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Matthew Schwartz
Lecture 7:
Music
1 Why do notes sound good?
In the previous lecture, we saw that if you pluck a string, it will excite various frequencies. The amplitude of each frequency which is excited will be proportional to the coefficient in the Fourier decomposition. In this lecture we will start to understand how different frequencies com- bine to produce music. This lecture is best studied alongside the Mathematica notebook music.nb on the isite.
In the first section “playing notes” of the notebook, you can listen to a pure frequency (C4 = middle C = 261 Hz). It sounds pleasant, but not particularly interesting. Now play the “square wave” version of middle C – you should notice that it sounds somewhat tinny and unpleasant. These are the same notes, but different sounds. Why do they sound different? One way to understand the difference is to compare the Fourier decomposition of the sine wave and the square wave:
0.0 0.2 0.4 0.6 0.8 1.
0.0 0.2 0.4 0.6 0.8 1.
Sine wave – one frequency only Square wave – lots of frequencies, sounds tinny Figure 1. Comparison of Fourier modes for sin and square wave
For both the sine wave and the square wave, the dominant mode is the fundamental. How- ever, the square wave has lots of other modes which make the note sound less pure. One way to understand why the square wave sounds worse is that has many high frequency notes with sig- nificant amplitude. It is hard for our brains to process all these high frequency notes, so we find it jarring. In fact, if all frequencies are present at once we get so-called white noise. White noise is perhaps as unmusical as you can get.
Now consider playing two notes at once. In the “playing pairs of notes” section, play the 250 Hz and 270 Hz notes at the same time. It doesn’t sound great. Why?
The problem is that you hear a rattling around 20 times a second. This 20 Hz rattling is the beat frequency between the 250 Hz and 270 Hz. Indeed,
cos(270 Hz 2 πt) + cos(250 Hz (2πt)) = 2 cos(10 Hz 2 πt) cos(260 Hz 2 πt) (1)
This the sum of those two notes oscillates at 10 Hz and at 260 Hz. The 10 Hz oscillation (which we hear as a 20 Hz beat frequency) is jarring – your mind tries to process it consciously. Fre- quencies as high as 260 Hz do not have this effect.
Thus there seem to be two reasons sounds appear unmusical:
- Too many frequencies are present at once
- Beating occurs at frequencies we can consciously process.
This is a physics class, not a biology class, so we will not try to explain why these facts hold. We merely observe that whenever each criteria is satisfied, sounds appear unmusical. However, now that we have defined the problem, we can start to study music scientifically.
2 Dissonant and consonant note pairs
Now we’re ready to study music. If we had a pure sine wave at 300 Hz, then it sounds nasty when played at the same time as a sine wave of 320 Hz. However, if we play it with a sine wave of 580 Hz it does not sound so bad. That is because
cos(300 Hz 2 πt) + cos(580 Hz 2 πt) = 2 cos(140 Hz 2 πt) cos(440 Hz 2 πt) (2)
The beat frequency 2 × 140 Hz = 280Hz is not low enough to be harsh – it is just a note (try the Mathematica notebook). On the other hand, if we played 300 Hz and 580 Hz on an actual instrument it would sound horrible.
We can see why from studying our plucked string example. Recall that for a string plucked near the end, the relative Fourier coefficients scale like (^1) n. So the dominant frequency n = 1 (the
fundamental) has only twice the amplitude of the first harmonic (n = 2). Thus playing 580 Hz along side a plucked string would give
f(t) = cos(580 Hz 2 πt) +
n=
n
cos( 300 n Hz 2 πt) (3)
Writing T = Hz 2 πt to clean up the equation and expanding the sum
f (t) = cos( 580 T ) + cos( 300 T ) +
cos( 600 T ) + ··· (4)
Let us combine the 580 Hz oscillation with the 600 Hz oscillation using trig sum rules. Using
cos( 580 T ) +
cos( 600 T ) =
cos( 10 T )cos( 590 T ) +
sin( 10 T )sin( 590 T ) (5)
we find
f(t) = cos( 300 T ) +
cos( 10 T )cos( 590 T ) +
sin( 10 T )sin( 590 T ) +
n=
∞ 1 n
cos( 300 nT ) (6)
Now we see beating at 2 × 10 Hz = 20Hz, which is audible and jarring. There is beating between the 580 Hz note and the first harmonic of the plucked string. The point is that with pure sine waves, no harmonics are excited, but with real instruments they are.
