Waves and sound · Quantitative
Frequency, pitch and loudness
Two dials on a signal generator, and neither one does anything the other one does. One sets how high the note is. One sets how loud it is. Getting them the wrong way round is the single most common slip in this topic.
Start here
Two strings, one guitar, and only one of them is wrong.
Tighten a guitar string a little and the note goes up. Pluck the same string harder and the note gets louder and stays exactly where it was. Two different changes; two different results; and students routinely swap them.
A loudspeaker plays a steady note. You turn the volume up and touch nothing else. What has changed about what the cone is doing?
Louder means the cone moves FURTHER each time — a bigger amplitude — and it makes that trip exactly as often as before. The frequency is untouched, which is why the note does not change. How far and how often are two separate measurements, and neither one sets the other.
A vibrating object goes to and fro over and over, and two separate things can be measured about that. How often it goes to and fro is its frequency, measured in hertz: 1 Hz is one complete vibration each second. How far it moves each time is its amplitude, measured in millimetres.
Frequency decides the pitch you hear — a higher frequency is a higher note. Amplitude decides the loudness — a bigger movement pushes the air harder and sounds louder. Neither one sets the other. A quiet high note and a loud high note have the same frequency; a quiet high note and a quiet low note have the same amplitude.
At the bench · a signal generator, a loudspeaker and an oscilloscope
Two dials. Two different things happen.
Change a control to begin
The oscilloscope draws how far the cone is from its rest place against time, over a window of 20 milliseconds. Set how often the cone goes to and fro, and set how far it moves each time.
Commit first. You double the frequency of the signal and leave the volume dial alone. What happens to the trace on the screen?
300 Hz
1.0 mm
Frequency
—
complete vibrations each second
In this 20 ms window
—
complete vibrations drawn
How far the cone moves
—
What you would hear
—
Writing it down · the shape of this relationship
Number of vibrations = frequency × time
The triangle
Cover the one you want
N = f × t
f = N ÷ t
t = N ÷ f
Two things side by side means multiply. One thing over another means divide.
N · number of complete vibrations · no unit
f · frequency · Hz
t · time · s
1 Hz is one complete vibration each second, which is why the seconds and the hertz cancel to leave a plain count.
Worked example · one step at a time
A string is filmed and makes 1500 complete vibrations in 5.0 seconds. What is its frequency?
Step 0 of 5
Convert
1500 vibrations stays 1500 · 5.0 s stays 5.0 s
A hertz is counted per second, and the time is already in seconds, so there is nothing to convert.
Formula
frequency = number of vibrations ÷ time
Cover f on the triangle: N sits over t, so you divide.
Insert
frequency = 1500 vibrations ÷ 5.0 s
The count has no unit of its own; the second is what makes it a hertz.
Fine-tune
1500 ÷ 5.0 = 300
Vibrations divided by seconds leaves vibrations each second.
Answer
frequency = 300 Hz
Three hundred hertz, because 1 Hz is one vibration each second.
Worked example · one step at a time
A tuning fork makes 15 000 vibrations in 0.50 minutes. What is its frequency?
Step 0 of 5
Convert
0.50 min × 60 = 30 s
A hertz is counted per second, so the minutes have to become seconds first.
Formula
frequency = number of vibrations ÷ time
Cover f on the triangle: N sits over t, so you divide.
Insert
frequency = 15 000 vibrations ÷ 30 s
The converted time goes in. The 0.50 never does.
Fine-tune
15 000 ÷ 30 = 500
Vibrations divided by seconds leaves vibrations each second.
Answer
frequency = 500 Hz
Divide by 0.50 instead of 30 and the answer comes out 30 000 Hz — sixty times too big.
Your turn · the same five steps
Your note: 300 Hz. How many complete vibrations in 3.0 seconds?
Write all five lines before you check. The frequency is the one your own bench is showing.
The five lines, marked
Convert
the count stays a count · the time is already in seconds
A hertz is counted per second, and the bench times in seconds, so there is nothing to convert.
Formula
N = f × t
Cover N on the triangle: f and t sit side by side, so you multiply.
Insert
N = 300 Hz × 3.0 s
The frequency is the one your slider is set to; the time is in seconds already.
Fine-tune
300 × 3.0 = 900
Vibrations each second, multiplied by seconds, leaves vibrations.
Answer
N = 900 vibrations
A plain count, not a frequency — the seconds have been used up.
The five lines above give 900 complete vibrations in 3.0 s.
