Semitones, sharps and flats

Look at a keyboard. The black keys come in groups of two and three, with gaps between the groups: there is no black key between E and F, and none between B and C. A pianist stops noticing this in the second week, but the question is still there. Why? Why not put a black key after every white one, neat and even?

And a second question, left over from chapter 10. The staff has lines and spaces for the seven white keys: C, D, E, F, G, A, B, and C again. The five black keys got no place of their own. How do you write a note that has no line or space to sit on?

One word answers both puzzles, and the word is step. Start by measuring the keyboard.

The smallest step

Below is one octave, from middle C to the next C up. Press any two keys one after the other, and the card tells you how many steps lie between them. Then drag the "View" slider to the right and watch the keyboard flatten into a ruler.

On the ruler all thirteen keys are the same width, and that is not the artist's whim: it is how the ear hears them. Each key, black or white, is higher than the one before it by the same amount. That amount is called a semitone. Colour doesn't come into it. From C to the black key just to its right is a semitone. From E to F is a semitone too, it is just that both keys happen to be white.

Two semitones make a whole tone. C to D is a whole tone, with a black key sitting between them. E to F is a semitone, with nothing between them at all.

Now look at the arcs under the keys. They mark the steps between neighbouring white keys: W for a whole tone, H for a half step, which is a semitone. Here is the whole secret of the gaps:

C W D W E H F W G W A W B H C

The white keys are unevenly spaced. Wherever two white keys are a whole tone apart, there is room for one more note, and a black key fills it. Wherever they are already a semitone apart, there is nothing to fill: E and F are next-door neighbours as it is. The black key between E and F isn't missing. There is simply no room for one.

Press "All 13 keys in a row", then "White keys only". The first climbs in even steps. The second is the familiar scale of C major, and its steps are not all alike: two big ones, a small one, three big ones, a small one. The ear picks this up at once, even if it can't put a name to it. Why the white keys are laid out in exactly this pattern is something you will find out at the end of the chapter.

Check that you can hear the difference. Two notes play one after the other: is the step a semitone or a whole tone? In chapter 13 these steps get their interval names, the minor and the major second. For now "small" and "bigger" will do.

If you keep mixing them up, play both steps yourself on the widget above: C to C♯ and C to D. A semitone sounds cramped, as if the second note were pressed up against the first. A whole tone has more air in it. It is the ordinary step of a tune.

Sharp, flat, natural

Every black key has two neighbours, a white one on the left and a white one on the right. A black key takes its name from a neighbour. Rather than invent five new letters, music made do with two small signs.

A sharp, ♯, raises a note by a semitone. F♯ is the key a semitone above F, the black one just to its right. A flat, ♭, lowers a note by a semitone. B♭ is the key a semitone below B, the black one just to its left. The third sign, the natural, ♮, moves nothing. It cancels a sharp or a flat that was in force and gives the note back its plain form. Together these signs are called accidentals.

On the staff the sign goes to the left of the note, on the same line or in the same space. The note stays in its letter's place: F♯ sits exactly where F does, and the sign beside it tells you to play the key to the right.

Switch between notes and signs. Nearly every time, a sharp or a flat takes you to a black key. But try E♯. There is no black key to the right of E, so a semitone up lands you straight on F. E♯ is the white key F under another name. In the same way B♯ is C, C♭ is B, and F♭ is E. A sign doesn't say "press a black key". It says "move one semitone".

Now look at the labels under the picture: every black key has two names. The key between F and G is both F♯ and G♭. Raise F by a semitone or lower G by a semitone, and you arrive at the same place.

Find the named black keys on the keyboard. You will get both sharps and flats, and any octave counts.

Now the other way round: a key lights up and you choose its name. The answers come in pairs, as in the labels under the widget: "C♯ = D♭".

One last test of your attention. Every answer here is a white key, even though each note carries a sign.

Double sharps and flats

Now and then you will meet a double sharp, 𝄪, or a double flat, 𝄫. They shift a note by a whole tone at once. F𝄪 is the key G, and B𝄫 is the key A. They turn up in keys with lots of sharps or flats, where a letter already has a sharp and has to be raised once more. You won't need them in this course, but if you ever see a sign like a small × in front of a note, that is what it is.

One key, two names

If F♯ and G♭ sound the same and are played with the same finger, why have two names? Couldn't everyone agree to write, say, sharps all the time?

Try it. Below is the F major scale: the same "W W H W W W H" ladder as C major on the white keys, only starting from F. For now it is written with sharps only.

