Lipps-Meyer Law
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Lipps%E2%80%93Meyer Law

The Lipps-Meyer law, named for Theodor Lipps (1851-1914) and Max Friedrich Meyer (1873-1967), hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number two or a power of two",[1] in the frequency ratio between notes (see octave).

Perfect fifth.

"The 'Lipps-Meyer' Law predicts an 'effect of finality' for a melodic interval that ends on a tone which, in terms of an idealized frequency ratio, can be represented as a power of two."[2]

Thus the interval order matters -- a perfect fifth, for instance (C,G), ordered <C,G>, 2:3, gives an "effect of indicated continuation", while <G,C>, 3:2, gives an "effect of finality".

This is a measure of interval strength or stability and finality. Notice that it is similar to the more common measure of interval strength, which is determined by its approximation to a lower, stronger, or higher, weaker, position in the harmonic series.

The reason for the effect of finality of such interval ratios may be seen as follows. If ${\displaystyle F=h_{2}/2^{n}}$ is the interval ratio in consideration, where ${\displaystyle n}$ is a positive integer and ${\displaystyle h_{2}}$ is the higher harmonic number of the ratio, then its interval can be determined by taking the base-2 logarithm ${\displaystyle I=12log_{2}(h_{2}/2^{n})=12log_{2}(h_{2})-12n}$ (3/2=7.02 and 4/3=4.98). The difference of these terms is the harmonic series representation of the interval in question (using harmonic numbers), whose bottom note ${\displaystyle 12n}$ is a transposition of the tonic by n octaves. This suggests why descending interval ratios with denominator a power of two are final. A similar situation is seen if the term in the numerator is a power of two.[3][4]

## Sources

1. ^ Meyer, M.F. (1929). "The Musician's Arithmetic", The University of Missouri Studies, January.
2. ^ Robert Gjerdingen, "The Psychology of Music", (2002). The Cambridge History of Western Music Theory, Th. Christensen ed., p.963. ISBN 978-0-521-62371-1.
3. ^ Krumhansl, Carol L. Cognitive Foundations of Musical Pitch. New York: Oxford UP, 2001. 122. Print
4. ^ Wright, David. Mathematics and Music. Providence, RI: American Mathematical Society, 2009. 53. Print.