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PHYS 140
Evaluating Equal-Tempered Scales
The standard scale used on piano keyboards is the chromatic scale—the pattern of black and white keys
with 12 semi-tones comprising an octave. This choice of 12 semi-tones is a convention, and one could
imagine using other scales.
The 12-semitone scale happens to be a reasonably good one because the frequency ratios of many of its
intervals (when tempered equally) are close to ratios of simple whole numbers. For an “ideal” scale—
one in which the intervals are the most consonant (sound the best)—the frequency ratios would be
those given in the chart below.
Name of
Frequency ratio in “ideal” Decimal value
Cents
Interval
system
1200*lg(freq. ratio)
Minor 3rd
6/5
1.20
315.6
rd
(major) 3
5/4
(major) 4th
4/3
th
(major) 5
3/2
minor 6th
8/5
th
(major) 6
5/3
Fill in the decimal value of each ratio to two decimal places; then calculate the corresponding cents.
Let’s see how well the 12-tone interval does in producing “ideal” intervals. Recall that the frequency
ratio between adjacent tones is the 12th root of 2, or 1.05946.
Number of semi-tones
in interval
1
2
3
4
5
6
7
8
9
10
11
Frequency ratio in
equal-temperament
1.059
1.122
1.189
Number of cents
100
200
300
Fill in the rest of the rows in the 2nd column. Remember that you are just raising the 12th root of 2 to
some power.

Which intervals in the equal-tempered 12 semitone scale match the “ideal” intervals to within 5
cents? These are the intervals that sound the most consonant.

Which match to within 6- 15 cents? Those intervals would sound OK but not great.
The following exercises are adapted from “The Physics of Music and Musical Instruments,” by David Lapp.
Create a musical scale
Now it’s your turn to build a musical scale. Pick a scale that has no fewer than 5 semitones, and no
more than 18 semitones.
Make a chart like the one for the 12-tone scale. (You can add more rows if needed to the chart below.)
Calculate all the frequency ratios.

Previously, we used the 12th root of 2, or 1.059, to find the frequency ratios. What number will
you be using for your scale?
For the frequency ratios that match one of the ideal ratios within .01, also calculate the cents.
Number of semi-tones
in interval
1
2
3
4
5
6
7
8
9
10
11
Frequency ratio in
equal-temperament
Number of cents

Which intervals, if any, match within 5 cents?

Which intervals, if any, match within 15 cents?
Other Equal-Tempered Scales
In addition to the 12-tone scale, there are 3 other scales that seem to work well with equal
temperaments.
For each of the following scales, calculate the number cents between each note. Keep in mind that an
octave is 1200 cents. (This was chosen so that 1200 divided by the number of semitones (normally 12)
would give 100 cents. Here, you’re still going to assume an octave is 1200 cents, but you’ll be dividing it
by different numbers of semitones.) I’ve done a sample calculation for you.
a) 19 tone
1200 cents / 19 semitones = 63.16 cents per semitone.
b) 31 tone
c) 53 tone
Choose 2 of the 3 temperaments and fill in the charts below. To find how many steps in a scale make up
a given interval, try dividing the cents in the “ideal” interval by the number of cents per note on the
scale.
For example, in a 12-tone scale, to find the number of steps for a minor 3rd, you’d divide 315.6 cents
(see chart at top of 1st page) by 100 cents, giving 3 (plus some remainder.) So the number of steps that
give you a minor 3rd in a 12-tone scale is 3. Multiplying 3 by 100 cents is 300, so the deviation from ideal
is 315.6-300 = 15.6 cents.
Now do the same for 2 of the 3 scales below.
Circle the intervals that are within 5 cents of the ideal interval.

Which scale that you evaluated (12 tone or one of the others) has the most consonant intervals?

Why do you think we use 12 tones, instead of 19, 31, or 53?
Pick one of the scales you evaluated (19, 31 or 53) and choose the interval that was the closest to the
ideal interval. For this interval, play it in Mathematica in both that scale, the 12-tone equal-tempered
scale, and the Pythagorean scale (for the latter you might need to consult a previous handout.)
Example: If you want to play an interval with frequency ratio of 1.63, you could use the following code:
f = 262 (or whatever number you like—this is middle C)
Play[Sin[2 Pi f t] + Sin[2 Pi f *1.63*t],{t,0,1}]
Then you would repeat this for the other 2 temperaments.


