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Paul Erlich (born 1972) is a
guitarist A guitarist (or a guitar player) is a person who plays the guitar. Guitarists may play a variety of guitar family instruments such as classical guitars, acoustic guitars, electric guitars, and bass guitars. Some guitarists accompany themsel ...
and
music theorist Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the " rudiments", that are needed to understand music notation ( ...
living near
Boston Boston (), officially the City of Boston, is the state capital and most populous city of the Commonwealth of Massachusetts, as well as the cultural and financial center of the New England region of the United States. It is the 24th- mo ...
, Massachusetts. He is known for his seminal role in developing the theory of regular temperaments, including being the first to define pajara temperament Accessed 2013-10-29.. and its decatonic scales in 22-ET. He holds a
Bachelor of Science A Bachelor of Science (BS, BSc, SB, or ScB; from the Latin ') is a bachelor's degree awarded for programs that generally last three to five years. The first university to admit a student to the degree of Bachelor of Science was the University o ...
degree in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
from
Yale University Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the w ...
. His definition of harmonic entropy, a refinement of a model by van Eck influenced by Ernst Terhardt has received attention from music theorists such as
William Sethares William A. Sethares (born April 19, 1955) is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of conson ...
.Sethares, William (2005).
Tuning, Timbre, Spectrum, Scale
', p.371. Springer Science & Business Media. . "Harmonic entropy is a measure of the uncertainty in pitch perception, and it provides a physical correlate of tonalness the closeness of the partials of a complex sound to a harmonic series" one aspect of the psychoacoustic concept of dissonance....high tonalness corresponds to low entropy and low tonalness corresponds to high entropy."
It is intended to model one of the components of dissonance as a measure of the uncertainty of the virtual pitch ("missing fundamental") evoked by a set of two or more pitches. This measures how easy or difficult it is to fit the pitches into a single harmonic series. For example, most listeners rank a
harmonic seventh The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...
chord as far more
consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced w ...
than a \tfrac chord. Both have exactly the same set of intervals between the notes, under
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
, but the first one is easy to fit into a single harmonic series (
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s rather than undertones). In the harmonic series, the integers are much lower for the harmonic seventh chord, , versus its inverse, . Components of dissonance not modeled by this theory include
critical band In audiology and psychoacoustics the concept of critical bands, introduced by Harvey Fletcher in 1933 and refined in 1940, describes the frequency bandwidth of the "auditory filter" created by the cochlea, the sense organ of hearing within the in ...
roughness as well as tonal context (e.g. an
augmented second In classical music from Western culture, an augmented second is an interval that, in equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone.Ben ...
is more dissonant than a
minor third In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
even though both can be tuned to the same size, as in 12-ET). For the nth iteration of the Farey diagram, the
mediant In music, the mediant (''Latin'': to be in the middle) is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant note ...
between the jth element, f_j=a_j/b_j, and the next highest element: :\frac is subtracted by the mediant between the element and the next lowest element: :\frac From here, the process to compute harmonic entropy is as follows:
(a) compute the areas defined by the normal (Gaussian) bell curve on top, and the mediants on the sides
(b) normalize the sum of the areas to add to 1, such that each represents a probability
(c) calculate the entropy of that set of probabilities
See external links for a detailed description of the model of harmonic entropy.


Notes


References


External links


Some music theory from Paul Erlich
, ''Lumma.org''.
A Middle Path: Between Just Intonation and the Equal Temperaments
, ''DKeenan.com''.
Harmonic Entropy on the Xenharmonic Wiki
, ''en.xen.wiki'' {{DEFAULTSORT:Erlich, Paul Living people 1972 births American music theorists American male guitarists 21st-century guitarists 21st-century American male musicians