How Scales are Created: Music in Three Worlds
World culture and our own European musical culture recognise countless different scales. Despite this diversity, all scales also have similarities. How can similarities and differences be explained?
- No musical scale without the octave.
- Resonance plays an important role
- The perception of the octave in the mental world
- Real constraints for musical scales
- The overtone series is not a musical scale
- “Breaking down” the fifth
- Fractions and Resonances govern the scales
- Criteria for attractive musical scales (summary)
- First musical scales found with these criteria
- Standard pentatonic scales
- The major scale introduces tension to the resonances
- Pure and impure temperament
- How does the pythagorean comma come about
- The distribution of tones within the octave
- Tones for the two gaps
- How the tempered scales came into being
- The equal temperament
- Why resonance also works with imprecision
- Advantages of the equal temperament
What does all this have to do with Penrose’s three worlds?
-> Musical scales are part of all three worlds
Resonance and Music: An Overview
Scientists and musicians live in different worlds. However, the topic of resonance is relevant for both worlds. Here you will find a brief overview:
Appendix
- Calculationg with frequencies and intervals
- Expressions around waves and sine waves
Youtube-Demo to the Quintenexperiment (resonant fifth, in German)
The fifth is not an overtone. Nevertheless, there is a strong resonance over this interval. You can find a short resonance demonstration over a fifth here (in German).
The explanation of the scales in LDC
LDC is a new way to visualise complex issues and discussions. See how it can be used to explain the theory of scales (in German).
Philosophical Background: Three worlds
Reality can be viewed from different perspectives. Roger Penrose speaks of three worlds:
- The Platonic world contains timeless objects that are not located at a specific point, e.g. mathematical elements and abstract patterns.
- The physical world has space and time and can be described precisely from outside,
- The mental world is the world in our heads, i.e. what we think, feel and experience.
It is clear that all three worlds play a role in music. This is obviously also the case for scales.
The Author as a Pianist
You may know me as a computer scientist and wonder why I am writing about music theory here. In fact, I have always spent a lot of time at the piano. Here are two audio examples:
“Billie’s Bounce”
Live at Restaurant Zeppelin, Konstanz on 5. Dez. 2008.
Tenor saxophone: Bruce Roberts, UK.
Vibraphone: Christoph Mantels, DE.
Piano: Hansruedi Straub.
Double bass: Samuel Forrer.
Drums: Urs Heinrich.
“Body and Soul”
Live at Bühler, AI, 2015.
Piano: Hansruedi Straub.
Double bass: Samuel Forrer.
Drums: Urs Heinrich.