Math in the Real World: Music (9+)

CEMC

Math in the Real World: Music (9+) CEMC 1 / 21

The Connection

Many of you probably play instruments! But did you know that thefoundations of music are built with mathematics?

One of the first things you learn when you start with instruments issomething called a scale. The scales that we play have evolved over timeand vary by region.

The fundamental mathematics behind these scales? Fractions!

Math in the Real World: Music (9+) CEMC 2 / 21

The Connection

Many of you probably play instruments! But did you know that thefoundations of music are built with mathematics?

One of the first things you learn when you start with instruments issomething called a scale. The scales that we play have evolved over timeand vary by region.

The fundamental mathematics behind these scales? Fractions!

Math in the Real World: Music (9+) CEMC 2 / 21

The Connection

Many of you probably play instruments! But did you know that thefoundations of music are built with mathematics?

One of the first things you learn when you start with instruments issomething called a scale. The scales that we play have evolved over timeand vary by region.

The fundamental mathematics behind these scales? Fractions!

Math in the Real World: Music (9+) CEMC 2 / 21

Frequency

The sound of a note is based on itsfrequency.

In terms of a string, like the onesplucked on a guitar or struck by a ham-mer in a piano, this means the numberof vibrations up and down per second.

This is measured in a unit called theHertz (Hz). One Hertz means onevibration per second.

Math in the Real World: Music (9+) CEMC 3 / 21

Frequency

The sound of a note is based on itsfrequency.

In terms of a string, like the onesplucked on a guitar or struck by a ham-mer in a piano, this means the numberof vibrations up and down per second.

This is measured in a unit called theHertz (Hz). One Hertz means onevibration per second.

Math in the Real World: Music (9+) CEMC 3 / 21

Frequency

The sound of a note is based on itsfrequency.

In terms of a string, like the onesplucked on a guitar or struck by a ham-mer in a piano, this means the numberof vibrations up and down per second.

This is measured in a unit called theHertz (Hz). One Hertz means onevibration per second.

Math in the Real World: Music (9+) CEMC 3 / 21

An Introduction to Scales

A scale starts with a base frequency and ends on a note that is twice thefrequency (the scale runs through an octave).

We will focus on three types of Western scales, as they evolved throughtime.

Our story begins with the ancient Greeks.

They noticed that strings in the ratio 32 sounded quite good together.

Thus, the ‘Pythagorean Scale’ was created.

Math in the Real World: Music (9+) CEMC 4 / 21

An Introduction to Scales

A scale starts with a base frequency and ends on a note that is twice thefrequency (the scale runs through an octave).

We will focus on three types of Western scales, as they evolved throughtime.

Our story begins with the ancient Greeks.

They noticed that strings in the ratio 32 sounded quite good together.

Thus, the ‘Pythagorean Scale’ was created.

Math in the Real World: Music (9+) CEMC 4 / 21

An Introduction to Scales

A scale starts with a base frequency and ends on a note that is twice thefrequency (the scale runs through an octave).

We will focus on three types of Western scales, as they evolved throughtime.

Our story begins with the ancient Greeks.

They noticed that strings in the ratio 32 sounded quite good together.

Thus, the ‘Pythagorean Scale’ was created.

Math in the Real World: Music (9+) CEMC 4 / 21

An Introduction to Scales

A scale starts with a base frequency and ends on a note that is twice thefrequency (the scale runs through an octave).

We will focus on three types of Western scales, as they evolved throughtime.

Our story begins with the ancient Greeks.

They noticed that strings in the ratio 32 sounded quite good together.

Thus, the ‘Pythagorean Scale’ was created.

Math in the Real World: Music (9+) CEMC 4 / 21

An Introduction to Scales

A scale starts with a base frequency and ends on a note that is twice thefrequency (the scale runs through an octave).

We will focus on three types of Western scales, as they evolved throughtime.

Our story begins with the ancient Greeks.

