# Frequencies of Notes on a Piano: Learning object

## Applet Description

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 Cis 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(1382.3*t*)

Copyright © www.intmath.com

Credit: Loosely based on CSS Tricks and Chris Lowis

### Don't miss...

See the background to the above learning object in:

What are the frequencies of music notes?

Frequency of notes on a piano - interactive learning object (some basic analysis of what you are seeing above)

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