2 Section 2
Are there more notes which are harmonious? Yes. Consider the note with ν 5 = 391Hz. This note has 2 ν 5 = 783Hz. Thus the second harmonic of ν 5 matches the 3rd harmonic of ν 0. We call the note ν 5 = 32 ν 0 the perfect fifth
ν 5 =
ν 0 = perfect fifth = G 4 (9)
This is the G above middle C. In the same way, consider ν 4 = 348 Hz. The 3rd harmonic of ν 4 agrees with the 4rd harmonic of ν 0. We call this the perfect fourth
ν 4 =
ν 0 = perfect fourth = F 4 (10)
And so on.
It is easy to see that any rational number ratio of frequencies will be consonant. Many of these ratios have names
ν ν 0 1 2
3 2
4 3
5 4
6 5
5 3
8 5 name fundamental octave perfect fifth
perfect fourth
Major third
Minor third
Major sixth
Minor sixth example C4 C5 G4 F4 E4 E♭ A5 A♭ 5 Table 1. Notes names and ratios in the just intonation scale
There are an infinite number of rational numbers. So where do we stop? The answer is that the lower the numbers in the ratio (that is, the 3 and 2 in 32 are lower than the 8 and 5 in 85 ),
the more consonant they will be. That’s because for something like 11 17 , one would need the 17th harmonic of one note to match the 11th harmonic of another note. By such high harmonics, the amplitudes are no longer large, and the spectrum is messy (as you can see in Figure 2 ). Also, it
is more likely for frequencies with a ratio of 1117 to give harmonics which are close but not equal, generating dissonant beating, before generating the harmonic consonance. Thus, for numbers large than about 5 in the ratio, notes are no longer appreciated as harmonic.
3 Scales
If we have a given note, say C 4 , we can define all the other notes so that they will be harmonic with C 4.
3.1 Just intonation scale
The most harmonic notes will have the smallest integers in the ratio, as in Table 1. This is a particular choice of tuning known as the just intonation scale. The just intonation scale is in a sense the most harmonic choice for the frequencies of notes in a scale (it is default tuning for some non-Western instruments, such as the Turkish Baglama). But it is just a choice.
Note that if we pick a set of notes that sound harmoic with C 4 , the same set of notes will generically not sound harmonic with another note, like D 4. Thus if we’re playing a song, the set of notes we want to use is determined relative to some starting note. This starting note is called the key. For example, if you are in the key of C, the notes C, G and F sound good. But if you are in the key of D, the notes C, G and F will generally not sound as good.
4 Section 3
On some instruments, such as a violin which has no frets, there are no predefined notes. Thus on a violin, if you work in the key of C, you can play all the harmonics in exactly the right place (if you have a skilled enough ear and hand). Thus you can play the just intonation scale in any key. There are an infinite number of notes you can play on a violin. The same is true on most instruments actually. For example, even though an oboe has fixed holes corre- sponding to notes, oboe players can easily move the notes up or down by manipulating the reed. Controlling the precise frequency of a note with your mouth is critical to playing any woodwind instrument well. On other instruments, like a piano or a stringed instrument with frets like a guitar, the notes are essentially built in to the instrument. You can sometimes tweak the notes if you are skilled, or tune the instrument to a different key, but there are a finite number of notes which can be played in a giving tuning. Unfortunately, it’s impossible to have an instrument with a finite number of notes be capable of playing the most harmonic notes in every key. To see the problem, suppose you want your piano to be in the just intonation scale in the key of C. That means that you want all the other notes to be defined so that they are related to C by rational numbers with low integers. For example, the whole notes can be defined as
note C D E F G A B C ν ν 0 1
9 8
5 4
4 3
3 2
5 3
15 8 2 decimal 1 1.125 1.25 1.333 1.5 1.666 1.875 2 Table 2. Notes names and ratios in just intonation
This defines the note D as having the frequency 98 times the frequency of the C. Now, where
is the 5th of D? This should be at 3 2 ×^
9 8 =^
27 16 =^ 1.688 times^ ν^0. This note is somewhere between the A and the B, but it is not exactly a note in the key of C. It is not hard to see that to get an instrument which could play any note in any key, you would need an enormous number of available notes. Please make sure you understand this point, as it is key to understanding scales. So what can we do? There are two options: the first is you can tune your instrument to the key you want to play in. Stringed instruments can do this. But it is not so appealing of an option if we want to play music in different keys without retuning every time. The other option is to compromise. Most of our ears are not sensitive enough to distinguish close but slightly dif- ferent notes. Thus we can choose scales which are not exactly correct in any key, but close to correct in all keys.
