Download Lecture 5: Tuning Systems In Lecture 3, we learned about perfect

Math in Music
Spring 2009
Lecture 5: Tuning Systems
In Lecture 3, we learned about perfect intervals like the octave
(frequency times 2), perfect fifth (times 3/2), perfect fourth (times
4/3) and perfect third (times 4/5). When tuning the strings of a
musical instrument like a guitar or a piano, it’s preferable to use
these perfect intervals, because two strings with fundamental
frequencies a perfect interval apart will share many harmonics. Thus,
we can tune them to a perfect interval by using beats to make sure
that the harmonics line up.
So, perfect intervals are a natural choice for dividing the octave into a
musical scale. How do we do this? That is, if you set the frequency of
middle C, how should the frequencies of A, B, D, E, F, and G be
chosen?
The following construction of an ancient Greek version of the 7-note
major scale, called Pythagorean tuning, is based on a discussion in
Measured Tones.
The Greeks began their scale on what is now the note D in the
standard C-major scale. Suppose that this starting note has frequency
1; it won’t really, but the frequency of every other note will be
obtained by multiplication from this one. We want to construct a
scale of notes whose frequencies are between 1 and 2 (an octave
above). Let’s take as our basic note-construction procedure going up
or down a perfect fifth; this means multiplying the frequency by
either 3/2 or 2/3. If a note gets too high or too low, we’ll use octave
equivalence (i.e., multiplying or dividing by powers of 2) to put the
frequency back in the interval [1, 2].
First, we add a note A a perfect fifth above D, with frequency 3/2,
and a note G a perfect fifth below D, with frequency 2/3:
note
G
D
A
frequency
2/3
1
3/2
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Spring 2009
Next, we add notes E a perfect fifth above A and C a perfect fifth
below G. The notes C and G are below the frequency of our starting
D, so we multiply their frequencies by 2 enough times to get them into
the desired octave. We do the same to the frequency of E, bringing it
down an octave. Now we have a scale with 6 notes from D to D:
note
D
E
G
A
C
D
frequency 1
9/8
4/3
3/2
16/9
2
This is known as a pentatonic scale because it has 5 distinct pitch
classes. This scale occurs with surprising frequency in both Western
and Asian folk music. For example, the use of the pentatonic scale is
so universal in Chinese folk music that they just use numerals 1 though
5 to stand for notes, and dispense with staves.
To get the full Pythagorean scale, we just need to go one perfect fifth
above the E (to get B) and one perfect fifth below C (to get F). Then
we have the following 7-note scale:
note
D E
F
G
A
B
C
D
frequency 1 9/8
32/27
4/3
3/2
27/16 16/9
2
This scale has some nice properties:
First, there are only two sizes of steps from one note to the next.
The steps D-E, F-G, G-A, A-B and C-D all come by multiplying by 9/8.
The steps E-F and B-C come by multiplying by 256/243. Moreover,
since (256/243)^2 ≈ 1.11 while 9/8 =1.125, two small steps are
approximately the same as one large step. So, it makes approximate
sense to call them half-steps and whole-steps.
Note that the scale we have constructed is called the Dorian mode;
other modes of ancient music come by taking the Dorian mode and
shifting the starting point to a different note. In particular, the
Aeolian mode is the analogue of the modern C-major scale. We can
construct a Pythagorean major scale, starting on a note C with
frequency 1, by taking appropriate whole- and half-steps up from C:
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Spring 2009
note
C D
E
F
G
A
B
C
frequency 1 9/8
81/64
4/3
3/2
27/16
243/128 2
If we wanted to tune a piano according to this Pythagorean major
scale, this tells us how to tune all the white keys, but what about the
black keys? In other words, if we’ve divided the octave into 5 whole
steps and 2 half-steps, then we want to find the notes that fit inside
the whole steps.
To continue with our basic procedure, we go up a perfect fifth down
from F, to get a note B♭ with frequency 16/9, between A and B.
Similarly, we add a note F# between F and G, by going a perfect fifth
up from B. Since 243/128 =3^5/2^7, this F# has frequency 3^6/2^9.
Next, go a perfect fifth down from B♭ to get a note E♭, between D
and E, with frequency 32/27=2^5/3^3
Then, go a perfect fifth up from F#, to get a note C# between C and
D, with frequency 3^7/2^11.
Next, go a perfect fifth down from E♭to get a note A♭, between G
and A, with frequency 2^7/3^4=1.5802
Then go a perfect fifth up from C# to get a note G#, also between G
and A, with frequency 3^8/2^12=1.6018.
