Now that we have learned all about harmonics/ overtones, let us put our knowledge to work. We know that given a fundamental
frequency, we will be able to calculate the overtones of that frequency. Those overtones relate to fundamental frequencies of other notes that when played
with the first fundamental note, will sound consonant together. These are called Pentatonic Scales and usually include the first fundamental note with four
of its overtone notes for a total of five notes, hence "Penta-" meaning five.
Now it is your job to discover your own Pentatonic Scale. Open this website
Piano Player 7 and make sure to enable your Adobe flash
player. After it has loaded you should see something like this:
Click on the button with the # to have the labels for the keys show.
If you click on a key, then in the right screen on the keyboard you should see what key number you have selected, what note
it is (ex. A3, or C5), and the frequency of that note in hz (ex. 220hz, or 523hz). These frequencies are rounded for easy use. If you want a list of the
actual frequency of each note, click here 6 .
Now, select a note you would like to be the fundamental of your pentatonic scale (I suggest picking a white key for easier
playing later on). For this example I have chosen C3 which has a frequency of 131hz.
Using our knowledge of calculating overtones, we can easily fill out the table below (for the last column, use the online
keyboard to determine which note goes with which frequency. The frequencies won't always perfectly coordinate with a note so pick the one that is closest
to it):
Frequency
Note
Fundamental
f =131hz
C3
1st Overtone
2(f )=2(131)=262hz
C4
2nd Overtone
3(f )=3(131)=393hz
G4
3rd Overtone
4(f )=4(131)=524hz
C5
4th Overtone
5(f )=5(131)=655hz
E5
Now that we have our Pentatonic Scale, let's play it! You can play the white keys of the keyboard using your computer keyboard. Here is a picture of what
computer keys match up with which piano keys. The blue numbers on the computer keys correspond to the numbers on the piano keys.
Find what keys you need to press then go back to the online piano keyboard and play away!
Congratulations! You've calculated and played your very own Pentatonic Scale! :)
Pentatonic Scales
As was mentioned earlier, a Pentatonic Scale is the collection of five notes starting with a fundamental note and four
other notes that correlate with the frequencies of fundamental note's overtones. Pentatonic Scales are universal and are imbedded into our brains.
A musician by the name of Bobby McFerrin figured this out and now knows how to play... an audience. Watch the video to see how 5 .
Famous pianist Assaff Weisman explains the mathematics behind Bobby McFerrin's seemingly magical show 10 .
Overtones are what give timbre
This video helps emphasize the point pianist Assaff Weisman was trying to make about how an overtone of one note can
actually cause the strings of another note (whose frequencies match up with the overtones of the fundamental note) to vibrate without striking them
(I personally think you can hear it better in this video) 8 .
The video also talks about how adding the functions of the overtones will change the function of the fundamental.
This applet shows what the functions of the eight notes in the 4th octave would look like if we were to add the harmonics of each function. Select or deselect a note
by clicking the box to the left of it, then use the slider to add more harmonics.
Lastly, the video also discusses how overtones give an instrument its "brightness or color" 8 . This is called timbre, and is the reason
why a C on the piano and a C on a flute both sound like a C but have distinct sounds that tell your brain, "Hey, that is a piano" and "Hey, that is a flute."
I hope you had as much fun learning about how the Harmonic Series as I did! :)
