Harmonics and Tuning
If the 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, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. 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 31 harmonics and the intervals of 12-tone equal temperament (12TET), transposed into the span of one octave. Tinted fields highlight differences greater than 5 cents (1/20th of a semitone), which is the human ear's "just noticeable difference" for notes played one after the other (smaller differences are noticeable with notes played simultaneously).
| Harmonic | 12tET Interval | Note | Variance cents | ||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 4 | 8 | 16 | prime (octave) | C | 0 |
| 17 | minor second | C♯, D♭ | +5 | ||||
| 9 | 18 | major second | D | +4 | |||
| 19 | minor third | D♯, E♭ | −2 | ||||
| 5 | 10 | 20 | major third | E | −14 | ||
| 21 | fourth | F | −29 | ||||
| 11 | 22 | tritone | F♯, G♭ | −49 | |||
| 23 | +28 | ||||||
| 3 | 6 | 12 | 24 | fifth | G | +2 | |
| 25 | minor sixth | G♯, A♭ | −27 | ||||
| 13 | 26 | +41 | |||||
| 27 | major sixth | A | +6 | ||||
| 7 | 14 | 28 | minor seventh | A♯, B♭ | −31 | ||
| 29 | +30 | ||||||
| 15 | 30 | major seventh | B | −12 | |||
| 31 | +45 | ||||||
The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see just intonation). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), causes a listener to perceive a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz); that is, an octave below the lower (actual sounding) 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. Note that the lowest combination tone (100 Hz) is a 17th (2 octaves and a major third) below the lower (actual sounding) note of the tritone. All the intervals succumb to similar analysis as has been demonstrated by Paul Hindemith in his book The Craft of Musical Composition.
Read more about this topic: Harmonic Series (music)