## Sharps and flats and so on

When last we left the piano keyboard we neglected to discuss the names of the black keys. In fact, we did not even give all of the possible names of the white keys. The distance between one A and the next A on the keyboard is one octave. And out of all of the seven white keys and five black keys within an octave, I assert that all but one of them have three different names. One unfortunate key only has two names, and we’ll find out which one that is and why.

### Modifiers

There are things in music called sharps and flats, and there are also double-sharps and double-flats. These act as “modifiers” for the standard letter names.

Based on what we’ve said so far, you could expect there to be a total of thirty-five different names for the notes in an octave. You know there are seven letter names, A through G, and there are five variations of each using the four modifiers. For example, in the “A family” there’s A, A-sharp, A-flat, A-double-sharp, and A-double-flat. Total names: 5 x 7 = 35.

Also, the keyboard has seven white keys and five black keys in each octave, making twelve altogether. If each key had three names, there would be thirty-six names. But I asserted that one key has only two names, so we actually have only thirty-five names. Total names: 12 x 3 – 1 = 35.

Here’s a wacky stretched-out keyboard where the twelve keys are equally spaced. We’ll use it to help visualize how sharps and flats work.

Wacky stretched piano keyboard

### Sharps and flats

A sharp raises the pitch by a half step, and a flat lowers the pitch by a half step. If you start with C and raise it a half step, you arrive at the black key between C and D. So that key is called C-sharp. But if you start with D and lower it a half step, you also arrive at the black key between C and D. So that same key is also called D-flat. Following this pattern, we can come up with two names for each of the five black keys by sharping the white key to its left or flatting the white key to its right. Starting with the key we just named and moving to the right, we have:

• C-sharp or D-flat
• D-sharp or E-flat
• space with no black key
• F-sharp or G-flat
• G-sharp or A-flat
• A-sharp or B-flat
• another space with no black key

What happens in those two spaces with no black keys? Suppose we’re wondering what E-sharp is (since it didn’t appear in the list above). Raising E by a half step, we arrive at F. So E-sharp is another name for F. Using this technique gives us four names for white keys involving sharps and flats:

• C-flat is the same as B
• B-sharp is the same as C
• F-flat is the same as E
• E-sharp is the same as F

This completes the flats and sharps. Let’s take stock of where we are at this point. Note that the pound-sign-looking symbol means “sharp” and the small-b-looking symbol means “flat”:

Wacky stretched piano keyboard with sharps and flats

We’ve used twenty-one names so far. One way to figure this out is that for each of the seven letters, we added a flatted version and a sharped version; total: 7 x 3 = 21.

Another way: seven names were given to the white keys from the letters A through G directly. Each of the five black keys has two names, and four of the white keys have one extra name in addition to the single-letter name. Our total: 7 (A to G) + [5 x 2] (black keys) + 4 (additional white-key names) = 7 + 10 + 4 = 21.

So far we’ve named all of the twelve keys at least once (and nine of them twice). So from here on in, we’ll be adding additional names to keys that we’ve named already.

### Double-sharps and double-flats

A double-sharp raises a note by two half steps (or, equivalently, one whole step), and a double-flat lowers a note by two half steps (or one whole step).

Let’s consider the note D. If we start at C and go up by two half steps, we travel right through the C-sharp–D-flat key and land on D. Therefore C-double-sharp is the same note as D. We can also start at E and go down by two half steps, landing once again on D. Therefore E-double-flat is also the same note as D. In this way we can assign two additional names to all of the white keys that are between two black keys, and one additional name to the ones that are not:

• C is also D-double-flat
• D is also C-double-sharp and E-double-flat
• E is also D-double-sharp
• F is also G-double-flat
• G is also F-double-sharp and A-double-flat
• A is also G-double-sharp and B-double-flat
• B is also A-double-sharp

Some of the white keys now have three names from the list above. Some of them only have two names in the list above, but notice that they are C, E, F, and B, and these four have already been given an additional name (e.g., C is B-sharp and also D-double-flat). So now all of our white keys have three names. Whichever key gets gypped is going to be one of the black keys.

After the sharps and flats we had accounted for twenty-one names. We added at least one additional name for each white key in the list above. Three of the white keys got two additional names. So we added seven (one for each white key) plus three more (for the white keys that got yet another name). Total: 21 + 7 + 3 = 31. Four to go!

