Notes on music FYI

Music 1 - Notes and Harmonics

In western music, we use an "equal tempered (or well tempered) scale."  It has a few noteworthy characteristics;

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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Recalling harmonics

Let us recall "harmonics", visible on a string (as demonstrated in the recent lab).  Harmonics are wave shapes produced that have a maximum amplitude under given conditions (tension in string, length of string, composition of string, etc.).  Every stretched string has a particular lowest frequency at which it will naturally resonate or vibrate.  However, there are also higher frequencies that will also give "harmonics" - basically, pretty wave shapes (also known as "standing waves").  These higher frequencies are integer multiples of the lowest frequency.

So, if the frequency of the lowest frequency is 10 Hz (for an n = 1 harmonic), the next harmonic (n = 2) occurs at 20 Hz.  N = 3 is at 30 Hz, and so on.

For those of you who play guitar, you know that you get harmonics by lightly touching the strings at certain locations.  In the exact center of the neck (12th fret) you get a harmonic (the 2nd one) and the frequency is twice that of the open string - one octave above, as we will discuss.

A block problems to work on IN CLASS Tuesday

Wave practice!

Work out in your notes.  These will be reviewed on Thursday.

Recall also:

v = (frequency)(wavelength)

c = 3 x 10⁸ m/s (speed of light, including any type of electromagnetic wave)

1.  What is the wavelength of the radio station 97.9 ("98 Rock").  Keep in mind that the number refers to the frequency in MHz, and note that MHz means 'million (x 10⁶) Hz."

2.  The visible range of human eyesight is 700 nm to 400 nm.  What are the frequencies associated with this, and which end is red and which is violet?

3.  Find the speed of a 500 Hz wave with a wavelength of 0.4 m.

4.  What is the frequency of a wave that travels at 24 m/s, if 3 full waves fit in a 12-m space?  (Hint:  find the wavelength first.)

5.  Approximately how much greater is the speed of light than the speed of sound?

6.  Harmonics

a.  Draw the first 3 harmonics for a wave on a string.

b.  If the length of the string is 0.5-m, find the wavelengths of these harmonics.

c.  If the frequency of the first harmonic (n = 1) is 15 Hz, find the frequencies of the next 2 harmonics.

d.  Find the speeds of the 3 harmonics.  

7.  (Review) Differentiate between mechanical and electromagnetic waves.  Give examples.

8.  The note C vibrates at 262 Hz (approximately).  Find the frequencies of the next two C’s (1 and 2 octaves above this), and the frequency of the C below it.

9.  A red LED has a wavelength of 662 nm.  What is the frequency of light emitted from it?  (nm refers to x 10^-9 m)

10.  Concert A is defined as 440 Hz.  What are the frequencies of:
a.  A#, which is one semi-tone above A?
b.  C, which is three semi-tones above A?
c.  A-flat, which is one semi-tone below A?

(Recall that a semi-tone is the same as a piano key or guitar fret.)

More quiz practice - answers at bottom

Hey everyone,

By popular request, here are a few more quiz practice problems.

1.  How long should a pendulum be so that it has a 1.5 second period?  Also, what is the frequency of oscillation of this pendulum?

2.  Consider a 0.5-kg box sliding down a frictionless incline.  If it drops a total of 0.75 during the descent, find the following:

a.  initial PE relative to the bottom of the slide
b.  KE at the bottom of the slide
c.  speed at the bottom of the slide
d.  speed half-way down the slide

3.  A Telecaster guitar string is approximately 0.64-m long from bridge to "nut" (where the strings connect to the headstock).  Perform calculations for the first 3 harmonics, assuming that the lowest frequency (n=1) is 320 Hz.  Fill in the table below:

n          harmonic drawing          wavelength          frequency          speed

1                                                                                 320

2

3


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Answers:

1.  0.56 m

2.

3.7 J
3.7 J
3.8 m/s
2.7 m/s

3.  with rounding, answers will be close to these:

1     1.28 m     320 Hz     410 m/s
2     0.64 m     640 Hz     410 m/s
3     0.43 m     960 Hz     410 m/s

Wave stuff

So - Waves.....  

We spoke about energy.  Energy can, as it turns out, travel in waves.  In fact, you can think of a wave as a traveling disturbance, capable of carrying energy with it.  For example, light "waves" can have energy - like solar energy.  Ocean waves can certainly carry energy.  

