Consonance and Dissonance

The question addressed here is, what happens when two sounds are played together? At issue is, when does a musical chord "sound good", or is consonant, and when does a musical chord "sound bad", or is dissonant. These judgements of things sounding "good" and "bad" have little to do with physics. They generally are quite subjective, meaning different people will have different opinions. Despite this, there is a physiological response to two sounds played together that is common to most people. Today we'll try and understand that response.

We will need two tools to develop this understanding. The first tool is to review the notion of "critical bands" in the human auditory system. The second tool is to introduce the "musician's notation" for sounds, or the musical scale.

Critical Bands

Humans can hear sounds having frequencies in the range 20 < f < 20,000 Hz. As with loudness, our perception of pitch is "logarithmic" meaning that a "pitch scale" has adjacent sounds in the scale with frequencies differing by a multiplicative factor, rather than by an additive constant. This will be more clear when we talk about the musical scale below.

Our perception of pitch is related to the distribution of nerve cells along the "organ of corti" lying atop the basilar membrane inside the cochlea. The motion of the cilia (hair cells), in response to sounds of different frequency, excite different nerve cells, resulting in the first level of distinction of pitch. Which cilia end up moving depends on the pitch of the sound. This is the basis for dividing our pitch perception into critical bands.

The frequency range of audible sounds is divided into roughly 24 critical bands. Each critical band has a central frequency (f0) and a bandwidth, corresponding to the difference between the highest and lowest frequency sound that excites that critical band. For sounds with frequency greater than 1,000 Hz, the critical bandwidth is a fixed fraction (roughly 15%) of the central frequency of the band. This means that for the central frequency of a critical band, f0=10,000 Hz, the critical bandwidth is 1,500 Hz, meaning that all sounds with frequencies between 9,250 Hz and 10,750 Hz fall onto this critical band. A lower frequency critical band is centered at 2,000 Hz and has a bandwidth of ~280 Hz. Below 1,000 Hz, the critical bands have a constant bandwidth of 100 Hz.

Why the notion of critical bands is relevant, is that when two sounds have their frequencies within the same critical band, the physiological response results in a perception of "roughness" for the resulting sounds, and we identify the two sounds played together as being dissonant. Hence, to identify what is consonant or dissonant will involve looking at differences in the frequencies of the two sounds.

In reality, to determine the degree of consonance or dissonance of two sounds played together is not as simple as determining if their frequencies fall within the same critical band. There are a few complications. First, if the frequency difference between the two sounds is less than ~10 Hz, then the two sounds are "fused" meaning they sound sufficiently alike that we do not sense any dissonance. Instead, we hear a modulation of the amplitude of the resulting sound, called beats. Two sounds with a frequency difference equal to ~30% of the appropriate critical bandwidth are maximally dissonant. For larger frequency differences, the two sounds are more consonant. Once the two sounds no longer fall within the same critical band, then any sensation of dissonance disappears.

The Western Musical Scale

We perceive two sounds having frequencies equal to f and 2f as sounding very similar in pitch. Sounds differing by a factor of two in frequency are said to be an octave apart. For example, middle C on the piano has the frequency of its fundamental equal to 261.6 Hz, and the C note one octave higher has the frequency of its fundamental equal to 523.2 Hz. In Western culture, musicians divide an octave into twelve semitones. Two sounds differing by a semitone have the frequencies of their fundamental in the ratio of 21/12, or the twelfth root of 2. At this point, it might be useful to define the western musical scale, with the labels for the notes used by musicians, to illustrate this point:

Musical NoteFrequency (Hz)fNote/fA3
A3 220.0 1
A sharp/B flat 233.1 21/12
B3 246.9 22/12
C4 261.6 23/12
C sharp/D flat 277.2 24/12
D4 293.7 25/12
D sharp/E flat 311.1 26/12
E4 329.6 27/12
F4 349.2 28/12
F sharp/G flat 370.0 29/12
G4 392.0 210/12
A sharp/B flat 415.3 211/12
A4 440.0 212/12

The subscripts on the musical note indicate the position of the note relative to middle C on the piano (C4). This means, for example, that A3 is the first A key below middle C. The column labeled frequency gives the frequency of the fundamental for the musical note. Remember that the complex sounds produced by musical instruments will involve a superposition of pure tones from a harmonic series built on the appropriate fundamental frequency. Finally, the last column gives the ratio of the frequency for the particular musical note to the frequency of the A3 note. This column illustrates the twelve semitone division of the musical scale.

Analyzing Musical Chords

In general, a musical instrument playing a particular note will produce a complex sound (rather than a pure tone) having its pitch determined by the fundamental frequency for that note, and its timbre determined by the weighting of the higher harmonics built on that fundamental. To establish the degree of consonance or dissonance for a chord, or a pair of musical notes played together, we have to examine the entire harmonic series for each note, and look at frequency differences to see if they fall within the same critical band.

Let's see how this works by looking at two chords that span the spectrum of dissonance and consonance. Let's first examine striking a chord on a piano, corresponding to the two notes C4 and C5. These two keys are an octave apart. The first step is to write down all of the frequencies of the harmonic series for the two notes.

Note1st2nd3rd4th
C4261.6 Hz523.2 Hz784.8 Hz1046 Hz
C5523.2 Hz1046 Hz1570 Hz2093 Hz

In the table, the columns label the harmonic number. We can see that for this chord, the second (and fourth,sixth,...) harmonic of C4 has exactly the same frequency as the first (and second,third,...) harmonic of C5. Since the frequency difference between those harmonics is identically zero, which is clearly less than ~10 Hz, the pure tone sounds from these harmonics from the two different notes are fused. Otherwise, the difference in frequency between all other harmonics of C4 and C5 is large enough that the pure tones fall on different critical bands, meaning there is no dissonance between the pure tones of any harmonic for C4 and any harmnoic for C5. The end result is that this chord is maximallly consonant.

At the other end of the spectrum is the maximally dissonant sound when two keys, one semitone apart, are played together. Let's consider striking the keys F4 and F sharp (one semitone higher) together. We'll do this for the keys that are just higher than middle C on the piano.

Note1st2nd3rd4th
F4349.2 Hz698.4 Hz1048 Hz1397 Hz
F sharp 370.0 Hz740.0 Hz1110 Hz1480 Hz

Again, in the table, the columns label the harmonic number. Looking at the frequency differences between the harmonics of F4 and those of F sharp shows the following: for the first harmonic of each note, the frequency difference is 21 Hz; for the second harmonic of each note, the frequency difference is 42 Hz; for the third harmonic of each note, the frequency difference is 62 Hz; and so on. In each case, the pure tones from the notes have a frequency difference larger than ~10 Hz, meaning that the sounds are not fused. As well, it is clear from the differences, that for each harmonic, the pure tone sounds lie within the same critical band. Furthermore, for many of the harmonics the frequency difference is close to being equal to 30% of the critical bandwidth. This explains why this chord is so unpleasant to hear (dissonance).

There are many other chords that could be analyzed. Below, is an ordering of two-note chords in increasing dissonance: The relative degree of dissonance of these different chords can be established by examining frequency differences between the harmonic series of each note. In general, if there are two pure tones having a frequency difference greater than ~10 Hz (meaning the sounds are not fused), but within the same critical band than there is some level of dissonance. If two pure tones from the higher harmonics of the two notes are within the same critical band, the negative physiological response is not as strong, since the amplitudes of the higher harmonics for any given note get smaller with increasing harmonic number.

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Last updated: 1 Dec 1999
Comments: bland@indiana.edu