**scale**), you can probably understand a good part of what's coming.

Okay. So what is this diagram? Starting from the really low C on the left, if I double its frequency, I will hear the C one octave above it. If I triple the frequency, it would turn out to be the G third from the left, etc. So each multiple of the original frequency will produce a note that gets increasingly less harmonious with the note before it. Of course, 1 and 2 are extremely harmonious: the octave. 2 and 3 form the perfect fifth, 3 and 4 the perfect fourth, 4 and 5 the major third, 5 and 6 the minor third, 6 and 7 the major second, 11 and 12 the semitone. You can see for yourself that the harmonies get less and less pleasant. Dissonances like the tritone occur at 8 and 11 (terrible ratio, 8:11), and augmented fifth (8:13) etc. etc.

We can construct the harmonic series from any starting note of course. Let's try it from A (concert pitch)

A: 440 Hz

A': 880 Hz

E'': 1320 Hz

A'': 1760 Hz

C#''': 2200 Hz

E''': 2640 Hz

G''': 3080 Hz

Okay. Now let us do something here. If G''' is 3080 Hz, then G'' must be 1540 Hz, G' 770 Hz, and G itself 385 Hz. Let us construct a harmonic series from this G:

G: 385 Hz

G': 770 Hz

D'': 1155 Hz

If D'' is 1155 Hz, then D' must be 577.5 Hz, and D must be 288.75 Hz. Now, let us construct a harmonic series from this D:

D: 288.75 Hz

D': 577.5 Hz

A': 866.25 Hz

This means that the A constructed from this harmonic series is 433.125 Hz, a whole 7 Hz different from the original A at 440 Hz!!

What has happened? Long ago, musicians realised this problem: that starting a harmonic series from one note will form notes that are unique to that harmonic series. If you wish to change keys, say from A Major to D Major, if we insist on using the notes formed in the harmonic series of A to play in D Major, the sound would be horribly off key. Modulation was rendered impossible (not to mention to problems this caused for the french horn and other brass instruments: a separate topic, but well worth exploring!), and without modulation, music got boring fast.

Until a certain Johanne Sebastian Bach came into the picture. He had a wonderful solution to this problem: why not make every single note equally out of tune in such a way that ALL keys will sound equally out of tune? Of course, this sounds disagreeable: if it's out of tune, wouldn't we know it's out of tune and hate the music being played?

Turns out that isn't true. He wrote the Well-Tempered Clavier, a set of 24 preludes and fugues in all 24 keys to show that it was possible to do such a thing. Initially, I would think people were rather uncomfortable with the poor tuning, but today we are so used to his system that what we think is in tune today, would probably have been considered badly out of tune in his time!

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