MUSIC: tuning, scales, fractions, ratio, harmonics :MATH ! Part deux.

Recap from previous post. here is a readable table of the notes and relative frequencies for the scale of C:

beat harmonic scale
Notice that the whole tone intervals are not all the same size. There are two distinct sizes, with frequency ratios 9/8 and 10/9. Only the two semitone intervals EF and BC’ are the same. Check the others ! Use the slightly not obvious relationship between ratios  P/Q = (P/R)/(Q/R)

beat tone semitone

This was the situation faced by J. S. Bach when he wanted to write music in any key. He figured that if two intervals had the same relative size then the two notes from the first, played together, would have the same musical “feel” as the two notes from the second interval. He wanted the major chord Bflat-D-F to sound like the major chord C-E-G, and so on. Not only that, but the scale devised so far is lacking four of the five “black” notes, and hitherto these had been fudged in. If one pursues the harmonic approach one finds that for example F-sharp and G-flat are different. See later for details.

So he figured that as there were 12 semitones in the full octave the frequency of each note would have to be a fixed multiple of the previous note, so multiplying 1 by this fixed number 12 times must get you to 2. In other words the magic number is the twelfth root of 2. Its value is 1.059463094 from my calculator.

The table below shows the comparison between the new Equal Temper scale and the harmonic scale.

beat pitch comparison

In 1722 Bach composed twelve preludes and fugues for keyboard called Das Wohltemperierte Klavier

If the link doesn’t work here it is:

https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier

Part three will look at beat frequencies and reasons for preferring the harmonic system, though not for keyboard instruments.

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Filed under fractions, math, music, musical scales, teaching

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