Understanding the Physics of Music: Frequencies, Overtones, and Resonance

[BandGeek13]

I just want to make sure I know these things correctly for an upcoming test, so if anyone can let me know, correct me (especially on the bolded stuff), or give me some extra info, that'd be great!

-Music has a constant waveform (sinusoidal), meaning it consists of musical notes with constant frequencies. Noise, on the other hand, is a bunch of random frequencies and therefore has no constant waveform.
-Lower frequencies have a lower pitch and longer wavelengths. Higher frequencies are higher pitches with shorter wavelengths.
-Notes have a higher consonance the smaller the ratio between them.
-The scientific scale starts on 256 Hz and uses simple ratios. This works best for violins.
-The musician's scale is equitempered and is made up of 12 notes. It starts on A 440 Hz.
-The frequency of a string depends on four things: length of the string (inversely proportional to f), the tension of the string (square root proportion to f), diameter of the string (inversely proportional to f), and the density of the string (inverse square root proportion to f).
-Nodes, points where there are no movement, always occur at fixed ends of a string. These are points of destructive interferences.
-Equally spaces antinodes are points of max constructive interference.
-When a string vibrates fully, in it's fundamental mode, its lowest possible frequency, the fundamental frequency, is produced (f₀).
-When the string vibrates in divided segments, overtones are produced. These harmonics are whole-number ratios of the f₀ because there must always be fixed nodes at the end. I do not understand why overtones emit more than one frequency.
-What exactly is resonance? Can someone please explain the difference of semi-open and open tubes (other than the obvious. haha.) I don't understand why semi-open tubes are 1,3,5.. and open tubes are 1,2,3... What do these numbers mean?
-Sound waves are transverse, while a wave created in a rope, for example, is longitudinal. What is the difference about how they are created and how they look on a graph?This is not a homework question.. I'm just confused.
-What is a standing wave pattern and how is it made?

Thank you so much!

 

[DaveC426913]

-Music has a constant waveform (sinusoidal), meaning it consists of musical notes with constant frequencies.

Just one nitpick: a sinusoidal waveform is unnatural; it will only be produced electronically: a boring, robotic, pure beep.

All instruments will have a unique tone that is determined by the shape of its waveform. It might be http://en.wikipedia.org/wiki/Waveform" .

 

[BandGeek13]

All instruments will have a unique tone that is determined by the shape of its waveform. It might be square or triangular, sawtooth or other.

Okay. But it is periodic, right?

 

[atyy]

-Music has a constant waveform (sinusoidal), meaning it consists of musical notes with constant frequencies. Noise, on the other hand, is a bunch of random frequencies and therefore has no constant waveform.

Music has noisy waveforms such as the onset of a flute note, or the distortion of an electric guitar. Musical pitches have periodic waveforms. Any periodic waveform can be considered a sum of sinusoids whose frequencies are integer multiples of a fundamental frequency.

-Lower frequencies have a lower pitch and longer wavelengths. Higher frequencies are higher pitches with shorter wavelengths.

Yes, for sinusoidal waveforms. There are interesting effects in which a combination of high sinusoidal frequencies have a low pitch. Eg. 4f,5f,6f played together will have the same pitch as a single sinusoid of frequency f.

-Notes have a higher consonance the smaller the ratio between them.

Tricky. http://music-cog.ohio-state.edu/Music829B/music829B.html

-The scientific scale starts on 256 Hz and uses simple ratios. This works best for violins.

No. There is no such thing as the scientific scale. Two scales which use simple ratios are pythagorean tuning and just intonation. All the common scales are based on frequency ratios, and do not involve any absolute frequency such as 256 Hz.

-The musician's scale is equitempered and is made up of 12 notes. It starts on A 440 Hz.

The equal tempered scale is used by musicians who use "common practice harmony", which includes the music of Bach, Beethoven, Brahms. A 440 Hz has nothing to do with equal temperament. It's just a recent convention.

-The frequency of a string depends on four things: length of the string (inversely proportional to f), the tension of the string (square root proportion to f), diameter of the string (inversely proportional to f), and the density of the string (inverse square root proportion to f).

-Nodes, points where there are no movement, always occur at fixed ends of a string. These are points of destructive interferences.

Yes. Nodes can also occur in the middle of a string.

-Equally spaces antinodes are points of max constructive interference.

Yes.

