I think your question might be better worded as...
why did the composer or arranger of this sheet music use an
A7 chord label for chord tones
E G #C when the purported chord root
A is missing?
The simple answer is: in this case the chord is regarded as an incomplete chord.
Strictly speaking you would say the first chord is a
C# diminished chord in first inversion. You could label it like
C#dim/E with jazz labeling or
Dm: viio6/3 with Roman numeral analysis.
Why would someone treat it as an incomplete dominant seventh rather than an leading tone triad? I think two main reasons:
- Dominant chords
V and leading tone chords
vii0 are often treated as theoretically parts of a larger dominant harmony. They are sort of equivalent in dominant function.
- Root progression by descending perfect fifth is regarded as the "strongest" chord movement.
vii06/3 i and
V4/3 i are sort of functional equivalent, the labeling of
V using an incomplete chord is a bit more straight forward in showing the dominant to tonic progression.
Another way to understand the omission of the root in this progression is through looking at the voice leading. Here are two basic version of the harmony with the
A omitted and included and also with the
D doubled at the octave...
The essential voice movements are the leading tone moving to the tonic (
C# D) and the subdominant moving to the mediant (
G F.) The supertonic to tonic (
E D) is a bit subordinate. But notice the treatment of the dominant (
A.) That tone doesn't move. There is a kind of redundancy that allows for an omission.
Play the two progressions with and without the
A and they sound more or less the same, because they are more or less the same from a voice leading perspective.