General questions in acoustics
General questions in music acoustics
Questions related to string instruments
Questions related to wind instruments
- Waves in pipes: open and closed, cylindrical and conical.
- How does a sound wave reflect at an open end of a pipe?
- What is an end effect or end correction?
- What happens when we warm up wind instruments?
- How important are the materials from which instruments are made?
- Can you mute a flute?
- How do clarinet players do that big glissando in Rhapsody in Blue?
- Why is a saxophone in Eb and a trumpet in Bb? Wouldn't it be simpler if they all played the same music?
- How do I change a C trumpet to Bb and vice versa?
- What are multiphonics? How can you play two notes at a time on a woodwind instrument?
- What is an undertone in flute playing and what causes it?
- What is the end correction for a wind instrument? And what causes it?
- Do the end corrections have the same effect on different registers?
- What is the difference between whistle tones and edge tones on a flute?
- What is the process of homogenisation? What does it do to wind instruments?
- Fat pipes vs thin pipes: why does the pitch of a fat pipe of an organ depend more on the blowing pressure?
- Why does a narrower tube, for the same length, favour the upper register?
- A cone overblows at the octave, a cylinder at the 12th. What happens as we make our cone closer and closer to being a cylinder?
- Why doesn't the cone angle affect the overblowing interval on woodwinds?
- Why does the clarinet produce even harmonics?
- Why does a flute not need a bell?
- In woodwinds, how do you get the higher resonances in tune?
- What harmonics are present when you jump a register?
- Why does an oboe jump registers more easily than say a bagpipe chanter?
- On a woodwind, why do the 'short tube' notes bend more easily than the 'long tube' notes?
- When you jump a register, are the reeds (or lips) moving in the same mode, or do they get modes with a node, too?
- On saxophone and clarinet, is there a correct note that the reed and mouthpiece (alone) should sound?
- Inside the mouth of saxophonist or clarinettist, is there a pressure node or antinode near the reed?
- Why are the tone holes of the clarinet of different size?
Questions related to singing
Questions related to electronic instruments, loudspeakers etc
General questions in acoustics
- Space isn't quite empty. Can it transmit sounds?
This question is more subtle than it appears.
Sound is carried by pressure waves. Imagine the variations in pressure in a sound wave in air: a pressure maximum occurs where the density is highest. Due to molecular collisions, molecules tend to move from high to low pressure. That, plus the momentum of the molecules, produces the wave.
Pressure, however, is a macroscopic concept. We don't talk of the pressure of a few molecules, we talk about the forces that molecules exert during interactions. To talk of pressure, we need a significant number of molecules. In space, to have a considerable number of molecules/ atoms/ ions, you have to consider large volumes.
First let's consider just the atoms and molecules in space. The typical distance between them is about a centimetre. So we need a large volume to have a significant number. But it's more subtle than that. A sound wave propagates by intermolecular (or interatomic) collisions. Molecules in air only travel a nanometre or so (their mean free path) before they collide. What is the mean free path (m.f.p.) in space? Just from dimensional considerations, we can guess that it is roughly
m.f.p. ~ [(atomic cross-sectional area)(number density)]−1
(There's also a numerical factor in there, of order 1, but we shan't need it for this approximate calculation.)
So, taking the atomic area as ~ 10−20 m2, and number density (using the separation value quoted above) ~ 106 molecules m−3, we get a m.f.p. of about 1014 m or 0.01 light years. Collisions are rare. So, to talk of sound waves, we'd need to consider wavelengths of longer order than this. Very low frequencies.
In a plasma, things are more complicated, because the ions interact by electric and magnetic fields. (Strictly, I don't know whether a pressure wave in a plasma should be called a sound wave, but let's not quibble.) However, while the medium was a plasma, the radiation and matter were strongly coupled, so the temperature of the radiation was about the same as that of the matter.
The speed of sound is roughly proportional to the square root of the ratio of the temperature to the atomic or molecular mass. In the early universe, temperatures were very high and, early enough, the speed of sound was comparable with the speed of light!
