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Beautiful music is capable of reaching deeply into our souls, imparting peace and tranquillity.
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There is something satisfying about musical sounds that blend and harmonize.
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Some combinations of sound, however, are anything but pleasant to our ears.
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We are going to examine some of the characteristics of our ear/brain hearing system to find out why some intervals sound lovely and some sound raucous and unpleasant.
Music involves such highly complex sounds that one tends to be overwhelmed when trying to analyze just what is going on. For this reason, we must once more fall back on pure tones as we probe into the question of why some tonal combinations sound good and some do not.
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| We shall base this exercise on a tone of 500 Hz at a comfortable listening level:
We are now going to add a second tone, an identical 500 Hz of the same amplitude:
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Notice that there is considerable increase in loudness because the two sine waves have a time relationship described as being in phase.
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Because the two tones are in phase, they add constructively and the resulting combination has twice the amplitude of either tone alone. Those knowledgeable in electronics will recognize this as producing a 6-dB increase in signal level.
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Now, let’s see what happens if the same two 500-Hz sine waves are combined out of phase or in phase opposition, that is, when one waveform goes positive, the other goes negative, and vice versa.
First, one signal alone:
And now the second tone will be added momentarily and then taken away:
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| That four-second dead spot in the middle was caused by adding the second equal 500-Hz tone out of phase. When the two signals are in phase opposition, one cancels the other out and the resultant output is zero.
So far, we have been talking about signals with the same frequency. When two tones of differing frequency are combined, some very interesting things happen that have a direct bearing on musical sounds. A tone of 500 Hz added to a tone of 501 Hz sounds like this:
The two tones, only 1 Hz apart, alternately combine in phase and in phase opposition to produce a 1-Hz beat. By holding the 500-Hz tone constant and changing the frequency of the second tone, the beat frequency can be varied at will: |
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| The frequency of the beat is determined by the difference between the frequencies of the two tones which are beating together. Thus, if a tone of either 490 Hz or 510 Hz is combined with the 500-Hz tone, a beat of 10 Hz is produced.
As the difference between the two tones is increased so that the beat frequency increases to about 20 Hz, the ear becomes unable to discern the individual beats: | |
| As the beat frequency is increased beyond 20 Hz, a harsh, rattling sound is heard:
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Note this roughness well! It is the secret ingredient of what we consider to be unpleasant musical effects, as we shall see later.
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As with so many other factors of human hearing, the critical band seems to be involved in how we hear two tones which are sounded together. If the two tones are a critical bandwidth apart, they are heard not as beats or roughness but resolved harmoniously as two separate tones. In the following example, notice the transition from beats, through roughness, to a more pleasant sound, as the two combined tones are increasingly separated in frequency:
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To avoid the distraction of the beats and the region of roughness, and for the ear to separate the two tones, they must be at least a critical bandwidth apart.
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Critical bandwidth calculated from:
ERB = equiv. rectangular band
ERB = 6.23 f2 + 93.39 f + 28.52 (Hz)
f = frequency in kHz
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All this leads us to the conclusion that when several tones are sounded simultaneously, the result may be considered as either pleasant or unpleasant. Another way of describing these sensations is with the terms consonant and dissonant. In this psychoacoustical context, when we say consonance, we mean tonal or sensory consonance. This is distinguished from the musician’s use of the word, which is dependent on frequency ratios and musical theory. Here, we are referring to human perception. Of course, in an ultimate sense, the two definitions must come together. The audibility of these roughness effects does not depend on musical training.
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Ref.: Moore and Glasberg, J. Acous. Soc.
Am., 74, 3, 750 (1983)
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| Now, instead of frequency, let us consider the same beating effect between two tones in terms of their separation in fractions of a critical bandwidth. As we have seen, at 500 Hz, the critical band is somewhere around 100 Hz wide. Let us define 100 Hz as unity and consider fractions of that band.
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When two tones have zero separation, they sound as a single tone which has maximum consonance and minimum dissonance:
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That is point number one on our curve.
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And now, here are the two tones separated by about one-fourth of a critical bandwidth:
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This is the least consonant, or the most dissonant, sound.
