Intervals

In chapter 0 you played "Frère Jacques" from different notes, and each time you knew the tune at once. Yet every single note had changed. So your memory does not keep a tune as a list of notes. It keeps something else, something that stays the same when the whole tune moves.

That is this chapter's wall. To work out a tune by ear, play it from another note or write one of your own, you need to hear and name what the ear actually remembers: the distances between the notes.

A tune made of steps

Here is "Ah! vous dirai-je, maman", a French song from the 18th century. Mozart wrote twelve variations on it, and in English it is "Twinkle, Twinkle, Little Star". Above the keyboard are its steps: how many keys, counting the black ones, the tune goes up (plus) or down (minus) from one note to the next. Press any key and the tune starts from it.

Whichever key you start on, the chain "0, +7, 0, +2, 0, −2…" stays the same, and so does the tune. Now pick one step and move it by a semitone. Everything after it shifts by a semitone, and all the other steps stay as they were. Even so, the tune has gone wrong: the ear hears a single wrong step straight away.

Try the same with "Ode to Joy". Almost all its steps are small: 0, +1, +2, 0, −2… Beethoven built the theme almost entirely from neighbouring notes, which is why it is so easy to play with one finger.

Counting semitones

The distance in pitch between two notes is called an interval. The handiest way to measure it is in semitones, steps to the very next key, white or black, as in chapter 11. From C to G is seven semitones: C♯, D, D♯, E, F, F♯, G.

The ruler below does the counting for you. Press one key, then another: on the screen, on your computer keyboard or on a MIDI keyboard. Every key between them gets a step number, and the name of the interval appears above. The "narrower" and "wider" buttons move the upper note by a semitone.

Twelve semitones give twelve intervals within the octave. Each has a full name and a short one that musicians use:

Semitones	Short	Name	Up from C
0	P1	unison	C
1	m2	minor second	C♯
2	M2	major second	D
3	m3	minor third	E♭
4	M3	major third	E
5	P4	perfect fourth	F
6	TT	tritone	F♯
7	P5	perfect fifth	G
8	m6	minor sixth	A♭
9	M6	major sixth	A
10	m7	minor seventh	B♭
11	M7	major seventh	B
12	P8	octave	the next C up

Where the names come from

The names count letter names, not semitones. A third is C, D, E: three letters, counting both ends. A fifth is C, D, E, F, G. "Octave" comes from the Latin for "eighth". That is why C–E and C–E♭ are both thirds: both span three letter names. The number of semitones differs, and a word tells them apart: major (4 semitones) and minor (3). Seconds, thirds, sixths and sevenths each come in two sizes. Fourths, fifths and octaves usually come in one, and they are called perfect. "Perfect" is only a name: it says nothing about tuning, and you will soon meet the keyboard's perfect fifth that is not quite pure.

So on the keys you count an interval in two ways at once. The number (second, third) comes from the white keys, that is, from the letter names. The word "major" or "minor" comes from the semitones. The ruler shows both counts when both notes are on white keys. Play E–F and B–C: these are minor seconds, only a semitone, even though both notes are white.

Start with two intervals that sound very different. The octave sounds like "the same note, only higher": you know that from chapter 0. The fifth is smooth and a little hollow too, but it is a different sound: the opening of "Twinkle".

If it isn't working, listen to the second note: in an octave it is "the same one", in a fifth it is not. Now add the major third. It sounds brighter and warmer than the fifth, like a major chord with one note missing.

And the other way round: you are given the interval, and you find it on the keyboard. Counting semitones helps here, but in time your hand remembers the shape: a fifth up from a white key is almost always four white keys higher.

Go up three semitones from E. Which note do you land on, and what is the interval?

G, a minor third F♯, a major second G♯, a major third

E → F (a semitone, there is no black key between them) → F♯ → G: three semitones. Three letter names, E, F and G, make it a third, and three semitones make it a minor one. F♯ is only two semitones up, and G♯ is four.

Smooth and rough

On the ruler, play C and G together, and then C and C♯. The first sounds calm and blended. The second is prickly, as if the two sounds were rubbing against each other. Intervals that sound blended and stable are called consonances, and the tense, rough ones are dissonances. A dissonance is not a mistake. Without it music stands still: tension and release are what drive it forward. But where does the difference come from?

Beats

Take two notes that almost match. Turn the sound on and move the slider. One note stays at 440 Hz, the other is slightly higher.

With a difference of 1–3 Hz the loudness sways slowly: the sound swells, then fades. These are beats, which you met in chapter 1. The two waves keep falling into step and adding up, then drifting apart and cancelling out. However many hertz apart the notes are, that is how many beats you hear each second:

$$f_{\text{beats}} = |f_1 - f_2|$$

Slow beats do no harm. Instruments are tuned by them: a tuner turns the pin until the swaying stops. Towards 15–30 beats a second, though, the trembling blurs into roughness and the sound turns muddy. A semitone near A is a difference of about 26 Hz, right in that zone. That is where the minor second gets its sting.

