Pitch and intervals

Sing the first line of "Happy Birthday". Now sing it again, starting a little higher. Every note you sang is different from the first time, and nobody at the party would notice. But sing one note wrong in the middle and the whole room winces.

So a tune can't be its notes. It has to be something that survives when every note moves, and breaks when one note moves alone. The phrase below lets you find out what that something is.

Before you tap Fourth up: if every note goes up by five semitones, will you hear the same tune, a different tune, or the same tune with a different mood? Make a guess, then tap it.

What survived the move?

You probably heard the same little tune, just brighter and higher. Look at the line under the buttons: the note names all changed. What stayed is the list of steps written on the notes: +2, +5, -3.

That list is the tune. Each number is an interval, the distance from one note to the next, counted here in semitones. A semitone is the step from one key to the very next key on a piano, black keys included. C to C♯ is one semitone, C to D is two, and C to E is four, even though only two white keys separate them.

Move	Distance
C4 to D4	+2 semitones
D4 to G4	+5 semitones
G4 to E4	-3 semitones

Read it like directions: start anywhere, go up two, up five, down three. Follow the same directions from F and you get F, G, C, A. That move, every note shifted by the same interval, is transposition. In MIDI numbers it's plain addition: 60, 62, 67, 64 plus five is 65, 67, 72, 69, and the gaps between them can't change.

This is why a singer can ask the band to "take it down a tone" and nobody has to relearn the song. It's also the trick behind the big lift at the end of so many pop songs: Beyoncé's "Love on Top" climbs through four key changes in its final choruses, each a semitone higher than the last: the same song, higher and more urgent every time.

Now change one note instead

Here's the same phrase, but this time you can move only one note.

Lower the last note by one semitone. Three out of four notes are exactly where they were, and yet the ending changes colour: it goes from open and bright to shadowed. The last step went from -3 to -4. One number in the directions changed, so you're now following a different tune.

That's the answer to the opening puzzle. Moving every note keeps every interval. Moving one note changes two intervals (the one into it and the one out of it). Your ear tracks the distances, so the second edit is the one it notices.

Names for the distances you'll meet most

Musicians name the common intervals so they can talk about them without listing notes. Many people learn them through the openings of familiar tunes:

Semitones	Name	Example from C	Sounds like the start of
1	Minor second	C to D♭	The "Jaws" theme
2	Major second	C to D	"Happy Birthday" (the move onto "birth")
3	Minor third	C to E♭	"Greensleeves"
4	Major third	C to E	"When the Saints Go Marching In"
5	Perfect fourth	C to F	"Here Comes the Bride"
7	Perfect fifth	C to G	"Twinkle, Twinkle, Little Star"
12	Octave	C to the next C	"Somewhere Over the Rainbow"

The names carry spelling as well as size. C to F♯ and C to G♭ are the same keys on a piano but get different names because they play different roles in harmony. For moving a phrase around, the plain semitone count is all you need. Count the moves between keys, not the keys themselves: from C, the first step to C♯ is one.

Why an octave doubles instead of adding

An octave above 110 Hz is 220 Hz, 110 Hz higher. An octave above 440 Hz is 880 Hz, 440 Hz higher. Same interval, very different number of hertz.

Pitch works in ratios. An octave is a doubling. On a piano tuned the ordinary way, twelve-tone equal temperament (12-TET), the octave is split into twelve equal ratio steps, so each semitone multiplies the frequency by the twelfth root of two, about 1.0595. Twelve of those make exactly 2.

When you transposed the phrase up five semitones, every frequency was multiplied by the same five-semitone ratio. The ratios between the notes stayed put, and those ratios are what you hear as the tune.

Same idea, different place: a frequency shifter adds the same number of hertz to every part of a sound instead of multiplying. The ratios break, and a melody comes out metallic and strange rather than higher. It's this chart, done the wrong way on purpose.

Where the same interval stops sounding the same

An interval number doesn't fix how it sounds. In the first lab, transpose the phrase down a full octave (-12), then imagine going down another. Low down, a major third that sounded sweet at C4 starts to blur, because the harmonics of the two notes crowd into the same range and rub against each other. That's why bass players rarely play thirds low down, and why piano left hands spread out.

Context changes it too. C and G together sound like a sturdy, open fifth. Put an A♭ in the bass beneath them and the same two notes become part of something far more restless. The calculation tells you what you changed. Listening tells you what it does in this piece.

A keyboard where every interval has one shape

On a piano, a major third looks different depending on where it starts: C to E is two white keys, E to G♯ involves a black one. Guitarists don't have this problem. On a guitar neck a chord shape slides up the frets and stays the same chord, which is transposition you can feel in your hand.

An isomorphic layout does that for keys: every interval is the same physical move wherever you start. Maru's Isomorphic Keyboard lays notes on square and hexagonal grids this way, so learning one melody shape teaches you that melody in every key.

Many traditions build this idea into their names for notes. In Hindustani music, the home note Sa isn't a fixed pitch: the singer chooses the Sa that suits their voice, and Re, Ga, Ma and the rest are heard relative to it. Movable-do solfège (do, re, mi) works the same way. Both name the relationship, not the key.

Make it yours

Make up a four-note phrase of your own on the keyboard in the first lab, or hum one. Write it as a starting note and a list of signed steps, like "start on E, then +2, -4, +5".

Now rebuild it from two other starting notes. Which register lets it sing? A phrase that sparkles high on an electric piano might sit awkwardly in your voice. Then change one step by a single semitone and listen to how much the character moves.

Keep the step list. It's your tune in every key at once, and it's the one thing you need to hand it to another instrument.

Quick check

1. Your phrase is 60, 62, 67, 64 in MIDI numbers. Move it up three semitones. What do you get, and what stays the same? 63, 65, 70, 67. The steps +2, +5, -3 are unchanged, which is why it's still the same tune. If you added three to only some notes, the steps would change, and so would the tune.
2. C to E crosses two white-key steps. Is it two semitones? No, four. Counting white keys is tempting because it works for some pairs, but it skips the black keys between them. Count every key, black or white.
3. A friend says an octave is "always 220 Hz up". When is that right? Only when you start at 220 Hz. An octave doubles whatever frequency you start from, so above 880 Hz it's 880 Hz up. Equal musical steps are equal ratios, not equal hertz.

Next steps

• Go deeper: Rhythm, meter and groove. A tune is a pattern of distances in pitch. A groove is a pattern of distances in time, and it can bend in ways pitch can't.
• Use it: Melody and motifs. Take one step pattern and grow a whole melody out of it.