## Tuning as an approximation

Previously, we said that the notes written on the staff were approximations of the pitches of the harmonics. But what kind of sense does that make?

## Intervals and ratios

Let’s look at the picture we had indicating harmonics, their frequency ratios and the equivalent notes on the staff:

Notes with frequencies multiplied by various numbers

This picture is claiming that when you play note 1 (a.k.a. C2) on the piano, the strings for that note vibrate at some frequency (as it happens, roughly 65.4

## Overtones and harmonics

We have discussed the circle of fifths as a way in which the various pitches, major keys, and minor keys can all be arranged in a circle. But certainly there must be something else special about the interval of a perfect fifth to make it a worthy basis for all of this discussion. Sure,

## Understanding key signatures

Once you have the major keys arranged in the circle of fifths, and you know why they only go around the circle seven steps either way, so they have up to seven flats or seven sharps, it’s only a small leap to understanding key signatures.

The basic concept is that if your music is in a major key, most of the notes are going to be the notes in the major scale. Suppose, for example, you have a piece in the key of D-major. Now, the D-major scale has two sharped notes, F-sharp and C-sharp. So your piece is going to have a lot of F-sharps and C-sharps and not so many F-naturals and C-naturals. So wouldn’t it be great to remove some excess clutter from your music? Why not just remind the musician at the beginning that all Fs and Cs are going to be sharped unless otherwise specified? Then you don’t have to sprinkle all those extra sharp-symbols all over your music. You’d save ink, help the environment, and contribute to saving our planet by decreasing your carbon footprint.

If you’re not a musician, this probably sounds like a huge nuisance. It would be hard enough to read those notes off the page, and now you have to keep remembering extra sharps (or flats) that aren’t written! Nevertheless, this is what musicians do. At the left of every staff there’s a key signature which reminds you of these “default” accidentals that are to be assumed unless otherwise specified.

So without further ado, here’s a circle of fifths with the key signatures included. The names of the major keys are given, and also the names of the minor keys (which we will discuss momentarily):

The circle of fifths, with major and minor keys indicated

## Some keys are more equal than others

You may wonder: what should you do

## Notes on the staff

The time has finally come to show some notes on a musical staff.

The scheme is rather straightforward. The notes on the staff follow the white keys on the piano keyboard. Each successive white key goes up half of a staff line. A note can be on a staff line or in the space between staff lines. The range of notes is not limited by the height of the staff; notes can go right off the top or bottom of the staff. In these cases, you draw in little pieces of the staff lines (ledger lines) that would have been there if only the staff were taller. A picture should make this perfectly clear:

The notes on the staff

But how do you know what notes they actually are? You could guess that higher notes on the staff represent higher pitches, but unless I told you a letter name of one of the notes, you’d still be lost. Once you knew one note’s name, however, you could figure out the rest easily enough by just going through the letters from A to G until you reached your note. For example, if I told you that the note straddling the bottom line of the staff was E, and asked you what note straddles the top line of the staff, you could count up the lines and spaces: E, F (space between the bottom two lines), G (second line from the bottom), A, B, C, D, E, F. So the note straddling the top line of the staff would be F.

By the way, generally the lines are counted from bottom to top, so if someone says “there was a note on the first line of the staff” they mean that the note was on the bottom line. I can imagine jazz musicians using this terminology all the time: “So, dig this, man, I went to this jam session, and they gave me the chart, and there was, like, this note on the first line of the staff.”

So is the bottom line’s note really E? It depends. This is where clefs come into play. Essentially, a clef is a goofy-looking symbol at the left of the staff, and some key feature of the clef’s symbol tells you what one of the notes is. Here is a picture of some clefs:

Musical clefs and their names

On the left we have the treble clef, also referred to as a G-clef.

## More about the Circle of Fifths

Now we can explain why the circle of fifths looks the way it does. Recall that we had an ambiguous area from 5 o’clock to 7 o’clock:

A clock with the circle of fifths

We’re going to examine the circle of fifths

## A review

Previously we discussed a clock in which the numbers going clockwise from the 12 o’clock position went up by sevens. Of course, we want to keep the numbers between 1 and 12, so when the number gets larger than 12, we subtract the 12 out. We start with the number 7 at the 1 o’clock position. When we add 7 to that, we get 14, but that’s bigger than 12, so we subtract 12 from 14 and the next number becomes 2. Notice that adding 7 and subtracting 12 is the same thing as subtracting 5, so we could also have explained what we’re doing by saying that for each step around the clock face, we either add 7 or subtract

## Up to six hours at a time

Suppose I have a clock and it reads 12 o’clock. If I set the clock forward one hour, then it will say 1 o’clock. If I do it again, the clock

The