pitch and harmony

Perception of pitch can be likened to attempting to sing an imitation of the sound you hear.

This is relatively easy when a sound is comprised of just one frequency or a series of harmonics (such as a guitar or clarinet sound). In these cases the harmonics of the voice can be varied until they form a one to one relationship with the sound. Often we don't actually vocalise this imitation, it simply remains as a response or perception in our mind.

In some cases we can't hear the complete sound because certain frequencies are not loud enough, or are masked by other sounds occuring simultaniously. When this occurs our perceptual systems simply assume the missing frequencies are there based on experience. This causes the perceptual phenomenon known as virtual pitch, where the pitch of a sound is attributed to the fundamental tone of the perceived harmonics, even though the fundamental was never actually sounded. This happens when you listen to a double bass recording through a small loudspeaker that can't produce low frequency sounds.

The pitch perception of inharmonic sounds arising from flexural vibrations is more complex. Try pitching your voice to the sound of a cymbal or the gong sound downloaded from the preceeding section.

In the example of a cymbal you may find many pitches you can sing that to some extent match the sound. Since the cymbal sound doesn't have harmonics (unless by some rare accidental relationships) you will be picking out the loudest individual overtones to sing to. The gong sound has fewer overtones than a cymbal, and if you look closely at the spectrum you will notice that there are some overtones at frequencies in almost harmonic relationships above a missing fundamental at about 240Hz. Therefore this sound has a strong pitch perception centered around 240Hz as well as a number of other weaker pitch perceptions.

This graph shows a computer model of pitch perception resulting from a small gong (produced by Denzil Cabrera's Psy-sound software). The 243, 162 and 121 Hz virtual percepts are subharmonics of the 486 Hz spectral percept and the 234 virtual is a subharmonic of the 702 spectral percept. All the spectral percepts are close to harmonics of 243 Hz which becomes the strongest pitch percept after about 0.5 seconds. Other partial frequencies visible in the spectrum shown earlier are not audible because they are masked by the stronger partials shown here.

Of course bells also vibrate flexurally and so until the recent invention of the harmonic bell, also had inharmonic overtones and confusing pitch percepts.

If we confine ourselves to thinking about the relative consonance and dissonance between certain sounds we can reveal a relationship between a sound's timbre and the pitches of that sound that produce harmony when heard simultaniously. Dissonance is caused when frequencies in one sound are only slightly different from the frequencies of another.

Slight differences in frequency between two sounds produce beating as the two sounds move from being 'in phase' to 'out of phase' with each other. This is heard when tuning string instruments. In phase sounds have concurrent peaks and valleys and so add their amplitudes. Out of phase sounds have peaks concurrent with valleys and so cancel out each others amplitudes. If two sounds have slightly different frequencies they move in and out of phase and so cause beating at a frequency equal to the frequency difference between the two sounds. When the beating becomes to quick to be heard it produces a perception of roughness or dissonance between the two sounds.

Comparison of 200Hz and 300Hz harmonic sounds.

The minimum dissonance (or maximum consonance) between sounds with harmonic overtones occur when one sound is in a simple frequency ratio with the other.

In this example two harmonic sounds are at a frequency ratio of 3/2 (an interval of a fifth in the Western scale). Alternate harmonics of Sound 2 are either at the same frequency or half way between the harmonics of Sound 1, thus minimising dissonance between the two sounds. In certain circumstances these two sounds may appear to fuse producing a sound with many of the harmonics of a 100 Hz harmonic sound. The fundamental and prime number harmonics above 3 are missing from the harmonic series starting at 100 Hz and so this composite sound is an example of virtual pitch.

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An example of a dissonance curve showing minima at consonant intervals (with thanks to Terry McDermott).

Just tuning is a musical tuning system that uses integer frequency ratios of harmonic sounds to maximise consonance. Harry Partch (see Genesis of a Music, Da Capo Press ) describes a tuning system that uses all the possible ratios of integers up to 13. Most musical systems that use instruments producing harmonic sounds have evolved through using integer ratio tunings.

Recently William Sethares devised a method of calculating tuning ratios that minimised the dissonances between inharmonic sounds (see Tuning, Timbre, Spectrum and Scale, Springer-Verlag, 1998). This was achieved by calculating and summing the roughness between each overtone of the sounds for incremental changes in the tunining ratios. In his book he provides a theory for the tuning systems used in Indonesian gamelan derived by minimising the dissonances between gongs and the human voice.

Harmony in Western music has been complicated by the desire to modulate chords and melodies; that is, to play the same set of ratios starting at different fundamental frequencies. The logical conclusion of modulation is to produce an equally spaced array of fundamental frequencies known as the equally tempered scale. This scale, while mathematically very simple (each step is the twelth root of 2 apart) has imbedded in its nomenclature and traditions all the complexities of the many centuries of its evolution. Unfortunately, consonance suffered for the simpicity of the Western scale, and only the intervals of the forth and fifth remain at minimum possible dissonances.

Bells have generally been excluded from Western instrumental ensembles because their inharmonic timbres produced dissonances with other instruments. Carillons (large bell instruments) have had similar problems which will be discussed in the next section. The invention of the harmonic bell has solved all of the problems associated with achieving the desired consonance using bells.