Math in the Real World: Music (9+)
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1 Math in the Real World: Music (9+) CEMC Math in the Real World: Music (9+) CEMC 1 / 21
2 The Connection Many of you probably play instruments! But did you know that the foundations of music are built with mathematics? Math in the Real World: Music (9+) CEMC 2 / 21
3 The Connection Many of you probably play instruments! But did you know that the foundations of music are built with mathematics? One of the first things you learn when you start with instruments is something called a scale. The scales that we play have evolved over time and vary by region. Math in the Real World: Music (9+) CEMC 2 / 21
4 The Connection Many of you probably play instruments! But did you know that the foundations of music are built with mathematics? One of the first things you learn when you start with instruments is something called a scale. The scales that we play have evolved over time and vary by region. The fundamental mathematics behind these scales? Fractions! Math in the Real World: Music (9+) CEMC 2 / 21
5 Frequency The sound of a note is based on its frequency. Math in the Real World: Music (9+) CEMC 3 / 21
6 Frequency The sound of a note is based on its frequency. In terms of a string, like the ones plucked on a guitar or struck by a hammer in a piano, this means the number of vibrations up and down per second. Math in the Real World: Music (9+) CEMC 3 / 21
7 Frequency The sound of a note is based on its frequency. In terms of a string, like the ones plucked on a guitar or struck by a hammer in a piano, this means the number of vibrations up and down per second. This is measured in a unit called the Hertz (Hz). One Hertz means one vibration per second. Math in the Real World: Music (9+) CEMC 3 / 21
8 An Introduction to Scales A scale starts with a base frequency and ends on a note that is twice the frequency (the scale runs through an octave). Math in the Real World: Music (9+) CEMC 4 / 21
9 An Introduction to Scales A scale starts with a base frequency and ends on a note that is twice the frequency (the scale runs through an octave). We will focus on three types of Western scales, as they evolved through time. Math in the Real World: Music (9+) CEMC 4 / 21
10 An Introduction to Scales A scale starts with a base frequency and ends on a note that is twice the frequency (the scale runs through an octave). We will focus on three types of Western scales, as they evolved through time. Our story begins with the ancient Greeks. Math in the Real World: Music (9+) CEMC 4 / 21
11 An Introduction to Scales A scale starts with a base frequency and ends on a note that is twice the frequency (the scale runs through an octave). We will focus on three types of Western scales, as they evolved through time. Our story begins with the ancient Greeks. They noticed that strings in the ratio 3 2 sounded quite good together. Math in the Real World: Music (9+) CEMC 4 / 21
12 An Introduction to Scales A scale starts with a base frequency and ends on a note that is twice the frequency (the scale runs through an octave). We will focus on three types of Western scales, as they evolved through time. Our story begins with the ancient Greeks. They noticed that strings in the ratio 3 2 sounded quite good together. Thus, the Pythagorean Scale was created. Math in the Real World: Music (9+) CEMC 4 / 21
13 Pythagorean Scale To create this scale, start with a base frequency. For simplicity, give this a value of 1. Math in the Real World: Music (9+) CEMC 5 / 21
14 Pythagorean Scale To create this scale, start with a base frequency. For simplicity, give this a value of 1. As we know, an octave ends at double the frequency, which in this case is 2. Math in the Real World: Music (9+) CEMC 5 / 21
15 Pythagorean Scale To create this scale, start with a base frequency. For simplicity, give this a value of 1. As we know, an octave ends at double the frequency, which in this case is 2. As the ratio 3 2 is nice, we multiply the base frequency by 3 2 top frequency by 3 2. and divide the Math in the Real World: Music (9+) CEMC 5 / 21
16 Pythagorean Scale To create this scale, start with a base frequency. For simplicity, give this a value of 1. As we know, an octave ends at double the frequency, which in this case is 2. As the ratio 3 2 is nice, we multiply the base frequency by 3 2 top frequency by 3 2. and divide the This leaves us with four missing notes in our eight note scale. Math in the Real World: Music (9+) CEMC 5 / 21
