  Presented is a two-tier analysis of the location of the real roots of the
general quartic equation $x^4 + ax^3 + bx^2 + cx + d = 0$ with real
coefficients and the classification of the roots in terms of $a$, $b$, $c$, and
$d$, without using any numerical approximations. Associated with the general
quartic, there is a number of subsidiary quadratic equations (resolvent
quadratic equations) whose roots allow this systematization as well as the
determination of the bounds of the individual roots of the quartic. In many
cases the root isolation intervals are found. The second tier of the analysis
uses two subsidiary cubic equations (auxiliary cubic equations) and solving
these, together with some of the resolvent quadratic equations, allows the full
classification of the roots of the general quartic and also the determination
of the isolation interval of each root. These isolation intervals involve the
stationary points of the quartic (among others) and, by solving some of the
resolvent quadratic equations, the isolation intervals of the stationary points
of the quartic are also determined. Each possible case has been carefully
studied and illustrated with a detailed figure containing a description of its
specific characteristics, analysis based on solving cubic equations and
analysis based on solving quadratic equations only. As the analysis of the
roots of the quartic equation is done by studying the intersection points of
the "sub-quartic" $x^4 + ax^3 + bx^2$ with a set of suitable parallel lines, a
beautiful Pythagorean analogy can be found between these intersection points
and the set of parallel lines on one hand and the musical notes and the staves
representing different musical pitches on the other: each particular case of
the quartic equation has its own short tune.
