The first element of a modulation is the distance traveled. What does that mean in musical terms?
The primary measure of relatedness between keys is the number of notes shared in common: The greater the overlap, the more closely related the keys. Each Major key contains seven notes: Thus, the greatest possible overlap is six notes. Because there are twelve pitches in the chromatic scale, the minimum overlap is two notes (five distinct for each key plus two shared=all twelve pitches of the chromatic scale.)
In the West, hearing and vision are often correlated: For instance, we speak of pitches as “high” and “low” and of melodies “rising” and “falling.” This cross-domain mapping is not universal: As ethnomusicologists have shown, to the ancient Greeks, pitches were “sharp” or “heavy,” in Bali they are “small” or “large,” to the Saya people of the Amazon “young” and “old.” In Zimbabawe, what we call low pitches are “crocodiles,” whereas high ones are “those who follow crocodiles.” (Eitan and Timmers, “Beethoven’s last piano sonatas and those who follow crocodiles,” 9th International Conference on Music Perception and Cognition, 2006).
To Western ears, our musical-spatial framework is so ingrained, it is hard to realize it is only a cultural metaphor: To us. the piccolo is “above” the tuba, the singer “reaches” for her high note and voices may move in “parallel” or “contrary motion.”
While individual pitches are aligned on a vertical plane—up and down--keys tend to be “visualized” on a horizontal plane — near and far. Closely related keys are perceived as ”neighboring,” whereas those that are not are perceived as “distant.”
For instance, the pitches B and C lie very close together:
Meanwhile, there is a wider interval between C and G:
With respect to keys, however, the opposite is true: Because the keys of C-Major and G-Major share six notes in common, they are perceived as neighboring.
Meanwhile, C-Major and B-Major—which share only 2 of 7 notes—are heard as far apart.
The circle of fifths is an iconic diagram of keys arrayed in circle, like the face of a clock: The more notes two keys share in common, the closer they lie on the circle.
Each Major key is paired with the minor one whose “natural” form shares the same scale: For instance, the keys of C-Major and a-natural minor share exactly the same seven pitches. These are called the relative Major and minor, because their pitch content is so closely related.
The circle of fifths is the primary way of visualizing harmonic distance. It is common to speak of “traveling” around the circle. Note that, just as with the twelve-hour time cycle, twelve steps around the circle of fifths returns you to your starting point.