As the universe itself expanded, the wavelengths of the pressure variations of the early universe have been expanded by a similar factor – and complicated by gravity. So the sound waves have now ultralow frequencies. And are difficult to measure. However, the temperature variations in that early plasma, when the universe was only a few hundred thousand years old, those variations are the subject of intense research effort. They are called the cosmic microwave background (CMB). The wavelength of these (electromagnetic) waves has also expanded, which is why it is now microwaves rather than light.
These waves are of interest to cosmologists, including Mark Whittle of the University of Virginia. He came to our lab to talk about it and one of the postdocs here, Alex Tarnopolsky, made some sound files for him, transposed them up into the audible range. Mark now uses it in his seminars and calls it the "primal scream of the infant universe".
You can hear the first million years (transposed upwards by 50 octaves) on his website, where he has a rather nice popular account of these acoustics aspects of cosmology.
General questions in music acoustics
- When we were playing two recorders, we heard a low buzzing sound, sometimes making a chord with our notes. What is that?
It is called the Tartini tone. It is most often noticeable when the two notes being played are sustained, about equally loud, not too low in pitch (say treble clef and above) and moderately loud. I've put a section about it on a page called Interference beats and Tartini tones where you can hear sound files made with pure sine waves that show the effect quite well. There is also a page called Tartini tones and temperament: a musician's introduction.
If you listen for the Tartini tone, you can use it to tune your chords: the tuning you will arrive at is Just Intonation, and not Equal Temperament. In sustained chords, the former usually sounds better.
(A multimedia introduction is at Tartini tones, consonance and temperament.)
The effect may be due to nonlinear effects (heterodyning) in the ear, or perhaps due to higher processing in the auditory cortex, or (more likely, I think) both of these.
The strong Tartini tone that we sometimes hear is usually generated in the ear. In the case of the two recorders (above), there is no vibration in the air at the difference frequency. However, when two notes are played simultaneously on the same instrument -- as played here on a violin by John McLennan -- it is possible to produce a difference tone via small non-linearities in the instrument. This is explained in more detail in Interference beats and Tartini tones and in Tartini tones and temperament: a musician's introduction. |
- What are heterodyne and comination tones? How do they relate to beats and difference tones? And where do they come from?
Nonlinear systems, in response to two signals with frequencies f and g, produce heterodyne or combination signals, with frequencies mf +/- ng, where m and n are integers. To me, the two terms have the same meaning, although heterodyne is used more frequently in radio and TV.
One of the heterodyne terms, that with frequency f-g, is called the difference tone in acoustics and the Tartini tone in music. (I'm choosing f > g.) If (f-g) is rather smaller than f or g, then the phenomenon is called interference beats or just beats: the combined signal is an oscillation with frequency (f+g)/2, amplitude modulated by the beat frequency (f-g). When the beat frequency is less than a dozen Hz or so, it can be clearly heard as a variation in amplitude, and is commonly used to tune instruments. (Beats at mf-ng are also thus used.) See Interference beats and Tartini tones for examples and sound files.
Where do such terms come from?
Non-linear systems inherently produce them. I shan't go through the algebra here (though it's not difficult) but one can expand a non linear response using a Taylor series. The series includes higher order terms, which become more important as the signals get larger. Simple trigonometric identities substituted in the product terms give the heterodyne terms.
In electronics, the nonlinear response of diodes was traditionally used for modulation and demodulation (producing heterodyne terms).
In music, nonlinearities occur in several places. The basilar membrane in the inner ear is nonlinear, and it is plausible that Tartini tones are generated here. I have read, however, that Tartini tones can be heard when one note is input to one ear and another to the other, via headphones. I've not experienced this myself but, if it happens, it suggests that they are produced in the neural processing. (Neural processing is highly nonlinear.)
