The Well-Tempered Tune

Temperament

Temperament refers to an adjustment of the 12 intervals which comprise an octave. This is not a trivial matter since it is impossible to have both perfect intervals (other than the octave) and an equally spaced twelve tone scale. Equal temperament and perfect intervals are incompatible.

Perfect intervals are those for which there is perfect overlap (or coincidence) of overtones for the two notes leading to a sound which is generally pleasing to most people's ears due to the reduction or elimination of noticeble beats. A prime example is the octave where the first overtone is an octave above the fundamental, and therefore it coincides with a note an octave above the fundamental. Another is the perfect fifth where the third partial of a lower note coincides with the second partial of a higher note, e.g. the third partial of A coincides with the second partial (octave) of E. E4 is therefore a fifth above A3.

The incompatibility of perfect fifths and octaves can be seen if we ask how many perfect fifths we need in a series to achieve the same result as a series of octave steps. For example, if we start on A0, the lowest note of the piano, how many fifths do we need before we return to an A that could be achieved by a series of octaves. The table below shows a series of fifths and octaves starting on A0:

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Fifths	A0	E1	B1	F#2	C#3	G#3	D#4	A#4	F5	C6	G6	D7	A7
	27.50	41.25	61.88	92.81	139.22	208.83	313.24	469.86	704.79	1057.19	1585.79	2378.68	3568.02
Octaves	A0	A1	A2	A3	A4	A5	A6	A7
	27.5	55	110	220	440	880	1760	3520

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Starting on A0, we arrive after increasing by a series of 12 perfect fifths at A7. However, just because we call it an A7 doesn't mean that it really is a unique A7. If each fifth is a perfect fifth, each step corresponds to an increase in frequency by 3/2. So the final frequency of this A7 should be 27.5*(3/2)^12, or 3568.02 Hz.

The problem is that we get a different A7 if we move up the keyboard in perfect octaves from A0. We increase the frequency by a factor of 2 for each step, so the frequency of A7 achieved by 7 perfect octave steps from A0 is 3520.

There is a significant difference between the two A7's achieved by a series of perfect fifths and octaves: 48 Hz or 23.44 cents.

But the A7 on the keyboard can only have one frequency. One solution is to change what we mean by a fifth. The equal tempered scale is composed of 12 equally space intervals or semitones. Each corresponds to a change in frequency by the 12th root of two, so that after making 12 steps by this amount we double the frequency (an octave is 12 steps). The 12th root of 2 is 1.05946.

A fifth is now defined as an increase in frequency due to 7 semitone steps: e.g. A, A#, B, C, C#, D, D#, E
or 1.05946 multipled seven times: 105946^7 = 1.49831, not 1.50000. The interval has been shrunk a bit, and is no longer perfect. But walking up the scale with a series of 12 of these fifths will exactly equal 7 octaves, both increasing the frequency by 128 times!

This works very nicely, and the solution is what we call equal temperament. But the fifth is no longer perfect, and some claim that even though it almost sounds the same, it becomes tiring and irritating with time. The temperament doesn’t wear well.

So there have been many ways proposed to keep some intervals perfect at least over small stretches (i.e. repetitive intervals can never be perfect). For example, the fifth from A to E can be made perfect, as long as we adjust some of the semitones above E to bring us back to A and a perfect octave. Now the "distance" of the semitone depends on where we are on the keyboard. The result is that some keys will sound very nice, and some will not. This is the joy of well temperaments.

Show below is the tuning of the temperament octave (A3-A4) on a Baldwin F in equal temperament. At the bottom the magnitude of the deviation of the overtones which should be coincident if the intervals were "perfect".