You may have once heard someone tune a guitar using the harmonics, but probably have never thought about how important they really are. In all the root position chords and their inversions diagrammed in the previous chapter, a perfect 5th interval exists between two of the notes. Use the diagram on the following page to play the harmonics on the guitar. If the frets are removed from the fretboard, the harmonics would still exist in the exact same locations. They are a phenomenon of nature and physics located directly above the 5th, 7th, and 12th frets and the 12TET octave is derived from the exact locations of the harmonics. A perfect 5th interval is equivalent to seven semitones and a perfect 4th interval is equivalent to five semitones. All root position chords have a perfect 5th above the root note. And all 2nd inversions contain the perfect 4th interval above the bass note. All 1st inversion chords have either a P5th or a P4th in them, and the location of these is always the middle of the three intervals in the chord formula. Below, the sus4 chord has both a perfect 5th and a perfect 4th interval. On a guitar, the frets are not spaced equally. Instead, the frequency of the notes is derived mathematically using fractional exponents. The exact locations of the harmonics were known 2,500 years ago by Pythagoras, and were used to space the frets, such that they create twelve equal-sounding steps in pitch. However, Pythagorean tuning based on the harmonics led to increasingly complex frequency ratios between the notes, and chords such as the min3 and maj3 were considered unlistenable in Pythagorean tuning. 12TET, which came around the 1600’s created twelve actual equal steps in pitch in an octave using fractional exponents, a different mathematical system than Pythagoras used, and simplified the approximate frequency ratios of the intervals. |