They noticed that strings in the ratio 32 sounded quite good together.

Thus, the ‘Pythagorean Scale’ was created.

Math in the Real World: Music (9+) CEMC 4 / 21

Pythagorean Scale

To create this scale, start with a base frequency. For simplicity, give this avalue of 1.

As we know, an octave ends at double the frequency, which in this case is 2.

As the ratio 32 is nice, we multiply the base frequency by 3

2 and divide thetop frequency by 3

2 .

This leaves us with four missing notes in our eight note scale.

To fill in the last four, we iterate by multiplying by 32 , and every time we

obtain a value greater than 2, we halve it. We do this because we do notwant any notes in our scale higher than an octave above the base. Try itout!

Math in the Real World: Music (9+) CEMC 5 / 21

Pythagorean Scale

To create this scale, start with a base frequency. For simplicity, give this avalue of 1.

As we know, an octave ends at double the frequency, which in this case is 2.

As the ratio 32 is nice, we multiply the base frequency by 3

2 and divide thetop frequency by 3

2 .

This leaves us with four missing notes in our eight note scale.

To fill in the last four, we iterate by multiplying by 32 , and every time we

obtain a value greater than 2, we halve it. We do this because we do notwant any notes in our scale higher than an octave above the base. Try itout!

Math in the Real World: Music (9+) CEMC 5 / 21

Pythagorean Scale

To create this scale, start with a base frequency. For simplicity, give this avalue of 1.

As we know, an octave ends at double the frequency, which in this case is 2.

As the ratio 32 is nice, we multiply the base frequency by 3

2 and divide thetop frequency by 3

2 .

This leaves us with four missing notes in our eight note scale.

To fill in the last four, we iterate by multiplying by 32 , and every time we

obtain a value greater than 2, we halve it. We do this because we do notwant any notes in our scale higher than an octave above the base. Try itout!

Math in the Real World: Music (9+) CEMC 5 / 21

Pythagorean Scale

To create this scale, start with a base frequency. For simplicity, give this avalue of 1.

As we know, an octave ends at double the frequency, which in this case is 2.

As the ratio 32 is nice, we multiply the base frequency by 3

2 and divide thetop frequency by 3

2 .

This leaves us with four missing notes in our eight note scale.

To fill in the last four, we iterate by multiplying by 32 , and every time we

obtain a value greater than 2, we halve it. We do this because we do notwant any notes in our scale higher than an octave above the base. Try itout!

Math in the Real World: Music (9+) CEMC 5 / 21

Pythagorean Scale

To create this scale, start with a base frequency. For simplicity, give this avalue of 1.

As we know, an octave ends at double the frequency, which in this case is 2.

As the ratio 32 is nice, we multiply the base frequency by 3

2 and divide thetop frequency by 3

2 .

This leaves us with four missing notes in our eight note scale.

To fill in the last four, we iterate by multiplying by 32 , and every time we

obtain a value greater than 2, we halve it. We do this because we do notwant any notes in our scale higher than an octave above the base. Try itout!

Math in the Real World: Music (9+) CEMC 5 / 21

Pythagorean Scale

We get the values below:

1 32

98

2716

8164

243128

43 2

Arranging these values from lowest to highest gives us the following scale:

1 98

8164

43

32

2716

243128 2

This scale is commonly labeled as:

C D E F G A B C

You will notice that there are two different step sizes; the tone which is 98

and the semitone which is 256243 . The pattern of this scale is ttsttts.

Image source: http://www.piano-keyboard-guide.com/wp-content/uploads/2015/05/tones-and-semitones-in-a-major-scale.png

Math in the Real World: Music (9+) CEMC 6 / 21

Pythagorean Scale

We get the values below:

1 32

98

2716

8164

243128

43 2

Arranging these values from lowest to highest gives us the following scale:

1 98

8164

43

32

2716

243128 2

This scale is commonly labeled as:

C D E F G A B C

You will notice that there are two different step sizes; the tone which is 98

and the semitone which is 256243 . The pattern of this scale is ttsttts.

