A Musical Explanation of One of Einstein's Key Ideas

The only instruction from my childhood that I can remember in music was what my paternal grandmother taught me before I was a teenager. On the organ that my grandfather played, she taught me the scale from C to the next C (CDEFGABC): do re mi fa sol la si do. I learned that the spaces on the stave are FACE and the lines are EGBDF. I learned about whole notes, and half notes, and quarter notes etc. I learned about sharps and flats. That is all.

Recently, I bought a little keyboard that works with my laptop and fits in my suitcase. I started looking up musical terms using my web based subscription to the Oxford English Dictionary (OED). The dictionary talks about a full 'tone' as an interval between notes, and, at first, I thought it was a time interval. Then I realized it is a difference in frequency. The scale I learned seems to be called diatonic, and it goes like this:

C tone D tone E half-tone F tone G tone A tone B half-tone C

There are twelve of these scales - each starts with one of the keys from C to the next B inclusive. They are shown in the table below. Larger pictures of these scales are shown on the bottom part of this page. I have used the first key of each scale as a label.

I seem to have a good ear and worked out how to play, with the right hand, the national anthems of New Zealand, Australia, and the USA. Germany's was more difficult, however, and I needed a recording and the score. With pencil and paper and the internet, I copied the score which appears on the left side of the next picture. Above the double line on the right are my notes through which I think I came to understand the scales. The dictionary talks about 'thirds' and 'fifths' and so on in terms of diatonic 'degrees' - but the degree is just the frequency interval from one note to the next in the scale, and both notes are counted. This means that the 'first' is the first note in the scale, and so on. At the bottom of the right side of the next picture is the German national anthem written in terms of the key numbers in the scale.


German National Anthem (left) ... and its interval form (bottom right)

If we write the score for each scale, we will get twelve different scores, but the interval representation seems to be the same for all twelve scales. Now we come to Einstein's laws and maths: He said the laws of physics, expressed as mathematical equations, must have the same form in all 'coordinate' systems - which are just different ways of associating each point in space with an ordered set of numbers. There is one number for each dimension in the space. The score is like an equation written for a particular coordinate system.

At the time, the mathematics that Einstein used was called the absolute differential calculus. From 1915 to the 1960's it was called tensor analysis, and today it is part of differential geometry. The invariant mathematical equation expressing a physical law is called the law's tensor form. It's like the interval form of the anthem in the picture.




C







C sharp







D







D sharp







E







F







F sharp







G







G sharp







A







A sharp







B




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