Frequency ratios

Remember the monochord from chapter 1: half the string gives the octave, frequencies in the ratio 2:1, and two thirds give the fifth, 3:2. The simpler the fraction, the more overtones of the two notes coincide. In an octave every overtone of the upper note coincides with an overtone of the lower one. In a fifth every second overtone of the upper note meets every third overtone of the lower. In a minor second, 16:15, the first overtones do not coincide at all, but many of them sit close together and beat.

The widget below shows eight overtones of each note: the lower note at the bottom, the upper note at the top. A solid green line means two overtones coincide. A dashed line means they sit close and beat. Pick an interval and listen.

Now switch from "Pure" to "As on the keyboard". On a keyboard all semitones are equal: each step multiplies the frequency by $\sqrt[12]{2} \approx 1.0595$. This tuning, equal temperament, came up in chapter 1 and chapter 11. Its fractions almost never come out exact. The octave does: exactly 2:1. The keyboard's fifth is 2 cents narrower than a pure one: the G above middle C sounds at 392.00 Hz instead of 392.44. You will hardly notice: the overtones that coincided now drift less than 1 Hz apart, one slow beat a second. The major third is another matter. It is 14 cents wider than pure, and the overtones that coincided in a pure third beat about 10 times a second on the keyboard. Press "Compare" on the third: the pure one sounds still and softer, the keyboard one has a slight shimmer.

The tritone

Six semitones, exactly half an octave, make the tritone, "three tones". In pure tuning it is usually written as the fraction 45:32, and among the first eight overtones of its two notes nothing coincides at all. The tritone sounds tense and unsettled, like a question left unanswered.

The tritone has a nickname, "diabolus in musica". It is often credited to the medieval Church, which supposedly banned the interval. No one has found any formal ban. The phrase "mi contra fa est diabolus in musica" first appears in print in 1702: the German organist and theorist Andreas Werckmeister quotes it as a rule handed down from older authorities, and applies it not only to the tritone but also to F clashing with F♯. Medieval theorists really did advise steering a tune round the tritone because it sounded harsh, but that was a rule of composition, not a church ban. In Danse macabre (1874) Saint-Saëns deliberately has the solo violinist tune the top string down from E to E♭, so that the open strings A and E♭ give a tritone. This is the fiddle of Death, who calls the dead to dance at midnight.

Now the intervals together rather than one note after another. Consonances blend into a single sound, dissonances sound like two notes that don't get on.

The fifth on a keyboard sounds slightly different from a pure 3:2 fifth. Why?

On a keyboard all semitones are equal, and seven of them give a ratio of 1.4983 instead of 1.5 A keyboard goes out of tune over time, while a pure fifth is tuned afresh A pure fifth is seven and a half semitones

Equal temperament divides the octave into twelve identical steps, and $2^{7/12} \approx 1.4983$, a little less than 1.5. The difference is only 2 cents, so the keyboard's fifth is almost impossible to tell from a pure one. In return you can play in any key: every fifth on the keyboard is the same.

Together and apart, up and down

You can play an interval in two ways. If the notes sound one after the other, it is a melodic interval, the way it lives in a tune. If they sound together, it is a harmonic interval, the way it lives in chords. It is the same interval, but the two are heard differently: the ear takes a melodic interval as movement and a harmonic one as a colour. Try both on the ruler with the "up", "down" and "together" buttons.

A melodic interval also has a direction. A fifth up and a fifth down are the same distance, but they sound different, and you have to learn to recognise each of them separately. The start of "Twinkle" is a fifth up. A fifth down is the first step of the famous Minuet in G from Anna Magdalena Bach's notebook: D → G.

In chapter 10 you read tunes on the staff by their shape: a step, a skip, a leap. Now they have names. A step is a second, minor or major. A skip is a third. A leap is a fourth or wider. "Ode to Joy" moves almost only by steps, and "Twinkle" opens with a leap of a fifth. Tunes that are easy to remember usually mix the two: a leap catches the ear, and the steps after it bring the tune back down to earth.

Small intervals in both directions. Seconds and thirds are what most tunes are made of.

And now with leaps. You will hear a short tune of four notes: play it back on the keyboard. The first note is marked. Don't try to guess the notes one by one. Listen to the steps, up or down, near or far, and count the white keys.

And building downwards. This is noticeably harder: your hand is used to building upwards. Count the semitones to the left of the marked key.

Your songbook

Musicians with a good ear for intervals don't count semitones as they listen. They recognise an interval the way you recognise a familiar word. The quickest way there is a reference tune: for each interval, a tune you know from its very first notes. A perfect fourth up is the wedding march from Wagner's Lohengrin, "Here Comes the Bride", the one the organ plays as the bride walks in. A fifth up is "Twinkle". A major third down is the opening of Beethoven's Fifth Symphony. You hear the interval and at once recognise the start of a tune you know.

