Pythagoras, who is most famous for the theorem of right-angled triangles, also discovered the mathematical ratios of the frequencies of the notes in musical intervals in just temperament.
A musical interval is simply a label musicans give to a ratio of frequencies of two notes that are heard. Consonant intervals (ie the nice sounding ones, such as octaves and perfect fifths) have low integer ratios and the dissonant intervals (the not-so-pleasant ones, such as major sevenths and minor seconds) do not.
The ratios for the common intervals in just temperament are:
Minor 2nd: 16:15
Major 2nd: 9:8
Minor 3rd: 6:5
Major 3rd: 5:4
Perfect 4th: 4:3
Perfect 5th: 3:2
Minor 6th: 8:5
Major 6th: 5:3
Minor 7th: 9:5
Major 7th: 15:8
For example, if we had a tone of 880Hz and 440Hz (concert A these days) we would have a frequency ratio of 2:1, giving us an interval of an octave.
However, it is now more common practise to use equal temperament tuning instead of just temperament. This was because keyboard instruments tuned in just temperament were limited to one set of tuning and would only sound good in the key it was tuned to. The ratios above would be applied to a principle note. If that note was C for a piano, it would be called a just C piano. However, in unrelated keys the ratios for the above intervals would be different and thus sound dissonant and out of tune.
Equal temperament is a compromise. Each octave would be divided into 12 equally separated tones. This makes each key equally tolerable to the ear (or intolerable depending on your point of view) The distance would be the same, and the frequency of any note can be found using the formula:
f = P x 2n/12
f = The frequency of the note that you wish to find out
P = The frequency of the note that you know already
n = The number of semitones above the note you know already
For example, to find the frequency of the note B:
- You know that A is 440 Hz (concert pitch). P=440
- You know that B is two semitones above A. n=2
f = 440 x 22/12
f = 493.88 Hz
Thus the note of B has a tone of 493.88 Hz in equal temperament
So how are they different? Well, if you put the decimal ratios of both tunings side by side, the numbers vary slightly.
Interval Just Ratio Equal Ratio
Unison 1.000 1.000
Minor 2nd 1.064 1.059
Major 2nd 1.125 1.122
Minor 3rd 1.200 1.189
Major 3rd 1.250 1.260
Perfect 4th 1.333 1.335
Perfect 5th 1.500 1.498
Minor 6th 1.600 1.587
Major 6th 1.667 1.682
Minor 7th 1.800 1.782
Major 7th 1.875 1.888
Octave 2.000 2.000
The difference, although small, does become rather noticable when you try playing F# minor scales on an equally tempered and just C piano. Listening carefully will make the difference more obvious than attempting to understand the mathematical reason, however finding a non-equal tuned instrument in modern times is rather difficult.
Secretly tuning someone's piano to just temperament makes a good practical joke, because most pianists won't have a clue why their piano sounds exceptionally nice in some keys and nasty in others...