PHYS 140
Temperaments & Scales
- Handout -
Modern pianos and most other instruments are tuned to an “equal temperament” scale. This means
that there is an equal spacing between the pitches of each consecutive semi-tone. But are they equal in
ratios of frequencies, or equal in difference between frequencies? In this exercise, you’ll discover
exactly what “equal temperament” means.
1) Pick any two notes that are off by a single semi-tone. (Example: C4 and C4#, or B4 and C5.)
Calculate (a) The ratio of their frequencies, with the larger on top, and (b) the difference
between their frequencies. [Consult the chart on the backside of this handout.]
2) Repeat this for 2 other pairs of consecutive notes.
3) Which are “equal”—the ratio between the frequencies for the various pairs, or the difference
for the various pairs?
So now you know that an “equal temperament” tuning means that a piano is tuned so that adjacent
keys are separated by equal ratios of frequencies, and that this ratio is: _____________.
Why this number? Let’s derive it with a calculation. We’ll start with a simplistic example first.
Example
Suppose you had a musical system with only 3 notes: A, B, and C, and nothing in between (no semitones.) So a ‘piano’ would have A1, B1,C1 then A2,B2,C2, etc. You want A2 to have double the
frequency of A1. You also want all the ratios of consecutive notes to be equal: B1/A1 = C1/B1 etc.
Let’s call this ratio “x,” so that B1/A1 = x.
1) What must the product of B1/A1 and C1/B1 and A2/C1 be equal to, in terms of x?
2) Note that if you multiply symbolically the 3 ratios above, everything cancels out except for
A2/A1. Check for yourself:
So, you know that A2/A1 =__________ (answer from previous question).
3) But A2/A1 =2. Thus, you should have an equation that looks like 2 = x3. To solve this equation,
you take the cube root of each side, so you get x = cube root of 2. To figure out what number
this is, you can type this as 2^(1/3) on a calculator or Google.
What number do you get? ____________. That would be the frequency ratio of successive
notes in this hypothetical example.
Actual Calculation
Now let’s go back to the real world. How many semi-tones are in an octave? (Hint: not 8).
As in the previous example, we want an octave to have a ratio of 2. Write an equation that you
can use to solve for x, where “x” is the ratio of consecutive semi-tones.
Solve for x. Is it the same as you deduced from the keyboard chart? If not, double-check your work.
Why are pianos tuned this way?
If you decided to tune a piano the other way—by having equal differences of frequencies instead of
equal ratios—what spacing between frequencies would you choose? Here’s one way to decide: The
frequencies of the A notes are standardized across different tunings, which means no what system of
temperaments you’re using, A4 is taken to be 440 Hz exactly.
1) If A4 is 440 Hz, what should A5 be? (Hint: They’re an octave apart.)
2) What is the difference in frequency between A4 and A5?
3) If you wanted to divide the A4 – to – A5 octave into equal semi-tones, this would mean dividing
this difference into 12 equal parts (since there are 12 semi-tones in one octave.) For the A4 – to
– A5 octave, what would the frequency difference for one semi-tone be?
4) If every semi-tone on the piano was separated by the frequency difference you calculated
above, what frequency would you end up getting for A3?
[Hint: Start with A4 = 440 Hz and subtract 12 semitones.]
What frequency would you get for A6 (starting with A5 and adding semitones)?
5) Do those frequencies that you calculated in #4 satisfy that requirement that going up an octave
doubles the frequency?
6) Explain why an “equal temperament” tuning uses equal ratios of frequencies instead of equal
differences.
Check in with the instructor before continuing.
The Pythagorean Hypothesis
The ancient Greek Pythagoras (to whom we also owe the Pythagorean theorem) posited that:
The intervals that sound the most harmonious are those that are related
by the simplest integer ratios (like 3:2, or 2:1).
Recall that two notes having the ratio 3:2 are called fifths. What are notes with a 2:1 ratio called?
Taking Pythagoras’ idea seriously, one might try to tune a piano so that fifths have exactly the ratio 3:2.
