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PHYS 140
The Cent Scale – Handout
Logarithms: Base 2
Recall: Log10 10n = n.
Example: Log 103 = ________
Base 2: Log2 2n = n.
Log2 24 = 4.
You do these ones:
Log2 22 = ______.
Log2 8 = ______.
Log2 29 = ______.
Log2 16 = ______.
Log2 4 = ______.
Log2 2 = ______.
To use Google to evaluate base 2 logarithms, use the command lg. Example: lg(4) = 2.
Cents
The Cents scale divides an octave (ie, a 2:1 frequency ratio) into 1200 cents.
Multiplication of frequency ratios corresponds to adding cents.
Example
C3 -> C4 -> C5 -> C6
Every time you go up an octave, you increase the frequency by a power of 2.
C6:C3 is a frequency ratio of 23
C6:C3 corresponds to adding 1200 cents three times. 1200 + 1200 + 1200 = 3600 cents.
Note that you take the base 2 logarithm of the frequency ratio to get the cents difference:
# of cents = 1200 log2(frequency ratio)
A good musician (like a violinist) should be able to play a note accurate to about 5 cents.
In some tunings, there are distinctions between notes that in an equal temperament tuning would be
the same note.
Example: In the Pythagorean tuning, A flat and G sharp are different.
Example: Fifths
Pythagoras demands that a fifth have a ratio of 3:2. What number of cents is this?
Example 2: Going backwards
(a) What frequency ratio would 1200 cents correspond to? (If you’ve been paying attention, you should
be able to guess the answer!)
Check: To get rid of a base-2 logarithm, raise each side of the equation as a power of 2.
(b) In a 1/6 comma temperament, if C is 0 cents, then D is 197 cents and E is 393. What is the frequency
ratio between
 D to C
 E to D
Is this an equal tempered scale?
You’ll do the rest of this handout in your group. Refer to previous handouts as needed.
1) In the Pythagorean temperament,
(a) How many cents separate a perfect third?
(b) How many cents separate the A flat and G# that you calculated in your previous handout?
[Hint: The frequency ratio Aflat:G# can be found by multiplying Aflat:C4 and C4:G#.]
(c) How many cents would separate A flat and G# on an equal temperament tuning?
2)
In the equal-temperament tuning, how many cents are in a semi-tone? (Assume all semitone
steps correspond to the same number of cents and that you have 1200 cents per octave.)
3) Why are they called cents?
4) In a Pythagorean system, are the number of cents between semitones always equal? Try
calculating the following (using the chart on page 3 of the previous handout and your knowledge
of how to convert frequency ratios to cents):
(a) The number of cents between A4 and A4# [minor 2nd]
(b) The number of cents between A4 and B [major 2nd]
(c) Subtract (a) from (b) to get the # of cents between A4# and B on an A-major tuning for the
Pythagorean system.
(d) Is your answer to (c) the same as (a)? If not, what does this tell you?
Summary:
1. Pythagorean temperaments have exact1 frequency ratios but inexact (and unequal) additive
spacings (cents). Equal-temperament systems have exact _______________________ but
inexact _______________________________________.
2. Would a modern keyboard ever need to have different keys for sharps and flats (like one key for
C sharp and another for D flat)? Why or why not?
The Wolf Interval
Not to be confused with Wolf Tones, which occur as a result of a slight mismatch between the note
played on a string instrument and the resonant vibration of the body.
In the ¼ Comma Meantone temperament, if you take C to be 0 cents, E flat is 310 cents and G# is 773
cents.
(a) What is the difference (in cents) between E flat and G#?
(b) Find the frequency ratio that your answer to (a) corresponds to.
(c) If G# had frequency of 415.00 Hz, what would the frequency of E flat be? [check: you should get
a smaller number]
(d) Play these notes together in Mathematica. This is called the wolf interval. Can you hear the
howling?
Note that this interval sounds terrible—composers avoided it.
Have the instructor check off your work before you leave.
1
In the sense of being a ratio of whole numbers