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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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Distribution of E-Book Sales on Amazon

For e-books on Amazon the relationship between the daily sales rate s and the rank r is approximately given by:

s = 100,000 / r

Such an inverse proportional relationship between a ranked quantity and the rank is called a Zipf distribution. So a book on rank r = 10,000 can be expected to sell s = 100,000 / 10,000 = 10 copies per day. As of November 2013, there are about 2.4 million e-books available on Amazon’s US store (talk about a tough competition). In this post we’ll answer two questions. The first one is: how many e-books are sold on Amazon each day? To answer that, we need to add the daily sales rate from r = 1 to r = 2,400,000.

s = 100,000 · ( 1/1 + 1/2 + … + 1/2,400,000 )

We can evaluate that using the approximation formula for harmonic sums:

1/1 + 1/2 + 1/3 + … + 1/r ≈ ln(r) + 0.58

Thus we get:

s ≈ 100,000 · ( ln(2,400,000) + 0.58 ) ≈ 1.5 million

That’s a lot of e-books! And a lot of saved trees for that matter. The second question: What percentage of the e-book sales come from the top 100 books? Have a guess before reading on. Let’s calculate the total daily sales for the top 100 e-books:

s ≈ 100,000 · ( ln(100) + 0.58 ) ≈ 0.5 million

So the top 100 e-books already make up one-third of all sales while the other 2,399,900 e-books have to share the remaining two-thirds. The cake is very unevenly distributed.

This was a slightly altered excerpt from More Great Formulas Explained, available on Amazon for Kindle. For more posts on the ebook market go to my E-Book Market and Sales Analysis Pool.

Analysis of Viewers for TV Series

I analysed the number of viewers of all the completed seasons for the following tv shows: Fringe, Lost, Heroes, Gossip Girl, Vampire Diaries, True Blood, The Sopranos, How I met your Mother, Glee and Family Guy. The data was taken from the respective Wikipedia pages.

My aim was to find simple “rule-of-thumb” formulas to estimate key values from the number of premiere viewers and to see if there’s a pattern for the decline of a show. Below you can see the main results from the analysis.

Result 1: Finale vs. Premiere

The number of finale viewers is about 85 % the number of premiere viewers.

Result 2: Average vs. Premiere

The average number of viewers during a season is about 83 % the number of premiere viewers.

finaleaverage

Result 3: Decline Pattern

The average number of viewers during a season is about 93 % the average number of viewers during the previous season.

averageprevious

This last result implies that the decline in popularity is exponential. If the average number of viewers for the first season is N(1), then the expected number of viewers for season n is: N(n) = N(1) * 0.93^(n-1). We can also express this using a table:

Average season two = 93 % of average season one

Average season three = 86 % of average season one

Average season four = 80 % of average season one

Average season five = 75 % of average season one

Average season six = 70 % of average season one

etc …

Of course, this is all just the sum of the behaviour of all the analyzed shows. Individual shows can behave very differently form that.