17 Pythagorean Scale To create this scale, start with a base frequency. For simplicity, give this a value of 1. As we know, an octave ends at double the frequency, which in this case is 2. As the ratio 3 2 is nice, we multiply the base frequency by 3 2 top frequency by 3 2. and divide the This leaves us with four missing notes in our eight note scale. To fill in the last four, we iterate by multiplying by 3 2, and every time we obtain a value greater than 2, we halve it. We do this because we do not want any notes in our scale higher than an octave above the base. Try it out! Math in the Real World: Music (9+) CEMC 5 / 21
18 Pythagorean Scale We get the values below: Math in the Real World: Music (9+) CEMC 6 / 21
19 Pythagorean Scale We get the values below: Arranging these values from lowest to highest gives us the following scale: Math in the Real World: Music (9+) CEMC 6 / 21
20 Pythagorean Scale We get the values below: Arranging these values from lowest to highest gives us the following scale: This scale is commonly labeled as: C D E F G A B C Math in the Real World: Music (9+) CEMC 6 / 21
21 Pythagorean Scale We get the values below: Arranging these values from lowest to highest gives us the following scale: 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 9 8 and the semitone which is The pattern of this scale is ttsttts. Image source: Math in the Real World: Music (9+) CEMC 6 / 21
22 The Problem Some of the intervals between notes didn t sound great, thanks to some of the large or awkward ratios of this scale. Math in the Real World: Music (9+) CEMC 7 / 21
23 The Problem Some of the intervals between notes didn t sound great, thanks to some of the large or awkward ratios of this scale. Why are intervals important to us? Math in the Real World: Music (9+) CEMC 7 / 21
24 The Problem Some of the intervals between notes didn t sound great, thanks to some of the large or awkward ratios of this scale. Why are intervals important to us? Well, a large part of western music is the chord or triad. The triad is essentially made up of three alternating notes in the scale. Math in the Real World: Music (9+) CEMC 7 / 21
25 The Problem Some of the intervals between notes didn t sound great, thanks to some of the large or awkward ratios of this scale. Why are intervals important to us? Well, a large part of western music is the chord or triad. The triad is essentially made up of three alternating notes in the scale. To make these sound good, we want to replace the awkward fractions 81 64, 27 16, and Thus, the Classical Just Scale was created. Math in the Real World: Music (9+) CEMC 7 / 21
26 Classical Just Scale We want the three notes in the chord to be in a clean ratio, which is usually 4 : 5 : 6. Math in the Real World: Music (9+) CEMC 8 / 21
27 Classical Just Scale We want the three notes in the chord to be in a clean ratio, which is usually 4 : 5 : 6. Let s examine the chord made up of the first, third, and fifth notes of the scale; replacing the third note, 81 64, with an unknown: 1 : x : 3 2 Math in the Real World: Music (9+) CEMC 8 / 21
28 Classical Just Scale We want the three notes in the chord to be in a clean ratio, which is usually 4 : 5 : 6. Let s examine the chord made up of the first, third, and fifth notes of the scale; replacing the third note, 81 64, with an unknown: 1 : x : 3 2 We can make the first and third numbers look like 4 and 6 by multiplying by 4 to get 4 : 4x : 12 2 Math in the Real World: Music (9+) CEMC 8 / 21
29 Classical Just Scale We want the three notes in the chord to be in a clean ratio, which is usually 4 : 5 : 6. Let s examine the chord made up of the first, third, and fifth notes of the scale; replacing the third note, 81 64, with an unknown: 1 : x : 3 2 We can make the first and third numbers look like 4 and 6 by multiplying by 4 to get 4 : 4x : 12 2 Equating the middle term to 5 we find 4x = 5 x = 5 4 Math in the Real World: Music (9+) CEMC 8 / 21
30 Classical Just Scale We want the three notes in the chord to be in a clean ratio, which is usually 4 : 5 : 6. Let s examine the chord made up of the first, third, and fifth notes of the scale; replacing the third note, 81 64, with an unknown: 1 : x : 3 2 We can make the first and third numbers look like 4 and 6 by multiplying by 4 to get 4 : 4x : 12 2 Equating the middle term to 5 we find 4x = 5 x = 5 4 So 5 4 is our new third noteṁath in the Real World: Music (9+) CEMC 8 / 21