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When the two tones are separated by about one-half a critical bandwidth, the roughness has partially receded to give us about 40 percent of full consonance:
At a separation of about three-fourths of a critical bandwidth, a further improvement in consonance to about 80 percent is noted:
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| When two notes separated by a full critical bandwidth are combined, 100 percent consonance results:
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This puts the effect of combining two tones in proper perspective. If their frequencies are separated by a critical bandwidth or more, the effect is consonant. If less than a critical band separates the tones, varying degrees of dissonance are heard. The most dissonant (that is, the least consonant) spacing of two tones is about one-fourth of a critical bandwidth.
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Musicians define an octave as a musical interval whose two tones are separated by eight scale tones:
Tones separated by an octave have an essential similarity recognized by everyone. There is a very good reason for the octave’s consonance, which directs our attention once more to the critical band.
An octave represents a frequency ratio of two to one. For example, the frequency of the C an octave above middle C is twice the frequency of middle C:
This means that the harmonics of both are either well separated or coincident up through the audible spectrum when the two are played together:
In fact, the sound of the higher note reinforces that of the lower one. The result is consonance—full, rich, and complete.
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| Let us try another two-note interval, middle C:
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And the G above middle C:
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Playing C and G together gives a very pleasant effect:
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Musicians call this interval the perfect fifth because the two notes are five notes apart:
The perfect fifth is only slightly less pleasant than the octave interval. Let us examine the reasons for such a high degree of consonance.
The fundamentals of C and G are separated by 170 Hz, which is far greater than the critical bandwidth in that frequency range. So far, so good, as this meets the criterion for consonance.
The second harmonics of C and G are 265 Hz apart, a separation some three times the width of the critical band. This also is on the side of consonance.
As the harmonics of C and G are examined closely, we note that they are either separated more than a critical bandwidth or they are at essentially the same frequency, both factors contributing to consonance.
After all, if two harmonics are at the same frequency, they really are a single tone. If they are only a few Hz apart, a beat is superimposed on that frequency which has a positive musical value.
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| With a renewed sense of getting close to something really significant musically, let’s apply these critical band criteria to another musical interval, a minor seventh. The minor seventh can involve middle C again:
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And the B-flat above middle C:
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Playing these two notes together gives us a minor seventh interval:
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While considered by musicians a less consonant interval than the perfect fifth, it is nonetheless a useful tool in the composer’s toolbox.
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Let’s see if we can discover the reason why the minor seventh is considered less consonant than the perfect fifth.
Comparing the frequencies of the fundamentals and harmonics of middle C and B-flat, we fail to find coincident pairs as we did with the perfect fifth interval.
For example, the second harmonic of C and the fundamental of B-flat are 57 Hz apart. The critical bands are about 75 Hz wide in this frequency range. This means that this interval is about three-fourths of a critical bandwidth. It will only have about 80 percent of full consonance.
The fourth harmonic of C and the second harmonic of B-flat are separated 114 Hz, or about nine-tenths the width of a critical band. It is not fully consonant but close.
The fifth harmonic of C and the third harmonic of B-flat are about 90 Hz apart. This means a separation of less than a critical bandwidth—in fact, about 50 percent of a critical band. This interval contributes only half of the full consonance, which means half dissonance as well.
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| For a fourth example, we can compare the seventh harmonic of C with the fourth harmonic of B-flat. We find them only 34 Hz apart or about 15 percent of a critical bandwidth. This pair is close to being totally dissonant and would certainly contribute some roughness to the combination.
For a minor seventh interval, we find numerous harmonics of C and B-flat close enough together to result in some roughness. Evaluating the separation of harmonics, we find many near misses which are not coincident but less than a critical bandwidth apart contributing to the roughness. Of course, there are many other harmonics which are spaced far more than a critical bandwidth which are fully consonant.
We see that the perfect fifth is close to perfect—that is, close to the consonance of the octave interval. The minor seventh has some intervals separated less than a critical bandwidth, hence, somewhat dissonant.
We conclude that the critical band approach has value in explaining, or even predicting, the degree of consonance an interval exhibits.
Is dissonance necessarily bad? We offer no value judgment on that here. Dissonance can be considered another dimension of musical creativity to be explored. The music of some ethnic groups is more dissonant than ours, and even within our own musical culture, some composers are noted for the degree of dissonance in their works.
Our purpose in this analysis is only to relate consonance and dissonance to the critical bands of the human auditory system.
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