Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.

**Harmony**

The note one octave above C4 is C5 and it has double the frequency, so C4 and C5 have frequency ratio 1 : 2. If we take the frequency of Middle C as a unit, then C5 has a frequency 2 and we can plot the combination (C4 – C5) as the graph of the function sin( t ) + sin (2 t + φ ) [the phase φ here is acoustically unimportant; we tweak it to make the pictures below pretty]. We can also plot the two components on orthogonal axes (the x and y axes). This gives what is called a Lissajous curve:

C4 (Middle C) and G4 (the G above C4) have frequencies in the ratio 2 : 3 and combine to produce a chord called a Perfect Fifth (we count inclusively from C to G or *do-re-mi-fa-**so*). Since the numbers in the ratio are small, the combination (C – G) is harmonious. Here are the graph and the Lissajous curve of a Perfect Fifth:

Now we look at the chord (C – F or *do*–*fa*) which is a Fourth, with frequency ratio 3 : 4.

Once again, a harmonious effect is produced. Next we look at the important interval of a Third (C – E or *do-mi*). The frequency ratio is now 4 : 5 and the graphs are as follows:

We notice that the waveform and Lissajous plots are getting gradually more complex, but the effect is still harmonious.

We now drop the upper note down one semitone to E-flat to get a Minor Third, with a frequency ratio 5 : 6, looking like:

In the opposite direction, moving up from G4 to A4, we have a chord called a Major Sixth (C – A or *do-la*) with a frequency ratio 3 : 5 and it looks like this:

**Discord**

We have not yet combined the tonic note C with D or with B. Now D is one tone above C and B is a semitone below. Both combinations yield discordant sounds. To illustrate the problem, we look at the combination of C and C♯ (separated by a semitone). The result is a shrill sound, not of itself pleasing, although it often occurs in music only to be relieved or resolved by a more concordant sound. Here are the graphs:

We see from the graph that there is a beating effect: when two notes of closely similar frequencies are combined, they result in sounds with the sum and difference of the two notes, that is, with very high and very low pitches. We can see this in the graph.

All the ratios we have given above are for the tuning system called *Just Intonation*. The actual tuning of a piano follows a different scheme called *Equal Temperament*, where the ratios are irrational numbers; however, they are close to the ratios we have used above (See an earlier Blog Post on Temperamental Tuning.

The Lissajous curves that have appeared above are all closed curves, of relatively simple form. The Lissajous curves for pairs of oscillations having a frequency ratio that is an irrational number no longer close upon themselves but densely fill the square region so that nothing much can be discerned from the plots.

**The Harmonograph**

In the nineteenth century, a mechanical contraption for drawing diagrams like those above became very popular. It consisted of a set of pendulums mounted on a table and was called a **Harmonograph**. A wide range of patterns could be produced by changing the lengths of the pendulum. We may return to this topic.

Simple computer programs can now emulate the action of a harmonograph. The illustration above shows a selection of Lissajous curves for different frequency ratios and phases.