# The equal temperament

In equal temperament, every pure third (702 C) is narrowed by one-twelfth of the Pythagorean comma (i.e. approximately 2 cents). The difference between equal fifths and pure fifths is very difficult to hear. The major thirds are far from perfect (386 C), but nevertheless approximately 4 C narrower than Pythagorean thirds (404 C). In the timbre of the piano, for example, equally tempered thirds have become the standard over the past two centuries. It also appears to be that the so-called pure intervals (the octave, the fifth and the fourth) are less tolerant to deviations from the natural timbre than thirds, seconds, sixths and sevenths.

The ratios of equally tempered intervals are always multiples of hundreds, potentially creating the impression that they are the “pure intervals”. With the exception of the pure octave (1200 C) this is, however, not the case. In the case of the piano it must, moreover, be noted, that the octaves are pure only in the middle part of the instrument's range. Going down on the scale, the tuning level falls (compared to ordinary equal temperament) more and more, while going up the scale mean a rising tuning level. A_{2} might be -40 C and C^{5} +20 C compared to the tuning level of the middle range of the piano.

The diatonic scale that was mentioned in connection with the Pythagorean tuning system has formed the foundation for music in the West since medieval times. Western instruments (with the exception of some percussion instruments) produce a harmonic spectrum. Many tone positions of the scale can be taken directly from the harmonic series: if the 1^{st} harmonic is taken as the first degree of the major scale (the *C* in C major), the 9^{th} harmonic then corresponds with the second degree (*d*), the 5^{th} and the 10^{th} harmonics with the third degree (*e*) and the 3^{rd} and the 6^{th} harmonic with the ^{5th} degree (*g*). The seventh degree (*b*) corresponds with the 15^{th} harmonic, but the fourth and the sixth degrees are not found until long into the harmonic series. It is, however, not impossible to conceptualise the diatonic scale as having been created by inserting additional tones into a scale that has originally been formed from the harmonic series. The notion of an acoustic scale is also rooted in this concept: the acoustic C scale included the tones *C, D, E, F sharp, G, A* and *B flat*, because the seventh and the eleventh harmonic are closest to *B flat* and *F sharp*.

The overtone singing (“throat singing”) practised in some cultures, such as the Mongolian Tuvan tribe, is based on using the harmonic overtones (from the 7^{th} to the 12^{th}, for example) produced from a low tone in the air passage formed by the mouth, the throat and the nasal cavities. This results in an ascending scale with decreasing intervals, where the intervals nevertheless correspond approximately to whole tones.

## About the addition of interval ratios

Natural intervals have simple integer ratios. We know that the octave can be divided into pure fifths and fourths, but how can this be expressed this using ratios? 2/3 + 3/4 is by no means the same as 1/2.

The calculation holds true, however, when the ratios are multiplied together instead of added together:

2/3 · 3/4 (the thirds are cancelled) = 2/4 = 1/2.

The fifth can be divided into a natural major and minor third:

4/5 · 5/6 = 2/3

A natural major third can be divided into a major and a minor tone:

9/8 · 10/9 = 4/5

A major tone can be found, for example, between do and re on the pure scale, while a minor tone can be found between re and mi. Recognising this fact or, in other words, achieving pure intonation, is possible in choral singing, for example. In equally tempered instruments, the size of every whole tone is usually identical.

All intervals called pure intervals can be found in the harmonic series. The division ratio comes directly from the ordinal number: the interval between the 5^{th} and the 8^{th} harmonic is a minor sixth, giving a ratio of 5/8.

The Pythagorean tuning system includes only major tones: two consecutive fifths minus an octave constitutes a major tone: 2/3 · 2/3 · 2 = 8/9.

The difference in size between some of the pure intervals can be illustrated using the following diagram:

## The tuning coordinate system

The following diagram shows the pitch names along an equal-tempered scale ordered by fifths so that each tone is placed on its own “ground level”, not in proportion to the first note of the scale.

If the objective is to create pure fifths, the difference between consecutive fifths must always be approximately 2 C. For example: if *c* = 0, then *g* = +2, *d* = +4, and so on. What follows is that the chain of pure fifths inevitably pushes the tones *G flat* and *G sharp* approximately 24 cents apart (the Pythagorean comma). The line ascending from the bottom left represents this type of tuning. The "tone boxes" of the Pythagorean tuning system correspond to the line.

If we want to create pure major thirds, the *e* must be approximately 14 cents lower than *c*. In order to establish pure thirds between all tones of the scale, the tone positions must be placed along the line starting from the top left. In this case, the difference between *G flat* and *F sharp* would be even greater (approximately 41 C, or one fifth of a whole tone). The minor thirds would also be very narrow in comparison with the pure thirds (269 C versus the 316 C of the “natural minor third” at the ratio of 5:6). Moreover, every fifth would be approximately 5 C too narrow.

The coordinate system can be used to illustrate the fact that the equal temperament is a compromise: no interval is “natural”, but no interval deviates enough to be disturbing. Many tempering methods (which are still being developed) bring “liveliness instead of flatness” to the intervals.

## Tuning system applet

Use the applet below to compare different tuning systems. The keyboard can be used with the mouse or the computer keyboard. The bottom line of the keyboard (z, x, c, v, etc) corresponds with the white keys, while the black keys can be played with the line above it (s, d, g, h and j). The octave can be switched to the upper octave by pressing the shift key. The waveform can be selected using the lower radio buttons.