frequency

Overtones – What They Are And How To Compute Them

In theory, hitting the middle C on a piano should produce a sound wave with a frequency of 523.25 Hz and nothing else. However, running the resulting audio through a spectrum analyzer, it becomes obvious that there’s much more going on. This is true for all other instruments, from tubas to trumpets, basoons to flutes, contrabasses to violins. Play any note and you’ll get a package of sound waves at different frequencies rather than just one.

First of all: why is that? Let’s focus on stringed instruments. When you plug the string, it goes into its most basic vibration mode: it moves up and down as a whole at a certain frequency f. This is the so called first harmonic (or fundamental). But shortly after that, the nature of the vibration changes and the string enters a second mode: while one half of the string moves up, the other half moves down. This happens naturally and is just part of the string’s dynamics. In this mode, called the second harmonic, the vibration accelerates to a frequency of 2 * f. The story continues in this fashion as other modes of vibration appear: the third harmonic at a frequency 3 * f, the fourth harmonic at 4 * f, and so on.

String Vibrating Modes Overtones

A note is determined by the frequency. As already stated, the middle C on the piano should produce a sound wave with a frequency of 523.25 Hz. And indeed it does produce said sound wave, but it is only the first harmonic. As the string continues to vibrate, all the other harmonics follow, producing overtones. In the picture below you can see which notes you’ll get when playing a C (overtone series):

(The marked notes are only approximates. Taken from http://legacy.earlham.edu)

Quite the package! And note that the major chord is fully included within the first four overtones. So it’s buy a note, get a chord free. And unless you digitally produce a note, there’s no avoiding it. You might wonder why it is that we don’t seem to perceive the additional notes. Well, we do and we don’t. We don’t perceive the overtones consciously because the amplitude, and thus volume, of each harmonic is smaller then the amplitude of the previous one (however, this is a rule of thumb and exceptions are possible, any instrument will emphasize some overtones in particular). But I can assure you that when listening to a digitally produced note, you’ll feel that something’s missing. It will sound bland and cold. So unconsciously, we do perceive and desire the overtones.

If you’re not interested in mathematics, feel free to stop reading now (I hope you enjoyed the post so far). For all others: let’s get down to some mathematical business. The frequency of a note, or rather of its first harmonic, can be computed via:

(1) f(n) = 440 * 2n/12

With n = 0 being the chamber pitch and each step of n one half-tone. For example, from the chamber pitch (note A) to the middle C there are n = 3 half-tone steps (A#, B, C). So the frequency of the middle C is:

f(3) = 440 * 23/12 = 523.25 Hz

As expected. Given a fundamental frequency f = F, corresponding to a half-step-value of n = N, the freqency of the k-th harmonic is just:

(2) f(k) = k * F = k * 440 * 2N/12

Equating (1) and (2), we get a relationship that enables us to identify the musical pitch of any overtone:

440 * 2n/12 = k * 440 * 2N/12

2n/12 = k * 2N/12

n/12 * ln(2) = ln(k) + N/12 * ln(2)

n/12 = ln(k)/ln(2) + N/12

(3) n – N = 12 * ln(k) / ln(2) ≈ 17.31 * ln(k)

The equation results in this table:

k

n – N (rounded)

1 0
2 12
3 19
4 24
5 28

And so on. How does this tell us where the overtones are? Read it like this:

  • The first harmonic (k = 1) is zero half-steps from the fundamental (n-N = 0). So far, so duh.
  • The second harmonic (k = 2) is twelve half-steps, or one octave, from the fundamental (n-N = 12).
  • The third harmonic (k = 3) is nineteen half-steps, or one octave and a quint, from the fundamental (n-N = 19).
  • The fourth harmonic (k = 4) is twenty-four half-steps, or two octaves, from the fundamental (n-N = 24).
  • The fifth harmonic (k = 5) is twenty-wight half-steps, or two octaves and a third, from the fundamental (n-N = 28).

