as i understand it every sound is made of a waveform which not only contains the fundamental wavelength but also a load of harmonics of that wave too. ie, one sound is actually more than one wave.
but in the case of a body with a mass vibrating at certain pitch, say a guitar string, does the string itself change the wavelength of its vibration or is it con/destructive interference or another force acting with the produced sound wave that cause the dissimillation into the various harmonics?
sorry for any spelling mistakes, its late where I am.
thanks. Mawk
I’m not sure what you mean by every sound being composed of a fundamental waveform and a bunch of harmonics, but the case of a vibrating string - guitar, violin, piano, zither or what have you is fairly easy to descirbe.
Consider an open string. It is fixed at each end but free to move everywhere in between. The longest standing wave that can exist on the string will have two nodes (points where the amplitude of vibration is zero) - one at each end, and one anti-node, where the amplitude is at a maximum, in the middle. Thus the wavelength for this lowest, or fundemental mode, is twice the length of the string. Lets call the frequency of this mode F[sub]0[/sub].
The next mode has three nodes, one at each end and one in the middle. It’s wavelength is equal to the length of the string and so it’s frequency will be F[sub]1[/sub] = 2F[sub]0[/sub].
The next mode has four nodes, one at each end and one each a third of the way in from the ends. The wavelength is two thirds of the string length and the frequency is F[sub]2[/sub] = 3F[sub]0[/sub]. Higher harmonics follow the same pattern.
You ask why these harmonics exist. The short answer is that there is nothing to stop them from arising once the string is plucked, or bowed or hammered. Each of them is an allowed state of vibration for the system. If you are careful to pluck the string exactly at its mid-point, you can expect the fundamental harmonic to dominate, however even the very best of instruments is not a perfect system, the vibration of the strings transfers energy to the body of the instrument, the strings are not perfectly rigid at the endpoints, the strings are not perfectly uniform. All of these perturbations help to give rise to the other harmonics.
And I suspect that the resonant chamber in the body of the instrument plus the sound post that modifes its characteristic emphasizes certain harmonics so as to produce a pleasing sound.
I do know that a sinusoid produces a rather harsh tone while a square wave (findamental plus odd hamonics) produces quite a mellow tone.
I learned it differently when I was studying synthesizer. The sine wave is the fundamental with no overtones to the pure sine wave. With no overtones at all it sounds naked and electronic, but not harsh, just blank. With just a slight touch of overtones it’s like a flute timbre. The cylindrical bore of the flute helps to reduce the overtones. I think a flute in its lower registers sounds quite mellow, although the upper register of a piccolo sounds quite piercing. The conical bore of the clarinet produces a distinctive set of overtones that give the clarinet its particular sound. You’re right that the square wave, mathematically, uses the odd-numbered harmonics. As for subjective impressions, I’m not sure if mellow is exactly the word to use; it does sound to me like it has more texture. The square wave approximates the harmonics within a clarinet timbre. The really harsh-sounding one is the sawtooth wave.
Music theory was awhile back, but here goes,
although it’s tough to visualize, the guitar string (I’ll go with what I know) is vibrating at all the different wavelengths at the same time. The fundamental (full length) is the loudest, the harmonics get progressively quieter at higher frequencies. The blending of the harmonics of notes played together causes reinforcement or cancellation of certain frequencies, this causes the consonace or dissonance that gives music it’s tension or resolution. The quality of a musical insturment is largely a case of even, consistent harmonics; a garbage can lid vs a cymbal. If you touch a vibrating string at exact midpoint, you cancel the fundamental and all harmonics not multiple of 2. If you touch it at 1/3 you cancel all but multiples of 3. The harmonics exist unless acted on by outside forces.
Larry
A couple of oddball facts … first, all of the harmonics sound pleasing except the 11th harmonic, which does not fall on a note, but in between two notes.
For this reason, piano keys strike the strings 1/11 of the way from the end, to minimize that harmonic.
Second oddball fact … the reason that we have 12 evenly-spaced notes in an octave, rather than say 11 evenly spaced notes, is that it fits the harmonics better. As we increase the number of notes in an octave above 12, the first improvement on the fit comes with 24 notes, a scale which is used in some Eastern music …
A sound is a vibration of the air. A single impulsive vibration like a gunshot is a sound, but cannot really be said to have a fundamental frequency: only continuous vibrations have a lowest (fundamental) frequency.
