To sort-of answer v the OP’s question, “Daedalus” once suggested, in his mostly- tongue-in-cheek-but-a-little-serious way that the reason that we like harmonic overtones (such as those generated by striking a bar on a xylophone, glockenspiel, or marimba) is because they indicate that the piece of wood or metal is well-constructed and without breaks or flaws. Broken or flawed pieces “ring” with non-harmonic frequencies, and to our ears they sound “bad”. He goes on to suggest our tree-swinging primate ancestors could tell sound tree limbs from cracked and broken (and therefore dangerous) ones by the sound they made.
Extremely off-the-cuff and way-out speculative, but the only thing approaching an answer I’ve ever heard. Explanations about chords being in near-integer ratios are important, but don’t try to address the “why” we find such tones pleasing.
*“Daedalus” was the pen name David E. H. Jones used for his odd science column in New Scientist and, later, Nature. Worth looking up.
Yes. In fact I’ve always struggled with playing guitar because I find the slight out of tune sound of a major third to be distracting. I think it’s because not only can I hear the beat frequency but I can feel it through my hands. If I record myself playing and listen back to it, I don’t notice it so much.
Your models are off, at least for comparison to the case of a simply-supported bar (not clamped at one end, nor completely free), which is the case for the xylophones, etc. I mentioned above. In that case the harmonics are simple multiples of the fundamental.
I admit that a tree branch looks a lot more like a clamped-at-one-end bar, though.
I apologise. I had no idea about this and it has no negative connotations that I am aware of on this side of the Atlantic. I am sure that most readers will realise that I was referring to people from the Far East, mainly China and Japan, where much of their music sounds discordant to us Westerners.
No, the modes are as I described. See here (and references therein) for more details. It turns out that vibrational modes of an object are different depending on whether the restoring force is due to an external source of tension (like a string), due to the compressibility of the material (like an air column), or due to the bending of the object itself (like the bending of a xylophone bar.) The first two models yield modes that are in a nice harmonic series, but the last one model does not. On a mathematical level, the differential equation governing bending modes is the Euler-Bernoulli equation, not the wave equation.
I’ll also admit that I always thought the modes of a xylophone bar were the same as those of a string, until about a year ago when I taught a course on the subject. It seems to be a pretty common misconception.
7th chords are used after the dominant chord (the V) to draw the listeners ear back to the tonic chord (I).
For example in the key of G, a player will use a D7 before going back to G. Its like a reminder that the music needs to go back to that tonic G chord. A trick of the ear and mind.
Hope this isn’t getting too technical. A lot of Western music is based on three chord songs. The tonic (I) dominant (V) subdominant (IV) chords or more commonly I IV V.
This is what I meant by drawing the ear. Music pulls us along, and we anticipate where its going. It’s considered discordant when it deviates from that path.
No need to apologize. I am aware that many people are unaware that it is regarded as offensive in some places. As I said, my remark was mainly to avoid others taking issue with it and derailing the thread.
I did vibratiung simply-supported bars for my thesis. Unlike the two examples you gave (cvlamped at one end and unsupported, simply-supported bars do not have hyperbolioc sine or hyperbolic cosinme components. The gfrequencies are integral multiples of the fundamental. All courtesy of the same equation you cite, with appropriate boundary conditions.
See Timoshenko, or any of the classic mechanical engineering texts.
The shapes of a simply-supported beam are sines and cosines, just like those of your plucked string (despite the fourth-order equation). Here’s one treatment:
A xylophone isn’t simply supported, though. The bars are supported a distance in from the ends. I’ve always assumed they are supported at the nodes of the fundamental mode, but I don’t know this.
I think you are looking at it backwards; the combinations sound good, so we assigned the term “chord” to describe how to reproduce it.
What might interest you is 16-tone music, which sounds odd because we are accustomed to 12-tone (doh-re-mi) music. Here is a sample: 16-tone Music - YouTube
Here are some lecture notes on the physics of music, talking about xylophones and glockenspiels. (The toy xylophones little kids have are really glockenspiels. Xylophones have (usually) wooden bars.)
