key as coordinate system

In geometry, we measure points against a coordinate system. The results of such measuring are the coordinates which are given to each measured point. Once we have the coordinates, we can easily duplicate the points in some other coordinate system. 

In music, we measure notes and chords against a key. The results of this measuring are the scale degrees. When we have the scale degrees we can easily transpose the notes and chords in some other key.

Scale degrees of notes

In chapter Scale degrees we learned about the scale degrees of a scale. We can use this knowledge to determine the scale degrees for any set of notes, in relation to any scale. When analyzing compositions, we determine the scale degrees in relation to the current key. It is important to understand that in the rest of the book, whenever we speak about the scale degrees in relation to the current key, we mean on the scale degrees of the major scale or natural minor scale that correspond to the current key. A major scale is used for a major key while a natural minor scale is used for a minor key. 

As an example, we can do that for the melody at the beginning of Chopin’s Prelude No. 7 in A major. Since it is written in the A major key, we shall use the A major scale when determining the scale degrees. The next figure shows the results. The symbol # appearing next to some of the numbers means that the scale degree is raised. 

Figure: The scale degrees of a melody

Thereafter, we can use the scale degrees to transpose the melody into any other key. The next figure shows the results of choosing the C major key.

Figure: The transposed melody has the same scale degrees

This was analogous to measuring a set of geometric points in a particular coordinate system and then duplicating the points in another coordinate system by using the same coordinates.  

The next task is learning to measure the chords using a key. That is, learning to determine the scale degrees of chords.

Scale degrees of chords

The scale degree of a chord is the scale degree of its chord root. Thus, knowledge of determining the scale degrees of notes also gives us the ability to determine the scale degrees of chords. Scale degrees of chords are always annotated in Roman numerals. 

For instance, the G major triad chord has G as the chord root and thus its scale degree in the G major key is I, while its scale degree in the C major key is V.