They may also be produced in a nonlinear source. For instance, when they are produced by double stopping on a string instrument, one would expect the two notes to interact through the hair of the bow and through the bridge. The bow-string interaction is highly nonlinear, so this can produce heterodyne terms. This is shown in the example given above. When wind players produce multiphonics, they also produce heterodyne signals. To take one particular example that we have studied in detail: when a didjeridu player vocalises at a pitch different from that of the drone he is playing, strong heterodyne terms are produced.
Finally, it is possible to produce heterodyne terms in a microphone. No microphone is perfectly linear and so sufficiently strong signals could, in principle, produced heterodyne terms in the microphone itself.
Tartini tones sound unusual because one doesn't have a sense of direction associated with them. One can imagine that they are being produced 'inside one's own head', which may in fact be true. However, it's worth noting that, even in a slightly reverberant space, the same may be said of a pure tone. Listen to the sound files on Interference beats and Tartini tones and see if you agree.
- What is the frequency, and how does it relate to pitch? Where do harmonics fit in? Does the frequency of the lowest harmonic determine the pitch?
The frequency f is the number of vibrations per unit time. For example, when a tuning fork sounds the note A4, its tines vibrate 440 times per second. Its frequency f is 440 cycles per second, which is usually written as 440 Hertz or 440 Hz. (The unit of frequency is named for Heinrich Hertz, a pioneer of electromagnetic radiation.) The pitch of a musical note is determined by the frequency of its sound wave.
Pitch differences depend on the ratio of frequencies. If the frequency is doubled, the pitch rises by an octave, independent of starting frequency. If it increases by a factor of 3/2, the pitch rises by a fifth. (It follows that pitch is proportional to the logarithm of the frequency.)
The duration of each cycle of a vibration is called its period, T. If there are f cycles in one second, then each cycle must last 1/f seconds. In other words, the frequency is the reciprocal of period, or f = 1/T. So, for the note A4 at 440 Hz, the period is 2.27 milliseconds.
To relate pitch to frequency and back, see notes, frequencies and MIDI numbers.
We give an introduction to pitch, loudness and timbre in Quantifying Sound.
Musical tones usually comprise vibrations that are periodic. Such tones may be considered as the sum of pure tones from the harmonic series. So a note with frequency f usually contains also components with frequencies 2f, 3f, 4f etc. These components are called harmonics and the component with frequency f, the highest common factor, is called the fundamental. This is discussed in more detail in What is a sound spectrum?
We usually hear a pitch corresponding to the fundamental. However, we hear that pitch even if the fundamental is absent -- called the missing fundamental. For periodic tones, our sense of pitch is determined by the spacing of harmoncs in the region of several hundred Hz. So, for example, a small loudspeaker is very inefficient at 40 Hz. Consequently, when the sound of a bass playing its lowest note (E1 at 40 Hz) is played on such a speaker, the radiated sound has strong high harmonics, but almost no fundamental. Nevertheless, the pitch we hear is E1. The radiated sound might include the tenth, eleventh, twelfth etc harmonics at 400, 440, 480 Hz etc, and the spacing between these is 40 Hz.
- What is all the fuss about temperament? Is it just an academic problem for musicologists? And what does it have to do with the circle of fourths?
Litres of ink have been spilt on this topic, and I expect that there are many web sites. But it is by no means a purely academic problem, so let me give a quick introduction. For example, suppose a violist tunes up nice pure fifths C3, G3, D4, A4, removing interference beats as he does so. These will give intervals with frequency ratios of 3:2. The violinist tunes in unison to the last three of these notes, ie G3, D4, A4, but then adds E5. So the ratio of the frequency of the violinist's E string to the violist's C is (3/2)*(3/2)*(3/2)*(3/2) = 5.063. If they played these open strings together it would sound (to many people, anyway) uncomfortably out of tune. The fifth harmonic of C3 at 5.000 times the fundamental clashes and produces beats with the open E5 at 5.063 times that fundamental. Of course in practice, while the violist must play the C3 as an open string, the violinist will rarely play E5 on open string, so in musical context the violinist solves it by adjusting the position on the fingerboard and/or adding some vibrato (or by glaring at the violist!).
