Lesson 13: Mathematics and Music

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.title[
# Lesson 13: Mathematics and Music
]
.author[
### Fei Ye
]
.date[
### May, 2024
]

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## Unit 11A: Mathematics and Music

*Primary Source:* PPT for the book "Using & Understanding Mathematics".

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## Sound and Music (1/2)

- Any vibrating object produces sound. The vibrations produce a wave.

- Most musical sounds are made by vibrating strings, vibrating reeds, or vibrating columns of air.

- One basic quality of sound is pitch . The shorter the string, the higher the pitch.

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## Sound and Music (2/2)

- The **frequency** of a vibrating string is the rate at which it moves up and down. The higher the frequency, the higher the pitch. Measured in *cycles per second* (cps).

- The lowest possible frequency for a particular string, called its **fundamental frequency**, occurs when it vibrates up and down along its full length.

- Waves that have frequencies that are integer multiples of the fundamental frequency are called **harmonics**.

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## String Vibrations

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![](data:image/png;base64,#img/StringVibration.png)
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## Music Scales and Mathematics (1/2)

- Raising the pitch by an **octave** corresponds to a doubling of the frequency.

- Pairs of notes sound particularly pleasing when one note is an octave higher than the other note.

- An octave is the interval between, say, middle C and the next higher C. For example, middle C has a frequency of 260 cps, the C above middle C has a frequency of `$2\times260 \text{ cps}= 520 \text{ cps}$`.

- The musical tones that span an octave comprise a **scale**.

- The Greeks invented the 7-note (or diatonic) scale that corresponds to the white keys on the piano.

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## Music Scales and Mathematics (2/2)

- Johann Sebastian Bach adopted a 12-tone scale, which corresponds to both the white and the black keys on a modern piano.

- On the 12-tone scale, two consecutive notes on the piano keyboard are separated by a **half-step**.

- For each half-step, the frequency increases by some multiplicative factor `$f$`.
.center[
![:resize , 50%](data:image/png;base64,#img/pianokeys.png)
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.footmark[
  Image Source: [Understanding Half Steps and Whole Steps](http://www.fundamentalsofmusic.com/melody-melodic-intervals-half-step-whole-step.html)
]

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## Music Scales and Mathematics

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![](data:image/png;base64,#img/MusicScalesMathematics.png)
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## Example: The Dilemma of Temperament (1 of 2)

Because the whole-number ratios in the table are not exact, tuners of musical instruments have the problem of *temperament*, which can be demonstrated as follows.

Start at middle C with a frequency of 260 cps. Using the whole-number ratios, find the frequency if you raise C by a sixth to A, raise A by a fourth to D, lower D by a fifth to G, and lower G by a fifth to C.

Having returned to the same note, have you also returned to the same frequency?

**Solution:** According to the table, raising a note by a sixth increases its frequency by a factor of approximately 5/3. For example, the frequency of A above middle C is
`$$5/3 \times 260 \text{ cps} = 433.33 \text{ cps}.$$`

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## Example: The Dilemma of Temperament (2 of 2)

Raising this note by a fourth increases its frequency by 4/3, producing D with a frequency of
`$4/3 \times 433.33 \text{ cps} = 577.77 \text{ cps}.$`

Lowering D by a fifth (a factor of 2/3) to G gives a frequency of
`$2/3 \times 577.77 \text{ cps} = 385.18 \text{ cps}.$`

Lowering G by another fifth (a factor of 2/3) puts us back to middle C, but with a frequency of
`$2/3 \times 385.18 \text{ cps} = 256.79 \text{ cps}.$`

Note that, by using whole-number ratios, we have not quite returned to the proper frequency of 260 cps for middle C.

The problem is that the whole-number ratios are not exact. That is, `$5/3 \times 4/3 \times 2/3 \times 2/3 = 80/81$` is close to, but not exactly 1.

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## Musical Scales as Exponential Growth

An exponential function can be used to find any frequency on the scale.
If `$Q_0$` is the initial frequency, then the frequency of the note `$n$` half-steps higher is given by
`$$Q = Q_0  \times f^n \approx Q_0 \times 1.05946^n.$$`

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## Example: Exponential Growth in Musical Scales (1/2)

Use the exponential growth law to find the frequency of

1. the note a fifth above middle C,
2. the note one octave and a fifth above middle C, and
3. the note two octaves and a fifth above middle C.

**Solution:** We let the frequency of middle C be the initial value for the scale.
Set `$Q_0 = 260 \text{ cps}$`.

The note a fifth above middle C is G, which is seven half-steps above middle C. Therefore, we let `$n = 7$` in the exponential law and find that the frequency of G is
`$$Q \approx Q_0 \times 1.05946^7 \approx 390 \text{ cps}.$$`

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## Example: Exponential Growth in Musical Scales (2/2)

The note one octave and a fifth above middle C is `$12 + 7 = 19$` half-steps above middle C. Letting `$n = 19$`, we find that the frequency of this note is
`$$Q \approx  Q_0 \times 1.05946^{19} \approx 779 \text{ cps}.$$`

The note two octaves and a fifth above middle C is `$(2 \times 12) + 7 = 31$` half-steps above middle C. Letting `$n = 31$`, we find that the frequency of this note is
`$$Q \approx Q_0 \times 1.05946^{31} \approx 1558 \text{ cps}.$$`

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## Practice: Notes of a Scale

Find the frequency of the 12 notes of the scale that starts at the F above middle C; this F has a frequency 347 cps.

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## Practice: Notes of Scales Above or Below A

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  <iframe src="https://www.myopenmath.com/embedq2.php?id=679310&amp;seed=2021&amp;showansafter" width="100%" height="400px" data-external="1"></iframe>
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<a href="https://www.myopenmath.com/embedq2.php?id=679310&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a>
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## Practice: Exponential Growth and Scale

Starting at middle C, with a frequency of 260 cps, find the frequency of the following notes:

1. seven half-steps above middle C
2. a third (four half-steps) above middle C
3. an octave and a fifth (seven half-steps) above middle C