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Presented is a two-tier analysis of the location of the real roots of thegeneral quartic equation $x^4 + ax^3 + bx^2 + cx + d = 0$ with realcoefficients and the classification of the roots in terms of $a$, $b$, $c$, and$d$, without using any numerical approximations. Associated with the generalquartic, there is a number of subsidiary quadratic equations (resolventquadratic equations) whose roots allow this systematization as well as thedetermination of the bounds of the individual roots of the quartic. In manycases the root isolation intervals are found. The second tier of the analysisuses two subsidiary cubic equations (auxiliary cubic equations) and solvingthese, together with some of the resolvent quadratic equations, allows the fullclassification of the roots of the general quartic and also the determinationof the isolation interval of each root. These isolation intervals involve thestationary points of the quartic (among others) and, by solving some of theresolvent quadratic equations, the isolation intervals of the stationary pointsof the quartic are also determined. Each possible case has been carefullystudied and illustrated with a detailed figure containing a description of itsspecific characteristics, analysis based on solving cubic equations andanalysis based on solving quadratic equations only. As the analysis of theroots of the quartic equation is done by studying the intersection points ofthe "sub-quartic" $x^4 + ax^3 + bx^2$ with a set of suitable parallel lines, abeautiful Pythagorean analogy can be found between these intersection pointsand the set of parallel lines on one hand and the musical notes and the stavesrepresenting different musical pitches on the other: each particular case ofthe quartic equation has its own short tune.

Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes