When talking about notes in a piece of music, it's important to be able o refer to them within the context of the key centre. In essence this means that being able to describe each note with reference to the parent major scale.
As an example, consider the C major scale:
C D E F G A B
Each degree of this scale can be given a unique number:
C D E F G A B 1 2 3 4 5 6 7
Thus, if you came across a piece of music which you knew was in the key of C major, and wanted to refer to an E note, then you would say that it was the 3rd degree. Straight forward enough in the case of something diatonic to a major scale, but what about other types of key and non-diatonic notes?
Say, for example, that you had a piece of music in the key of C minor. In this case, you're more likely to see an Eb than an E. Because the peice of music is an a 'C-type' of key, you would refer to the note in question in the context of the C major scale: E is the 3rd degree of C major, and Eb is a semitone lower, so you would refer to Eb as the b3rd degree.
You always use a number which corresponds to a degree with the same letter in the parent major scale. For example, if you saw an Eb in a piece of music in the key of C major, you would refer to it as a b3rd; if you saw d# (which is enharmonically equivalent) you would refer to it as a #2nd. In the first instance Eb is an 'E-type' of note (therefore some kind of 3rd of C major) whilst in the second case D# is a 'D-type' of note (some kind of 2nd of C major).
To illustrate this further, look at the following notes, many of which are not diatonic to C major:
Take care not to fall into the trap of thinking that a scalar tone must be some sort of 'sharp' or 'flat' just because it has an accent in its name. The 'sharp' or 'flat' of a degress number denotes any deviation from the notes of the parent major scale. For example, consider D major:
D E F# G A B C# 1 2 3 4 5 6 7
In this case, F# is the 3rd degree, not the #3rd (the #3rd would be F##, i.e. the 3rd degree of the parent major scale raised by a semitone). Likewise the b3rd degree would be F (for all there is no 'b' in the note name) because the 3rd degree of the parent D major scale has been lowered by a semitone.
To illustrate this further, look at the Eb major scale:
Eb F G Ab Bb C D 1 2 3 4 5 6 7
Here, the 4th degree is a note which has an accent (a flat sign in this case). The b4th degree would actually be Abb (the 4th degree lowered by a semitone) and the #4th degree would be A (the 4th degree raised by a semitone).
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