-When a string vibrates fully, in it's fundamental mode, its lowest possible frequency, the fundamental frequency, is produced (f₀).

Yes.

-When the string vibrates in divided segments, overtones are produced. These harmonics are whole-number ratios of the f₀ because there must always be fixed nodes at the end. I do not understand why overtones emit more than one frequency.

A string can vibrate with multiple sinusoidal frequencies.

-What exactly is resonance?

When you drive a system with a sinusoidal forces of varying frequency, the system will vibrate with the frequency of the force. At the resonant frequency, the force will cause the largest displacement of the system. The pendulum has a single resonant frequency. Musical instruments have multiple resonant frequencies.

Can someone please explain the difference of semi-open and open tubes (other than the obvious. haha.) I don't understand why semi-open tubes are 1,3,5.. and open tubes are 1,2,3... What do these numbers mean?

At closed ends the air cannot move, so these are nodes. At open ends, the air moves maximally, so these are antinodes. When you draw sinusoids that fit these constraints, you will find that an open tube permits frequencies 1f,2f,3f, where 1 is the lowest frequency satisfying the constraints.

-Sound waves are transverse, while a wave created in a rope, for example, is longitudinal. What is the difference about how they are created and how they look on a graph?This is not a homework question.. I'm just confused.

In a transverse wave, the displacement of the rope is perpendicular to the direction in which the wave travels. In a longitudinal wave, the displacement is back and forth along teh direction in which the sound wave travels.

-What is a standing wave pattern and how is it made?

Add up two sinusoids traveling in opposite directions. In each traveling sinusoid, the peaks change position over time. When you sum them, the peaks in the resulting standing wave do not change position over time - these peaks are the antinodes.

 

[BandGeek13]

No. There is no such thing as the scientific scale. Two scales which use simple ratios are pythagorean tuning and just intonation. All the common scales are based on frequency ratios, and do not involve any absolute frequency such as 256 Hz.
The equal tempered scale is used by musicians who use "common practice harmony", which includes the music of Bach, Beethoven, Brahms. A 440 Hz has nothing to do with equal temperament. It's just a recent convention.

okay, well it says this in my textbook and we've done classwork using these, but maybe it's just a way of explaining easier?

In a transverse wave, the displacement of the rope is perpendicular to the direction in which the wave travels. In a longitudinal wave, the displacement is back and forth along teh direction in which the sound wave travels.

i'm having trouble visualizing this...

thank you so much for all your awesome help, DaveC426913 and atyy!

 

[mikelepore]

I do not understand why overtones emit more than one frequency.

There is a procedure in mathematics called the Fourier series. Any periodic wave, no matter how how complicated its shape, can be expressed as the sum of an infinite number of sine or cosine functions. In general, each term in the series will have a frequency that is a fundamental frequency multiplied by some integer, and a different amplitude and phase angle. Add all of these infinite number of sine waves by superposition and you get the sound of the violin, tuba, piccolo, or whatever. The overtones of a musical sound are numerous terms of the series, the ones with the loudest amplitudes being the ones that we can hear. For some reason the human ear and brain evolved to act as a frequency analyzer, by which I mean, after many waves have been added by superposition, so that the separate waves are no longer there individually, we can sense the individual waves that were superimposed to make up that sum. That ability allows us to hear many sound sources at the same time, the individual notes in a chord, and the overtones of a single note that has a complicated waveform.

 

[slider142]

okay, well it says this in my textbook and we've done classwork using these, but maybe it's just a way of explaining easier?


i'm having trouble visualizing this...

thank you so much for all your awesome help, DaveC426913 and atyy!

Here is a visual of the longitudinal waves in open and closed cylinders (ie., flutes and clarinets).
Also, note that equal temperament tuning does not mean that all the intervals are equally tuned in all instruments. Unfortunately, due to the Pythagorean comma, most instruments (excepting classical stringed instruments) have to detune some intervals (most commonly thirds or fifths) in order to accommodate another interval (usually the octave). Ie., if you were to tune a piano by perfect fifths, the resulting "octaves" will be larger than the true octaves, they have to be detuned by the small limma (leftover frequency) by sacrificing one or more of the intervals inside the octave. These fixes are studied in tuning theory.
Also note that while it may look like it, bowed instruments generally do not have plain transverse waves in their strings like plucked instruments. Their periodic motion is a little more interesting.