Musical intervals

Look at this piano keyboard. We’ve written the note names on the white keys. Do you notice how the note names repeat as you go up and down the keyboard?

keyboard A to A long keys

Have you ever wondered:

question markWhy do the names of notes A, B, C, D, E, F and G repeat?

How can 2 different notes be called the same? Isn’t that a bit confusing?

Why do we have just 7 note names A, B, C, D, E, F and G?

These are all very good questions. The answer is down to mathematics and physics!

Music is made of sounds.

Sound is created when something vibrates and sends waves of energy (vibrations) through the air into our ears. Sound can also travel through other things like water or even a solid wall!

In a brass musical instrument, we buzz our lips to create vibrations, and these resonate inside the instrument, travel out of the bell and through the air to reach peoples’ ears.

sound wave reaching ear

Sound changes depending on how fast or slow an object vibrates. Fast vibrations produce a high sound and slow vibrations produce a low sound. We call this the “pitch” of the sound.

In a brass instrument, if we vibrate our lips fast we produce a high-pitched sound, and if we vibrate our lips slowly we produce a low-pitched sound.

If we vibrate our lips twice as fast we get a note that sounds very like the first note, but higher in pitch. They sound very similar because the vibrations are so closely related.

octaves

For example, look at the wave diagram above. The higher A is vibrating twice as fast as the lower A. So the two As sound like the same note, but one is pitched higher and the other lower. This is why we give these notes the same name A.

This is true of all notes with the same name – if you compare their vibrations, one note will be vibrating 2 times (or 4 times, or 8 times etc) the other one.

We call the difference between one note and another an “interval”. If we start with the note A, then the next note, B, is the second, so we say that the interval between A and B is a “2nd”. The third note is C, so the interval between A and C is a “3rd”, and so on.

G is the seventh (7th). The next note after G is another A, so we say that this higher A is the eighth note or “octave” (written as “8ve”) above the starting note A.

This gives us a musical scale starting on A. We call it the scale of “A minor”. It has 8 intervals:

Scale of A minor

So why are there 7 notes? Well, in fact if you look at a piano keyboard (see below) there are more than 7 notes. There are the 7 white notes: A, B, C, D, E, F and G, but there are also 5 black notes between the white ones.

keyboard A to A with sharps and flats

The black notes are named after the white notes either side of them, but with the word “sharp” or “flat” added. For example, the black note between A and B is called either “A sharp” (written A#) or “B flat” (written Bb). This is because it is a little higher in pitch, or sharper, than A, and a little lower in pitch, or flatter, than B.

Altogether there are 12 different notes that repeat as you travel along the keyboard and get higher in pitch (count them!). But why these 12 notes? Why not more than 12, or less?

Well, there is a very good reason, and it isn’t just because some ancient musician thought 12 was a good number.

It turns out the 12 notes are all related to one another! Let’s see how…

We now know that the note A and the A an octave higher are related because the higher one is vibrating twice as fast as the lower one, but are there any mathematical relationships between the notes in between?

Well, yes there are!

Musical notes sound good together if their vibrations are closely matched, like the low A and the higher A, where the mathematical ratio between them is 2:1 (2 to 1). The next best ratio is 3:2 (3 to 2) where every 3 vibrations of the higher note last as long as 2 vibrations of the lower note, i.e. the higher note is vibrating 1.5 (one and a half) times as fast.

If our lower note is A, and we play a note that vibrates 1.5 times as fast, we get the note E, which is the fifth note in the musical scale of A!

Harmonics A and E a fifth

See also our page on Harmonics.

So it turns out that the notes A and E are mathematically related by the ratio 3:2. What about the all the others? Well now we are going to meet one of the most amazing and beautiful ideas in musical theory: the “Circle of Fifths”.

The Circle of Fifths goes like this:

Let’s start with A and find the note that vibrates 1.5 times as fast as A. As we have seen this is E.
Now start with E and find the note that vibrates 1.5 times as fast as E. This turns out to be B!
Do the same with B, and we get to F# (F sharp, one of the black keys on the piano keyboard).
Note that F# is also called Gb (G flat).

We can continue this, each time starting with the new note we have arrived at, and applying the 3:2 ratio to it.

This is what we get:

Circle of fifths in a line

We have got back to A! And, on the way, we have found all 12 of the notes on the piano keyboard, white and black! And each time we have found the “fifth” note in the musical scale of the previous note! (Terms and conditions apply – see below).

This is called the “Circle of Fifths”:

Circle of Fifths annotated

Does your brain hurt yet? Best go and have a lie down while you absorb the wonder of the Circle of Fifths!

Terms and Conditions (the small print)

Note that in practice, applying the ratio 3:2 to define our 12 notes has limitations, because the circle does not quite get back to exactly the same place it started. You can show this on a calculator. If you multiply the ratio 3:2 or 1.5 by itself 12 times you get 129.75 (going round the circle of fifths from A back to A), but if you multiply 2 by itself 7 times you get 128 (all higher As vibrate 2 times the A below), and 128 does not equal 129.75! That means the A we end up with is vibrating slightly faster than it ought to, and the note is slightly higher or sharper than A!

To get round this problem on a piano, we use something called “equal temperament” which means slightly adjusting all the notes so that they sound ok together. It’s a compromise that has worked well for over 300 years!

For brass players, you can use your ears to make sure the notes sound good together. In practice, it means that any note we play might need to played slightly higher (sharper) or slightly lower (flatter) depending on what musical scale we are using, and what everyone else in the band is playing!

Also, if we keep multiplying by 1.5 we get a big number (we calculated it as 129.75 above). If we played a note vibrating that fast no one would hear it, not even your dog, because it would be so high in pitch! But that’s ok, because whenever the number gets too big we can just halve it to get the same note but an octave lower.

If we do this for the Circle of Fifths, we get the following ratios: Start with A as 1, then E is 1.5. B is 1.5 * 1.5 = 2.25, but divide it by 2, and we get B = 1.125 or 1 and 1/8th. In this way, we can keep all our notes between the starting A and the final A an octave higher

This subject is complicated even for an advanced music student, but if you want to understand more, look up “Musical Tuning” on the internet. E.g. Musical tuning on simple wikipedia.