Intervals

An interval is a difference between two frequencies.

In music, the smallest possible difference is 6%, which is the difference between two piano keys that are next to each other. This smallest step is called a semitone.

There is one semitone between G3 and G#3. There are four semitones between A#2 and D3. Moving 3 semitones lower means moving 3 keys to the left. Moving 6 semitones higher means moving 6 keys to the right, and so on.

Octaves

A difference of 12 semitones is called an octave. Moving an octave higher = moving 12 semitones higher = doubling the frequency. There is a difference of one octave between B3 and B4. There are three octaves between D2 and D5, and so on.

We can use these terms even with non-musical pitches. If the lowest you can sing is 100 Hz and the highest is 800 Hz, you have a voice range of 3 octaves (100 to 200 is one octave, 200 to 400 is another, and 400 to 800 is another).

A regular piano has 88 keys (87 semitones between the first and the last key), which is a range of over 7 octaves. A guitar can play tones from E2 up to D6, which is almost four octaves (with two more tones, D#6 and E6, it would be four octaves: E2 to E6). You can see ranges of various instruments here.

Transposing

Try to play the beginning of "Happy Birthday To You": G2 G2 A2 G2 C3 B2. Now, play it one octave higher: G3 G3 A3 G3 C4 B3. Does it still sound similar?

Now, play a totally different piece: C#3 C#3 D#3 C#3 F#3 F3. Does it sound familiar? It is the original piece, moved six semitones higher!

We can see that a piece of music can be "shifted" (transposed) any number of keys lower or higher, and it will still sound "the same" in a way. Actual tones are not important, as long as we keep the same intervals between these tones.

Instead of starting at G2, G3, or C#3, we can start at any key. And instead of mentioning specific keys, we can write a "pattern": 0 0 2 0 5 4 - each number is an interval between the needed key and the key where we started (in semitones).

A melody of "Baby Shark" can be written as (0 2 5 5 5 5 5 5 5) repeated 3x, then 5 5 4. You can start, e.g., at D4 and play D4 E4 G4 ...

Harmony

What if two different tones sound simultaneously? Do they always sound nice together? It depends on an interval between them! Some intervals sound nicer than others.

0 semitones

When two instruments play the same tone, e.g., 165 Hz (E3), it always sounds nice. Try to play a tone and sing the same tone (you can just do "mmmm" with a closed mouth).

12 semitones - Octave

When one frequency is 2x higher than another, they still sound very nice together. Try to press the same keys above and below and compare it.

7 semitones

7 semitones correspond to increasing the frequency by 50%. When A2 is 110 Hz, E3 (E3 = A2 + 7 semitones) is 110 + 55 = 165 Hz. It is the third most beautiful interval.