Nor are the other simultaneous patterns of vibration (modes) necessarily called “harmonics”: that only applies where the frequencies of those other patterns an exact multiple of the fundamental frequency. And there will always be slight deviation from harmonicity in real systems (mainly from stiffness, which idealisations usually treat as zero). Consider the example of a church bell: the reason it sounds a little strange is because its higher frequencies are anharmonic, ie. not multiples of the fundamental. However, strings are usually flexible enough that their higher frequencies can be treated as harmonically related to the lowest.
Swing your arm from the elbow like a pendulum. OK? That’s one pattern (“mode”) of vibration, having a certain frequency. Now keep your arm still and waggle your wrist - that’s a second pattern, having a higher frequency, yes? But of course, you can do both at the same time, can’t you? Swinging and waggling your arm, you have superposed the two motions: your arm is vibrating in both pattern, at both frequencies.
Now that’s a pretty crappy example for all kinds of reasons, but the general principle applies to strings: they vibrate in lots of patterns at once. Added together, the string executes the familiar “flattening triangle” motion. But the point is that this motion can be decomposed into the different patterns of the harmonics, all happening at once.
See the first four separate harmonics hafway down this page, followed by the flattening triangle you get by adding them together simultaneously.
Could you explain what you mean by that, please? In all the pianos I have seen, all the keys strike their chords in the same place, regardless of the length of the chord. They look like this:
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were | would be the chords, . the places where the strikes occur, and ### the keys themselves. What I get from your description would look like this:
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were '. would be the places were the strikes occur, arrayed so they’d be at a specific relative distance on the chord.
This is true in a practical sense but not necessarily a musical one. A sound doesn’t have to have contributions from other harmonics, but in a practical sense, any system that generates a musical soun can’t help having at least a little mixture of higher harmonics. It wasn’t until we came up with things like synthesizers that it was possible to generate virtually perfect single frequency tones.
On the other hand, a lot of systems have only slight amouns of overtones in them. A simply-supported bar struck in the middle is almost all fundamental (think xylophones and marimbas and the like). I was surprised to find, on looking at the result of whistling into a microphone hooked up to an oscilloscope, that the output looked like a pure sine wave. Nevertheless, I know the higher harmonics are in there – I sis an undergrad thesis on the simply-supported block.
It falls between two notes relative to the 12-pitch, even-tempered division of the octave that has become the standard in the Western European classical tradition.
I’ll go and re-check my acoustic piano, but I’m quite sure that last time I was in there, the hammers were positioned to strike all the strings the same distance from the bottom of the soundboard. The length of the strings varies across the soundboard, though. So some hammers may strike a string at a point x/11 of the string’s length, but most don’t.
It does not fit the harmonics better. The even-tempered scale (12 tones, equally spaced) is in fact subtly mis-tuned from the natural harmonics of described in the OP. The perfect fifth and perfect fourth on an even-tempered instrument are not the same as the natural fifth and fourth that occur in the overtone series for a vibrating string.
It’s true that if you use the overtone series to generate the interval of a fifth, and then follow the circle of fifths (start at a pitch, go up a fifth from there, go up a fifth from there, etc.) you’ll get a series of pitches closely approximating the 12-tone scale. However, it won’t be an even-tempered scale. If you use the circle of fifths method, you’ll actually get a scale in which, for example, F sharp is not the same pitch as G flat.
To hear this, compare two recordings of any work for horn, one on a modern valved horn and the other on the natural horn. This discrepancy across the octave is what accounts for the flavor of the different keys - the placement of slightly narrow/slightly wide intervals is different depending on where you start the scale. This is most apparent on a “natural” instrument and largely lost on an even-tempered instrument (the variation in the size of intervals goes away).
The 24-note scale of several non-Western musics is largely theoretical. That is, the octave divided into 24 segments is used to explain how the music works, or should work. I can’t think of a classical music practice that actually uses a 24-note scale. It’s like the the 12-tone chromatic scale of Western classical music - the octave is divided into 12 half-steps, but music in any given key will rely on only 8 of those tones (with accidentals used as appropriate). The divisions of the octave simply allow you to start a scale at different points within the octave and to switch keys within a performance (which is really just moving the base of the scale for a moment).