Can you hear a difference between the 3 temperaments?
Does the one that sound the best correspond to the one that is closest to the ideal ratio?
Have the instructor check your work before continuing.
More Mathematica Investigations
On Moodle, I’ve made available a Mathematica notebook file adapted from The Mathematics of Tuning
Musical Instruments by Erich Neuwirth.
Open the file in Mathematica. It has a predefined command called PlayScale. You should also see the
command
PythagoreanMajorScale = {1, 9⁄8 , 81⁄64 , 4⁄3 , 3⁄2 , 4⁄3 ∗ 81⁄64 , 3⁄2 ∗ 81⁄64 , 2}
This scale is in the 12-tone system, but it only has 8 notes. This is because it’s the diatonic scale, a subset of the chromatic scale. If you multiply the list by 262, you can hear what this scale sounds like in the
key of C:
PlayScale[262 ∗ PythagoreanMajorScale,.5]
The “.5” refers to the duration of each note.
Fill in the missing notes in the Pythagorean Major scale to turn it into the complete 12-tone chromatic
scale. (To see which are “missing,” consult your previous handout that has the Pythagorean intervals
on it. There are 3 intervals on that chart that are missing from the scale. In addition, you should add the
minor 6th, which has a ratio of 128/81.) Play the scale.
Without erasing the Pythagorean one, create a scale that represents the 12-tone Equal-Temperament
scale. Here’s an easy way to do that:
EqualScale = Table[2^(𝑖 ⁄12), {𝑖, 0,12}]
Plug this “EqualScale” into your PlayScale function to hear what it sounds like.

Can you hear any difference between the Pythagorean and Equal-Temperament scales?
By modifying the command above, play a 19-tone scale, a 31-tone scale, and a 53-tone scale.

How do these scales sound to you?
One thing I find mystifying about temperaments with more than 12 tone is how you’d make a major
scale out of them. As we saw previously, a major scale in a 12-tone system selects 8 of the 12 tones.
This website: http://www.tonalsoft.com/monzo/19tet/19samba.htm suggests taking the following 10
notes out of a 19-tone scale. This is analogous to a D major scale on a 12-tone system.
19
17
16
14
13
11
8
6
3
0
12.00
10.74
10.11
8.84
8.21
6.95
5.05
3.79
1.89
0.00
D
C#
C
B
Bb
A
G
F#
E
D
(The first column shows how many semi-tones away from D the note is; the middle column, as far as I
can tell, is the number of cents divided by 100.) Thus, one can construct it in Mathematica as follows:
Major19Scale = {1, 23⁄19 , 26⁄19 , 28⁄19 , 211⁄19 , 213⁄19 , 214⁄19 , 216⁄19 , 217⁄19 , 2}
Play this scale in Mathematica, starting at middle D:
PlayScale[293.66 ∗ Major19Scale,.5]
Without erasing this scale, create another one that lacks the C and the B flat. (In other words, erase the
13th and 16th semitones.) This is what a D major scale would sound like in 19-tone temperament if you
only used 8 semi-tones.

Which of these (a scale with 10 tones, or one with 8 tones) sounds more natural, in your
opinion?
Now use the pre-defined 12-tone version of D major in Mathematica (this is already in the source file):
EqualMajorScale = 2^({0,2,4,5,7,9,11,12}⁄12)
Play a 12-tone D major scale followed by the 19-tone D major (with 8 semi-tones). Again, start at middle
D:
PlayScale[293.66 ∗ EqualMajorScale,.5]

Does the standard D major scale sound different from the 19-tone D major scale (using the
subset of 8 notes)? If so, which do you like better?
Creating a 19-tone piano keyboard
In the standard 12-tone system, the keyboard looks like:
To the right is shown the full 19-tone scale (borrowed from tonalsoft.com):

Which black keys would you have to split to get a 19-tone keyboard?

Which black keys that don’t exist at all in a 12-tone scale would have to
be added in?

How many black keys in total would you be adding?
Does this equal the difference between the number of semi-tones
in a 19-tone scale and the number in a 12-tone scale?

Draw what a 19-tone keyboard would look like.
(19 12.00
18 11.37
17 10.74
16 10.11
15
9.47
14
8.84
13
8.21
12
7.58
11
6.95
10
6.32
9
5.68
8
5.05
7
4.42
6
3.79
5
3.16
4
2.53
3
1.89
2
1.26
1
0.63
0
0.00
E )
Eb
D#
D
Db
C#
C
B#
B
Bb
A#
A
Ab
G#
G
Gb
F#
F
E#
E
Exploring the Pentatonic Scale with Isle of Tune
There’s a web app called Isle of Tune that you can download and play. It allows you to make songs by
building a road and lining the road with different notes.
The set of notes available is kind of weird—it seems to span a diatonic scale, but it’s missing two of the
notes. In fact, this is a popular way to limit a scale—it gives you a set of 5 notes in every octave that
sound pretty good together. This set of notes is called the Pentatonic Scale. (The actual number of
notes available in the sim is larger than 5, since it spans 2 octaves.)
Try this experiment: Build a road in isle of tunes (a loop works well) and line it so that on both sides of
the roads there are flowerpots. This produces intervals, because as the car drives past the flowers, it
sounds both flower pots. On one side of the road, make every note the same; on the other side of the
road, try out every different note. (It might help to make the notes very spaced out so you can analyze
them carefully as the car drives past.)

Keeping in mind that the full range of notes provided is 2 octaves, how many consonant [good
sounding] intervals do you find in a single octave? (Personally, I don’t think any of these
intervals really sound terrible—such is the magic of using the pentatonic scale. But you might
find some sound better than others.)

If this simulation included every note in the 12-tone chromatic scale (which it doesn’t), how
many consonant intervals would you expect to find? (Refer back to the first page. Count them
as consonant if they’re within 15 cents of the ideal interval.)
Get the instructor to check off your work.

Feel free to build a song of your own choosing.