Image source: http://www.piano-keyboard-guide.com/wp-content/uploads/2015/05/tones-and-semitones-in-a-major-scale.png

Math in the Real World: Music (9+) CEMC 6 / 21

Pythagorean Scale

We get the values below:

1 32

98

2716

8164

243128

43 2

Arranging these values from lowest to highest gives us the following scale:

1 98

8164

43

32

2716

243128 2

This scale is commonly labeled as:

C D E F G A B C

You will notice that there are two different step sizes; the tone which is 98

and the semitone which is 256243 . The pattern of this scale is ttsttts.

Image source: http://www.piano-keyboard-guide.com/wp-content/uploads/2015/05/tones-and-semitones-in-a-major-scale.png

Math in the Real World: Music (9+) CEMC 6 / 21

Pythagorean Scale

We get the values below:

1 32

98

2716

8164

243128

43 2

Arranging these values from lowest to highest gives us the following scale:

1 98

8164

43

32

2716

243128 2

This scale is commonly labeled as:

C D E F G A B C

You will notice that there are two different step sizes; the tone which is 98

and the semitone which is 256243 . The pattern of this scale is ttsttts.

Image source: http://www.piano-keyboard-guide.com/wp-content/uploads/2015/05/tones-and-semitones-in-a-major-scale.png

Math in the Real World: Music (9+) CEMC 6 / 21

The Problem

Some of the intervals between notesdidn’t sound great, thanks to someof the large or awkward ratios of thisscale.

Why are intervals important to us?

Well, a large part of western music isthe chord or triad. The triad is es-sentially made up of three alternatingnotes in the scale.

To make these sound good, we want toreplace the awkward fractions 81

64 ,2716 ,

and 243128 . Thus, the ‘Classical Just

Scale’ was created.

Math in the Real World: Music (9+) CEMC 7 / 21

The Problem

Some of the intervals between notesdidn’t sound great, thanks to someof the large or awkward ratios of thisscale.

Why are intervals important to us?

Well, a large part of western music isthe chord or triad. The triad is es-sentially made up of three alternatingnotes in the scale.

To make these sound good, we want toreplace the awkward fractions 81

64 ,2716 ,

and 243128 . Thus, the ‘Classical Just

Scale’ was created.

Math in the Real World: Music (9+) CEMC 7 / 21

The Problem

Some of the intervals between notesdidn’t sound great, thanks to someof the large or awkward ratios of thisscale.

Why are intervals important to us?

Well, a large part of western music isthe chord or triad. The triad is es-sentially made up of three alternatingnotes in the scale.

To make these sound good, we want toreplace the awkward fractions 81

64 ,2716 ,

and 243128 . Thus, the ‘Classical Just

Scale’ was created.

Math in the Real World: Music (9+) CEMC 7 / 21

The Problem

Some of the intervals between notesdidn’t sound great, thanks to someof the large or awkward ratios of thisscale.

Why are intervals important to us?

Well, a large part of western music isthe chord or triad. The triad is es-sentially made up of three alternatingnotes in the scale.

To make these sound good, we want toreplace the awkward fractions 81

64 ,2716 ,

and 243128 . Thus, the ‘Classical Just

Scale’ was created.

Math in the Real World: Music (9+) CEMC 7 / 21

Classical Just Scale

We want the three notes in the chord to be in a clean ratio, which is usually4 : 5 : 6.

Let’s examine the chord made up of the first, third, and fifth notes of thescale; replacing the third note, 81

64 , with an unknown:

1 : x : 32

We can make the first and third numbers look like 4 and 6 by multiplying by4 to get

4 : 4x : 122

Equating the middle term to 5 we find

4x = 5

x = 54

So 54 is our new third note.

