Ever wondered where we got the 8 note scale?

And how the fret positions of a guitar or banjo were invented?

This is how the whole thing evolved from primitive times up to modern instruments. I'll try to keep the math to a minimum, but you might want to grab a calculator now.

Simple Harmonics

I guess most people know the fundamentals of the ways in which a string vibrates but just for the record, lets start at the beginning.
When a string is plucked, it generally vibrates along its entire length. However, you can 'stop' a string at the mid-point and get a higher note, which still sounds pretty pleasant when combined with the first.
More experimenting shows that stopping at the one-third, quarter and one-fifth also works .

These pleasant sounding combinations of 1:2 1:3 1:4 and 1:5 are known as simple harmonics.

The basic note is called the Fundamental (or first harmonic) the half-length is the second harmonic, the one-third length gives the third harmonic then the 4th = 1/4 and 5th = 1/5.
Trying to stretch it beyond that, up to 6 and 7 doesn't really work, all you get is a squeaky noise.

Oh, and before we go any further. I want to point out something that had me confused for years. A fith harmonic is NOT the same thing as a 'fifth' interval in music - nothing whatever to do with it.

For practical purposes these notes are an awful long way apart. Wouldn't it be nice if we could find a way to sort of bring them closer together?
Well, it isn't so difficult.

The Pentatonic scale

Start with a string, say 100cm long. You can get the 3rd harmonic by plucking an identical but shorter string 33.33cm long. But, here's the clever bit, you can get a note that also 'harmonises' with that one by plucking a string that is twice as long (66.67cm). This is known as a subharmonic. In this case, the second subharmonic. ( keep in mind that shortening the string by a simple fraction gives harmonics, making it longer by a simple multiple gives subharmonics)

That's a neat trick, instead of producing a high note harmonic way above the basic note (the fundamental) we can produce a pleasant harmony with a note just a little above it.

Let's try for another The fourth harmonic (from a string 25cm) gives a second subharmonic at 50cm but we already have that, it's just the second harmonic of the original note. But the fourth also gives us its third subharmonic at three times the length at 75 cm - that's a new one.
You can get the 5th harmonic by plucking a 20cm string. So that gives us more notes at 60 and 80 cm, We ignore the one at 40cm because it is higher than the second harmonic and what we are aiming for is a scale running from the fundamental up to the second harmonic (50cm)

Here are the string lengths we just discovered.

Now we have something that could be called a scale, notes running in sequence from the fundamental note up to its second harmonic - what we call 'the octave' these days. And it works too. This five interval scale is called a 'pentatonic' scale and is the basis of most folk music (and much of the Beatles best stuff too!) Any note of this scale sounds pleasant when played with any one or combination of the others. You simply can't go wrong!

These string lengths are so important that they are marked on a guitar or banjo by 'dots' on the fretboard. They occur at the 4th, 5th,7th,9th and 12th fret (the 4th isn't marked - possibly because it is right next to the 5th)

The Diatonic scale

Can we find any more?
The notes we just found are all simple multiply/divide sums of the numbers 2,3,4 and 5. So these vaules we have are just fractions made by putting together the numbers 2,3,4 and 5 in some combination. We have used all the simple ones but there are a couple more we can squeeze in by using 3*3 and 5*3. i.e the third and fifth harmonics of the 3rd harmonic. In fact, we use the 8th subharmonic of these two rather high notes (NB 8=2*4 which are two of our allowed numbers).

100 * 8/(3*3) = 88.889cm
and
100 * 8/(3*5) =53.333cm

These two new notes are getting a bit 'iffy' as far as harmonising with the others is concerned but they are still pleasant sounding enough to be useful. And guess what, they fill in the missing notes of the scale. We now have a full 8 note (octave or diatonic) scale.

• 1 do
• 8/9 re
• 4/5 me
• 3/4 fa
• 2/3 so
• 3/5 la
• 8/15 te
• 1/2 do

This is what our basic one-octave string instrument looks like.

The Chromatic scale

Now this is all very well. But all we have so far is an instrument with 8 fret positions that can play a simple octave scale in one key. We want to be able to play in more than one key.

What happens if we try to play a scale starting on our second note (re)? Lets have a look at where the fret positions should be for a string that starts off with our note re. To make the calculations easier, start with a string that is 360mm long so that we know the fret positions for that string are at 320,288,270 etc as in the table above.