31 Classical Just Scale Now we examine a chord made up of the fourth, sixth, and eighth notes of the scale, replacing the sixth note, with an unknown: 4 3 : y : 2 Math in the Real World: Music (9+) CEMC 9 / 21
32 Classical Just Scale Now we examine a chord made up of the fourth, sixth, and eighth notes of the scale, replacing the sixth note, with an unknown: 4 3 : y : 2 We can make the first and third numbers look like 4 and 6 by multiplying by 3 to get 12 3 : 3y : 6 Math in the Real World: Music (9+) CEMC 9 / 21
33 Classical Just Scale Now we examine a chord made up of the fourth, sixth, and eighth notes of the scale, replacing the sixth note, with an unknown: 4 3 : y : 2 We can make the first and third numbers look like 4 and 6 by multiplying by 3 to get 12 3 : 3y : 6 Equating the middle term to 5 we find 3y = 5 y = 5 3 Math in the Real World: Music (9+) CEMC 9 / 21
34 Classical Just Scale Now we examine a chord made up of the fourth, sixth, and eighth notes of the scale, replacing the sixth note, with an unknown: 4 3 : y : 2 We can make the first and third numbers look like 4 and 6 by multiplying by 3 to get 12 3 : 3y : 6 Equating the middle term to 5 we find 3y = 5 y = 5 3 So 5 3 is our new sixth note. Math in the Real World: Music (9+) CEMC 9 / 21
35 Classical Just Scale The only note left to replace is the seventh note, So we will look at the chord made up of the fifth, seventh, and ninth notes of the scale. Math in the Real World: Music (9+) CEMC 10 / 21
36 Classical Just Scale The only note left to replace is the seventh note, 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, and therefore twice the first note (1), the ninth note is 18 8, which is twice the second note ( 9 8 ). Math in the Real World: Music (9+) CEMC 10 / 21
37 Classical Just Scale Replace the seventh note with an unknown: 3 2 : z : 18 8 Math in the Real World: Music (9+) CEMC 11 / 21
38 Classical Just Scale Replace the seventh note with an unknown: 3 2 : z : 18 8 We can make the first and third numbers look like 4 and 6 by multiplying by 8 3 to get 24 6 : 8z 3 : Math in the Real World: Music (9+) CEMC 11 / 21
39 Classical Just Scale Replace the seventh note with an unknown: 3 2 : z : 18 8 We can make the first and third numbers look like 4 and 6 by multiplying by 8 3 to get 24 6 : 8z 3 : Equating the middle term to 5 we find 8z 3 = 5 z = 15 8 Math in the Real World: Music (9+) CEMC 11 / 21
40 Classical Just Scale Replace the seventh note with an unknown: 3 2 : z : 18 8 We can make the first and third numbers look like 4 and 6 by multiplying by 8 3 to get 24 6 : 8z 3 : Equating the middle term to 5 we find 8z 3 = 5 z = 15 8 So 15 8 is our new seventh note. Math in the Real World: Music (9+) CEMC 11 / 21
41 Classical Just Scale Thus we end up with this scale: Math in the Real World: Music (9+) CEMC 12 / 21
42 Classical Just Scale Thus we end up with this scale: These notes lead to triads that are all in the same beautiful ratio! These chords are very pleasing to the ear. Math in the Real World: Music (9+) CEMC 12 / 21
43 The Problem Yet, the scale that we are used to today, the one that you would likely have tuned on your piano at home, is NOT the Classical Just. Math in the Real World: Music (9+) CEMC 13 / 21
44 The Problem Yet, the scale that we are used to today, the one that you would likely have tuned on your piano at home, is NOT the Classical Just. The two scales presented so far do not transpose well. That is, they sound strange when they are switched to a different key. Math in the Real World: Music (9+) CEMC 13 / 21
45 The Problem Yet, the scale that we are used to today, the one that you would likely have tuned on your piano at home, is NOT the Classical Just. The two scales presented so far do not transpose well. That is, they sound strange when they are switched to a different key. So, yet another scale was created - the Equal Temperament Scale. Math in the Real World: Music (9+) CEMC 13 / 21
46 Equal Temperament Scale The idea behind Equal Temperament is to have a consistent step size. Math in the Real World: Music (9+) CEMC 14 / 21
47 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 step and a tone to be 2 steps. Mathematically we set this up as t = s 2. Math in the Real World: Music (9+) CEMC 14 / 21
