# Equal Temperament Scale

Modern keyboards are tuned to a scale known as the Equal Temperament Scale. This is a compromise between playing something that sounds musically correct and the ability to play in any key. When you sing a major scale, you subconsciously use a scale called the Just Intonation Scale. This is the scale that sounds right, at least to people brought up on Western music. Unfortunately there are small peculiarities of the Just Intonation Scale which make it unsuitable for keyboards, which need to able to play in different keys.

For example, when you sing do re mi starting on C, you get C, D, E. When you start on D, you get D, E, F#. The E note produced by these two sequences is not the same. The difference between them is only very slight, but it is audible to a keen ear. When tuning the keyboard, which note should the tuner use?

The Equal Temperament Scale was devised about 200 years ago to overcome these difficulties and has been used ever since.

This entry will show the mathematical basis for the Just Intonation Scale, show the problems it introduces for keyboards, some of the other scales which attempt to overcome this and then show the Equal Temperament scale, which is the accepted solution to the problems.

### The Mathematical Basis for Pitch

The pitch of a note is given scientifically by a number called its frequency. Sound is a regular vibration of the air. Frequency is the number of vibrations per second. The more vibrations per second, the higher the frequency, the higher the pitch. The unit of frequency is hertz, abbreviated Hz. The note A, which is used for tuning orchestras, is 440 Hz, which means 440 vibrations per second.

If two notes are an octave apart, the higher note has a frequency twice that of the lower one. The A an octave above standard A has a frequency of 880 Hz, while the A an octave lower has a frequency of 220 Hz.

The 8 A's on a piano keyboard have the following frequencies in hertz:

22.5, 55, 110, 220, 440, 880, 1760, 3520

### The Mathematical Basis for Harmony

Notes that sound good together are said to be in harmony. The frequencies are found to be related to each other by simple ratios. We've already seen that an octave (do - do) is a frequency ratio of 2. The higher note is twice the frequency of the lower note. The next simplest harmony is the do - so. This is known as a perfect fifth, because on a piano, there are five white keys from do to so (do, re, mi, fa, so). The frequencies of the two notes are in the ratio 3:2. That is, the higher-pitched note has a frequency which is 3/2 times the frequency of the lower-pitched note.

The final chord we need is do - mi. This is called a major third (because of three notes in do, re, mi). The frequencies are in the ratio of 5:4. From these, we can calculate that the chord do-mi-so-do has frequencies in the ratio 4, 5, 6 and 8.

### The Just Intonation Scale

This scale is built up from a do-mi-so-do chord based on C, another based on G and a third based on F. All the other notes come out of this.

If we pick middle C = 264 Hz, we get:

 C E G C 4 5 6 8 264 Hz 330 Hz 396 Hz 528 Hz

We can also build a chord of do-mi-so on F, using the value for top C just found:

 F A C 4 5 6 352 Hz 440 Hz 528 Hz

We can build a chord of do-mi-so on G, using the value for G just found:

 G B D 4 5 6 396 Hz 495 Hz 594 Hz

Finally, we can divide the value for D by 2 to get the D an octave lower. This gives us the full major scale from do to do. Multiplying all the numbers by 2 gives the frequencies for the second octave.

 C D E F G A B C 264 297 330 352 396 440 495 528 528 594 660 704 792 880 990 1056

The interval between the notes C and D is given by 297/264 which is 9/8. This is the same as the interval between F and G and between A and B. This is known as a 'major whole tone'.

The interval between D and E is 330/297 which is 10/9. This is the same as the interval between G and A. It is known as a 'minor whole tone'.

The interval between E and F is 352/330 which is 16/15. This is the same as the interval between B and C. It is known as a 'semitone'.

### Problems with the Just Diatonic Scale

Because there are two different sizes of whole tone, we find that a note in one key is not exactly the same as the equivalent note in a different key. The interval between do and re is bigger than the interval between re and mi. Building a scale with do = G for example, we find the value of re = A is 396×9/8 = 445.5. This is 1% sharp compared with the value 440 used in the key of C.

Another problem is that although the intervals do - so, fa - do and so - re are all exact numeric ratios and therefore sound in tune, some other intervals are not so good: re - la should also be a perfect fifth, but is actually 440/297 = 1.481, more than 2% out.

### The Pythagorean Scale

The Pythagorean Scale is an attempt to overcome these problems. It builds the scale using only the interval of a perfect fifth, so all fifths are perfect. C = 1, G=C*3/2, D=G*3/2 and so on all the way to B#=E#*3/2. This means all whole tones are 9/8, so the semitone has to be reduced slightly to make everything fit in an octave. Thirds, which are two whole tones, sound sharp as a result. The ratios work out as follows:

1 9/8 81/64 4/3 3/2 27/16 243/128 2

The fourth and fifth are perfect, but the third (1.2656) sounds about 20% sharper than a true major third (1.2500). This difference is definitely enough to be audible. As a result this scale sounds bad and is rarely used.

It is interesting to note that string players in orchestras tune their strings in fifths, so that they are tuned to the notes of the Pythagorean scale. Cellists, for example, tune their top string to A and then tune the other strings in fifths below this. The lowest string, C, will be slightly out of tune with the highest, even though the fifths in between are perfect.

### The Mean Tone Scale

Another scale devised to overcome the problems of the Just Intonation Scale is the Mean Tone Scale. This has one size of whole tone and one size of semitone. The whole tone is chosen so that two of them together make up an interval of a perfect third, ratio 5/4. The semitone is chosen to fill in the gap so that five whole tones and two semitones make up an octave. The result is that fourths are about 5% sharp, fifths are about 5% flat, thirds are spot on.

The mean tone scale was popular with keyboard makers, because there was no problem with two different versions of the notes D or A. There was only one version of the notes in the C major scale, and one version of each of the flats, one version of each of the sharps. There was the problem, however, that the sharps were not the same as the flats. G# is one whole tone and two semitones lower than C, while Ab is two whole tones below C. Because a semitone is more than half a whole tone, G# ends up flatter than Ab. By setting the black keys as C#, Eb, F#, G# and Bb, music could be played in any of 6 different keys out of the possible 12.

One way to increase the number of musical keys in which music could be played was to have split black keys. Each black key could be split so that the front half played a sharp and the back half played a flat. The most common keys to be split were Eb, Ab and Bb. With this setup, music could be played in 9 of the 12 possible keys.

Some modern electronic keyboards have a mean tone scale setting, where the keyboard can be set into a particular key and will play all notes in the mean tone scale. This is supposed to sound a lot better.

### And finally, The Equal Temperament Scale

The equal temperament scale gets around the problems by compromising and playing all notes slightly out of tune. The octave is left as it is with a frequency ratio of exactly 2. This is divided into 12 equal semitones. A whole tone is defined as exactly two semitones. Since adding intervals involves multiplying frequency ratios, we don't just divide 2 by 12. Instead, we the take 12th root of 2. This is approximately 1.059463. It is slightly smaller than the semitone used in the just intonation scale. A whole tone is now (1.059463)^2 = 1.122462. A fifth is now 7 semitones, making it (1.059463)^7 = 1.498307. This is no longer perfect, but is only 0.1% out.

The Equal Temperament Scales plays equally well (or equally badly) in any of the 12 possible keys. With its introduction, composers were free to write music in any key they liked. This led J S Bach to compose "The Well-Tempered Clavier", in which he wrote a prelude and a fugue in each of the 12 keys.