Look at the staff. The fourth note of the scale is the black key you could call A♯ or B♭. Write it as a sharp, and two notes in a row land on the A space: A and A♯. B gets nothing at all. A scale like that is awkward to read. The notes should climb the staff like a staircase, one step at a time, and here one step has gone missing and another has doubled up.

Switch the spelling to "each letter once". Now the scale has all seven letters: F, G, A, B♭, C, D, E. Every note sits in its own place on the staff, the staircase is even, and only one sign is needed.

Here is the rule that settles almost every argument about the two names: a scale uses each of the seven letters exactly once. If the letter G is already taken, the black key next to it is F♯. If A is taken, the key above A is B♭. Try other scales: D major, E♭ major, B major. Compare the three spellings for each.

Two spellings of the same key are called enharmonic equivalents. On the keyboard F♯ and G♭ are indistinguishable. On paper the choice of name tells you what job the note is doing in the music: which step of the scale it is and where it is heading.

You have probably heard the word key, as in "a song in D major". It is the scale the music is built around, together with its main note. What major is, and why one note of a scale is in charge, is the subject of chapter 14. For writing music down, one thing matters right now: the key chooses the names. In D major the key between F and G is written F♯, in D♭ major it is G♭.

The D major scale: D, E, ?, G, A, B, C♯, D. The third note is the black key between F and G. How is it written?

F♯ G♭ It doesn't matter, they sound the same

The scale already has a G: it is the fourth note. It has no F. For each letter to appear once, the third note has to be F♯. The two sound the same, but G♭ would put two notes on one line and leave the F space empty.

Your turn. A scale plays, its keys are marked on the keyboard and one of them is highlighted. Choose the name that key goes by in this scale. The staff gives you a hint: there is a rest where the highlighted note should be, and you can see the notes on either side.

A sign lasts to the bar line

Picture a tune where F♯ comes eight times in a row. Eight identical signs, one before every note, are extra work and extra clutter on the page. So the convention is this: a sign written by a note holds for that note until the end of the bar, the stretch between bar lines you know from chapter 7. Every later note on the same line or in the same space in that bar takes the same sign, even if none is written. The bar line clears everything, and in the next bar the note is back to plain again.

In the example the second note has a sharp. The fourth note is also an F, with no sign, but it sounds as F♯ because the bar isn't over yet. In the second bar the F sounds without a sharp, even though there is no natural or any other sign. Press "Listen" and follow the labels.

Now play with it. Put a natural on the fourth note and it becomes a plain F. Put a sharp on the fifth note, the G, and listen to the last note of the bar. Remove the bar line: the two bars merge into one, and the sharp from the beginning carries through to the end.

Two fine points. First, a sign applies only at the same pitch, on the same line or space. An F♯ in F4 does not make the F an octave higher sharp: that one needs its own sign. Second, printed music is full of reminders, called courtesy or cautionary accidentals: a natural or a sharp in brackets, or simply a redundant sign in the next bar, so that the player doesn't slip. They don't change the rule. They are just a safety net.

Read notes with accidentals. Answer with the keys: on the screen, on the computer keyboard or on a MIDI keyboard.

Key signatures

The bar rule saves signs within a bar. But if a whole piece is in D major, F♯ and C♯ will turn up in every bar, and you would be writing signs endlessly again. So there is a second shortcut, shorter still.

The sharps or flats of the key are written once at the start of every line, right after the treble clef. This is the key signature. It applies to every note with that letter, in any octave, for the whole line and the whole piece, until it is changed. A sharp on the F line at the start of a line means: every F here is played as F♯, unless a natural stands before it.

A tune in D major: two sharps at the start of the line, F and C. The notes themselves carry no signs, but what you hear is F♯ and C♯.

A key signature tells you where you are: at a glance, a musician knows what key the piece is in. The twelve major keys fit neatly round a circle.

This is the circle of fifths. C major is at the top, with no sharps or flats. One step clockwise is a fifth up, and the key gains a sharp: G major has one, D major two, A major three. One step anticlockwise is a fifth down and one more flat: F major has one, B♭ major two. At the bottom of the circle the two paths meet: F♯ major with six sharps and G♭ major with six flats are the same scale on the same keys, written two ways.

Switch "Accidentals" to "on every note" and count the signs on the staff. B major has five sharps in its key signature, once per line. Without them you would have to write the same five sharps in almost every bar, and a long piece would run to hundreds. A key signature is a shorthand, and a very good bargain.

The sharps and flats in a key signature always come in the same order. Sharps: F, C, G, D, A, E, B. Flats, the other way round: B, E, A, D, G, C, F. A signature with one sharp is always F♯, with two it is F♯ and C♯. So you don't need to learn each key's signature separately: know how many there are, and the order tells you which. Notice, too, that the order of sharps is itself a chain of fifths: a fifth up from F is C, a fifth up from C is G. Fifths come back later in this chapter, and very usefully.