Math in the Real World: Music (9+) CEMC 8 / 21

Classical Just Scale

We want the three notes in the chord to be in a clean ratio, which is usually4 : 5 : 6.

Let’s examine the chord made up of the first, third, and fifth notes of thescale; replacing the third note, 81

64 , with an unknown:

1 : x : 32

We can make the first and third numbers look like 4 and 6 by multiplying by4 to get

4 : 4x : 122

Equating the middle term to 5 we find

4x = 5

x = 54

So 54 is our new third note.

Math in the Real World: Music (9+) CEMC 8 / 21

Classical Just Scale

We want the three notes in the chord to be in a clean ratio, which is usually4 : 5 : 6.

Let’s examine the chord made up of the first, third, and fifth notes of thescale; replacing the third note, 81

64 , with an unknown:

1 : x : 32

We can make the first and third numbers look like 4 and 6 by multiplying by4 to get

4 : 4x : 122

Equating the middle term to 5 we find

4x = 5

x = 54

So 54 is our new third note.

Math in the Real World: Music (9+) CEMC 8 / 21

Classical Just Scale

We want the three notes in the chord to be in a clean ratio, which is usually4 : 5 : 6.

Let’s examine the chord made up of the first, third, and fifth notes of thescale; replacing the third note, 81

64 , with an unknown:

1 : x : 32

We can make the first and third numbers look like 4 and 6 by multiplying by4 to get

4 : 4x : 122

Equating the middle term to 5 we find

4x = 5

x = 54

So 54 is our new third note.

Math in the Real World: Music (9+) CEMC 8 / 21

Classical Just Scale

We want the three notes in the chord to be in a clean ratio, which is usually4 : 5 : 6.

Let’s examine the chord made up of the first, third, and fifth notes of thescale; replacing the third note, 81

64 , with an unknown:

1 : x : 32

We can make the first and third numbers look like 4 and 6 by multiplying by4 to get

4 : 4x : 122

Equating the middle term to 5 we find

4x = 5

x = 54

So 54 is our new third note.

Math in the Real World: Music (9+) CEMC 8 / 21

Classical Just Scale

Now we examine a chord made up of the fourth, sixth, and eighth notes ofthe scale, replacing the sixth note, 27

16 with an unknown:

43 : y : 2

We can make the first and third numbers look like 4 and 6 by multiplying by3 to get

123 : 3y : 6

Equating the middle term to 5 we find

3y = 5

y =5

3

So 53 is our new sixth note.

Math in the Real World: Music (9+) CEMC 9 / 21

Classical Just Scale

Now we examine a chord made up of the fourth, sixth, and eighth notes ofthe scale, replacing the sixth note, 27

16 with an unknown:

43 : y : 2

We can make the first and third numbers look like 4 and 6 by multiplying by3 to get

123 : 3y : 6

Equating the middle term to 5 we find

3y = 5

y =5

3

So 53 is our new sixth note.

Math in the Real World: Music (9+) CEMC 9 / 21

Classical Just Scale

Now we examine a chord made up of the fourth, sixth, and eighth notes ofthe scale, replacing the sixth note, 27

16 with an unknown:

43 : y : 2

We can make the first and third numbers look like 4 and 6 by multiplying by3 to get

123 : 3y : 6

Equating the middle term to 5 we find

3y = 5

y =5

3

So 53 is our new sixth note.

Math in the Real World: Music (9+) CEMC 9 / 21

Classical Just Scale

Now we examine a chord made up of the fourth, sixth, and eighth notes ofthe scale, replacing the sixth note, 27

16 with an unknown:

43 : y : 2

We can make the first and third numbers look like 4 and 6 by multiplying by3 to get

123 : 3y : 6

Equating the middle term to 5 we find

3y = 5

y =5

3

So 53 is our new sixth note.

Math in the Real World: Music (9+) CEMC 9 / 21

Classical Just Scale

The only note left to replace is the seventh note, 243128 . So we will look at the

chord made up of the fifth, seventh, and ninth notes of the scale.

