The history of present-day tuning begins with Pythagoras, who was the first to define a tuning as a mathematical system.  Pythagoras discovered that a combination of two pitches was most pleasing to the ear, when the frequency ratio between the two pitches was as simple as possible.  Thus, unison, with a ratio of 1:1 would be considered most harmonious, followed by the octave at 2:1, followed by the fifth at 3:2 and so on.

The problem is that as a scale grows in size, the number of possible ratios increases exponentially, along with the number of notes.  The purity of every ratio can not be maintained simultaneously, because some ratios are incompatible.  For instance, the products of a fifth and an octave, which are both prime number ratios, can never be equal, resulting in what is really a spiral of fifths, as opposed to the circle of tempered fifths that we know today.

Pythagorean tuning creates two major problems when applied to a scale of twelve semitones.  One is that the discrepancy between twelve fifths and seven octaves is about one quarter of a semitone (23.46 cents).  This is known as the Pythagorean Comma and creates one fifth (the “wolf” fifth) that will always be out of tune.  The other major problem is that while the fifths are kept pure, both the major and minor thirds are made relatively harsh.  It is believed that the initial efforts to soften the harsh thirds of Pythagorean tuning that led to the first formal efforts at tempering this scale.

The reason that equal temperament was not adopted for so long a time is that in equal temperament not a single ratio is kept pure, with the exception of the octave.  For this reason, a number of unequal temperaments were experimented with between the 14th and 19th centuries, all of them sacrificing certain ratios for the sake of others and having certain keys in better tune than others.  For a long time, different keys had different harmonic characteristics, a feature that was lost with equal temperament.  As people became more accepting of dissonance, equal temperament was eventually adopted as the tuning standard, since it allowed unlimited modulation from key to key, by transforming the spiral of fifths into a circle.  Ironically, equal tempered thirds are considered to be one of the harshest intervals in the system, which was one of the initial problems of Pythagorean tuning.

Here is a tuning table for Anderas Werckmeister’s III (V), one of the last unequal temperaments.  Note the theoretically small differences in cent values, between this table and equal temperament.

SEMITONE	POSITION	DIFFERS FROM EQUAL TEMPERAMENT
1 (C)	0 cents	None (unison)
2 (C# or Db)	96 cents	- 4 cents
3 (D)	204 cents	+ 4 cents
4 (D# or Eb)	300 cents	None
5 (E or Fb)	396 cents	- 4 cents
6 (F or E#)	504 cents	+ 4 cents
7 (F# or Gb)	600 cents	None
8 (G)	702 cents	+ 2 cents
9 (G# or Ab)	792 cents	- 8 cents
10 (A)	900 cents	None
11 (A# or Bb)	1002 cents	+ 2 cents
12 (B or Cb)	1098 cents	- 2 cents
13 (C)	1200 cents	None (octave)

It is still debated whether Bach used one of the unequal, called “Well Temperaments”, or Twelve-Tone Equal Temperament for the composition of his Well-Tempered Clavier.

To see and hear examples of tuning systems from non-Western cultures, click here to go on to the next page.