How the tempered scales came into being

Musical scales before tempering

Natural musical scales

The musical scales of human cultures developed naturally, i.e. without any conscious mathematical considerations whatsoever, in the course of millennia. The fact that there is a great deal of mathematics behind them nonetheless has something to do with the resonances between the scale tones and the fundamental tone. These resonances strike us as attractive, and music that is based on such resonances is capable of uniting human communities.

Mathematically, resonances can be traced back to fractions with as low numbers as possible, and we were able to deduce mathematically which nine intervals have to display the most distinctive resonances. It is not by chance that all the musical scales commonly used worldwide – i.e. the standard pentatonic scales, our major and our minor scales, the ecclesiastical modes and many more – exclusively consist of a selection of five or seven of these nine tones.

The fundamental tone and the tension

What can also be observed in all cultures, and what we cannot imagine being otherwise, is the fact that all musical scales have a clearly defined basis, i.e. a fundamental tone, to which the other tones are always related. This is the fundamental tone in relation to which each scale tone builds up its resonance; the stronger it is, the more harmonious the tone sounds beside the fundamental tone, and also inside a melody. On the other hand, the higher the numbers in the fractions, i.e. the poorer the resonance between a tone and the fundamental tone, the tenser the melody tone appears to us. The sharpest tone in our major scale is the major seventh (for instance H in C major), which is a semitone below the octave. This tone has the strongest tension of all tones in the major scale; it calls for its resolution to the octave, thus leading the melody from tension to relief. This tension is only possible because the fundamental tone resonates audibly or inaudibly in the course of a melody and the resonance of the major seventh to it is tense.

All this, however, still has nothing to do with tempering and works in pure major.

Why were the musical scales tempered?

Two yardsticks: linear and exponential

Tempering means that the frequencies of the scale tones are slightly altered, reduced or increased – i.e. tempered. At first sight, this slightly weakens the resonances; nevertheless, tempering has prevailed in Europe’s musical culture as a matter of course.

To understand tempering, it is helpful to understand that we measure intervals against two different yardsticks: one of them is linear, the other is exponential. A precise explanation of the reasons for and consequences of these two yardsticks can be found here. In simple terms, this is about the fact that all the intervals are relative. Thus if I build a musical scale on the basis of C, the scale tones are related to C; if I choose another fundamental tone, for instance D, then the frequency of the E in the key of C is different from that of the E in the key of D. Let’s have a closer look at this surprising fact:

An example of the relativity of two yardsticks

In C major, the tone E is a major third above the fundamental tone C; the frequency of E amounts to 5/4 of the fundamental tone C. A tone D is a major second above the C and thus has 9/8 of the frequency of the C. If we now choose this D (=9/8) as the fundamental tone, then E also occurs in D major, but this time as a major second. The crucial point is that this E is not precisely as high as was the E of C major previously:

Tonality (key)	Frequency in relation to the fundamental scale tone	Frequency in relation to C major	Function of the tone in C major and D major
C major	C = 1	1	Fundamental tone
	D = 9/8	9/8	Major second
	E = 5/4	5/4	Major third
D major	D = 1	9/8	Fundamental tone
	E = 9/8	9/8 x 9/8 = 81/64	Major second

Table 1: Relativity of the frequencies in relation to the fundamental tone

In Table 1 you see that the seemingly identical E has different pitches in the two musical scales:

E in C major = 5/4 = 1.25
E in D major = 81/64 = 1.266

When I tune a string to the E of C major, the string will not completely accord with the expected E in D major. The difference is slight, but it can be measured and is perfectly audible to keen ears.

The pure intonation only works for a defined fundamental tone

As soon as the fundamental tone changes, all the scale tones relate to the new fundamental tone with regard to resonance, and the pitches of the earlier key are not all the same any longer. As outlined before, the reason for this is the two different ways of measuring the intervals: once in a linear way (auditory perception) and once exponentially (physical frequencies).

The three worlds

Again, this is about the three worlds according to Penrose: the auditory perception takes place in the mental world, the frequencies are part of the physical world, and the ideal world is that of the mathematics of the integer fractions. In music, all three worlds interact with each other in an extremely interesting way.

The objective of tempering

The point of tempering is the fact that the fundamental tone can thus be freely exchanged without the resonant musical scales known from before having to be given up. Tempering is an ingeniously chosen compromise, which is really capable of uniting both goals.

Historical development

The fascination with resonant acoustics is typical of all human cultures. This is how the highly resonant musical scales came into being: the standard pentatonic scales and the major scales, various minor scales and the Doric mode frequently used in Gregorian chant. The music which previously used to be played with these musical scales always had a constant tonality, i.e. one fundamental tone that is not changed as long as the melody is being played. All the tones of the melody resonate with the fundamental tone, and the fact of how strongly the tone of the melody resonates with the fundamental tone indicates the tension which the melody has with the individual tones.

To support the fundamental tones, polyphonic instruments from earlier cultural epochs often have additional strings (drone strings) or pipes (bagpipe) whose pitch cannot be changed to ensure that precisely this tension between the melody tone and the fundamental tone is emphasised. Whereas the tones of the melody vary, the fundamental tone (drone) is sounded throughout the piece and provides it with a solid base.

However, certain novelties gained ground in the Renaissance. Thus people began to change the fundamental tone during the piece. This allows for greater diversity in music. As long as the keys were related to each other, only slight frequency deviations were caused; in the case of really closely related keys, they also merely concerned one tone. The further the keys were apart from each other, however, the more difficult things became. Thus it sounded very unpleasant, for example, if someone tried to play F sharp major on an organ tuned to C. This marked the beginning of a period in Europe of a wide variety of experiments with slightly altered temperaments which attempted to reconcile the incompatibility of the fundamental tone shift with the purity of the intervals in various compromises. What ultimately prevailed in the late Baroque period was the equal temperament, which constitutes a really convincing compromise and enabled the rich development of harmonics in classical music and modern jazz.

Hans Rudolf Straub