How and why did the Greeks combine observation of nature with mathematics in their art and architecture?

The first definite mathematical influence on architecture we mention is that of Pythagoras. Numbers for Pythagoras also had geometrical properties. The Pythagoreans spoke of square numbers, oblong numbers, triangular numbers etc. Geometry was the study of shapes and shapes were determined by numbers.

including the octave, the answer would 8 notes in a major scale with one note being played twice. ex) c major scale includes: c, d, e, f, g, a, b, c

notice that c is played twice

however, if you were to ask me how many different notes are in a major scale- the answer would be 7.