Here you can build a reference book like that. Each row holds tunes that start with that interval or lean on it. All of them are in the public domain, and almost all are instrumental pieces by composers from the 18th to the early 20th century. Press a tune: it plays, and your choice is remembered. The best-known examples for the sevenths and the octave are 20th-century songs still under copyright, so those rows are empty. Fill them in yourself: press "Record a tune's opening" and play the start of a tune you know on the keys. The songbook works out the first step and puts the tune in the right row.

In this exercise, after each answer, buttons appear with your reference tunes: one for the right interval and, if you got it wrong, one for the interval you chose. Compare how they differ. Don't forget to record a reference for the octave yourself.

Reference tunes are crutches, and that is fine: in a few weeks you will no longer need them, and the interval will be recognised on its own. The Intervals trainer speeds this up. It has seven levels, from the fifth and the octave to every interval up and down and played together, and it gives you more of the ones you get wrong. For your hands there is Build an interval. Five minutes a day does more than an hour once a week.

Can someone have no ear at all?

Can someone really have no ear at all?

Time to answer this question from chapter 0. While intervals still get mixed up, it is easy to decide "I'm tone-deaf". Before you read the answer, place your bet.

An inability to perceive music in someone with normal hearing and normal intelligence does exist. It is called amusia. It can be acquired, after a stroke or a head injury, or congenital, present from birth. People with congenital amusia understand speech normally and recognise everyday sounds, but they may miss a wrong note in a familiar tune, fail to tell one tune from another, and not hear a note move up or down by a semitone.

How many people are like this? For a long time the usual figure was about 4%, from a 1980 British study by Kalmus and Fry, the Distorted Tunes Test, in which people were asked to spot "spoilt" notes in familiar tunes. In 2017 Peretz and Dominique Vuvan analysed online tests taken by 16,625 adult volunteers. Of these, 214 failed both pitch tests while scoring normally on the rhythm test: 1.3% of the participants. The authors themselves give an estimate of 1.5%. Counting by one pitch test instead of two would have given 4.2%. The estimates differ because the tests and the line drawn for "normal" differ, but the order of magnitude is clear: one to four people in a hundred.

Another figure from the same study is more interesting. Among the participants in whom the tests found no amusia at all, about one in five answered "yes" to the question "Do you think that you lack a sense of music?" Many more people believe they are tone-deaf than really are. Often what lies behind the phrase is a lack of practice, or a bad experience in childhood, when the "ear" was tested with a different skill altogether.

The ear can be trained, and this has been tested. In 2006 Christophe Micheyl, Andrew Oxenham and their colleagues compared how 30 musicians and 30 non-musicians tell apart two notes very close in pitch. At first the non-musicians' thresholds were on average more than six times larger. Then eight new non-musicians trained for 14 hours, and after 4–8 hours of practice their thresholds matched the musicians'.

Even amusia responds to training better than was thought. In 2018 Kelly Whiteford and Andrew Oxenham gave 20 people with amusia four sessions: some compared the pitch of two tones, others judged which side a noise came from. Afterwards 11 of the 20 no longer met the criteria for amusia, and a year later the result had held. The group that trained on the noise improved too, so the gain most likely came not from the exercise itself but from getting used to the tasks and the test. This is one study, not the final word, but "amusia for life" has turned out not to be the rule.

The intervals in your tune

Let's look at your tune in the Notebook as a chain of steps. Thin grey lines are steps (seconds), coloured ones are skips (thirds), thick coloured ones are leaps, and the red one is the widest interval. If the Notebook is still empty, compare "Ode" and "Twinkle": one moves almost only by steps, and in the other the leap jumps out at once.

Notice where the biggest leap in your tune falls. In memorable tunes it often comes at the start of a phrase or at its peak, and after it the tune comes back by steps. We will return to this in chapter 18, when we write tunes.

Next

Now you can count intervals on the keys, tell a fifth, an octave and a third apart by ear, hear the difference between a smooth and a rough sound, and you know where it comes from. You have a songbook of reference tunes and a trainer that will keep bringing intervals back until you recognise them without thinking. And the question "can someone have no ear?" has an honest answer: yes, but rarely, and almost everyone's ear can learn.

The next wall lies in the notes of a tune themselves. Play "Twinkle" and stop on the note before the last: your hand wants to go on to the last one. Why do some notes sound settled, like a full stop at the end of a phrase, while others pull somewhere else? This is the mode, and it is the subject of chapter 14.

The Notebook can now read your tune as steps: under it there is a row of intervals between neighbouring notes (press any of them to hear it) and the biggest leap. In chapter 14 the Notebook works out the mode of your tune and can move it from major to minor.