Behind every note there is a number: the count of vibrations per second. Behind every interval there is a ratio of two such numbers. This page shows how vibrations turn into pitches.
New to intervals and scales? This page explains the physical foundation. How concrete scales are built from it, we show step by step. Scales explained →
The waves are animated – red is always what reaches the ear, grey the individual vibrations behind it.
A tone is produced when something vibrates: a guitar string, a column of air in a flute, a drum skin. The motion alternately compresses the air and pulls it apart. This pressure wave reaches our ear, and we hear a tone.
How often the vibration goes back and forth per second is called frequency. The unit is hertz, Hz for short. 220 Hz means: 220 times back and forth in one second. The faster the vibration, the higher the tone. How far the string swings only changes the loudness, not the pitch.
A note head in notation therefore says nothing more than which frequency should be played.
If you double the frequency, the same note sounds one octave higher – and carries the same name. 220 Hz is an a, 440 Hz is an a, 880 Hz is an a. This is the most important relationship in the whole notation system, and it holds in every musical culture in the world.
The reason can be seen in the animation: when doubled, every second vibration of the higher note lines up exactly with one of the lower. The two waves never drift apart, they mesh cleanly. To our hearing they almost merge into a single sound.
Because it is always a doubling and not a fixed amount in hertz, the distance between two notes keeps growing towards the top: from A to a it is 110 Hz, from a¹ to a² already 440 Hz – and yet both sound equally far apart.
The relationships between the notes are fixed by physics. Where the whole series sits in the frequency range is not. Some note therefore has to be given a fixed value by agreement, otherwise two orchestras could not play together.
This reference note is a¹ at 440 Hz, the concert pitch. It has been internationally recommended since 1939 and written down as a standard since 1975. All other notes are derived from it – including the values in the table further below.
But it really is only an agreement: baroque ensembles often tune to 415 Hz, many orchestras play a little higher at 442 or 443 Hz because the sound then feels brighter. The music stays the same, it just sits a bit higher or lower overall.
A real string never vibrates only as a whole. At the same time its halves, its thirds, its quarters vibrate too. Above the played root, further notes therefore always sound quietly along: the overtones. Their frequencies are whole multiples of the root – double, triple, quadruple.
Which overtones sound along and how strongly makes up the timbre. That is why a violin and a clarinet sound different even though the same note stands in the score. The waveform leaves the round shape of the pure vibration, but the pitch stays unchanged.
For music theory the decisive point is: the intervals are already contained in this series. Octave, fifth, fourth and major third appear as the first steps – before anyone has written down a scale. The notation system is not an arbitrary choice, it follows what a string does on its own.
| No. | Frequency | Multiple | Note | Interval from the root |
|---|
When two notes sound at the same time, their waves add up. Whether the result feels calm or restless depends on how quickly the shared pattern repeats. For the fifth, three vibrations of the upper note come for every two of the lower: ratio 3:2. After this short stretch both are back at the start, the pattern is easy to grasp, the blend feels stable.
The larger the numbers in the ratio, the longer it takes until it repeats – and the more tense the interval sounds. The tritone needs 32 vibrations; its picture looks restless to match. This is the physical core of consonance and dissonance.
| Interval | Ratio | Above a = 220 Hz | Repeats after | Effect |
|---|
All twelve semitones across three octaves, starting from the concert pitch a¹ = 440 Hz. From row to row the frequency is multiplied by the same factor, roughly 1.0595. That is why the gaps in hertz are small at the bottom and large at the top, even though it is a semitone everywhere. After twelve steps the frequency is exactly doubled: the octave.