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The Physics of Basic Music Theory

The construction of scales, chords, and complex series of musical notes started with the utilization of harmonics and overtones. A harmonic occurs when a frequency is multiplied by a whole number. For instance, the 2nd harmonic of the frequency 440 Hz is 880 Hz. The difference between harmonics and overtones is simply in their labeling. The 1st overtone of a frequency is the same as the 2nd harmonic (multiplied by 2). The 5th overtone of a frequency is the 6th harmonic (multiplied by 6), and so on. For our purposes, we will use the harmonic labeling because its identification number is the same as the whole number multiplier. In most forms of music, the 2nd harmonic is considered to be the same note with twice the frequency. This is called the octave. All harmonics of powers of 2 will be octaves of the fundamental note. The most important harmonics to the physics of music are the 1st, 2nd, 3rd, 5th, and their octaves and exponents. Another important interval is the inverse of the 3rd harmonic, found by dividing the frequency by 3 and multiplying by powers of 2 to remain in the same range.

One of the most influential concepts in the evolution of music was the Circle of Fifths. In this case, a "fifth" refers to an interval between notes and is virtually equivalent to one octave below the 3rd harmonic. (There is also a Circle of Fourths, which is the same thing, but backwards. The "fourth" refers to the inverse of the 3rd harmonic.) It was found that when this interval was repeated twelve times, with octave adjustments to remain within one octave, twelve equal intervals appeared that ended very close to an octave above the starting note. Thus, each interval was slightly altered to produce a twelve-note scale with equal pitch spacing between each note, known as the chromatic scale. Because equal pitch intervals do not mean equal frequency intervals, there is a specific number by which a frequency can be multiplied to obtain the next note in the chromatic scale. Since doubling the frequency will yield a note twelve equal steps above the note, the multiplier must be the twelfth root of 2.

Among the most common scales heard in music are the major scale and its modes. A mode is just a different version of a scale with a root note on one of the scale's other notes. For instance, the chromatic scale's modes would be exactly the same as the original scale because the intervals are equally spaced. A major scale or one of its modes can be found by taking the root note and multiplying or dividing its frequency by 3 to obtain six other notes. The major scale itself is found by taking the root, multiplying its frequency by 3 five times to get 5 different notes and dividing by three once obtaining seven notes total for the scale. It is easy to see that there are seven different modes of the major scale, starting on each one of its seven notes. Each of the seven modes has a different feeling to it. The major (Ionian) scale has a happier sound than any of the others. The minor (Aeolian) and Dorian scales (with roots on the 6th and 2nd notes of the major scale, respectively) have more of a sad or evil sound. Other simpler existing scales can be found using the same method but finding only four notes beyond the root. These scales are called pentatonic scales because of their five notes and are probably the most widely used scales in the world.

Another aspect of music employed by many of its forms is the chord. A chord is a combination of separate notes sounding at the same time, creating a larger unique sound. The main idea behind a chord is that the overtones of its notes and the intervals between them sound along with the chord's notes and are equal to them. For example, when a note and its octave are played together, the difference in their frequencies will sound, in the form of beats, along with the two notes and is equal to the lower of the two. The most fundamental chord is the major chord, or major triad. It consists of a root note, a major 3rd (f*5) and a perfect fifth (f*3). The frequency difference between the root and the major 3rd is equal to 2 octaves below the root. The frequency difference between the root and the perfect 5th is equal to 1 octave below the root, and between the 3rd and the 5th is again 2 octaves below the root. The harmonics of the notes, which still sound even when only the fundamental notes are played, combine into the same sort of pattern. Another common chord is the minor chord, which consists of the root, minor (lowered) 3rd and the 5th. The major 3rd from the major chord is lowered one step on the chromatic scale to get the minor 3rd that is present in the minor scale. This gives the chord the same sad or evil feeling as the scale. Scales and chords are by no means limited to these "rules." The beauty of music is that there are no rules. Most of the evolution in the forms of music occurs from coincidentally finding intervals that are pleasing to the ear rather than mathematically deriving a new form.

Joe Walker is a 17-year-old guitar player who has been playing for almost three years on a Fender standard Mexi-Strat.