Now, since the eighth note of the scale is 2, which is an octave above, andtherefore twice the first note (1), the ninth note is 18

8 , which is twice thesecond note (98).

Math in the Real World: Music (9+) CEMC 10 / 21

Classical Just Scale

The only note left to replace is the seventh note, 243128 . So we will look at the

chord made up of the fifth, seventh, and ninth notes of the scale.

Now, since the eighth note of the scale is 2, which is an octave above, andtherefore twice the first note (1), the ninth note is 18

8 , which is twice thesecond note (98).

Math in the Real World: Music (9+) CEMC 10 / 21

Classical Just Scale

Replace the seventh note with an unknown:

32 : z : 18

8

We can make the first and third numbers look like 4 and 6 by multiplying by83 to get

246 : 8z

3 : 14424

Equating the middle term to 5 we find

8z

3= 5

z =15

8

So 158 is our new seventh note.

Math in the Real World: Music (9+) CEMC 11 / 21

Classical Just Scale

Replace the seventh note with an unknown:

32 : z : 18

8

We can make the first and third numbers look like 4 and 6 by multiplying by83 to get

246 : 8z

3 : 14424

Equating the middle term to 5 we find

8z

3= 5

z =15

8

So 158 is our new seventh note.

Math in the Real World: Music (9+) CEMC 11 / 21

Classical Just Scale

Replace the seventh note with an unknown:

32 : z : 18

8

We can make the first and third numbers look like 4 and 6 by multiplying by83 to get

246 : 8z

3 : 14424

Equating the middle term to 5 we find

8z

3= 5

z =15

8

So 158 is our new seventh note.

Math in the Real World: Music (9+) CEMC 11 / 21

Classical Just Scale

Replace the seventh note with an unknown:

32 : z : 18

8

We can make the first and third numbers look like 4 and 6 by multiplying by83 to get

246 : 8z

3 : 14424

Equating the middle term to 5 we find

8z

3= 5

z =15

8

So 158 is our new seventh note.

Math in the Real World: Music (9+) CEMC 11 / 21

Classical Just Scale

Thus we end up with this scale:

1 98

54

43

32

53

158 2

These notes lead to triads that are all in the same beautiful ratio! Thesechords are very pleasing to the ear.

Math in the Real World: Music (9+) CEMC 12 / 21

Classical Just Scale

Thus we end up with this scale:

1 98

54

43

32

53

158 2

These notes lead to triads that are all in the same beautiful ratio! Thesechords are very pleasing to the ear.

Math in the Real World: Music (9+) CEMC 12 / 21

The Problem

Yet, the scale that we are used to to-day, the one that you would likely havetuned on your piano at home, is NOTthe Classical Just.

The two scales presented so far do nottranspose well. That is, they soundstrange when they are switched to adifferent key.

So, yet another scale was created - the‘Equal Temperament Scale’.

Math in the Real World: Music (9+) CEMC 13 / 21

The Problem

Yet, the scale that we are used to to-day, the one that you would likely havetuned on your piano at home, is NOTthe Classical Just.

The two scales presented so far do nottranspose well. That is, they soundstrange when they are switched to adifferent key.

So, yet another scale was created - the‘Equal Temperament Scale’.

Math in the Real World: Music (9+) CEMC 13 / 21

The Problem

Yet, the scale that we are used to to-day, the one that you would likely havetuned on your piano at home, is NOTthe Classical Just.

The two scales presented so far do nottranspose well. That is, they soundstrange when they are switched to adifferent key.

So, yet another scale was created - the‘Equal Temperament Scale’.

Math in the Real World: Music (9+) CEMC 13 / 21

Equal Temperament Scale

The idea behind Equal Temperament is to have a consistent step size.

