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The Elements of Harmony > Intervals (basic) > Consonance

The Physics of Consonance

When a sound wave propagates through air, it oscillates, and in so doing creates a pattern. This pattern may be reinforced, destroyed (or anything in between) by another wave propagating together with it. Two sound tones that produce little disturbance of each other's patterns would be considered 'consonant'. Consonant waves have a high proportion of nodes in identical positions, and would not interfere with each other at all at these points.

Two sound sources emitting an identical tone would certainly be consonant, so much so that the sound will be reinforced, and will therefore be louder. But because the two sounds have the same frequency, one is indistinguishable from the other: we cannot determine any interval between them. There is no interval between unison tones.

But if one tone's frequency is doubled, the two sounds become distinguishable, even though the higher sound has many of the same characteristics as the lower sound. They are extremely 'consonant' and are said to be an 'octave' apart.

In order to construct harmonies that are consonant, we must first define a set (or scale) of notes that contains a large number of consonant pairs. These are chosen within the range of an octave. This set of notes can then be 'mirrored' in upper and lower octaves.

Furthermore, in order to create smooth melodic motion, the intervals between the notes of this scale should be as evenly spaced as possible.

After the frequency ratio 2:1, corresponding to the interval of an octave, the ratio 3:2 to produces the next most consonant pair within the range of the octave. We can also include the ratio 4:3. (Pythagoras considered these intervals consonant back in 530 BC, when his attention was drawn to "sounds fully concordant in combination with one another" as he heard hammers and anvils resounding together in a blacksmith's shop). But as we increase the integer values making up the ratio, mutual wave interference becomes more noticeable. We experience this as dissonance.

If we consider the ratio 5:4, we obtain yet another note, lower than that that from 4:3, and also considered acceptably consonant.

Let's call the note from 3:2 'G', the note from 4:3 'F' and the note from 5:4 'E'. Let's call the fundamental frequency, with which we wish to 'harmonize' the higher frequencies, 'C'.

We can thus combine C and G and obtain a very consonant pairing. C and F will also give a reasonably consonant pairing. C and E will also be acceptably consonant. Furthermore, we can experiment with the interval between E and G, and this should be acceptably consonant too, since the proportional increase of E to obtain G is 6/5, which is the next ratio (6:5) in the above ratio series. Let's call the note from the ratio 6:5 'E-flat'.

Now, if we combine the sound of three notes, we obtain a so-called triad, and would expect the following combinations to be consonant, as none involve any dissonant pair:

C - E - G

C - E flat - G

Even the interval E flat - G involves the same proportional increase as C - E, so no excessive dissonance is introduced.

These two triads are considered the pillars of Western harmony.

But we need to select additional notes within the octave, to make it more 'even' and melodically smooth. At the same time, we must respect the criterion of consonance between any pair within the set of notes we are defining. Therefore, we need to relate the additional notes to F and G, our original two most consonant notes.

Two notes consonant with G and with each other would have the ratios 3:2 and 5:4 with respect to the frequency of G, which itself has a ratio of 3:2 to starting note C. Let us call the resulting notes:

D, with ratio 3:2 x 3:2 = 9:4 (this note is in the higher octave!)

B, with ratio 5:4 x 3:2 = 15:8

Similarly, two notes consonant with F and with each other would have the ratios 3:2 and 5:4 with respect to the frequency of F, which itself has a ratio of 4:3 to starting note C. Let us call the resulting notes:

C', with ratio 3:2 x 4:3 = 2 (this note is in the higher octave!)

A, with ratio 5:4 x 4:3 = 5:3

So we may sound the following triads, and obtain fully consonant sounds (or chords), since all possible pairs of notes are fully consonant (their interval structure is identical with C - E - G, above):

G - B - D

F - A - C

Now if we select all the notes that we have derived (with the exception of E-flat, for the moment) and place them in ascending order (and lowering D by an octave - just divide by 2), we obtain:

pitch names: C D E F G A B C'

ratios to C: 1 9/8 5/4 4/3 3/2 5/3 15/8 2

This is, theoretically, a scale that achieves our initial goal, that is:

1. To produce the maximum number of consonant triads. With the exception of D - F and D - A (D having been lowered by an octave), all non-adjacent pairs are fully consonant.

2. To define a series of notes that provides a smooth, step-by-step flow.

This scale, derived from the consideration of simple frequency ratios, is called the Just Diatonic Scale.

We can now create an even smoother scale by inserting notes between C & D, F & G, A & B). If we also add the E-flat, which we obtained earlier, and insert a note between G & A, corresponding to the 3:2 ratio to E-flat, and call it A-flat, we obtain the so-called 12-note, chromatic scale:

C C# D E-flat E F F# G A-flat A A# B

This is the scale upon which all modern Western music is based, and allows for the use of 'chromaticism', i.e. the smoothest possible melodic connections between the notes of the Just Diatonic Scale. The interval between any two adjacent notes in the chromatic scale is called a semi-tone. Note, however, that some semi-tone intervals are larger than others. If we calculate the proportional increase from one note to the next, we will see that the step from E to F (and from B to C') is just larger than half the size of the interval from C to D (and F to G, and A to B). Therefore, the intervals C - C# and C# - D cannot both be made equal to the interval E - F. This problem was solved by Chu Tsai-Yu (in 1584) in China, and Simon Stevin (in 1585) and Marin Mersenne (in 1636) in Europe, by dividing the octave into 12 equal semi-tones (i.e.

equal frequency ratios). This tuning method is called equal temperament.