The equal temperament has prevailed in our occidental music culture – despite the obvious shortcoming that its intervals are not pure any longer. This was only possible because some substantial advantages offset the flaw of impurity: 1.  One single tuning serves all keys: the fundamental tone is freely selectable. In pure intonation, instruments basically have to be retuned for each key and each fundamental tone. With a harpsichord, this concerns a few strings, but with an organ, this is really a great undertaking in view of the vast number of registers and pipes. The further the keys are apart from

Starting point: two gaps In the previous post, we saw that in the sequence of the ten scales tones found so far, there are two gaps. Can we find resonant tones there, too? We already know the following: We already know the ten most resonant intervals in the octave. These ten intervals serve to constitute the five standard pentatonic scales and our major and minor scales. There, the gaps are not obtrusive; they are only conspicuous in the distribution of all the ten potential scale tones. Intervals do not occur on their own, either in a chord or in a melody. Thus

The ten most resonant tones within the octave In this series of texts, we examine musical scales from the perspective of the three worlds. All three worlds are involved, as we saw, for example, when we answered the question as to why the musical scales of all musical cultures always cover precisely one octave. This cannot be explained in purely mathematical or physical terms. It is only through the involvement of the third world, namely our mental world, that the significance of the octave becomes evident. The selection of the tones used in a musical scale is determined by all three worlds through

On this page, I will explain some rules which are applicable when we calculate with intervals and their frequencies. Intervals are fractions An interval ranges from a lower tone to a higher one. The fraction of the interval is calculated by dividing the frequency of the higher tone by the frequency of the lower tone, for instance E  =  330 Hz A  =  440 Hz 440/330 = 4/3. This is a fourth. The interval of the fourth is always 4/3: in the fourth, the higher tone is precisely 4/3 times as fast as the lower tone. What counts here are