According to this, the glockenspiels are supported at the nodes of the fundamental, which tends to dampen the overtones. Wooden bars on xylophones have material removed in the center, to alter the relative frequencies of the fundamental and overtones, to make them more harmonic. I’m not an expert on this stuff, so read the notes.
You’d likely be more familiar with Goodall’s work through his collaborations with Rowan Atkinson and Richard Curtis with the themes and incidental music for TV series including Blackadder, Mr Bean, Red Dwarf and Vicar of Dibley which would be evidence that this is not a dry, technical dissertation.
The problem with many physical realisations of instruments is that the harmonics are not really all that harmonic. Second order effects - usually related to the object being three dimensional rather than two mean that the harmonics are often a bit out. This can cause all sorts of grief, and handling it part of the the instrument builder’s art. Pianos are one of the worst. Because the strings are not of zero diameter, the frequencies of the modes of vibration are just a little bit out from a perfect harmonic of the fundamental. Some of this gives the piano its characteristic sound, but because the piano has such a wide range it also makes life difficult. You have the problem that the second harmonic of a note is not the same frequency as the note an octave higher. So the instrument is actually out of tune with itself. To preserve the harmonic capabilities of the instrument tuners will actually stretch out the octaves a little. In the middle octaves the stretch is quite small, but the further you get from the middle of the keyboard the more the stretch needed. Smaller instruments have a bigger problem as their strings are shorter, but not so much thinner. Concert sized instruments have the least stretch, but there is a bit. Stretch tuning is also often done on guitars, and I suspect harps. There are other evil things that can be done when tuning, sometimes varying from a strict equal temperament.
One thing that has occasionally occurred to me - pianos are notoriously one of the hardest instruments to record and reproduce. I wonder if the production of harmonics in the recording and reproduction chain are a more than usual problem. Usually even, and low order odd harmonics are not too disturbing when below a reasonable level, but with a piano, even the second harmonic will actually be out of tune with the instrument by a subtle amount. This may be enough to cause the reproduction to sound just that little bit off.
Interesting, Francis, thanks. What makes a tack piano (honky-tonk?) sound delightfully out of tune, even when just a single note is played? Is it deliberate violation of that second-harmonic thing, or is it simply deliberate mistuning among strings for those keys which have multiple strings? (In other words, do the lower-tone, single-string piano key notes sound just fine even on honky-tonk pianos?).
That is the very definition of “simply supported” – it means that the bar rests on those points, but is not clamped.
Those points, in turn define the position of the endmost nodes.
When you say that glockenspiels “are supported at the end, which tends to dampen the overtones”, it means that they aren’t quite “simply” supported. In a truly simply supported beam, the overtones aren’t dampened at all. Many xylophone=type instruments have the bars resting as freely as possible on the supports, with some light restraints to keep the bars from jumping or falling off. Glockenspiels, especially those carried in marching bands, have to be a little more robust, since they’re moving around, but they’re still kept as free as possible. That characteristic ringing tone they produce, that cuts through a lot of other sound IS the fundamental plus overtones. I find it hard to believe that the overtones are significantly dampened.
If you taught a course on vibrating bars and left out the “simply supported” case, then you left out an important and instructive case. If you told your students that a xylophone-type instrument does not have a fundamental and overtones that are multiples of the fundamental, you have misinformed them.
Your page lists as freely vibrating bar. You should be aware that this is NOT identical to a Simply Supported bare., The boundary conditions are different. The Simply Supported bar is constrained to zero displacement at two points close to the ends, and to no clamping. The solutions are sines and cosines, with no contribution from the hyperbolic sine and cosine solutions that can occur in cases with different boundary conditions. A freely vibrating bar (like one tossed into the air) is not constrained ast those points, giving different boundary conditionsd