There are several wave characteristics (applicable to most conventional waves) that are useful to know:

amplitude - the "height" of the wave, from equilibrium (or direction axis of travel) to maximum position above or below

crest - peak (or highest point) of a wave

trough - valley (or lowest point) of a wave

wavelength (lambda - see picture 2 above) - the length of a complete wave, measured from crest to crest or trough to trough (or distance between any two points that are in phase - see picture 2 above).  Measured in meters (or any units of length).

frequency (f) - literally, the number of complete waves per second.  The unit is the cycle per second, usually called:  hertz (Hz)

wave speed (v) -  the rate at which the wave travels.  Same as regular speed/velocity, and measured in units of m/s (or any unit of velocity).  It can be calculated using a simple expression:

There are 2 primary categories of waves:

Mechanical – these require a medium (e.g., sound, guitar strings, water, etc.)

Electromagnetic – these do NOT require a medium and, in fact, travel fastest where is there is nothing in the way (a vacuum). All e/m waves travel at the same speed in a vacuum (c, the speed of light):

c = 3 x 10^8 m/s

First, the electromagnetic (e/m) waves:

General breakdown of e/m waves from low frequency (and long wavelength) to high frequency (and short wavelength):

Radio

Microwave

IR (infrared)

Visible (ROYGBV)

UV (ultraviolet)

X-rays

Gamma rays

In detail, particularly the last image:

http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html

http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html

http://www.unihedron.com/projects/spectrum/downloads/full_spectrum.jpg

Mechanical waves include:  sound, water, earthquakes, strings (guitar, piano, etc.)....

Again, don't forget that the primary wave variables are related by the expression:

v = f l

speed = frequency x wavelength

(Note that 'l' should be the Greek symbol 'lambda', if it does not already show up as such.)

For e/m waves, the speed is the speed of light, so the expression becomes:

c = f l

Note that for a given medium (constant speed), as the frequency increases, the wavelength decreases.

Quiz next week and practice HW

On the quiz:

1.  Energy:  Potential (mgh) and kinetic (0.5 mv^2).  How to calculate these and use the conservation of energy principle.

2.  Period of pendulum formula use

3.  Parts and properties of a wave

Quiz dates:

A:  2/15
E:  2/16

A draft of the lab will be due the class AFTER the quiz.

Practice problems:

Consider a 5-m long pendulum:

1.  What is the period of this pendulum?
2.  What is the frequency of oscillation of this pendulum?
3.  Where in its swing does it have maximum PE?
4.  Where does it have maximum KE?
5.  Where are the PE and KE equal?
6.  Where is the speed greatest?

Imagine that this pendulum has a 4-kg ball and it is lifted to a point 2-m above the lowest point in the swing.

7.  How much PE does it have at this location?
8.  How much KE will it have at the lowest point in the swing?
9.  What will the speed be at the lowest point in the swing?

Wave question

10.  Consider a string that is 0.5-m long.  If its n=3 harmonic has a frequency of 15 Hz, what is the wavelength and speed of this wave?

Things to include in your lab report

1.  Title - you make up one

2.  Your name and name(s) of lab partners

3.  Date(s) performed

4.  Purpose(s) of experiment

5.  Data table, including headings and units

6.  Analysis where you try to answer these questions:

- How is frequency (f) related to harmonic number (n)?
- How is speed (v) related to harmonic number (n)?
- How is speed (v) related to tension (provided by weight, W)?

Anything else that you noticed in the experiment?  How is frequency related to tension?  This may remind you about certain musical instruments, etc.  Write about anything that is interesting and related.

This is where the bulk of the writing in the write-up occurs.

7.  Sources of error (threats to validity)

8.  General concluding remarks

A and E block lab prep

During our next class, you will collect data for the new "harmonics on a string" lab.  Here is what to expect:

Procedure:

1.  Assemble harmonics apparatus:  sine wave generator, oscillator, string, pulley, weight to hang over end.

2.  Record L, the distance from oscillator (metal tip that vibrates) to top of pulley.  Keep this fixed during the experiment.

3.  Starting from a frequency of 1 Hz, raise the sine wave generator until a clear n=1 harmonic is formed.  You may find that the frequency is not an integer number.  Record the frequency for the clearest harmonic you and your partner(s) can find.  Repeat this for the next several harmonics, as many as you can see.  Use a strobe light if helpful.

4.  In a table, record the following (though not necessarily in this order):  m (kg), W (N), L (m), n, f (Hz), wavelength (m), and v (m/s).

5.  Calculate the relevant weights, wavelengths, and speeds for all trials.

6.  Try a different hanging mass and repeat the experiment.  If you have time, do additional hanging masses.

Analysis

You have data and calculations.  What can you see (and say) about these relationships?  You'll need to write about each relationship, and supply a graph (if helpful to make your case).

- harmonic number and frequency
- speed and frequency
- speed and tension (supplied by the hanging weight)*

* You may need to obtain data from other groups, particularly if you are generating a graph here.

What things can you conclude about waves on a string?  Are these ideas applicable to sound waves in air?

This lab write-up will be due in 3-4 classes after you perform the experiment.