Read notes with a key signature. There are no signs by the notes themselves here, so look at the start of the line.

Reading with accidentals is a skill of its own, and it grows best with short daily sessions. The "Reading notes" trainer has the levels "With sharps and flats" and "In G major", and the "Notes on the keyboard" trainer has a "Black keys" level.

Why there are twelve notes

Why are there exactly twelve notes?

Chapter 1 gave you half the answer. After the octave, the sweetest-sounding interval is the fifth, a frequency ratio of 3 : 2. Climb in fifths, and after twelve steps you are almost back at the note you started from, seven octaves higher. Almost: you overshoot by 23.46 cents, the Pythagorean comma. Twelve is the first number at which the circle of fifths comes this close to closing.

The other half of the answer is about the names and colours of the keys. Play the fifths again, but this time start from F and keep an eye on the keyboard.

Keep pressing "+ a fifth". F, C, G, D, A, E, B: seven notes, six fifths, and that is every white key exactly once. The white keys are seven notes linked by a chain of fifths. The next five fifths land on the black keys: F♯, C♯, G♯, D♯, A♯. The letters have started a second round, and each one has picked up a sharp.

The five black keys are a chain of fifths as well, and five notes linked by fifths make a pentatonic scale, the scale behind many folk tunes all over the world. That is why you can play something that sounds like music on the black keys alone, almost without trying.

The twelfth fifth brings you to E♯. In pure tuning it is not F: the figure under the name shows it has drifted 23.46 cents higher. Here the names are more honest than the keys. E♯ and F are different letters, and in a world of pure fifths they really are different notes. Press "F and E♯ together" and listen to them clash: about five beats a second.

Now switch the tuning to "equal temperament" and go through the fifths again. The drift under the names disappears, and E♯ lands exactly on F. Equal temperament is an agreement to make every fifth 1.96 cents narrower than pure, so that the circle closes. Then all twelve steps of the octave are the same size: every semitone is a frequency ratio of $\sqrt[12]{2} \approx 1.0595$, or exactly 100 cents. Only in this tuning are E♯ and F, or F♯ and G♭, the same sound in every key. The enharmonic equivalents this chapter started with are a gift of equal temperament.

What equal temperament costs

Narrowing a fifth by two cents is barely noticeable. But the comma hasn't gone anywhere. It has been shared out among all the intervals, and some got a bigger share than others. Hear it for yourself. Below are two steady notes with a rich sound, the lower one on middle C. Press "Sound" and switch the tuning.

The pure fifth sounds smooth: the lower note's overtone at three times its frequency and the upper note's overtone at twice its frequency coincide exactly, at 784.9 Hz. The tempered fifth gives a slow wave of loudness, less than one beat a second. It is hard to notice unless you listen for it.

Now choose the major third, C and E, a ratio of 5 : 4. The tempered third is 13.7 cents wider than the pure one, seven times the error of the fifth, and the beats run fast: about ten a second on middle C. The sound turns rough and restless. The octave, though, stays pure in equal temperament: check, and you will find no beats.

That is the price. Equal temperament puts almost every interval except the octave slightly out of tune, but it does so equally in every key. C major sounds exactly like F♯ major, only higher. So you can move freely from key to key and shift a song to wherever it is most comfortable to play.

A pianist plays E♯ and then F straight after it on a well-tuned modern grand piano. What does the listener hear?

The same note twice Two different notes, E♯ slightly higher Two different notes, E♯ slightly lower

The piano is tuned in equal temperament, and on it E♯ is the same key as F. You can hear the 23.46-cent difference only in pure tuning, where E♯ is reached by twelve pure fifths: on a synth tuned in pure fifths, for example.

What's next

Every key now has a name, and you know how to choose it. You can hear the difference between a semitone and a whole tone, and you can see why there is no black key between E and F. You read sharps, flats and naturals, and you know that a sign lasts to the bar line and a key signature to the end of the piece. And you have an answer to the third big question: the twelve notes are twelve fifths that almost close the circle, and equal temperament closes it, giving one key two names.

So far all our notes have lived in the treble clef, from middle C upwards. But the left hand plays lower, and down there the treble clef sprouts clusters of ledger lines. Low notes need a clef and landmarks of their own: the bass clef. That is chapter 12.

The Notebook has learned about keys. Choose the key of your tune, and the Notebook writes the black keys with the right signs, F♯ in D major and B♭ in F major, and puts the key signature at the start of the line. In the next chapter the Notebook gets a second track, the bass, with a bass clef of its own.