Tone of Voice and Mind : The Connections between Intonation ...

52 Chapter 3
quency (Figure 3-3). If the F0 is 110 Hz, corresponding to a low A on the piano,
then there will also occur vibrations at 220 Hz (F1), 330 Hz (F2), 440
Hz (F3), 550 Hz (F4), and so on. In fact, the amplitudes (intensities) of the
higher harmonics drop off rapidly, so that it is difficult to hear more than one
or two higher harmonics unless the musical instrument is specifically designed
to amplify them, but the presence of higher harmonics is real and an inherent
part of all music using instruments that have vibrating strings or tubular
resonating bodies.
The upshot of the phenomenon of higher harmonics is that, in listening
to music, we rarely hear pure tones consisting of only one frequency and we
rarely hear pure intervals consisting of only two distinct fundamental frequencies.
Even if only two notes are sounded on the musical instrument, there are
at least faint echoes of the higher harmonics – and that means that our evaluation
of the pleasantness/unpleasantness or consonance/dissonance of two-tone
combinations is actually the evaluation of many tone combinations with a variety
of frequencies and amplitudes. A simple example is illustrated in Figure
3-4. There, we see that the “total dissonance” of an interval of 6 semitones (e.g.,
F3
F2
F1
F0
1045 Hz (C”) 1377 Hz (F#”) 1045 Hz (C”)
1568 Hz (G ” )
784 Hz (G’) 1007 Hz (B ” ) 784 Hz (G’)
1172 Hz (D ” )
523 Hz (C’) 738 Hz (F#”) 523 Hz (C’)
784 Hz (G’)
261 Hz (C) 369 Hz (F#”) 261 Hz (C)
392 Hz (G)
Tone 1 Tone 2 Tone 1
Tone 2
Example A Example B
Figure 3-4. In the evaluation of an interval of two pitches, the frequencies of the upper
partialscanhaveaninfluence. In example A, the tritone (6 semitone) interval between
the F0s (e.g., C and F#) is much larger than the interval of a minor third. On their
own they would be rather consonant, but the location of certain of the upper partials
(e.g., 784 and 738 Hz) is such that some dissonance is heard. In example B, the slightly
larger interval of a fifth (e.g., C and G) is notably more consonant – a fact that can be
explained by the absence of dissonant intervals among the upper partials. Solid lines
indicate consonant intervals; broken lines indicate dissonant semitone intervals.