Equal Temperament

Pipe Organ at Versailles Chapel

During the baroque period, a rather technical innovation in tuning was taking place that would have enormous consequences. Until this time, instruments had been tuned using the whole integer system discovered by Pythagoras. This was called just intonation. However, this system led to a mathematical curiosity called the Pythagorean Comma, a discrepancy that develops mathematically based on whether tuning is done by octaves (1:2) or by fifths (2:3).

Octaves (1:2 ratio)
A	A	A	A	A	A	A	A
110	220	440	880	1760	3520	7040	14080

Fifths (2:3 ratio)
A	E	B	F♯	C♯	G♯	D♯	B♭	F	C	G	D	A
110	165	247.5	371.25	556.88	835.31	1252.97	1879.45	2819.18	4228.77	6342.15	9514.73	14272.1

As a result of this system, the ratios between semitones varied widely.

To avoid this, musicians invented a tuning system called equal temperament. In this system, the octaves are tuned at a 1:2 ratio and the frequencies of the other eleven semitones are evenly distributed across the remaining space along the trend line of the Pythagorean intervals.

Technically, this makes every note and every interval (except the octaves) slightly out of tune. In practice, the difference is slight and it takes a skilled musician to notice the difference; singers and most instruments can easily adjust their intonation to the Pythagorean system as necessary regardless.

However, what equal temperament allowed was a keyboard instrument that could play equally well (or equally poorly, depending on your point of view) in all twelve keys. An excited Johann Sebastian Bach wrote "The Well Tempered Clavier," a series of 24 piano études, one for each major and minor key. This opened up a whole new range of possible harmonic vocabulary, as instruments were no longer restricted to closely related keys.