So indeed the formula produces the correct overtone series for any note. And for any note the same is true: The second overtone is exactly one octave higher, the third harmonic one octave and a quint higher, and so on. The corresponding major chord is always contained within the first five harmonics.

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Wavelength (And: Why Is The Sky Blue?)

A very important type of length is wavelength, usually symbolized by the Greek letter λ (in m). It is defined as the distance from crest to crest (one complete cycle) and can easily be calculated for any wave by dividing the speed of the wave c (in m/s) by its frequency f (in Hz):

λ = c / f

What are typical wavelengths for sound? At room temperature, sound travels with a speed of c = 343 m/s. The chamber pitch has a frequency of f = 440 Hz. According to the equation, the corresponding wavelength is:

λ = 343 / 440 ≈ 0.8 m ≈ 2.6 ft

Are you surprised? I bet most people would greatly underestimate this value. Bass sounds are even longer than that. The lowest tone on a four-string bass guitar has a frequency of f = 41.2 Hz, which leads to the wavelength:

λ = 343 / 41.2 ≈ 8.3 m ≈ 27 ft

So the wave coming from the open E string of a bass guitar doesn’t even fit in a common room. In the case of light, the situation is very different. As noted in the introduction, the wavelength of light ranges between 4000 A (violet light) and 7000 A (red light), which is just below the size of a bacterium.

Wavelength plays an important role in explaining why the sky is blue. When light collides with a particle, parts of it are deflected while the rest continues along the initial path. This phenomenon is known as scattering. The smaller the wavelength of the light, the stronger the effect. This means that scattering is particularly pronounced for violet and blue light.

Unless you are looking directly at the Sun, all the light you see when looking at the sky is scattered light coming from the particles in the atmosphere. Since blue light tends to scatter so easily, the sky ends up in just this color. But why not violet? This is a legitimate question. After all, due to its smaller wavelength, violet light is even more willing to scatter. While this is true, it is also important to note that the sun’s rays don’t contain all the colors in the same ratio. In particular, they carry much less violet than blue light. On top of that, our eyes are less sensitive to violet light.

(This is an excerpt from my Kindle book: Physics! In Quantities and Examples)

Quantitative Analysis of Top 60 Kindle Romance Novels

I did a quantitative analysis of the current Top 60 Kindle Romance ebooks. Here are the results. First I’ll take a look at all price related data and conclusions.

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  • Price over rank:

pricerank

There seems to be no relation between price and rank. A linear fit confirmed this. The average price was 3.70 $ with a standard deviation of 2.70 $.

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  • Price frequency count:

pricescount

(Note that prices have been rounded up) About one third of all romance novels in the top 60 are offered for 1 $. Roughly another third for 3 $ or 4 $.

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  • Price per 100 pages over rank:

pricerank

Again, no relation here. The average price per 100 pages was 1.24 $ with a standard deviation of 0.86 $.

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  • Price per 100 pages frequency count:

PPP1

About half of all novels in the top 60 have a price per 100 pages lower than 1.20 $. Another third lies between 1.20 $ and 1.60 $.

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  • Price per 100 pages over number of pages:

PPP2

As I expected, the bigger the novel, the less you pay per page. Romance novels of about 200 pages cost 1.50 $ per 100 pages, while at 400 pages the price drops to about 1 $ per 100 pages. The decline is statistically significant, however there’s a lot of variation.

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  • Review count:

reviewscount

A little less than one half of the top novels have less than 50 reviews. About 40 % have between 50 and 150 reviews. Note that some of the remaining 10 % more than 600 reviews (not included in the graph).

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  • Rating over rank:

rankreviews

There’s practically no dependence of rank on rating among the top 60 novels. However, all have a rating of 3.5 stars or higher, most of them (95 %) 4 stars or higher.

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  • Pages over ranking:

pagesrank

There’s no relation between number of pages and rank. A linear fit confirmed this. The average number of pages was 316 with a standard deviation of 107.

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  • Pages count:

pagescount

About 70 % of the analyzed novels have between 200 and 400 pages. 12 % are below and 18 % above this range.