Thinking back to the Pythagorean scale, consider a semitone to be 1 stepand a tone to be 2 steps. Mathematically we set this up as t = s2.

Thus, from the pattern ttsttts we have 12 steps.

Math in the Real World: Music (9+) CEMC 14 / 21

Equal Temperament Scale

The idea behind Equal Temperament is to have a consistent step size.

Thinking back to the Pythagorean scale, consider a semitone to be 1 stepand a tone to be 2 steps. Mathematically we set this up as t = s2.

Thus, from the pattern ttsttts we have 12 steps.

Math in the Real World: Music (9+) CEMC 14 / 21

Equal Temperament Scale

The idea behind Equal Temperament is to have a consistent step size.

Thinking back to the Pythagorean scale, consider a semitone to be 1 stepand a tone to be 2 steps. Mathematically we set this up as t = s2.

Thus, from the pattern ttsttts we have 12 steps.

Math in the Real World: Music (9+) CEMC 14 / 21

Equal Temperament Scale

The first note in our scale is always 1.

Then, the second note would be 1 × t = 1 × s2 = s2.

The third note would be s2 × t = s2 × s2 = s4.

The fourth note would be s4 × s = s5.

Continuing this pattern, the eight and last note, which we know is 2, wouldbe s12.

Math in the Real World: Music (9+) CEMC 15 / 21

Equal Temperament Scale

The first note in our scale is always 1.

Then, the second note would be 1 × t = 1 × s2 = s2.

The third note would be s2 × t = s2 × s2 = s4.

The fourth note would be s4 × s = s5.

Continuing this pattern, the eight and last note, which we know is 2, wouldbe s12.

Math in the Real World: Music (9+) CEMC 15 / 21

Equal Temperament Scale

The first note in our scale is always 1.

Then, the second note would be 1 × t = 1 × s2 = s2.

The third note would be s2 × t = s2 × s2 = s4.

The fourth note would be s4 × s = s5.

Continuing this pattern, the eight and last note, which we know is 2, wouldbe s12.

Math in the Real World: Music (9+) CEMC 15 / 21

Equal Temperament Scale

The first note in our scale is always 1.

Then, the second note would be 1 × t = 1 × s2 = s2.

The third note would be s2 × t = s2 × s2 = s4.

The fourth note would be s4 × s = s5.

Continuing this pattern, the eight and last note, which we know is 2, wouldbe s12.

Math in the Real World: Music (9+) CEMC 15 / 21

Equal Temperament Scale

The first note in our scale is always 1.

Then, the second note would be 1 × t = 1 × s2 = s2.

The third note would be s2 × t = s2 × s2 = s4.

The fourth note would be s4 × s = s5.

Continuing this pattern, the eight and last note, which we know is 2, wouldbe s12.

Math in the Real World: Music (9+) CEMC 15 / 21

Equal Temperament Scale

Then we can say:

s12 = 2

s = 2112

Then our scale is as follows:

1 216 2

13 2

512 2

712 2

34 2

1112 2

With this scale we get perfect transposition, however you will notice we loseour nice intervals (though they are actually quite close to nice, if you try itout!).

Some musicians still like to use the Classical Just scale as they consider itto have a much richer sound, whereas Equal Temperament is a trulymathematical scale!

Math in the Real World: Music (9+) CEMC 16 / 21

Equal Temperament Scale

Then we can say:

s12 = 2

s = 2112

Then our scale is as follows:

1 216 2

13 2

512 2

712 2

34 2

1112 2

With this scale we get perfect transposition, however you will notice we loseour nice intervals (though they are actually quite close to nice, if you try itout!).

Some musicians still like to use the Classical Just scale as they consider itto have a much richer sound, whereas Equal Temperament is a trulymathematical scale!

Math in the Real World: Music (9+) CEMC 16 / 21

Equal Temperament Scale

Then we can say:

s12 = 2

s = 2112

Then our scale is as follows:

1 216 2

13 2

512 2

712 2

34 2

1112 2

With this scale we get perfect transposition, however you will notice we loseour nice intervals (though they are actually quite close to nice, if you try itout!).

