Search IntMath
Close

# 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 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).

Combined signal: no yes

Note: A Frequency = 220 Hz

The function: y = sin(2π(220)t) = sin(1382.3t)

Credits: Loosely based on CSS Tricks and Chris Lowis' Playing multiple notes on Web Audio.

### 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)

Graphs of y = a sin bx and y = a cos bx

## Problem Solver This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.