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A melody can be written in a type of relative notation, where we don't think in notes but rather in relative intervals.
X - means choose any note
0 - means hit note again where you are
positive - means to raise current note by this many semitones
negative - means to lower current note by this many semitones
So the first few measures of the melody of happy birthday could be written as:
X 0 2 -2 5 -1 -4 0 2 -2 7 -2
Is there an official name for this type of notation?
In the theory and analysis of twentieth-century music, we have the notion of ordered pitch intervals. These intervals are measured in semitones (as in the system you gave) and, because they are "ordered," they specify ascending or descending.
As such, your "Happy Birthday" notation becomes:
0 +2 -2 +5 -1 -4 0 +2 -2 +7 -2
Similar concepts are used when programming music, or determine trends across a corpus; for instance, IMSLP's Search by Melody feature uses something like it. (In their case, they give middle C the value of 60.)
