Download Physics of Music PHY103 Worksheet #4 Setup for fretted monochord

Physics of Music PHY103 Worksheet #4
Setup for fretted monochord lab
1) The tempered scale.
a) Consider the possibility that you are playing jazz with baroque instruments.
Because they are baroque instruments you need to tune them to a flatter A of 420
Hz instead of the modern standard of 440 Hz. However since you are playing
jazz you need to be able to play chords in any key. Compute the frequencies for
the 4th octave of the tempered scale with the lowered (flat) A. The notes of the
tempered scale differ by a factor of
21/12  1.05946 .
b) Suppose you are building a guitar with frets and the length of the string is 1m
long. Where along the neck of the guitar would you place the frets in order to
have a tempered scale?
2) Pitch measurement and calculating with cents.
Pitches are commonly measured with respect to the frequencies of the tempered scale
with a concert A of 440Hz. These frequencies are listed in the table below. Tuners
usually give the nearest note on the tempered scale and the difference between this not
and the one you played. This difference is given in cents. Cents are defined in the
following way: There are 100 cents in each half tone, and twelve half tones in an octave.
So there are 1200 cents in an octave. An octave corresponds to frequency change of a
factor of two. In other words a second note that is an octave above a first note has twice
the frequency of the first. Consequently 1 cent
1
1200
corresponds to a factor of 2 . If you are sharp by
+21 cents you multiply the frequency of the nearest
21
1200
tempered scale note by 2
to calculate the actual
frequency of your note. If you are flat by 18 cents
Frequencies of Notes in the
Tempered Scale 4th octave
Note
Frequency
(Hz)
C4
261.63
C# (D♭)4
277.18
D4
D# (E♭)4
E4
F4
F#(G♭)4
G4
293.66
311.13
329.63
349.23
369.99
392.00
G#(A♭)4
415.30
a) What is the frequency of the note C4-10 (C4
and 10 cents flat)?
A4
A#(B♭)4
440.00
466.16
b) What is the frequency of the note G3+5?
B4
493.88
To predict the notes in the
octave above this multiple the
above frequencies by two. To
predict the notes in the octave
below this, divide the above
frequencies by two.
18
1200
you would multiple by a factor of 2 . Here is an
example of the format used by tuners: C4-10. The
first letter is the nearest note on the tempered scale.
The second note is the octave (4 is that begun by
middle C on the piano). The last note is the number
of cents the note is above or below the pitch of the
note. In this example the note is -10 cents below C4.