The sound of music

Tonight we will chat about the most well-known of mechanical waves:  sound.

Sound is a mechanical wave, meaning that it REQUIRES a medium through which to travel.  Whereas light (or other EM waves) can travel anywhere (more or less), and travel fastest (at the speed of light) through a vacuum, sound is restricted greatly.  It can only travel through a medium, which itself carries the vibrations that are sound waves.

The same characteristics previously discussed still apply:  frequency, wavelength, speed, amplitude, crest, trough.  However, sound itself is a longitudinal wave (jiggling "back and forth") rather than a transverse wave (like EM waves, which vibrate "up and down").
Wave definitions to recall
Frequency (f) - number of waves per second (measured in hertz, or Hz).  For sound, frequency refers to pitch.  Some frequencies are defined as notes
Wavelength (𝝺) - the length of the wave, from crest to crest, etc. (in m). The symbol for wavelength is lambda.
Wave speed (v) - how fast the wave energy travels (in m/s)
Amplitude - the height of the wave from equilibrium.  For sound, amplitude usually refers to volume.

 What does sound "look" like?

Animation of sound as a mechanical wave:

As long as you stay in the same medium:
  • The speed stays the same.
  • As frequency goes UP, wavelength goes DOWN.
  • Speed = frequency x wavelength



The Speed of Sound Consider the speed of sound in dry air:
337 m/s (at 10 deg C)
343 m/s (at 20 deg C)
Note that this is WAY less than the speed of light - about 1 millionth of it.
Speed of sound in helium is around 972 m/s - what can this predict?
In water, the speed of sound is approximately 1500 m/s.
In steel, it is approximately 5000 m/s.

 Hearing: The frequency range of average human hearing is approximately: 20 Hz - 20,000 Hz
This changes with age and repeated exposure to loud sounds.
Dogs can hear up to 45,000 Hz.
Mice, bats, dolphins, and cats have an even higher range.

Music


Octaves In music, doubling the frequency is defined as an octave.
An octave is the same note, but in a higher register.  
An octave lower is the same note, but half the frequency - in a lower register.
To find the frequency of a note 2 octaves higher, you double the frequency again.

Octaves are also the difference between the low “DO” and the high “DO” on a DO-RE-ME major scale.
 Chords
Chords are also produced when tones have simple ratios.  For example:
A major chord is produced when 3 particular notes of a major scale are played together:
  • The first note in the chord (root note), the third note in the scale, and the fifth note in the scale.  The ratios are:  5/4 (which is 1.25) and 3/2 (which is 1.5).
  • Other chords and musical intervals have different ratios.  
  • Basically, chords are made of notes that sound good together because their frequency ratios work out nicely - the waves add together well


Equal Tempered Scale

In western music, we use an "equal tempered (or well tempered) scale."  It has a few important characteristics to mention:

The octave is defined as a doubling (or halving) of a frequency.

You may have seen a keyboard before.  The notes are, beginning with C (the note immediately before the pair of black keys):

C
C#
D
D#
E
F

F#
G
G#
A
A#
B
C

(Yes, I could also say D-flat instead of C#, but I don't have a flat symbol on the keyboard.  And I don't want to split hairs over sharps and flats - it's not that important at the moment.)

There are 13 notes here, but only 12 "jumps" to go from C to the next C above it (one octave higher).  Here's the problem.  If there are 12 jumps to get to a factor of 2 (in frequency), making an octave, how do you get from one note to the next note on the piano?  (This is called a "half-step" or "semi-tone".)

The well-tempered scale says that each note has a frequency equal to a particular number multiplied by the frequency that comes before it.  In other words, to go from C to C#, multiply the frequency of the C by a particular number.

So, what is this number?  Well, it's the number that, when multiplied by itself 12 times, will give 2.  In other words, it's the 12th root of 2 - or 2 to the 1/12 power.  That is around 1.0594.

So to go from one note to the next note on the piano or fretboard, multiply the first note by 1.0594.  To go TWO semi-tones up, multiply by 1.0594 again - or multiply the first note by 1.0594^2.  Got it?

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