So, the Pythagorean procedure of continually adding new notes
eventually generates inconsistent ways of filling in the gaps. What
happened is that going up 6 perfect fifths from D and going down 6
perfect fifths from D generates two notes that are almost, but not
quite, octave equivalent. Put another way, going up a perfect fifth 12
times gets you to a note which is not quite 7 octaves above where you
started: (3/2)^12≈2^7, but (3/2)^12 is about 1.3% more than 2^7.
[see Benson, figure 5.3]
This ratio (3/2)^12 / 2^7 = 1.0136 is known as the Pythagorean
comma; it’s the ratio of G# to A♭ in the scale we just constructed.
All right, suppose you ignore these difficulties, because you just want
to play pieces in one key, without using exotic notes like G# or A♭.
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Spring 2009
Even so, there are still unsatisfactory features of the Pythagorean
scale. Because we tried to have as many perfect fifths as possible, we
don’t have perfect thirds or sixths. Recall that a perfect third is the
ratio 5/4 and a perfect sixth is 6/3. The interval from C to E in the
Pythagorean scale is 81/64 = 1.265 whereas 5/4=1.25. That’s about a
1% error, and the sixths are off by the same amount. This actually
makes an audible difference when you play a major triad.
[play Mathematica example: Pythagorean versus perfect CEG]
Again, it’s more convenient to have perfect 3rds so that you can easily
tune a musical instrument to the scale.
So, in order to have some perfect thirds and some perfect fifths, we
might choose a different compromise. Suppose we begin at C,
pretending it has frequency 1, and then require that notes E and G
above it be a perfect third and perfect fifth away:
note
C
E
G
C
frequency
1
5/4
3/2
2
Next, starting on G we construct another triad G-B-D that’s made
up of a perfect third and a perfect fifth. (Because D is beyond an
octave above C, we divide by 2 to put it in range.) Now we have
note
C
D
E
G
B
C
frequency 1
9/8
5/4
3/2
15/8
2
Of course, we get the same D as the Pythagorean scale, but the B is
different. Then, we construct another perfect triad that’s a perfect
fifth below C.
note
C
D
E
F
G
A
B
C
frequency 1
9/8
5/4
4/3
3/2
5/3
15/8
2
Now we have a 7-note scale like the Pythagorean major scale. This is
called the Ptolemaic major scale (see Johnston) and is the starting
point for a family of tuning systems known collectively as just
intonation. Notice the following interesting feature of this scale:
there are 3 different possible values for the ratio of one note to the
next one. The steps C-to-D, F-to-G and A-to-B are 9/8 ratios (same as
the Pythagorean whole tone) but the steps D-to-E and G-to-A are
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Spring 2009
10/9, a slightly smaller ratio. The ratio between the larger and
smaller steps is (9/8)/(10/9) = 81/80, and this is known as the
syntonic comma.
The smallest steps E-to-F and B-to-C are 16/15, a bit larger than the
Pythagorean semitone of 256/243.
[play scales in both systems]
There are many ways to fill out a 12-note chromatic scale starting
from here, some of which we will discuss later on. But I want to pass
on to the system in use today. Historically, the Ptolemaic scale (in
various forms) was commonly used in Western music up until the 18th
century. Notice that this scale is designed so that the three most
commonly used major chords---on C, G and F---sound perfect. But
what if you want to modulate, or play a song in some other key, using
the same instrument? What usually happens is that suddenly those
major chords which sounded good in the key of C sound increasingly
awful in other keys. For example, the interval D-to-A is
(5/3)/(9/8)=40/27=1.481, about 1.2% flatter than a perfect fifth, so
any attempt to play in the key of D will fall flat (so to speak). For
example, compare the triad on A (a pure minor chord) with the triad
on D:
[play in Mathematica]
The solution to this problem, which allows us to move freely among
keys, is to give up on having perfect intervals. How do we do this?
We saw when constructing the Pythagorean chromatic scale that going
up by 12 perfect fifths is not quite equal to 7 octaves. (The
difference is the Pythagorean comma.) So, as our scale-construction
ratio, let’s replace the 3/2 by a number whose 12th power will be
exactly 7 octaves, namely 2^(7/12), or the 7th power of 2^(1/12)
For convenience, let the letter r stand for 2^(1/12). In this system,
known as equal temperament, we go up a fifth by multiply by r^7.
