Hans Rudolf Straub, Author at Hansruedi Straub

The Distribution of Tones within the Octave

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

By Hans Rudolf Straub|2025-11-15T13:08:43+00:0016. December 2021|Categories: music, Theory of the Three Worlds, Music scales|Tags: musical scales, Three worlds|Comments Off

How does the pythagorean comma come about?

The Pythagorean comma The Pythagorean comma demonstrates that our tonal system is not perfectly consistent but has a gap whose form and cause I will describe in this post. The comma is relevant in terms of both ourpythagorean comma musical practice, since it has very specific effects, and of philosophy and science, since it is typical of the problems that we observe in the interplay of our three worlds (according to Penrose). Thus it is a topic that is not solely relevant to musicians but also to people who are interested in the question as to how mathematics (ideal world),

By Hans Rudolf Straub|2025-12-03T14:57:28+00:005. November 2021|Categories: music, Theory of the Three Worlds, Music scales|Tags: musical scales, pythagorean comma, Three worlds|Comments Off

Pure and impure temperament

The two diverging ideals of a theory Like every theory, the theory of music moves between two extremes. On the one hand, a theory enables us to summarise quite different observations and explain them in a simple manner – the simpler, the better. On the other hand, we also want to apply this explanation, if possible, to everything that we observe. Thus a theory is good if it is as simple as possible but also explains as much as possible. The challenge is to attain these two extreme objectives of every good theory at the same time. What is typical

By Hans Rudolf Straub|2025-11-15T13:10:36+00:0024. July 2021|Categories: music, Theory of the Three Worlds, Music scales|2 Comments

The major scale introduces tension to the resonances

The major scale The major scale (Ionian mode) is the most widespread musical scale both in Europe and globally. It is a heptatonic scale, i.e. a musical scale with seven tones. It is characterised by very special resonance ratios, which serve well to explain its worldwide appreciation. Below, I have listed the tones of the major scale of C, ascending from the bottom to the top, together with the intervals between each tone and the fundamental tone. Of course, it is these intervals that constitute the musical scale. We could also start the musical scale with any other tone and

By Hans Rudolf Straub|2025-11-15T13:10:59+00:0029. June 2021|Categories: Information, music, Theory of the Three Worlds, Music scales|Tags: musical scales, Three worlds|0 Comments

Standard pentatonic scales

As we have seen in the previous post, the tones C – D – E – G – A – C constitute the standard major pentatonic scale. All in all, another four pentatonic scales can be created with the simple criteria for resonant pentatonic scales. These five pentatonic scales are the five musical scales which according to our mathematical criteria allow for resonances among all their tones. We will see later on that we are able to create all the musical scales traditionally used in Europe with our pool of the nine most resonant tones. In the heptatonic scales, however, for instance

By Hans Rudolf Straub|2025-11-15T13:11:51+00:0026. May 2021|Categories: Information, music, Theory of the Three Worlds, Music scales|Tags: musical scales, Three worlds|0 Comments

Expressions around waves and sine waves

Sine waves play a crucial part for our considerations of resonance. On this page, I would like to explain the terms that I use in this respect. Wave A wave is a motion in time which oscillates around a baseline. A wave can have different shapes. For our considerations of resonance, we use pure sine waves; such a wave is shown in the graph above. Amplitude The amplitude is the deviation of a wave from the baseline. It does not play any primary role in our considerations. Period A period lasts as long as the wave takes to arrive in the same

By Hans Rudolf Straub|2025-11-15T13:12:24+00:009. May 2021|Categories: Information, music, Theory of the Three Worlds, Music scales|Tags: musical scales, Three worlds|0 Comments

Calculating with frequencies and intervals

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

By Hans Rudolf Straub|2025-11-12T10:44:22+00:006. May 2021|Categories: music, Theory of the Three Worlds, Music scales|Tags: musical scales, Three worlds|0 Comments

First musical scales

Starting position Do the criteria which we have postulated so far already enable us to create musical scales which are so attractive that they occur in reality? After all, the criteria look rather artificial and theoretical at first sight – can they nonetheless serve to explain musical scales that have grown naturally? Indeed they can. The mathematical criteria for resonance have obviously had an impact on the human ear and have for millennia prompted people again and again to invent music which is fundamentally structured by precisely those musical scales that we are able to derive mathematically with the help of

By Hans Rudolf Straub|2025-12-03T14:59:08+00:0029. April 2021|Categories: Theory of the Three Worlds|Tags: musical scales, Three worlds|0 Comments

Criteria for attractive musical scales (overview)

What is this about? According to the theory of the three worlds, mathematics (ideal world) plays a part in physical processes (physical world). Without our subjective perception (mental world), however, we would not be able to notice any of this. I examine the way in which these very different worlds converge in reality with the example of musical scales. There are some riddles here, for instance why human cultures have created thousands of different musical scales, but every one of them uses the octave. This constancy of the octave can easily be explained by resonance, which in the case of the octave is

By Hans Rudolf Straub|2025-11-12T10:44:52+00:0021. April 2021|Categories: Theory of the Three Worlds|Tags: musical scales, Three worlds|0 Comments

Fractions and Resonances

Resonance works through shared overtones Resonance occurs when two vibratory physical objects vibrate together. What matters is the two objects’ natural frequency: 1st degree resonance: both objects vibrate in the same frequency (f2 = f1). 2nd degree resonance: one object vibrates in an overtonefrequency of the other (f2 = n * f1). 3rd degree resonance: both objects vibrate in a shared overtone frequency (f2 = n/m * f1). The 3rd degree resonance reveals itself by the fact that the ratio of the two frequencies corresponds to a fraction of integers (n/m). This 3rd degree resonance is what interests us, for it has an effect