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# The chromatic scale.
We hear musical notes, and some may even sound pleasant. But what exactly are these notes?
Each musical note has a certain frequency. We typically measure frequency in hertz (Hz), where 1 Hz is 1 cycle per second. So a tone of 440 Hz is a sound produced by oscillating 440 times every one second.
Western music is predominately given in the 12 notes chromatic scale in one octave, given by
C, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, B
And it repeats again, starting the next octave with C, and so on.
![[1 teaching/summer program 2023/puzzles-and-problems/---files/Pasted image 20230830100704.png]]
The octaves are labeled with numbers, so C4 is the C note in the 4th octave, and C5 is the C note in the 5th octave. Beginners learn music often starting with C4 ("middle C").
Each octave starts with the note C. So for instance, we have the ordering of notes:
![[1 teaching/summer program 2023/puzzles-and-problems/---files/chromatic-scale 2023-08-30 10.16.24.excalidraw.svg]]
%%[[1 teaching/summer program 2023/puzzles-and-problems/---files/chromatic-scale 2023-08-30 10.16.24.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/puzzles-and-problems/---files/chromatic-scale 2023-08-30 10.16.24.excalidraw.dark.svg|dark exported image]]%%
But how are these notes related to each other, and what happens if it keeps on going to the next octave? This is what we will investigate.
First we need something that can detect the frequency of the musical notes. One can use a tuner that you find on your phone app store, or here is a rudimentary one: [bonsoon.net/dev/tuner/](https://bonsoon.net/dev/tuner/)
And something that can play a musical note. We can use Chrome music lab: [https://musiclab.chromeexperiments.com/](https://musiclab.chromeexperiments.com/) (and click on shared piano)
Let us find out what are all the frequencies of the notes from C3 to B4.
Use the piano player and the tuner to find out their frequencies below (take a few good measurements and use an average for each). The tuner should tell you the note name if you are close to that note. Record your findings below:
Measurements of Frequency in Hz:
$$
\begin{array}{c|c|c|c|c|c} \\
n&\text{note}&\text{trial 1} &\text{trial 2} & \text{trial 3} & \text{average} \\ \hline
1&C3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} &\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} &\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & \\\hline
2&C♯3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
3&D& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
4&D♯3&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
5&E3&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
6&F3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
7&F♯3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
8&G3&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
9&G♯3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
10&A3&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
11&A♯3&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
12&B3& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
13&C4& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
14&C♯4& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
15&D&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
16&D♯4& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
17&E4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
18&F4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
19&F♯4& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
20&G4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
21&G♯4& \phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}}& & & &\\\hline
22&A4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
23&A♯4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
24&B4&\phantom{aaaa+\frac{1}{a+\frac{1}{1+\frac{a}{a}}}} & & & &\\\hline
\end{array}
$$
(continued)
Now, go to [https://www.desmos.com/calculator](https://www.desmos.com/calculator), we shall plot and find a curve that best fit this data.
We arbitrarily name the notes with $n=1$ starting with C3, and $n=24$ ending with $B4$.
Using desmos, click the + button and add a table. ![[1 teaching/summer program 2023/puzzles-and-problems/---files/Pasted image 20230830104634.png]]
Change the names of the variables to $n$ and $f$:
![[1 teaching/summer program 2023/puzzles-and-problems/---files/Pasted image 20230830104722.png]]
And put in the average frequency you measured for each note.
To see your data better you can click the gear icon and change the x and y ranges to $-5 \le X \le 25$ and $0\le Y \le 500$
![[1 teaching/summer program 2023/puzzles-and-problems/---files/Pasted image 20230830104901.png]]
One you have done that, we can enter below to ask what model fits our data points the best, we will explore three:
(a) Linear: $f\sim An+B$
(b) Power: $f\sim An^{B}$
(c) Geometric: $f\sim AB^{n}$
Type each one in separately, like so:
![[1 teaching/summer program 2023/puzzles-and-problems/---files/Pasted image 20230830105644.png]]
And desmos will calculate a model based on those forms.
![[---images/---assets/---icons/question-icon.svg]] Which one looks the best? Record the $A,B$ value for each, and also the $R^{2}$ value.
The one that is best describing should have an $R^{2}$ value closer to 1.
![[---images/---assets/---icons/question-icon.svg]] With the best model you have above, what is the $B$ value? This number is actually somewhat special. Look it up on the internet and see if it matches your $B$.
![[---images/---assets/---icons/question-icon.svg]] What do you learn about the relationships between each consecutive note?
![[---images/---assets/---icons/question-icon.svg]] Can you predict what the frequency of $C5$ is? What about $C0$ ?