Why there can be no perfection

Steven Pemberton, CWI Amsterdam

Version: 2021-03-10.

A new job

You've been offered a new job in Rotterdam.

But since you like your house in Amsterdam, and don't fancy looking for a new house, you decide to commute by train.

But that means you will need a bike to get to the train station, and since you are not allowed to take bikes on the train in the rush hour, you'll have to get one for in Rotterdam as well, and you'll also have to find somewhere to park it there.

You know what? Folding bikes are allowed in trains in the rush hour, so you decide to buy one.


First you make a list of what you want from a folding bike. It must be:

There are probably a dozen such properties that you want a folding bike to fulfil.


Well, let’s just take three of them: light, strong, and cheap.

Unfortunately, search as you may, you won’t be able to find a folding bike that matches all three:

Just those three constraints are not satisfiable.

You could find strong, cheap and light, but then it wouldn’t be a bike: being a bike and being foldable are two constraints that are non-negotiable – it must be a bike, and it must be foldable.


So in other words, you will have to relax at least one of your negotiable constraints.

In any case, you're going to have to put up with non-perfection.

The perfect partner

You shouldn’t expect to find a perfect partner either. What might you want in a partner?

There are lots of possible constraints, and they will be different for different people.


Then there are probably a few that you wouldn’t think of mentioning, maybe because they are non-negotiable, like:

and so on, and so on.


The problem is, as you add each constraint, the pool of potential perfect partners gets smaller and smaller.

And then you have to add the constraint that it is someone you will actually get to meet in a social context...

This is why you should prepare yourself for not meeting the perfect partner: you need to relax some of your constraints to find someone who is at least satisfactory.


Which brings us, funnily enough, to music.


You probably know that a note in music is caused by air vibrating, at a different frequency for different notes.

For instance, internationally it has been agreed that the note A above middle C has a frequency of 440Hz.


When we say that a tone is a frequency, what that means is that very rapidly the air is compressed and decompressed: once every 1/440th of a second for the note A. Typically we draw it like this:

A sine wave

The distance between each peak is 1/440th of a second; a peak shows the maximum compression, and a trough the minimum.


If we play two notes simultaneously, it turns out that our ears like combinations of frequencies that are related in certain simple ways, as small ratios of frequencies.

Pythagoras apparently discovered this.

For instance, a note played an octave higher is just twice the frequency, that is to say they have a ratio of 2:1.

So the A one octave higher would be 880Hz.

A A# B C C# D D# E F F# G G# A A# B
440 880

and the A an octave lower would be 220Hz.

We hear a note an octave higher as almost the same note, only 'higher'.


So let's look at some combinations of tones.

What you'll see is that when you play two tones together, their compressions interact: they get added together.

If at a given moment they both have the same compression, you get twice the compression as a result. If one is compressed and the other decompressed, they cancel each other out.

Combination of waves

Noise cancelling

In fact this is how noise-cancelling headphones work: they take the sounds you are hearing and play the opposite compression to into your ear so that they cancel each other out.


When you play two tones together, you also hear the note that is the difference between the two frequencies.

You can best hear this by playing two notes very close to each other, such as 220Hz and 221Hz. You will hear a beat of 1Hz.

If you change it to 220Hz and 222Hz, you hear two beats a second, and so on.

This is the best way to tune an instrument: you play two (nearly) identical notes, and change one until you don't hear the beat any more.


After 2:1, the next-most simple ratio of frequencies is 3:2.

This is classically the relationship between the notes called a “perfect fifth”, between the first note on the major scale, and the fifth (which are seven semitones apart).

So for instance, a fifth from A is the note E, so E should have a frequency of 440 × 3÷2, which is 660Hz.

A A# B C C# D D# E F F# G G# A A# B
440 660 880


Of course, we want to be able to play a perfect fifth from any note, not just A, so the perfect 5th from E, which is B, would have a frequency of 660 × 3÷2, which is 990 Hz.

A A# B C C# D D# E F F# G G# A A# B
440 660 880 990

Since 990 is above 880, we can halve the number to take it down into the octave we are building up, between 440Hz and 880Hz. So the B is at 495Hz.

A A# B C C# D D# E F F# G G# A A# B
440 495 660 880 990


The fifth above that is F# which will have a frequency of 742.5:

A A# B C C# D D# E F F# G G# A
440 495 660 742.5 880


And so we can build up a whole octave based on the premise that from every note we can also play the perfect fifth and step through C#, G#, D#, A#, F, C, G, up to D, which has a frequency of 594.7Hz.

A A# B C C# D D# E F F# G G# A
440 469.9 495 528.6 556.9 594.7 626.5 660 704.8 742.5 792.9 835.3 880

The fifth above that takes us back to A, which will have a frequency of 594.67 × 3÷2, which is ... 892 Hz? But shouldn't A be 880Hz?

Yes it should.


In other words, it is impossible to make an octave of notes where you can play a perfect 5th from every other note.

(By the way, this (almost) ring of 5ths is why we have 12 notes in our octave, and not, say 16, or 20).

This is how the bells of the Westerkerk are tuned, for instance, and this tuning has been used in music for hundreds of years.

They didn't know how to solve the problem of the 'Wolf tone', and so, as the bellringer of Amsterdam says "There are some scales you just can't use".

A solution

It is clear hopefully that the octaves are non-negotiable. Any solution has to have the octave of any note as twice the frequency, otherwise you would get awful dissonance.

So maybe we can loosen our other constraints.

The solution was first discovered by the great Flemish mathematician and scientist, Simon Stevin, around the start of the 17th Century.

We replace our constraint with: the octave should be divided into 12 equal steps, equal in the sense that each semitone has the same frequency ratio with its neighbour. But what is that ratio?


Let’s call the ratio r.

Starting from A, the calculation 440×r should give us A#.

Then A#×r should give us B. This is the same as 440×r×r

Then 440×r×r×r should give us C, and so on all the way up to the next A:

440×r×r×r×r×r×r×r×r×r×r×r×r = 880 (that’s twelve r’s).


Writing this another way:

440×r12 = 880


r12 = 880÷440


r12 = 2

or in other words

r = 12√2

Well, we know how to calculate that: r is just under 1.06.


So if we calculate the resultant octave, it looks like this:

A A# B C C# D D# E F F# G G# A
440 466.2 493.9 523.3 554.4 587.3 622.3 659.3 698.5 740 784 830.6 880

You can see that for E where the ideal 5th from A would be 660, that the difference is very small, less than 1Hz.

In other words, all notes, except the octave, are very slightly out of tune, but the difference is so small that since we want to hear the right tuning, we think it is properly tuned (amusingly, this is a lack of dissonance due to cognitive dissonance).


So, the conclusion is, don’t expect perfection, but if you relax some of your requirements you might just find something that so nearly matches that you can’t tell the difference.