Some musicians still like to use the Classical Just scale as they consider itto have a much richer sound, whereas Equal Temperament is a trulymathematical scale!

Math in the Real World: Music (9+) CEMC 16 / 21

Equal Temperament Scale

Then we can say:

s12 = 2

s = 2112

Then our scale is as follows:

1 216 2

13 2

512 2

712 2

34 2

1112 2

With this scale we get perfect transposition, however you will notice we loseour nice intervals (though they are actually quite close to nice, if you try itout!).

Some musicians still like to use the Classical Just scale as they consider itto have a much richer sound, whereas Equal Temperament is a trulymathematical scale!

Math in the Real World: Music (9+) CEMC 16 / 21

A Comparison

Image source:http://www.piano-keyboard-guide.com

Today’s pianos are tuned around a notecalled A4, which has a value of 440Hz.

Without going too in-depth into musictheory, we are going to create what iscall the A-major scale.

A-major contains F, C, and G sharps(denoted by the # symbol).

We can create the three scales we’vetalked about previously around thisnote.

Math in the Real World: Music (9+) CEMC 17 / 21

A Comparison

Image source:http://www.piano-keyboard-guide.com

Today’s pianos are tuned around a notecalled A4, which has a value of 440Hz.

Without going too in-depth into musictheory, we are going to create what iscall the A-major scale.

A-major contains F, C, and G sharps(denoted by the # symbol).

We can create the three scales we’vetalked about previously around thisnote.

Math in the Real World: Music (9+) CEMC 17 / 21

A Comparison

Image source:http://www.piano-keyboard-guide.com

Today’s pianos are tuned around a notecalled A4, which has a value of 440Hz.

Without going too in-depth into musictheory, we are going to create what iscall the A-major scale.

A-major contains F, C, and G sharps(denoted by the # symbol).

We can create the three scales we’vetalked about previously around thisnote.

Math in the Real World: Music (9+) CEMC 17 / 21

A Comparison

Image source:http://www.piano-keyboard-guide.com

Today’s pianos are tuned around a notecalled A4, which has a value of 440Hz.

Without going too in-depth into musictheory, we are going to create what iscall the A-major scale.

A-major contains F, C, and G sharps(denoted by the # symbol).

We can create the three scales we’vetalked about previously around thisnote.

Math in the Real World: Music (9+) CEMC 17 / 21

Pythagorean A-major Scale

A B C# D E F# G# A440 Hz 495 Hz 556.875 Hz 586.667 Hz 660 Hz 742.5 Hz 835.3125 Hz 880 Hz

Math in the Real World: Music (9+) CEMC 18 / 21

Classical Just A-major Scale

A B C# D E F# G# A440 Hz 495 Hz 550 Hz 586.667 Hz 660 Hz 733.333 Hz 825 Hz 880 Hz

Math in the Real World: Music (9+) CEMC 19 / 21

Equal Temperament A-major Scale

A B C# D E F# G# A440 Hz 493.883 Hz 554.365 Hz 587.330 Hz 659.255 Hz 739.989 Hz 830.609 Hz 880 Hz

If you were to search “Piano Key Frequencies” on your favourite searchengine, you would discover the values listed in the Equal Temperamenttable!

Math in the Real World: Music (9+) CEMC 20 / 21

Equal Temperament A-major Scale

A B C# D E F# G# A440 Hz 493.883 Hz 554.365 Hz 587.330 Hz 659.255 Hz 739.989 Hz 830.609 Hz 880 Hz

If you were to search “Piano Key Frequencies” on your favourite searchengine, you would discover the values listed in the Equal Temperamenttable!

Math in the Real World: Music (9+) CEMC 20 / 21

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Math in the Real World: Music (9+) CEMC 21 / 21