Jonah Dempcy
Jazz Lessons
Overtone Series and Western Harmony
Harmonics and tuning
If the first 15 harmonics are transposed into the span of one octave, they approximate some of the notes in what the West has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, and in relation to that scale, many of the harmonics are slightly out of tune, and the 7th, 11th, and 13th harmonics are significantly so. In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships.
Below is a comparison between the first 20 harmonics and their equivalent frequencies in the 12-tone equal-tempered scale. Orange-tinted fields highlight differences greater than 5 cents, which is the "just noticeable difference" for the human ear. (Because physical characteristics of musical instruments cause significant variations from these theoretical values, they should not be used for tuning without adjusting for those variations.)
|
|
|
Some theorists believe that the overtone series is the basis of the experience of consonance and dissonance, for example, one writer states that
"If two different notes are played simultaneously, the composite sound includes the harmonics of both notes. In musical intervals, one or more of those harmonics are common to both notes. For example, if C3 and G3 are sounding together, their harmonic series intersect at G4 (2nd of G3, 3rd of C3) and G5 (4th of G3, 6th of C3). If these common harmonics are at the same frequency or nearly so in both notes, the composite sound will seem harmonious. If their frequencies differ significantly, we tend to hear the notes as "out of tune" with each other. Tuning involves changing the fundamental pitch of one of the notes to control the relationship between these common harmonics"
However, this theory neglects the fact that several intervals made up of electronically generated pure sine waves (with no overtone content) are still perceived as consonant or dissonant. The frequencies of the overtone series, being a range of integral multiples of the fundamental frequency, are naturally related to each other by small whole number ratios and it is these small whole number ratios that are the basis of the consonance of musical intervals, for example, a perfect fifth, say 200 and 300 Hz (cycles per second), produces a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz) that is, an octave below the lower note. This 100 Hz first order combination tone then interacts with both notes of the interval to produce second order combination tones of 200 (300-100) and 100 (200-100) Hz and, of course, all further nth order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When we contrast this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 we get, for example, 700-500=200 (1st order combination tone)and 500-200=300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains 4 notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. All the intervals succumb to similar analysis as been demonstrated by Paul Hindemith in his book, The Craft of Musical Composition.