Insights into Mathematics | The impossibility of a Pythagorean scale and "sqrt(2)" | Maths and Music | N J Wildberger @njwildberger | Uploaded 2 years ago | Updated 23 hours ago
Let's use the important cycle of fifths and the musical clock system based on the 12 tone chromatic scale to try to precisely create frequencies for a "perfect" scale. This attempt we know is doomed to failure, but it is instructive to see the same arithmetical issue encountered in the previous video manifesting itself in a different way.

Also the symmetrical role of the "augment fourth" or "diminished fifth" interval, which we now know is more properly described as a "6-step", becomes clearer, and the connection with the arithmetic conundrum around discussion of the non-existent "square root of 2" becomes clearer.

Isn't it amazing that the Pythagorean essentially bumped into the same arithmetical problem, at least implicitly, in two quite different fashions? From one point of view it is the geometric relation between the side and diagonal of a square, while from another it is the musical relation between a note and the interval which is in some sense "halfway" to the octave.

The cycle of fourths also manifests itself here: going clockwise by steps of 7 is really the same as going counterclockwise by steps of 5. Those two numbers 7 and 5 really play a crucial role in a proper understanding of modern music built up from the even tempered scale, as well as forming the bedrock pattern on which much of classical music, jazz, pop and rock'n roll are based.

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