonic series that gives an instrument its characteristic tone. Fletcher and Rossing's 1998 The Physics of Musical Instruments set out how vibrating strings and air columns generate these integer-multiple components and how their relative strengths shape timbre. The referent is a measurable frequency present in a real vibrating system, read off a spectrum, and it is not a function that solves a differential equation or a way of averaging numbers.[1]

Unit

One partial at an integer multiple of the fundamental frequency

Recovery means

Not applicable

Measured by

Located on a frequency spectrum as a peak at n times the fundamental, with its strength given by the peak's height

A smooth function of several variables whose second partial derivatives cancel out to zero, so that it satisfies Laplace's equation: such functions describe steady states with no sources, like the temperature across a plate once it has settled or the electric potential in empty space, and each value equals the average of the values around it. Axler, Bourdon, and Ramey's 1992 Harmonic Function Theory developed these functions as the centre of potential theory and their close tie to complex analysis. The referent is a mathematical object defined by a differential equation on a region, not a frequency you can hear in a sound or a single number that summarises a list.[2]

Unit

A twice-differentiable function whose Laplacian is zero on a region

Recovery means

Not applicable

Measured by

Checked by confirming that the sum of its unmixed second partial derivatives vanishes at every point of the region

A kind of average found by taking the reciprocal of each value, averaging those, and taking the reciprocal of the result: it is the right way to average rates and ratios, such as speeds over equal distances, and it is always the smallest of the three classical Pythagorean means, pulled down by the small values in a set. Ferger's 1931 The nature and use of the harmonic mean discussed what this average is and when to use it. The referent is a single summary number computed from a list of values, and it has nothing to do with a frequency in a vibrating body or a function that solves a differential equation.[3]

Unit

One averaged value: the reciprocal of the mean of the values' reciprocals

Recovery means

Not applicable

Measured by

Computed as the count of values divided by the sum of their reciprocals