The first thing to understand is that if you want to go up by a constant interval, you multiply the frequency by a particular number.
For example, to go up by an octave, you multiply the frequency by 2. Since multiplication by 2 is the simplest multiplication we can do, this sounds pleasing to the human ear - so pleasing, in fact, that we learn to hear the two notes as the same.
If we want to go up by two octaves, we multiply by 2 again, for a combined total of 4 times the original frequency. And so on.
But there are other nice numbers that we can multiply the frequency by. If we multiply by 3, for example, then we go up by an octave and a fifth. To get a fifth, we go back down the octave by dividing by 2, so a fifth corresponds to multiplying by a factor of
If we multiply by 5, then we go up by two octaves and a major third. So a third corresponds to multiplying the frequency by a factor of
Thirds, fifths and octaves are fundamental to Western music, and all other intervals are built from them. The reason that they sound so nice and concordant is because they are built up from very simple multiplications.
For example, if we start at
C and multiply by
5/4, we get to
E, and if we multiply again by
5/4 we go up another third to
G♯. Now if we divide by
3/2 to go down by a fifth, we get to
C♯. The total multiplier is
5/4 * 5/4 * 2/3 = 25/24 = 1.041666...
If instead we multiply by
2, we go up to a high
C. Now, if we divide by
3/2, we go down a fifth to
F. If we now divide by
5/4, we go down by a third to
D♭. The total multiplier is
2 * 2/3 * 4/5 = 16/15 = 1.06666...
Since these two numbers are so similar, it's easy to get confused between the notes
'Now, hang on!' I hear you say. '
D♭ aren't just similar notes - they are the same note! After all, they both occupy the same key on my piano keyboard!'
This is actually a very clever musical trick. In order for piano keyboards to make sense, they can't treat
D♭ as separate notes, at least not if they want to avoid something horrific like this:
this is known as a split-key keyboard, of the type used in the 16th century when they were still figuring this stuff out
Instead, we need to approximate notes so that we can make a scale using only twelve different tones. So we end up having one key for both
D♭. Pressing this key might play a
C♯, it might play a
D♭ or it might play something in between.
A choice of approximations is called a temperament, and there were many different temperaments used right up to the Classical period. The title of J. S. Bach's 'The Well-Tempered Clavier' refers to one such temperament.
Different musicians had different preferred temperaments. One common quality was that certain keys (normally 'white-note' keys, such as C major) would sound very pure and concordant, while others would sound more off-key and spicy. This was sometimes considered a desirable feature of a temperament: different keys had different characters.
The temperament used almost universally on modern pianos is much more boring, but also more versatile. It is called 'Equal Temperament', and its name means that all of the semitones on the keyboard are exactly the same interval apart. An equal-temperament semitone is exactly a 12th of an octave, so it corresponds to multiplying the frequency by
the twelfth root of 2 = 1.05946309436....
(notice how this comes in between the
1.0666 that we calculated earlier!)
Now, what does an equal-temperament fifth sound like? Well, it sounds like the twelfth root of 2 raised to the seventh power (since there are seven semitones in a perfect fifth):
2 ^ (7 / 12) = 1.49830707688...
By a brilliant mathematical coincidence, this is almost exactly equal to
3/2. So there is no audible difference between a fifth on a piano (
1.498...) and a fifth that you would naturally sing (
What about the major third? A major third is four semitones, which corresponds to
2 ^ (4 / 12) = 1.2599...
This is still fairly close to
5/4 = 1.25, but now the difference is audible (there are some sound recordings on https://en.wikipedia.org/wiki/Major_third that you can listen to). A major third on a piano is noticeably different from a major third that you would naturally sing.
For the most part, you don't have to worry too much about this when you are making music, but it's worth keeping in mind sometimes.