Take the note 're' - the string length for that note is 320mm. So if we take that as the fundamental for a new scale, what fret positions do we get? It's just 320 * 360/360 , 320 * 320/360 , 320 * 288/360 etc

• new do 320
• new re 284.44
• new me 256
• new fa 240
• new so 213.33
• new la 192
• - stop here, the next note is higher than the top note of the original scale so that's enough.

Now look at those numbers, some of them are quite close to the original do,re,me lengths

• new do is re
• new re sort of matches me
• new me is NEW
• new fa is the same as so
• new so is close to la
• new la is the same as te

Is this just a freak occurrence? Try a different start point. Let's start form 'fa' at 270mm. So me becomes our new fundamental. Now lets work out the string lengths for a scale starting here.

• second new do 270 * 1
• second new re 270 * 320/360 = 240
• second new me 270 * 288/360 = 216
• second new fa 270 * 270/360 = 202.5
• second new so 270 * 240/360 = 180
  - stop here, don't want to go higher than the top note of the original scale.

What have we got?

• second new re is the same as new fa is the same as so.
• second new me is the same as la - nearly the same as new so
• second new fa is NEW again.

It looks like there is a pattern of sorts, it isn't a perfect match. Maybe a picture will help You can see that a lot of the frets match up, those that don't are often close. Using the other scales has also filled in where there were obvious gaps in the original sequence. Notice that the first and third frets are filled in using the second subharmonic of other scale notes.

Notice something else? It isn't very neat is it? The fret positions don't look anything like those found on a real instrument. The gaps are irregular. What's going on?

Well, what we have here is a 'just' or 'true' chromatic scale. If you were playing music on a single string, this would be the fret pattern to use. It will give a very pure note sequence in any scale. But when you start adding strings to the instrument with different fundamental notes some way apart, things start to go wrong.

The tempered scale

We have all been brought up on the idea of a repeating musical sequence of 12 notes
C C# D D# E F F# G G# A A# B C' C#' D' etc...
And we have the idea that we can play a scale starting anywhere we like, providing it ends on the 'same' note an octave above. But in fact, this ideal does not fit in with harmonic reality.

Let me explain:
There are 12 notes in the sequence.
So if we start at C and count up 12 notes and we repeat that 7 times (the reason for choosing seven will come clear later), we should come to another C, 7 octaves above the starting note.
We should also be able to get to that same note by counting 7 notes 12 times, because 12 * 7 = 7 * 12.

Let's try that with real strings.
Start with the note C and play G, which is seven notes above. That means we are playing 'do' and 'so'.
Now 'so' is one of the harmonics (the second subharmonic of the third harmonic - check back at the do,re,me table above) - that's why it makes a pleasant sounding 'chord'
So we are playing first the fundamenal (say at 360mm again) and 'so' at 360mm *2/3. Now here's the trick. We do that again, starting from 'so' this time as the 'do' of a new scale, so you see we are going up 7 notes at a time just as in the calculation above. If we do that 12 times we get the sum:

360mm * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3 * 2/3
Which is 360mm * 0.0077073

Now try the other way, 7 steps of 12 notes.
12 notes is just 'do' to 'do' or 360 to 360 *1/2, do that 7 times and we get:

360mm * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2
That is 360mm * 0.0078125 - not the same number at all! In fact, the notes are about 1/4 tone apart, enough to produce a major dissonance.

So, the conclusion is: - In a strictly harmonic note sequence, the notes gradually get 'out of step' as you move away fron the starting octave. Chords formed from different octaves (different strings) don't match and you start hearing the errors quite quickly in the form of 'beats'.

This is a major problem. It's not so bad for instruments like the violin or cello, but for a piano or harpsichord, it's a disaster.

It turned out that the answer was to throw out the harmonic scales and replace them by something that followed roughly the same pattern but that was more even in the way that it treated the intervals.
Lets see if we can work it out.
We want a sequence of 12 steps that runs from L to L*1/2 i.e from the fundamental to the second harmonic length. It isn't an even division because we can see that the intervals are closer together as you go up. That sort of thing usually suggests a log or 'power' scale.

Lets try something really simple - L / 2n where n runs from 0 (which just gives us L) up to 1, (which gives L / 2) in 12 steps.