Pythagoras / Uccello
What Pythagoras did for music, by giving it a mathematical basis, could be said to have been done for painting by Paolo Uccello, and other Renaissance artists, with the discovery of optical perspective.
Perspective in general has to do with “point of view” -literally so, with artistic perspective. To be unambiguous, artistic perspective is linear perspective, as in "lines of a design drawing." It is a snapshot, or visual moment, as the painter sees it.
To draw a perspective drawing involves the use of a straight edge or ruler but, be it noted, not for measurement -unlike Pythagorean intervals, which are all about measurement -of the lengths of the strings of a harp, for instance. When we consider lines it is straight lines that we take into consideration. A “line” is implicitly straight.
The discovery by Pythagoras of the mathematical intervals was likely made upon string instruments, the strings of which are always stretched tight. The musical staff of modern musical notation is seen by musicians as the graphic representation of the strings of a string-ed instrument (such as a guitar.)
What I am suggesting is that the lines of a perspective drawing can be compared to the strings of string instruments, instruments such as the guitar, violin, or piano. Both the strings of a musical instrument and the lines of a perspective drawing are straight. Strings are lines. Lines are strings.
Pythagoras' discovery was likely made by listening to, and comparing the harmonies of, strings of different lengths when plucked side-by-side. The tuning of strings by an expert who is adept at tuning by tightening and loosening the strings by ear, as those of a piano (to give the best example), is a related, but different matter.
By luck (and arriving early), I once overheard the tuning of a harp before a performance in a music conservatory. It was spellbinding! I can well understand the fascination Pythagoras experienced listening to blacksmiths' hammering. The blacksmiths may have explained to Pythagoras that different pitches are produced by striking steel bars of different size and shape.
Curved lines are a related but different matter. The musical Circle of Fifths might be taken figuratively as analogous to curved lines in line drawing, but generally the mathematics of both harmony and perspective is elementary, basic arithmetic, as far as mathematics is concerned.
While the straight edge which is used for measuring, that is to say, “a ruler,” does not have to be ruled for perspective drawing, the principle of congruent angles comes into play and, therefore, measurements. For measuring angles a protractor is handy proof of congruent angles -but not needed for constructing a perspective drawing.
By definition, then, the point at which two lines intersect, forming congruent angles, is the vanishing point of perspective drawing.
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