This document discusses how the wave patterns of different musical notes interact to determine whether they sound harmonious together when played. It explains that notes with frequency ratios that result in their waves aligning periodically, such as the ratio of 3:2 between the notes C and G, will sound good together. Chord notes in a C major chord have frequency ratios that produce this alignment. The document also provides a formula to calculate the frequency in Hertz of a note from its MIDI pitch number.

1. Math InD’Aoust
By : Ashley
Music

2. • Do you wonder why some notes played
together sound so good?
• The answer to that is the wave pattern of a
note. Created by a guitar, piano or any other
instrument.
• The number of times per second each wave
hit our ear is called frequency, which is
measured in Hertz. (Hz)
• The more waves per second, the higher the
pitch.

3. Here is how to understand why some note
combinations sound better, first look at the wave
patterns of 2 notes that sound good together. Let’s
use middle C and the G just above it as an example..
If you can notice every 3rd wave of the G matches
up with every 2nd wave of the C.
(The G note is the Red Wave) and (C note is the Green Wave)

4. • But here in the second graphic of a C and an
F# they don’t make such a great sound
because there is no pattern.
(The F# note is the Red Wave) and (C note is the Green Wave)

5. Now let’s look at a chord, to find out why it’s notes sound good together. Here are the
frequencies of the notes in the C Major chord.
C – 261.6 Hz
E – 329.6 Hz
G – 392.0 Hz
• The ratio of E to C is about 5/4ths. This means that every 5th wave of the
E matches up with every 4th wave of the C. The ratio of G to E is about
5/4ths as well. The ratio of G to C is about 3/2. Since every note’s
frequency matches up well with every other note’s frequencies they all
sound good together.

6. C Major
• Here are the ratios of the notes in the C Major
key in relation to C: (this only applies to C major which is most common)
• For example, the D note. The ration means every 9th wave of the D, matches up with every 8th
wave of the C.
• C–1
• D – 9/8
• E – 5/4
• F – 4/3
• G – 3/2
• A – 5/3
• B – 17/9

7. • Hertz (number of vibrations a second) = 6.875 x 2 ^ ( ( 3 + MIDI_Pitch ) /
12 )
• The MIDI Pitch value is according to the MIDI standard, where middle C
equals 60, and the C an octave below it equals 48. As an example, let’s
figure the hertz for middle C:
• H = 6.875 x 2 ^ ( ( 3 + 60 ) / 12 ) = 6.875 x 2 ^ 5.25 = 261.6255
• H = Hertz.
• The next note up, C#, is:
• Hertz = 6.875 x 2 ^ ( ( 3 + 61 ) / 12 ) = 277.1826
• The next note, D, is:
• Hertz = 6.875 x 2 ^ ( ( 3 + 62 ) / 12 ) = 293.6648
• The jump between C and C# is 15.56 Hertz, the jump between C# and D
is 16.48 Hertz. Although the Hertz jump is not equal between the
notes, it is an equal jump in the exponent number and it sounds like an
equal jump to our ears going up the scale. This gives a nice smooth
transition going up the scale.

8. Mathematics in Music
• Pappas, Theoni. The Joy of Mathematics: Discovering Mathematics All around
You. San Carlos, CA: Wide World Pub./Tetra, 1989. Print.
• Heimiller, Joseph. "Where Math Meets Music." Voice-to-note MIDI Music Editor: Music
Masterworks Composing Software. May 2002. Web. 20 Mar. 2012.
<http://www.musicmasterworks.com/WhereMathMeetsMusic.html>.
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