An insect wing beats 7200 times in 2.0 minutes. What is its frequency?
This one needs the Convert line to do some work.
The five lines, marked
Convert
2.0 min × 60 = 120 s
A hertz is counted per second, so the minutes have to become seconds first.
Formula
frequency = number of vibrations ÷ time
Cover f on the triangle: N sits over t, so you divide.
Insert
frequency = 7200 beats ÷ 120 s
The converted time goes in. The 2.0 never does.
Fine-tune
7200 ÷ 120 = 60
Beats divided by seconds leaves beats each second.
Answer
frequency = 60 Hz
Divide by 2.0 instead of 120 and the answer comes out 3600 Hz.
The five lines give 60 Hz. The whole question turned on the first one.
Key fact
Frequency is the number of complete vibrations each second, measured in hertz: 1 Hz is one vibration per second. Frequency sets pitch and amplitude sets loudness, and the two are independent — turning a note up does not raise it, and raising it does not make it louder.
Think again
“A loud note is a high note.”
Loud and high are answers to two different questions. High is about how often the source goes to and fro; loud is about how far it moves each time. A double bass played hard is very loud and very low. A recorder played gently is quiet and very high. On the bench above the two dials do not talk to each other at all: move one and the other reading does not budge. The confusion is partly the fault of the English word big, which people use for both, and partly of turning the volume up on a speaker and hearing a thin sound get more present — which is more of it, not more of a higher note.
“A higher note travels faster, which is why you hear it first.”
Every note in the same air travels at the same speed, about 340 m/s, whatever its frequency. It has to: if high notes outran low ones, a brass band a hundred metres away would arrive scrambled, with the piccolo half a bar ahead of the tuba, and it does not. What frequency changes is how many times a second the air is squeezed as the sound goes past, not how quickly the squeezing travels.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A tuning fork of frequency 256 Hz is struck and rings for 4.0 seconds. How many complete vibrations does it make in that time?
Rung 2 · The one that catches people
A brass band plays a hundred metres away. A student says the piccolo reaches you before the tuba, because higher notes travel faster. Which statement is right?
Rung 3 · Explain
A loudspeaker plays a note. Explain what has to change about the cone to make the note higher, and what has to change to make it louder.
Rung 4 · Take it somewhere new
On a guitar, the thin strings sound higher than the thick ones, tightening a string raises its note, and pressing it down at a fret raises it too. Plucking any of them harder does none of those things. Explain what all three of the first changes have in common, and why the fourth is different.
Key note
Frequency is the number of complete vibrations each second, measured in hertz, where 1 Hz is one vibration per second. Frequency sets the pitch: more vibrations each second is a higher note. Amplitude is how far the source moves from its rest place each time, and it sets the loudness. The two are independent, and every frequency of sound travels at the same speed through the same material.
Going further
The numbers behind musical pitch are tidier than they look. Double a frequency and you get the same note one octave higher: 220 Hz, 440 Hz and 880 Hz are all the note A. The concert A that orchestras tune to is fixed at 440 Hz by international agreement, which is a convention rather than a fact about the universe — eighteenth-century instruments were tuned lower, and a few orchestras still play at 415 Hz to suit the music.
Loudness is the awkward one to put a number on, because your ear does not respond in proportion. Doubling the amplitude does not sound twice as loud, and doubling it again adds much less than the first doubling did. That is why loudness is measured on the decibel scale, which is built on multiplying rather than adding: every 10 dB is ten times the energy arriving, and roughly twice as loud to a listener. A whisper sits near 30 dB, a conversation near 60 dB, and a road drill near 100 dB — ten million times the energy of the whisper, and nothing like ten million times as loud.
Before this lesson
Connects to
At GCSE this becomes
- Period as the reciprocal of frequency, the wave equation, the audible range, and reading frequency and amplitude off an oscilloscope trace.
Where to next
Ask Mr Badmus AI
Not sure which dial does which?
The bench is a teaching model. The trace is a graph of the cone's displacement against time over a fixed 20 millisecond window, drawn to one scale throughout: 44 pixels per millisecond across, and 45 pixels per millimetre up. The cone is treated as making one pure frequency, which no real instrument does — a real guitar string or voice puts out a mixture, and that mixture is what makes a violin and a flute at the same pitch sound different. The words quiet, moderate and loud are bands set for this bench and are not decibel measurements. The speed of sound in air is taken as about 340 m/s, which is the value at about 20 degrees Celsius and rises with temperature.
Lesson content © MrBadmusAI.