# Frequency and Pitch

Frequency and pitch are discussed in a little more detail here. This is a background page to the multimedia chapters Sound and Quantifying Sound.

### Pitch depends on frequency but also on loudness and spectrum

In the introduction on sound, we saw that pitch depends primarily on frequency. However, it also depends, weakly on sound level. It depends, too, on harmonic content. We'll discuss these later: this page looks just at the frequency dependence. The relation between physical and perceptual properties of sound are shown here.

### Pitch depends (approximately) logarithmically on frequency

In the film clip below, we double the frequency from 500 to 1000 Hz. The pitch interval between these two frequencies is called an octave (for reasons we'll see later in Consonance and temperament). For nearly all listeners, the pitch interval between 1000 and 2000 Hz is same as the first. Any doubling of frequency corresponds to an octave increase in pitch, no matter what the initial frequency.

500 Hz corresponds (roughly) to the musical note B4, or the B above middle C, as shown on the piano keyboard. 1000 and 2000 Hz to B5 and B6. On a piano or similar keyboard, if you go up 12 keys (160 mm to the right), you double the frequency. (Exactly double on most electronic keyboards and organs, slightly more than double on pianos.)

### Frequency and scales

On many keyboard instruments, such as pianos, the pitch difference between any two adjacent notes is the same. It is called a semitone, for reasons we'll see later. This means that they must have the same frequency ratio. There are twelve intervales in the octave on a piano keyboard: let's call the frequency ratio r. If we ascend 12 steps, we increase the frequency by r12, and we've made an octave, so   r12  =  2  or   r  =  21/12. So the frequency ratio of a semitone on such a keyboard is the twelfth root of two, or 1.059, an increase of 5.9%.

In the 'do-re-mi' major scale, the steps are two semitones, as shown below. So the frequency ratio is r2  =  1.0592  =  1.122. (And of course r2  =  21/6.) So to play 'do-re-mi', we increase the frequency by 12%, then another 12%, no matter where we start. (After three 'do-re-mi's, I couldn't avoid adding another note to resolve the little tune. The interval between the last two notes is corresponds to the frequency ratio r  =  1.059.)

Does music really need a number as weird as the twelfth root of two? And why are there 12 semitones to the octave? And what about scales in which the semitones are not the same size? We'll discuss this later in Consonance, scales and temperament [page still to make].

### Hearing range, infrasound and ultrasound

Not all pressure waves are sound. If I wave my hand from side to side, I set up a pressure wave in air with a frequency of a few Hz. But this is too low for us to hear: it is called infrasound. A pressure wave whose frequency is too high to hear is called ultrasound. So what is the frequency range of the human ear? On our hearing test site, you can measure the range of your own hearing. Provided that you use headphones and you have reasonably good hearing, you'll find that you can hear frequencies above about 20 Hz. The upper limit will depend on your age, but also on the extent to which you have exposed your ears to very loud sounds, such as those provided by personal in-ear sound systems.

When we measured the speed of sound in air, we obtained a value of 340 m.s−1. So we can convert the limits to wavelength. 20 Hz gives a wavelength of   λ  =  340 m.s−1/20 Hz  ~  20 m: the length of a cricket pitch. The shortest length that we can hear (in air) is about one thousand times smaller: about 20 mm.

### Ultrasound and echolocation

We cannot hear ultrasound, but bats can: they use it for echolocation when hunting insects. Ultrasound is also used to image the internal parts of the body, especially to diagnose heart problems or to image the foetus in a pregnant woman.

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