Is this compatible with the equal temperament tuning? Let’s find out.
You should have found on page one that frequency ratio between adjacent semi-tones required for
equal temperament tuning is 1.05945.
Example. The frequency ratio for a third (4 semi-tones apart) in equal-temperament would be
(1.05945)^4 = 1.2599. In the Pythagorean system, the ratio is set to 81/64, or 1.2656, which is close to
the equal-temperament value, but not quite the same.
1) Predict the frequency ratio for a fifth (an interval 7 semi-tones apart) using the equal
temperament tuning. See example above.
2) How far off is this from 3:2? Calculate the percent different between the answer you got above
and the ratio 3:2 (or 1.5).
% difference = (theoretical # - actual #)/(theoretical #)x100%
Moral: You can tune a piano to equal temperaments, or to Pythagorean tuning (where the fifths
all have an exact 3:2 ratio), but you can’t have both at the same time.
Limitations of Pythagorean tuning
Even if you tune an instrument so that the fifths come out perfect, you still won’t get the other
intervals sounding very nice. A “third,” you’ll recall, is a set of notes separated by 4 semi-tones;
A “fourth” is separated by 5 semi-tones. Let’s see what these intervals come out to be in the
Pythagorean temperament.
The ancient Greeks had certain ideological beliefs about numbers. They believed strongly in
ratios of whole numbers. (If you wanted to annoy an ancient Greek, you could try teaching
them about the concept of irrational numbers—numbers that cannot be expressed as a ratio of
whole numbers.) So in a Pythagorean scale, you would start with one note (let’s use A4 = 440
Hz), and construct other notes by insuring that the intervals are ratios of whole numbers.
See the chart on the next page.
Here’s how this worked out in practice:
Name of
Separated by
Interval
how many
semi-tones
A
Unison
0
nd
A to A# Minor 2
1
A to B
(major) 2nd
2
A to C
Minor 3rd
3
A to C# (major) 3rd
4
A to D
(major) 4th
5
A to D# Augmented 4th 6
A to E
(major) 5th
7
Frequency ratio
Frequency of note
In Hz
1
(2/3)5*23
(3/2)2/2 = 9/8
(2/3)3*4
(3/2)4*(1/2)2 = 81/64
(2/3)*2 = 4/3
(3/2)6*(1/2)3
3/2
440.00
463.53
Fill in the frequencies for the “major” intervals (the major 2nd, 3rd, 4th, and 5th.) These are
calculated by taking 440.00 and multiplying by the relevant fraction.
1) Predict which intervals are going to sound the best, based on the Pythagorean Hypothesis
(re-read the text box on the previous page) and which would probably sound the worst.
Then go to Mathematica and play each of the major intervals. (Don’t delete the old ones as
you go along.) For example, to play a major 2nd based on the note A4, you could use the
following:
𝑓 = 440;
Play[Cos[2Pi 𝑓 𝑡] + Cos[2Pi (9/8) ∗ 𝑓 𝑡], {𝑡, 0,2}]
2) Why are the factors of 2 Pi in there?
3) Rank the intervals you played in Mathematica from least-harmonious sounding to mostharmonious sounding. (There is no “right” answer to this.)
Pythagoras (and the other ancient Greeks) considered thirds to be less important than fifths, so
perhaps the fact that thirds tend to sound really bad in a Pythagorean tuning didn’t bother them
very much. However, there’s another disadvantage of Pythagorean tuning, which will become
clear in the next example.
In the following questions, remember that when combining n equal intervals, you raise the frequency
ratios to the nth power.
Example: Assuming going up an octave doubles the frequency (a ratio of 2/1), going up three octaves
gives an overall frequency ratio of (2/1)^3, or 8/1. Note that everything is raised to the 3rd power.
The Pythagorean Comma
There’s a fundamental inconsistency in the Pythagorean system. Suppose you start at the lowest C on a
piano (C1) and go up an octave, then another octave, etc., until you reach C8.
1) What should the frequency ratio between C8 and C1 be (note that this is 7 octaves)?