Starting on C with frequency 1, G has frequency r^7, the D above has
frequency r^14 (which we divide by 2=r^12 to get r^2 for a lower D),
the next note A has frequency r^9, and so on. We get the following
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Spring 2009
chromatic scale:
note C C# D
D# E
F
F# G
G# A
A#
B
C
freq 1 r r^2 r^3 r^4 r^5 r^6 r^7 r^8 r^9 r^10 r^11 r^12=2
In the equal temperament system, every semitone is equal to the ratio
r, and every whole tone is equal to r^2. Comparing the semitones and
whole tones with the other systems, we have
Pythagorean
Ptolemaic
Equal Temp.
semitone
256/243=1.0535 16/15=1.06666
r=1.05946
whole tone
9/8=1.125
9/8 or
r^2=1.1225
10/9=1.11
... so that the scale steps for equal temperament are between those
for the previous two systems.
The real advantage of this compromise is that all the keys have
exactly the same interval ratios, given by powers of r. In every key, a
major third is r^4 (4 semitones) and a major sixth is r^9. So, the scale
does not change shape when you change keys.
The disadvantage is that basic intervals like the third and the fifth are
slightly discordant. The difference is
3/2 /(r^7) = 1.00113..., about a 10th of 1 percent. Can you hear the
difference?
[play examples]
==============================
Irrational Numbers
As you know, rational numbers are fractions of integers; these play a
key role in constructing the Pythagorean and just intonation tuning
systems. Irrational numbers are everything else. How do we know
such numbers exist? Because we can describe them, and prove that
they can’t be fractions of integers.
Here is the grandfather of all examples of irrational numbers: the
square root of 2. We prove that this can’t be a rational number by
assuming it is, and then reaching a contradiction.
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Assume that √2 = p/q, where p and q are relatively prime integers
(i.e., gcd(p,q)=1). Squaring both sides and clearing denominators
gives
2 q^2 = p^2.
Now, because 2 divides the LHS it must divide the RHS, i.e. 2|p^2.
This means that p^2 is an even number, so p must be an even number.
Hence, 4 divides p^2 and so 4 divides 2q^2. Hence 2 divides q^2, and
so 2 divides q. But we assumed that p and q have no common factors!
Now, any rational should be expressible as p/q with no common
factors; but somehow the specific equation √2=p/q contradicts this.
So, we must be incorrect in assuming that √2=p/q is possible for
integers p and q.
Aside: One important fact that’s behind the above argument is that
if 2 divides p^2 then 2 must divide p. This is a special case of the
following fact about prime numbers:
Proposition. If k is a prime number and k divides the product of
integers p and r, then k divides p or k divides r.
So, now we know that √2 is irrational. This in turn implies that the
ratio 2^(1/12) that’s at the heart of the equal temperament system is
also irrational. Again, we can prove this by contradiction:
Suppose that 2^(1/12) equals a rational number p/q. Then raising it
to the sixth power gives √2 = p^6/q^6. But this is impossible since we
already know that √2 can’t be equal to a fraction of integers!
==============
Cents
As you have seen, musical intervals are really ratios of frequencies,
and taking successive intervallic steps really means multiplying the
corresponding ratios. In order to turn these operations into addition,
we introduce a way of measuring intervals using logarithms, and these
are measured in units called cents. This is specifically adapted to the
equal temperament system, so that 100 cents is the same as a
semitone (i.e., the ratio 2^(1/12)); this is where the name cent comes
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from. If x is a ratio of frequencies, then we convert that into cents
using the formula
c = 1200 log_2(x),
where log_2 indicates “log base 2”. In terms of the usual base 10
logarithm, log_2(x) = log(x)/log(2). For example, a Pythagorean
semitone 256/243 corresponds to
1200 log(256/243)/log(2) = 90.2 cents,
which shows that the Pythagorean semitone is about 10% flatter than
the 100-cent semitone.
Assignment: Read Harkelroad, Chapter 3
1. Suppose you want to tune a musical instrument to a major scale
starting on the note A with frequency 220Hz. To what frequency
would you tune the note F# above A, in the Pythagorean, Ptolemaic
and equal temperament systems?
2. Calculate the values of the Pythagorean comma and the syntonic
comma in cents.
3. Suppose we fill out the Ptolemaic major scale to get a 12-note
chromatic scale in the following way: say that F# is a perfect third
above D, C# is a perfect third above A, E♭ is a perfect third below G,
A♭ is a perfect third below C, and B♭is a perfect third below F.
(a) Express the difference between E♭
in this system and in Pythagorean system in cents, and in syntonic
commas.
(b) Do the same for C#.
4. Write your own description (about 300 words long) describing the
differences between the Pythagorean, Ptolemaic and equal
temperament tuning systems.
5. Prove that √5 is irrational, and explain why this implies that the
“golden ratio” phi must also be irrational.