In music theory, the concept of "key" refers to the tonal center or the central pitch around which a piece of music revolves. In the key of C major, the notes typically used are C, D, E, F, G, A, and B. However, when discussing the role of B-flat in the key of C major, it is important to understand the concept of key signatures and the diatonic scale.

Key signatures are used to indicate the key of a piece of music and are represented by sharps or flats placed at the beginning of a staff. In the case of C major, the key signature is empty, meaning there are no sharps or flats indicated. This absence of sharps or flats implies that all the notes used in the key of C major are natural notes (i.e., no accidentals).

Now, let's consider the B-flat note in the key of C major. While B-flat is not a natural note in the key of C major, it is commonly used as a borrowed or chromatic note. Borrowed notes are notes that are borrowed from related keys or modes but are not part of the original key. In the case of C major, B-flat is borrowed from the key of F major, which has a key signature of one flat (B-flat).

In music, the use of borrowed notes adds color and variety to a composition while still maintaining the overall tonal center. The B-flat note in the key of C major creates a temporary tension or dissonance that resolves back to the tonic (C) to reaffirm the key. This is known as a chromatic alteration.

Here's an example to illustrate this concept: In the key of C major, if we encounter a chord progression such as C - F - G - C, we can introduce a B-flat to add a chromatic alteration. The progression could become C - F - G7 - C, where the G7 chord includes the B-flat note. This temporary dissonance created by the B-flat resolves back to the C major chord, reinforcing the tonal center.

References:

• Benward, B., & Saker, M. (2009). Music in Theory and Practice, Vol. I. McGraw-Hill.
• Piston, W. (1987). Harmony. W. W. Norton & Company.