- References
- External links
Paul Erlich (born 1972) is a guitarist and music theorist living near Boston, Massachusetts. He is known for his seminal role in developing the theory of regular temperaments, including being the first to define pajara temperament[1][2] and its decatonic scales in 22-ET.[3] He holds a Bachelor of Science degree in physics from Yale University.
His definition of harmonic entropy influenced by Ernst Terhardt[4] has received attention from music theorists such as William Sethares. 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 chord as far more consonant than a 
chord. Both have exactly the same set of intervals between the notes, under inversion, but the first one is easy to fit into a single harmonic series (overtones rather than undertones). Due to the least common multiple, the integers are much lower for the major chord, 
, versus its inverse, 
. Components of dissonance not modeled by this theory include critical band roughness as well as tonal context (e.g. an augmented second is more dissonant than a minor third even though both can be tuned to the same size, as in 12-ET).
References
2. ^{{webarchive |url=https://web.archive.org/web/20131105233354/http:/groups.yahoo.com/neo/groups/tuning/conversations/topics/33368 |date=05 November 2013 |title="Alternate Tunings Mailing List", Yahoo! Groups}}.
3. ^{{cite journal | last=Erlich | first=Paul | journal=Xenharmonikôn | title=Tuning, Tonality, and Twenty-Two-Tone Temperament | year=1998 | volume=17 | url=http://lumma.org/tuning/erlich/erlich-decatonic.pdf}}
4. ^{{cite book | title=Tuning, Timbre, Spectrum, Scale | year=2004 | last=Sethares | first=William A. | pages=355–357 | url=http://sethares.engr.wisc.edu/paperspdf/HarmonicEntropy.pdf}}
External links
- "Some music theory from Paul Erlich", Lumma.org.
- "A Middle Path: Between Just Intonation and the Equal Temperaments", DKeenan.com.
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