When two notes are played simultaneously, we hear the resulting sound as consonant or dissonant.  Consonant intervals are considered sweet and smooth, whereas dissonant intervals sound rough and harsh. Generally speaking, different pitches whose frequency multiples tend to reinforce each other are said to be consonant, whereas pitches whose frequency multiples clash are said to be dissonant.  In the case of the latter, the ear senses a regular swelling of sound or beating as the different frequencies go in and out of phase.

Throughout the various periods of musical history there has been a great deal of disagreement about which intervals are consonant and which are dissonant.  This is not particularly surprising since, from a physicist’s point of view, the difference is really one of degree rather than of kind.

Intervals with very simple mathematical ratios are called “perfect” consonances. The octave in particular sounds consonant because two waves in the upper note correspond exactly with one wave of the lower note, thus giving it a ratio of 2:1 [see the table on this page].  Other “perfect” intervals include the unison (1:1, i.e. the same note), the fifth (3:2 – i.e. three waves in the lower note equaling two waves in the lower note), and the fourth (4:3).  These intervals give the smoothest blend because the frequencies of the upper partials of their respective notes do not clash very noticeably. Such intervals were greatly favoured by Western composers in ancient and medieval times. To modern ears, however, they can sound hollow or even downright boring.

Since around 1250, major and minor thirds (5:4 and 6:5) together with major and minor sixths (5:3 and 8:5) have also been considered consonant.  They sound richer and sweeter than perfect fourths and fifths because there is some beating as well as some agreement between their lower harmonics.

Dissonant intervals include the major second (9:8), the semitone or minor second (16:15), the minor seventh (16:9) and the major seventh (15:8).  The greater the numerator and the denominator of a fraction representing the ratio of vibrations between two sounds, the more acute the resulting dissonance.  This is why, for example, a minor second sounds harsher to the ears than a major second does.  That said, it is also true to say that the twentieth century witnessed a considerable expansion in the range of intervals that were considered consonant.  The fact that the ear can get used to dissonant intervals and treat them as consonant is proved by the way we now accept tonic seventh chords as forming legitimate cadences in jazz music.

Intervals and Frequency Ratios

Interval	Frequency Ratio	Cents from staring point
		Just intonation	Equal Temperament
Unison	1:1	0	0
Semitone	16:15	111.731	100
Minor Second	10:9	182.404	100
Major Second	9:8	203.910	200
Minor Third	6:5	315.641	300
Major Third	5:4	386.314	400
Perfect Fourth	4:3	498..045	500
Augmented Fourth	45:32	590.224	600
Diminished Fifth	64:45	609.777	600
Perfect Fifth	3:2	701.955	700
Minor Sixth	8:5	813.687	800
Major Sixth	5:3	844.359	900
Harmonic Minor Seventh	7:4	968.826	1,000
Minor Seventh	9:5	1,017.597	1,000
Major Seventh	15:8	1,088.269	1,100
Octave	2:1	1,200.000	1,2

When reading this table, remember that "frequency ratio" refers to the number of waves in the upper note compared with the number of waves in the lower note. An octave, for example, has a ratio of 2:1. This means there are two waves in the upper note for every single wave in the lower note. If you are still confused, check out the section on the harmonic series. A "cent" is a convenient measure of the interval between any two notes. By definition, there are 1,200 cents in an octave. In the scale of equal temperament, a semitone therefore equals 100 cents.

The degree of dissonance actually heard can vary not only with the interval but also with the tone quality or timbre of the instruments involved. In some cases, the dissonance does not register very strongly because the clashing harmonics are either absent or of a very low amplitude. A dissonant interval played by two oboes, for example, will actually sound sweeter when played by two flutes. This is because the flute has a relatively "pure" sound with few partials, whereas the oboe produces a rich harmonic spectra.

For hundreds of years, audiences have been conditioned to believe that the sounding of a dissonant interval should be “resolved” by playing a consonant one.  In Bach’s time the urge to hear the resolution was particularly strong.  The story is told that one morning the great composer was awakened from his sleep by the sound of his second wife persistently playing a dominant seventh chord on the harpsichord in the music room.  Unable to bear the unresolved dissonance, he jumped out of bed, rushed to the harpsichord and played the appropriate resolution.

Reacting to the traditional rule of harmony that dissonance must be followed by consonance, some twentieth century composers such as Schoenberg went to the other extreme, creating works that glorified in the art of dissonance without using any consonance at all.  Nowadays, this approach has found a legitimate place in the music used for background effects in Hollywood-style movies [for example in creating atmosphere in horror movies]. The resulting cacophony of sound, however, rarely appeals to audiences in the concert hall.  If you do find yourself in charge of an advanced group of musicianswho wish to display their skills in this idiom at their next concert, I suggest you limit yourself to only one or two such pieces.   Asian audiences tend to have fairly conservative musical tastes.