### The home stretch

Let’s finish this off and consider the other four cases. If we start at E and move two keys to the right, we pass over F and end up on the black key to the right of F, which as we know is also called F-sharp. So E-double-sharp is the same key as F-sharp. Similarly, if we start at F and move two keys to the left we pass over E and end up on the black key to the left of E, so F-double-flat is the same key as E-flat. Similarly we can determine that B-double-sharp is the same key as C-sharp, and C-double-flat is the same key as B-flat.

It may seem complicated, but the following picture will clarify everything. The symbol for “double-flat” is two flat-symbols next to each other, and the funky-looking four-dots-with-an-X symbol means “double-sharp”:

Wacky stretched piano keyboard with all names

### Who got gypped

If you look carefully at the picture above, you’ll see that all the keys have three names except one: the key between G and A! This key only has two names: G-sharp and A-flat.

In a sense you could have predicted this by trying to guess which of the keys within an octave could possibly be unique. There are a bunch of white keys that are between two black keys, and there are two similar pairs of white keys without any black key between them. So none of the white keys appear to be unique. There are two groups of black keys, and both groups have a black key on the left and a black key on the right, but there’s only one black key that isn’t either the leftmost or rightmost in its group: the one in the middle of the group of three. So if you had an inkling that sharps and flats had something to do with keys that were next to one another, you might have guessed which key would be the odd man out.

## The piano keyboard

If you were to look straight down at a piano keyboard, you would see something like this:

The more eagle-eyed among you have noticed that there are black keys and white keys. The black keys are set back from the outer edge and are raised. The white keys are shaped like rectangles with cut-outs to allow the black keys to fit in.

Also, you’ll notice that some pairs of white keys have a black key between them, but not all of them. There’s a repeating pattern: a group of two black keys, then a space where there could have been a black key but isn’t, then a group of three black keys, and then another space. The black keys keep alternating in groups of two and three.

The seven letter names A through G are assigned to the white keys as shown in the picture. The seven names repeat right along with the repeating pattern of black keys. The letter names are the names of the notes you’d be playing when you press the white keys. For now we’ll ignore the names of the black keys’ notes.

Suppose you want to find the key corresponding to the note C. Knowing the keyboard, it’s clear that any white key to the left of a group of two black keys must be C.

## Steps

Now I’m going to assert that the distance in pitch between any pair of adjacent keys on the piano keyboard is the same. It is in fact a distance of a half step, also referred to as a semitone.

Now consider the keys nearest to each other labeled C and D. They’re not actually adjacent, because there’s a black key between them. The distance between the C and the black key just to its right is a half step, and so is the distance between that black key and D. So the total distance between the C and D is two half steps. This distance of two half steps is called a whole step or a whole tone.

You could imagine that someone looked at C and D and started off naming the distance between them a whole step, and then suddenly realized there was this extra black key between them.

But wait, you say, what about those white keys with no black key between them, like E and F, and also B and C? Well, they’re only a half step apart, since they’re adjacent and there’s no other key between them. So the white keys on the piano are not all the same distance apart in pitch. Some are a whole step apart, and some are only a half step apart.

Hold on, you say, isn’t that misleading? The white keys look like they’re all the same distance apart physically, but they’re not the same distance apart in pitch. OK, let’s fix that.

Wacky stretched piano keyboard

In this wacky stretched keyboard layout, the distance between the keys actually does correspond to the distance in pitch. And with this information we can begin to understand how musical transposition works.

## Transposing music

To transpose a piece of music is to raise or lower its pitch by some amount. All the notes need to be raised or lowered by the same amount.

Suppose we have a melody with the notes C–E–G and we want to transpose it up by a whole step. We already know that the distance between C and D is a whole step, so it seems that the first note of our transposed melody would be D. Let’s look at our wacky stretched keyboard. To transpose the melody on this keyboard, we slide it up by two keys. Since distance in the wacky keyboard corresponds to distance in pitch, you could imagine just putting your fingers on the C–E–G notes and sliding your hand two keys to the right. The C slides to D and the G slides two keys over to A, but when the E slides two keys to the right it ends up on a black key, the one between F and G. I’ll just tell you that in this context, that note would be called “F-sharp”.

So if we transpose C–E–G up by a whole step, we get D–F-sharp–A. On our wacky stretched keyboard, you transpose up a whole step by simply sliding everything two keys to the right.