Take the frequency of middle C to be f0 = 262 Hz and compute the ratios fn/f0 fo

Question

Take the frequency of middle C to be f0 = 262 Hz and compute the ratios fn/f0 for the frequencies fn corresponding to a) D, b) E, c) F, d) G, e) A, and f) B and g) F#. By referring to Table 12-3, you will see that for D, n = 2. For the work that you will turn in, do the following additional steps. Convert the decimals that your calculator reports to the nearest fraction (e.g. 1.49 -> 3/2). This does not work very well for F#, but do you recognize the number that your calculator gives you for this value? (This interval is called the tritone. It's one of the "blue notes" that figure prominently in Blues, Jazz, and Rock and Roll. What interval is present in this (12 tone) scale that the Greeks found especially harmonious? a)  b) c)  d)  e)  f)  g)

Explanation / Answer

Please note that the ratios of frequencies of various notes changes with the tuning system. Since nothing has been mentioned about it, I am going to assume the Equal Temperament tuning system which is, since the last decade, well accepted and widely used in all the musical instruments worldwide. In that tuning system the frequencies corresponding to the notes of the Chromatic scale (all the notes included, that is 12 per octave) when arranged in increasing order, form a GP series, that is Geometric progression. And the common ratio of that series is 21/12   , that is the twelfth root of two.

In other words, the ratio between two adjacent/consecutive notes is 21/12 .

So, if two notes are one semitone apart (adjacent), then the ratio of their frequencies is 21/12

if two notes are two semitones apart ( like D and C or G and F ), the ratio of their frequencies is 22/12

Similarly, if two notes are n semitones apart, the ratio of their frequencies will be 2n/12

So, understand that in our problem, the fn /f ois the ratio of two frequencies which are n semitones apart.

So, fn/fo = 2n/12

now coming to the problem:

a.) D and C are two semitones apart, so fn/fo = f2/fo = 22/12 = 1.1224620 ~ 9/8

b.) E and C are four semitones apart, so fn/fo = f4 /fo = 24/12 = 1.2599210 ~ 81/64

c.) F and C are five semitones apart. So, fn/fo = f5 /fo = 25/12 = 1.3348398 ~ 4/3

d.) G and C are seven semitones apart. So, fn/fo = f7 /fo = 27/12 = 1.4983070 ~ 3/2

e.) A and C are nine semitones apart. So, fn/fo = f9 /fo = 29/12 = 1.6817928 ~ 27/16

f.) B and C are eleven semitones apart. So, fn/fo = f11 /fo = 211/12 = 1.8877486 ~ 243/128

g.) F# and C are six semitones apart. So, fn/fo = f6 /fo = 26/12 = 1.4142135 ~ 729/ 512

F# and C are six semitones apart or three full tones apart(That is why it is called a tritone! )

Now, understand that the fractions in bold are the basis for the famous Pythagorean tuning system which was developed by the Greek mathematician Pythagoras. When the questioner talked about the harmonious nature that the ancient greeks found, he is undoubtedly talking about the Pythagorean tuning system. In this tuning system, the ratios of all intervals are based on the ratio 3:2 which is also known as the 'pure' perfect fifth. It is chosen because it is one of the most consonant and easiest to tune by ear. If you observe all the ratios of this Pythagorean tuning system (given in bold) are a power of 3 divided by a power of 2 or vice-versa. Some of these Pythagorean intervals are also used in other tuning systems. For example 'Just Intonation' tuning system also has the Pythagorean fifth and fourth. Not to go deep into the theory, in simple words, it is the ratio 3:2 which is the backbone of Pythagorean tuning system ( which the questioner refers to as the scale of Greeks) that gives it the harmonious nature. And yes, the values of fn /fo that we found out using the Equal temperament tuning system ( or the ratios of today's music! ) are not exactly equal to these Pythagorean ratios. That is why we call it Equal Temperament system. Because the ratios have been Equally Tempered or Modified. Why? There are some interesting reasons behind it. But that is a topic for another day ...

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