3.2 Pythagorean scale
One way to approximate the scale is by choosing the notes to be related by powers of 3 2 and octaves to be related by factors of 2, For example, 3 2 ν^0 is the perfect fifth (G4 when C4=ν^0 ). Then
2
ν 0 which is the fifth of the fifth, or the fifth of G4 which is D 5. The next note has ( (^3) 2
ν 0 or the fifth of D 5 which is A 6 and so on. We can bring any power of 32 back to the interval between 0 and 1 by dividing by 2 to some power. For example, since D5 = 94 ν 0 then D4 = 98 ν 0. We then get
note C D E F G A B C ν ν 0 1
9 8
81 64
4 3
3 2
27 16
243 128 2 decimal 1 1.125 1.266 1.333 1.5 1.688 1.898 2 Table 3. Notes names and ratios in Pythagorean tuning
This is called the Pythagorean tuning. Note that the octave, perfect fifth (G) and perfect
fourth (F) agree with their values in Table 2. Some notes do not agree: for example, E is 8164 ν 0 = 1.266ν 0 in this tuning. This ratio is close to 54 but not exactly. Thus if we play C and E it will
be close to a consonant sounding note, but not exactly.
Scales 5
Figure 4. Circle of fifths. Each note going clockwise is a perfect fifth above the previous note. Going counterclockwise, each note is a perfect 4th above the previous note. The circle only closes in the equal- tempered scale.
In this circle, each note is 1 fifth above the note clockwise. So G is a fifth above C, D a fifth above G and so on. Going counterclockwise, the intervals are fourths: C is a fourth above G and F is a fourth above C. Going up a fifth is the same as going down a fourth and adding an octave, If the notes are defined with the Pythagorean scale, the circle doesn’t close: going up by 12 steps, and normalizing back down to the original octave leaves you
2
2 −^8 = 1.014 times where
you started. Thus the circle doesn’t close by 1.4%. In the equal-tempered scale, it does exactly close, however, none of the notes have frequency ratios of exactly 23.
3.4 Summary
We discussed 3 scales. The just intonation scale chooses notes to be related by rational number ratios with integers as small as possible in the numerator and denominator. The Pythagorean scale has all notes related by 3 n 2 m^ for some m and n. Both just intonation and the Pythagorean scale require a key to start in. The third scale is the equal-tempered scale. Notes in the equal- tempered scale are related by 2
n 1 2 (^) for some n. Here is a comparison between the relative frequencies of the 3 scales in the key of C:
note C D E F G A B C just-intonation 1 98 54 43 32 53 158 2 Pythagorean 1 98 8164 43 32 2716 243128 2
equal-tempered 1 2
2 1 2 (^2)
4 1 2 (^2)
5 1 2 (^2)
7 1 2 (^2)
9 1 2 (^2)
1 1 1 2 (^2)
Table 5. Comparison of scales: exact ratios.
In decimals
note C D E F G A B C just-intonation 1 1.125 1.25 1.333 1.5 1.666 1.875 2 Pythagorean 1 1.125 1.266 1.333 1.5 1.688 1.898 2 equal-tempered 1 1.122 1.260 1.335 1.498 1.682 1.888 2
Table 6. Comparison of scales: decimal approximations.
Here is a graphical comparison of how far off the frequency is in the equal-tempered scale from the frequency in the just intonation scale
Scales 7
Figure 5. Difference between the equal-tempered frequencies νW T and the just intonation frequencies for whole notes C , D, E....C labeled as 1 to 8.
One thing we can see is that the perfect 4th and perfect 5th are very close to their optimal values, while the 6th and 7ths are not so close.
8 Section 3
