# Paul Erlich

Harmonic entropy for triads with lower interval and upper interval each ranging from 200 to 500 cents. Compare , , and . See full resolution for locations of the triads on the plot
Harmonic entropy: Dissonance may be explained as the uncertainty in determining a pitch caused by the relative closeness of complex ratios and distance around simple ratios. The space around intervals is shown above for the Farey sequence, order 50.

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 ${\displaystyle {\tfrac {4/5/6/7}{1}}}$ harmonic seventh chord as far more consonant than a ${\displaystyle {\tfrac {1}{7/6/5/4}}}$ 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, ${\displaystyle {\tfrac {4/5/6/7}{4}}}$, versus its inverse, ${\displaystyle {\tfrac {105/120/140/168/210}{105}}}$. 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).

For the ${\displaystyle n}$th iteration of the Farey diagram, the mediant between the ${\displaystyle j}$th element, ${\displaystyle f_{j}=a_{j}/b_{j}}$, and the next highest element:

${\displaystyle {\frac {a_{j}+a_{j+1}}{b_{j}+b_{j+1}}}}$[a]

is subtracted by the mediant between the element and the next lowest element:

${\displaystyle {\frac {a_{j-1}+a_{j}}{b_{j-1}+b_{j}}}}$

Distances, ${\displaystyle r_{j}}$, which are larger indicate less dissonance (more clarity) and smaller distances indicate more dissonance (more ambiguity).

## Notes

1. ^ The mediant of two ratios, ${\displaystyle {\tfrac {a}{b}}}$ and ${\displaystyle {\tfrac {c}{d}}}$, is ${\displaystyle {\tfrac {a+c}{b+d}}}$.

## References

1. ^ "Pajara", on Xenharmonic Wiki. Accessed 2013-10-29.
2. ^ ""Alternate Tunings Mailing List", Yahoo! Groups". Archived from the original on 5 November 2013. Retrieved 3 May 2019.CS1 maint: BOT: original-url status unknown (link).
3. ^ Erlich, Paul (1998). "Tuning, Tonality, and Twenty-Two-Tone Temperament" (PDF). Xenharmonikôn. 17.
4. ^ Sethares, William A. (2004). Tuning, Timbre, Spectrum, Scale (PDF). pp. 355–357.