The root (basse fondamentale) of a chord is the note upon which that chord is perceived or labelled as built or centered, the root of a chord in root position or normal form. This feeling of centeredness is readily perceivable aurally for the culturally trained, and is perhaps the pitch that would be most commonly sung if a chord was played and people were asked to sing a note in said chord, followed closely by the bass note or bottom note. When the chord is uninverted, or in root position, this is the bass note of the chord. Often the root is not actually the lowest pitch being played in a chord, in which case the chord is inverted.
Conventionally, the name of the note which is the root is used to denote the chord, thus a major chord built on C is a C Major chord. Since Rameau, the analysis and theory of tonal music usually treats the roots as the defining feature of chords and much information can be gained from a progression of roots even if chord types are unknown. Also, if the key is known then the chord forms are known for each root.
A root progression is the most familiar form of labelling chord progressions by their root, rather than bass if different, and is in contrast to an older pre-tonal conception of chords as sonorities, with a root position or first inversion triad being simply alternative and fairly equivalent ways of "filling in" the consonance between octaves, C (E G) C or C (F A) C.
In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x.
Per the fundamental theorem of algebra, there are two solutions to the equation defining the square roots of any number (although these roots may not be distinct, as in the square root of zero).
Square roots of non-perfect squares are always irrational numbers, i.e., numbers not expressible as a ratio of two integers.
In general, irreducible root systems are specified by a family (indicated by a letter A to G) and the rank (indicated by a subscript).
The Dynkin diagram can be extracted from the root system by choosing a base, that is a subset Δ of Φ which is a basis of V with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0.
Given two root systems with the same Dynkin diagram, we can match up roots, starting with the roots in the base, and show that the systems are in fact the same.
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