A Reflection on the Relational Systems of Musical Keys and Chords

I have always been fascinated by the mathematical relationships found in music. The idea that the tones chosen to compose music are not random, that the intervals of pitches inside a scale have an orbital pull around a key tone, that voices inside a smaller chordal system belong to a larger scalar system, and that there seems to be a harmonic force that holds them all together is endlessly pleasing to me.

As a youth, I would contemplate why we respond so differently to musical sound than we do to non-musical sounds, why one is pleasurable and the other simply provides noise data. I remember imagining that the phenomenon of musical frequencies deeply resonated with us as an innate higher form of communication, as some type of remembrance from a pre-existant life, very similar, as it turns out, to Gottfried Leibniz's theory of pre-established harmony—an intriguing philosophical theory about how all the substances in the world seem to causally interact with each other because they have been programmed by God in advance to "harmonize" with each other.

We owe much of our Western Music theoretical tradition to the ancient Greeks, in particular the mathematical and mystical philosophies and research of the Pythagoreans. The principle benefit of the Pythagoreans' investigations is the musical scale, based on their analysis of the various harmonic relations of the octave. For the Pythagoreans, music was more than a pleasurable or intellectual recreation, it was intrinsically connected to, and a manifestation of, the heavenly laws governing all of life. In their worldview, harmony of relation, or order, was the principle that regulated the whole universe.

The problem with creating a system of tones that work together in predictable relationships in order to produce an endless number of composition possibilities is quite a puzzle, especially when you consider that, theoretically, there are an infinite number of pitches, or division of frequencies, available to choose from. This creates an interesting paradox: in order for a scale to develop, or in other words, in order to systematize the selection of a palette of pitch tones used to create an unlimited number of musical compositions requires the employment of limiters to derive those tones. But you can't just choose any old pitches and produce a collection that is musically pleasing.

The western musical scale, as we have it today, is based on the interval of an octave—the distance between two pitches whose frequencies have doubled. The scale theory that developed was based on the striking of two lyre strings of the same length and observing their consonant, or pleasing sound—they harmonized with each other. Then it was observed that striking two strings, one of a certain length and the other exactly half the length of the first, would also produce a pleasing sound, where the frequency of the second string doubled in pitch frequency. They continued to follow this pattern, reducing the 3rd string by half, and so on, finding that each halving of string length doubled the frequency of the previous pitch, each doubling producing a consonant, harmonious sound. This doubling of frequencies, along with testing other harmonious ratios of string divisions, produced a series of pitches used today to form the Western, diatonic scale.

The diatonic scale is based on a fundamental pitch and its octave, where the ratio of the highest to the lowest pitch of the scale is 2:1. That octave is then divided into a fifth and a fourth, which have the ratios of 3:2 and 4:3 respectively, and so forth. When the fifth and fourth are multiplied together they make an octave.

This cosmology of harmonic relationships inside the octave gives rise not only to a musical scale, but to a bigger concept within which the scale operates: a musical key. When we speak of a musical key we are referring to a much larger concept than just a scale or a "key" note (tonic). Of course, a musical key includes a tonic and a scale (mode), but the psychological magic of a musical key is not found in memorizing scales and chords, it is found in discovering the family relationships between all the constituent parts of a chosen "key." For example, the harmonic qualities of chords, chord changes, intervallic movement in melody, and predictable tonal behaviors used in improvisation are all implied by the key.

A musical key describes a relational context for music; it is a structural system that creates meaning for any given note or set of notes used in the music--a palette of related tonal materials used for emotional communication. Using a musical key creates a context in which to measure movement, to create the sense of a musical journey, a point of departure for materials that are not in the key, and a sense of coming home at the completion of a journey. It is this movement within a specified key that creates emotional communication and connection.

A musical key can also be figured as a solar system of sorts where the tonic (key note) is the gravitational center of the key. The tones supplied by the scale are the building blocks for melody and harmony/chords within the key, where each tonal degree of the scale occupies a fixed intervallic distance away from the key center. When you understand the cosmos of the key, you understand how the movement of melodic tones, intervals, scale tones, chord tones relate and behave with each other.

Melodic tones travel in various tension and release movements in relation to their distance from the key note. Melodic tones are derived from the scale of the key and live inside of chords.

Chords are little subsystems inside the larger key and have their own tonal centers inside the chord structure. Each voice in the chord orbits around the central chord tone (chord root), while each chord system has an orbital relationship to the larger key center (tonic).

Interestingly, chords have tonal centers inside the chord structure, Each voice in the chord orbits around the central chord tone (chord root), while each chord system has an orbital relationship to the key center (tonic).

A key is the tonality that is central to a piece of music. What that means is that a key is based on a note and a structure built around that note. This structure is what provides context and meaning inside the music.

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