2) Here’s another way to get from C1 to C8. Under the Pythagorean system, going up a fifth means
multiplying the frequency by (3/2). There are 12 “fifths” between C1 and C8. (Actually, you could call
the last note made by the 12th fifth “B sharp,” but normally we just call that C.) If each fifth gets you a
frequency ratio of 3/2, what should the frequency ratio be after 12 fifths?
3) Express these ratios as decimal numbers so you can compare them. Did going up by fifths result in
the correct frequency ratio?
4) If the difference between this “B7 sharp” and “C8” was too small for the human ear to discern, it
wouldn’t be much of a problem. Try listening to the two notes in Mathematica. Can you tell the
difference? (One way to program this is to define a frequency “f” that is equal to C1 [consult your
frequency chart], and in one case multiply it by your answer to question 1, and in the other case multiply
it by your answer to question 3.)
One more example of Pythagorean mismatch
5) Suppose we want to play a third based on middle C: C4-E4. So you tune the piano so that E4:C4 has
a frequency ratio of 5:4.
To play a third based on middle E, you’d want E4-G4# to have the ratio G4#:E4 = 5:4 also. What
frequency ratio do you then end up with between C4 and G4#? Express it as a decimal.
[If you’re confused about how to do this calculation, consult the rule in italics at the top of the page.]
6) Now say you want to tune the third A4 flat –C5. Again, that means forcing C5:A4flat = 5:4. This
makes the ratio between A4 flat and C4 be
A4flat :C5 = (4/5)*2 = __________. (I inserted a factor of two because C4 has half the frequency of C5.)
Express your answer as a decimal.
7) In our modern equal-temperament system, Aflat is exactly same note as G#. Compare your values for
the Pythagorean system—how far off is the G4# ratio that you calculated in 5) from the A4 flat you
calculated in the previous question? Calculate the percent difference.
Play both frequencies in Mathematica—can you tell the difference?
This illustrates a more general problem in the Pythagorean system. In general, in a Pythagorean tuning,
there are these mismatches between the sharp of one key and the flat of next, so that A sharp sounds
different than A flat—you properly need two black keys there instead of one! In fact, old harpsichords
and organs had split black keys, so you could play whichever was appropriate for the key that you were
in. Alternatively, you could write music for which that problem never arises, by avoiding situations
where you’d have to play, say, the B flat. This was commonly done in the Renaissance, and in fact
Pythagorean tuning is considered appropriate for Medieval and Renaissance music.
Or, you could just pick one or the other—say A4 flat-- and just use it to represent both G4# and A4 flat.
Let’s see how well this works. In mathematica, we’ll play
(a) E and G# at the same time.
(b) C5 and G# at the same time.
(c) C5 at A flat at the same time.
Which do you predict will sound dissonant?
Test your hypothesis. Here’s the code I used:
middlec = 261.63;
e4 =(5/4)* middlec;
gsharp = 1.5625 middlec;
aflat = 1.6 middlec;
c5 = 2 middlec;
Play[Cos[2 Pi e4 t]+Cos[2 Pi gsharp t],{t,0,2}]
Play[Cos[2 Pi c5 t] + Cos[2 Pi gsharp t],{t,0,2}]
Play[Cos[2 Pi c5 t] + Cos[2 Pi aflat t],{t,0,2}]
Was your prediction confirmed?
Summary:

Describe one advantage and one disadvantage of the
Pythagorean tuning

Describe one advantage and one disadvantage of the
equal temperament tuning
Have the instructor check off your work.
If you have extra time, start the homework. Enter the answers on
Moodle, as always.
HW problem 1
In an equal temperament tuning, what is the frequency ratio of
an augmented fourth? (An augmented fourth is 6 semi-tones apart.)
Express it as a decimal number. Include two decimal places
in your answer.
HW problem 2
In Pythagorean tuning, what ratio would you want between E2
and C2? Express the ratio as a decimal number to 2 decimal places.
Construct the ratio so that it’s greater than 1 (E2:C2, not the other
way around.)