48 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 step and a tone to be 2 steps. Mathematically we set this up as t = s 2. Thus, from the pattern ttsttts we have 12 steps. Math in the Real World: Music (9+) CEMC 14 / 21
49 Equal Temperament Scale The first note in our scale is always 1. Math in the Real World: Music (9+) CEMC 15 / 21
50 Equal Temperament Scale The first note in our scale is always 1. Then, the second note would be 1 t = 1 s 2 = s 2. Math in the Real World: Music (9+) CEMC 15 / 21
51 Equal Temperament Scale The first note in our scale is always 1. Then, the second note would be 1 t = 1 s 2 = s 2. The third note would be s 2 t = s 2 s 2 = s 4. Math in the Real World: Music (9+) CEMC 15 / 21
52 Equal Temperament Scale The first note in our scale is always 1. Then, the second note would be 1 t = 1 s 2 = s 2. The third note would be s 2 t = s 2 s 2 = s 4. The fourth note would be s 4 s = s 5. Math in the Real World: Music (9+) CEMC 15 / 21
53 Equal Temperament Scale The first note in our scale is always 1. Then, the second note would be 1 t = 1 s 2 = s 2. The third note would be s 2 t = s 2 s 2 = s 4. The fourth note would be s 4 s = s 5. Continuing this pattern, the eight and last note, which we know is 2, would be s 12. Math in the Real World: Music (9+) CEMC 15 / 21
54 Equal Temperament Scale Then we can say: s 12 = 2 s = Math in the Real World: Music (9+) CEMC 16 / 21
55 Equal Temperament Scale Then we can say: Then our scale is as follows: s 12 = 2 s = Math in the Real World: Music (9+) CEMC 16 / 21
56 Equal Temperament Scale Then we can say: Then our scale is as follows: s 12 = 2 s = With this scale we get perfect transposition, however you will notice we lose our nice intervals (though they are actually quite close to nice, if you try it out!). Math in the Real World: Music (9+) CEMC 16 / 21
57 Equal Temperament Scale Then we can say: Then our scale is as follows: s 12 = 2 s = With this scale we get perfect transposition, however you will notice we lose our nice intervals (though they are actually quite close to nice, if you try it out!). Some musicians still like to use the Classical Just scale as they consider it to have a much richer sound, whereas Equal Temperament is a truly mathematical scale! Math in the Real World: Music (9+) CEMC 16 / 21
58 A Comparison Today s pianos are tuned around a note called A4, which has a value of 440 Hz. Image source: Math in the Real World: Music (9+) CEMC 17 / 21
59 A Comparison Today s pianos are tuned around a note called A4, which has a value of 440 Hz. Without going too in-depth into music theory, we are going to create what is call the A-major scale. Image source: Math in the Real World: Music (9+) CEMC 17 / 21
60 A Comparison Today s pianos are tuned around a note called A4, which has a value of 440 Hz. Without going too in-depth into music theory, we are going to create what is call the A-major scale. Image source: A-major contains F, C, and G sharps (denoted by the # symbol). Math in the Real World: Music (9+) CEMC 17 / 21
61 A Comparison Today s pianos are tuned around a note called A4, which has a value of 440 Hz. Without going too in-depth into music theory, we are going to create what is call the A-major scale. Image source: A-major contains F, C, and G sharps (denoted by the # symbol). We can create the three scales we ve talked about previously around this note. Math in the Real World: Music (9+) CEMC 17 / 21
62 Pythagorean A-major Scale A B C# D E F# G# A 440 Hz 495 Hz Hz Hz 660 Hz Hz Hz 880 Hz Math in the Real World: Music (9+) CEMC 18 / 21
63 Classical Just A-major Scale A B C# D E F# G# A 440 Hz 495 Hz 550 Hz Hz 660 Hz Hz 825 Hz 880 Hz Math in the Real World: Music (9+) CEMC 19 / 21
64 Equal Temperament A-major Scale A B C# D E F# G# A 440 Hz Hz Hz Hz Hz Hz Hz 880 Hz Math in the Real World: Music (9+) CEMC 20 / 21
65 Equal Temperament A-major Scale A B C# D E F# G# A 440 Hz Hz Hz Hz Hz Hz Hz 880 Hz If you were to search Piano Key Frequencies on your favourite search engine, you would discover the values listed in the Equal Temperament table! Math in the Real World: Music (9+) CEMC 20 / 21
66 Thank You! Visit cemc.uwaterloo.ca for great mathematics resources! Math in the Real World: Music (9+) CEMC 21 / 21
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