Similarly, most non-Western music, as performed, will rely on X number of tones appropriate to the mode. I can think of several musics which rely on scales with odd numbers of tones: there are 5 tone scales (which are not identical to the black keys on the piano), 7-tone scales (Indonesia), 9-tone scales, and plenty of 8-tone scales which happen not to use the same divisions as the Western classical 8-tone scale.
Fair and true, although you’re omitting a substantial body of work in that area if you limit yourself to the last 50 years. There are composers creating music for these scales, and creating music for modes that haven’t been previously documented in Western or non-Western musics, and there are composers creating music in which all tones in the mode are equally weighted.
However, I didn’t say “no one anywhere creates music for an octave divided into X.” I said that the division of an octave into X tones doesn’t mean that music for that octave weights all X tones equally. It was intended as a response to intention’s assertion that an octave of 12 equally spaced tones is somehow more “natural” than an octave of X tones
A periodic waveform can be expressed mathematically with a Fourier series, from which you can quite clearly see which frequencies are present and their relative amplitudes and phases.
From Fourier analysis, it can be seen that a square wave has harmonics in the series 1, 3, 5, 7, 9… n (at amplitudes 1, 1/3, 1/5, 1/7, 1/9… 1/n), a sawtooth has the harmonic series 1, 2, 4, 6, 8… n (at amplitudes 1, 1/2, 1/4, 1/6, 1/8… 1/n), and a triangle wave has the harmonic series 1, 1/3, 1/5, 1/7, 1/9… 1/n) - same as the square wave - but with amplitudes of 1, 1/9, 1/25, 1/49, 1/81… 1/n^2. A constant tone from a musical instrument can also be analysed in such a fashion.
Even harmonics sound mellower than odd harmonics, and the higher odd harmonics can be quite grating on the ears. So, in order of niceness, it goes sawtooth, triangle, square, with a sine wave (with no harmonics) being the purest sounding tone of all, albeit a little boring with it. This is why expensive thermionic valve (US: tube) audio amplifiers sound nicer than bipolar transistor amplifiers, despite the latters much lower distortion figures. The valve amp will generate relatively large amounts of even harmonics - making it sound even nicer to most ears - while the bipolar transistor amp will generate nastier sounding odd harmonics, and the higher orders of these produced by the crossover distortion will become fatiguing over prolonged listening periods. MOSFET transistors also tend to sound quite nice, as like valves they also produce mostly even harmonics as distortion residues.
The motion of a guitar string is surprisingly complex. When you pluck a guitar string, you start it vibrating at many different frequencies. One of which is dependent on the length from where the guitar pick touches the string to either end of the string, and although the string wants to look like a sine wave as it vibrates, you’ve got it started out as a very triangular wave. So there are all sorts of additional frequencies on the string. As time goes on, waves that don’t work out to a multiple of the fundamental frequency of the string die out pretty quickly. This is why guitar strings have a distinctive “pluck” then a long term “sustain” type of sound.
One neat trick to do with a guitar is to hit a string, then gently place your finger over the exact middle of the string (where there are usually 2 dots on the fretboard instead of one). If you do it carefully, you can dampen out the fundamental wave and only end up with the higher harmonics (mostly the 2nd harmonic).
Whistles, as well as the sounds of most woodwinds, are fairly close to pure sine waves. Brass instruments tend to make much more square-ish waves. An electric guitar gets its “electric” sound from intentional distortion called “clipping”, where the sine wave is fed through a circuit that intentionally can’t fully reproduce the full swing of the sine wave, so it clips the top and bottom of the wave off to make it more “square”.
The first partial in the natural harmonic series that sounds “odd” to us Westerners is the 7th partial, which sounds like a very flat minor seventh in relation to the fundamental.
Secondly, there are theoretically an infinite number of harmonics. So “all of the harmonics sound pleasing” is nonsense.
Yeah, I should have remembered this is the SDMB, no generalities need apply … here are the lower overtones, with their errors. Jpeg Jones is right, “all of the harmonics” is nonsense.
These are the errors, shown as an absolute percentage of the overtone frequency, of the distance from the overtone to the nearest note, for the lower overtones of an “A” (440 hz):
As Jpeg jones notes, the 7th overtone is the first one significantly out … but the 11th is more in error, so that’s the one that’s muted out on the piano.