# Sound questions about waves (Violin vs Flute at 256-Hz Tone)

1. Sep 10, 2008

### longestline1

I am confused by this,

if A Violin and a Flute each play 256-hz tones,
which wave would have a longer wavelength?
also would the wave differ in any other way?

Also:
Which would have a higher frequency, a trumpet or trombone (generally)

2. Sep 10, 2008

### chroot

Staff Emeritus
Neither a violin nor a flute actually play "pure" tones, with only one frequency component, though the flute is close. Both sounds include one frequency component, the fundamental, at 256 Hz, plus other tones in the instruments' overtone series. A 256 Hz wave produced by any instrument will have the same wavelength; wavelength and frequency are related by the speed of sound, and the speed of sound does not depend on which instrument you're playing.

The question "Which would have a higher frequency, a trumpet or trombone (generally)" is nonsense, because there is a wide overlap in the notes that are playable on each instrument. Furthermore, the range of the instrument is determined by the skill of the player.

- Warren

3. Sep 10, 2008

### rbj

two little twists of POV.

sometimes we mean a tone is "pure", if it's very periodic. BTW, a note at 256 Hz (around middle C) might not have any energy in its fundamental or 1st harmonic as long as it has sufficient energy in the other odd harmonics. if it's nearly zero energy in all of the odd harmonics, it will be better described as a 512 Hz tone.

not totally determined by the (level of) skill of the player. the sizes of the instrument and of the mouthpiece have an effect on the range. the larger instrument will normally have lower notes and, even if both instruments play the same note, the resonances of the larger instrument are at lower frequencies. i think it's safe to say that the trumpet "would have a higher frequency" in the teeny bit loose sense of the words.

4. Sep 11, 2008

### chroot

Staff Emeritus
The fundamental frequency is always taken to be that with the largest amount of energy. If an instrumental sound is said be 256 Hz, that's the frequency of the fundamental. By definition, you would not call a sound with no energy at 256 Hz a "256 Hz tone."

Sure, the trumpet in capable of higher notes which cannot be played on the trombone, but that certainly does not mean that any arbitrary trumpet will always be able to play higher notes than a better-skilled trombone player.

- Warren

5. Sep 11, 2008

### atyy

rbj was probably thinking of the musical definition of "note", and the perceptual "illusion" of the "missing fundamental". Instead of saying "a complex harmonic tone without a 256 Hz Fourier component that has the same pitch as a pure tone of 256 kHz", some people just say "256 Hz note" for convenience.

Jeans says that Baroque organ builders knew this and were able to produce low notes without using long pipes.

longestline1: the wavelength of a flute and violin playing a 256 Hz note will be the same. Both will produce a waveform that consists of a short segment repeating itself every 4 ms. The sounds will differ in the detailed shape of the short repeating segment.

Pitch is a biological property, and depends on how your brain processes sound. The trumpet can play notes with higher pitches than a trombone. But because a low note generally contains low and high frequency components, pitch alone tells us nothing about frequency range. The perceived pitch corresponds (roughly) to the difference between successive frequency components of a note.

Last edited: Sep 11, 2008
6. Sep 11, 2008

### uart

No it isn't. The fundemental frequency is the lowest frequency in a harmonic series. It often does have more energy than any of the higher harmonics, but not necessarily so.

The issue that rbj is referring to is the interesting fact that in our perception of tone the fundamental harmonic can be completely removed yet we still perceive the same tone based on the smallest frequency difference in the higher harmonics. For example the note “A below middle C” has a base frequency of 110Hz and a harmonic series of 110, 220, 330, 440 etc. Now if you completely removed the 110Hz component you might expect that this sound would now be perceived as “A above middle C” but surprisingly it isn’t. The clue is that the smallest frequency difference in the remaining series (220 330 440 550 etc) is still 110Hz and the note is still perceived as below middle C. (This is know as a “phantom” fundamental BTW).

7. Sep 11, 2008

### uart

It's a good question. I'm guessing the thing you're really trying to understand is why two different instruments can sound so different when playing a sustained note, when you might think that in theory they are both just producing sound waves in air of the same frequency.

The key factors in why different instruments sound different are.

1. Attack and Decay times (or envelope profiles)
2. Harmonics.

The attack and decay are best described in the time domain. Essentially this is just how quickly or slowly a note begins and ends. A plucked string for example has a noticeably different attack profile to a pipe organ, and this is one important cue to help us differentiate one instrument from another.

Attack and decay cant however explain why sustained notes on different instruments sound very different when playing the same note. This is down to relative strength of the various harmonics, something we often describe as the "timbre" of the note.

Last edited: Sep 11, 2008
8. Sep 11, 2008

### atyy

uart brings up a good point - the "attack" at the start of a note can determine what instrument we think we are hearing. If you time reverse a piano note, it sounds much more like an organ.

9. Sep 11, 2008

### rbj

i beg to differ. not only is there the present issue of tones with missing fundamentals, it is common to find "natural" (non-electronic) instruments where there is a resonant cavity or something that boosts the amplitude of the 2nd or 3rd harmonic to an amplitude higher than that of the 1st. the human singing voice is an example and Tuvan throat singers can tune in on much higher harmonics.

the fundamental frequency of a tone is the reciprocal of the smallest possible period.

$$f_0 = \frac{1}{T}$$

where

$$x(t+T) = x(t) \quad \forall t$$

since 2T, 3T, 4T can also substituted for T and satisfy the definition of periodicity (in other words, a 256 Hz tone could also be called a 128 Hz or 64 Hz tone if no other restriction was made to the definition, so we do not spuriously pick a fundamental frequency of a tone that has no odd-numbered harmonics at all), we pick the smallest possible positive value for T as the period that is inverted to be the fundamental frequency.

of course, because of Fourier:

$$x(t) = \sum_{n=-\infty}^{+\infty} \ c_n \ e^{i 2 \pi n f_0 t}$$

where

$$c_n = f_0 \ \ \int_{t_0}^{t_0+T} \ x(t) \ e^{-i 2 \pi n f_0 t} \ dt$$

for any arbitrary t0.

it cannot be true for that all odd n, that each cn is zero. if that were the case, we made an "octave error" and incorrectly chosen 2T as the period (or some other even multiple), instead of what we should have.

i agree with this statement, but for what you mean by "fundamental". what would you say is the fundamental if you heard an instrument (or a "tone", or if you don't like that semantic, a "waveform") that had nothing at 256 Hz, but had energy at 512 Hz, 768 Hz, 1024 Hz, 1280 Hz, and maybe some more multiples of 256 Hz? perhaps that "instrument" is electronic (but it wouldn't necessarily have to be, you should check out the waveforms that come out of a Wurlitzer Model 200 electric piano from the '70s) and the tone is synthesized, but it's still an "instrument" in my semantic.

yes, i would, if it had 512 Hz, 768 Hz, 1024 Hz, etc. even if there was no energy at 256 Hz.

didn't mean to imply differently. only to say that normally, however we can define "normally" (i didn't say "generally" which is what the OP said), you normally measure the trumpet to be outputting higher frequencies than the trombone. the "spectral centroid" would normally be higher for the trumpet than the trombone.

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