Frequencies of Notes on a Piano: Learning object
On the previous page, What are the frequencies of music notes?, we learned how to find the frequencies for a piano tuned by "equal-temperament". It turns out only 7 notes are actually "in tune" on a piano, and all the others are slightly out.
Let's explore that concept on this page. This is a practical example of the graphs we learned about in Graphs of y = a sin bx and y = a cos bx.
In the learning object below, there is a piano which you can play (one note at a time). As you play each new note, you'll see a graph demonstrating the frequency of that note. As the frequency goes up, the wavelength goes down.
Two important notes are indicated on the piano:
Middle C is marked in red (this is the note usually in the middle of an 88-key piano); and
A-440 (the A with frequency 440 cycles/second, or 440 Hz) is marked in light blue.
Each 1 unit on the t-axis represents one wavelength of the "fundamental" note for our applet, A below middle C. The frequency id 220 Hz, so the graph is given by (where f is the frequency):
y = sin(2πft) = sin(2π(220)t) = sin(1382.3t)
The time taken for each wave to pass our ear for A-220 is `1/220` second = `0.004545\ s`. This is what 1 unit represents on the t-axis below.
Things to do
1. Play the A on the far left. It has frequency 220 Hz.
2. Next, play some of the notes near to that A and notice how the graph changes as the frequency changes.
3. Now, play the higher A. It has a frequency of 440 Hz, and is one octave above the first A. Not how there are 2 wavelengths for A-440 in the space of one length of A-220.
4. Next read the descriptions (below the graph) for the notes that sound "nice" with A (that is C#, and E). These are simple multiples of the fundamental frequency. (Such notes are called "concordant" in music.) Where I say it "should be" a certain frequeny, I'm referring to the case where it was tuned so that A major sounded the best.
5. See what it says in the description for the "not-so-nice" notes. (Such notes are called "discordant" in music.)
6. When you choose "Combined signal", you'll see the graph of the addition of the A-220 signal and whatever note was just played (the component signals are shown in light grey), and will hear the 2 tones.
NOTE: The lower sounds may not play so well on a mobile phone speaker (if so, use earphones).
Note: A Frequency = 220 Hz
The function: y = sin(2π(220)t) = sin(1382.3t)
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See the background to the above learning object in:
Frequency of notes on a piano - interactive learning object (some basic analysis of what you are seeing above)