Wind and bowed string instruments can rapidly change the pitch by changing embouchure or finger position respectively and so, for sustained chords, can tune to eliminate beats or to achieve some other effect. A harpsichordist or a pipe organist does not have the option of changing the pitch (quickly) according to harmonic context. For these instruments, one must make a compromise between getting nice fifths and nice thirds. Not all organists, and very few harpsichordists, think that equal temperament is a sufficiently good compromise for music containing thirds because it favours the fifths far too much at the expense of the thirds. Just intonation is no good if you go beyond the home key, and even within that key the chord on the second note of the scale is very ugly.
Mean tone temperament gets the thirds right, and this spreads the dissonance over four perfect fifths. (e.g. tune C3 - E5 in the ratio 5:1 and make the fifths equal to the fourth root of 5, which is 1.495, which is close to the just fifth of 1.500. This is acceptable in itself, but it becomes much worse if you modulate into different keys.
A harpischordist might get away with mean tone thus: put all pieces in the sharp keys in the first half of the concert and then retune during interval to play the pieces with flats in the second half.
Many people opine that the best compromises are the so-called Well-temperaments like those of Vallotti & Young and those of Werckmeister, which spread the dissonance over more fifths. Some musicologists think that Bach wrote the Preludes and Fugues to demonstrate one of the Well-temperaments, perhaps Werckmeister.
See Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament.
Why don't pianists and guitarists bother with temperament?
Wind and bowed string instruments have nearly *exactly* periodic sounds, and thus their partials are almost exactly harmonic. For these instruments, equal tempered thirds in sustained chords don't sound great.
Some guitarists possibly do use equal temperament. Others can adjust the tuning according to the key that they are to play in, and can do so quickly. They may use several different temperaments without noticing it.
Some good players clearly do adjust the tuning by pushing the finger that stops one string parallel to the string, so as to increase or to decrease its tension.
Pianos have strong transients, which mean that they don't have periodic sounds. Further, they have three strings for most notes, and these are tuned slightly differently (to assist sustain) which gives chorus effects that disguise the problem. There is the further complication that the partials of thick steel strings are sharper than harmonics and thus their sounds are not exactly harmonic. Try it on a guitar using the very high harmonics, particularly with solid steel strings.
Pianists get used to equal temperament, and may even prefer it to others, but string quartets sometimes have trouble playing quintets with piano.
So what of harpsichords? They have thinner, softer strings than do pianos and so their partials are more nearly harmonic. Also, harpischord players are more likely to have studied different temperaments, and are more likely to be playing with wind and string players who are conscious of the issues.
See Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament. (By the way, I've recently written an orchestral overture called Circle of Fourths, which uses the circle as a chord, and might be considered as an experiment in orchestral temperament.)
- Whay do orchestras tune to A? And why to the obooe?
String players tune their open strings. Violins, violas, cellos, basses all have an A string. (So do guitars, mandolins and most others.) Further, A is the highest string on viola and cello and second highest on violin. (Brass bands usually tune to Bb, so that the trombones can tune their instrument with the slide all the way in.)
Why tye oboe? First, the oboe has a strong, readily identified timbre, especially in its bottom octave. Second, the oboe requires very little air, because the hole between the two halves of the double reed is very small. Consequently, a player can play a note for a long time relatively easily, if required.
- What is stretch tuning?
The frequency ratio of octaves on a piano are found to be slightly greater than 2:1, especially for the very high and very low notes, and especially for small pianos. The reason for this is that the octaves are usually tuned to eliminate interference beats between the fundamental of the upper note and the second partial (or resonance frequency) of the lower note. The second resonance of a struck string usually has a frequency that is slightly higher than twice that of the first, because of the finite bending stiffness of the string.
Playing an instrument that only sounds one note at a time, or singing, many musicians stretch octaves. However, playing chords with other instruments, they usually play 2:1 octaves.
- What causes broadening or finite width of harmonic peaks in a Fourier transform?
If a signal is perfectly periodic (i.e. repeats exactly after a period T) and infinitely long, we might expect its Fourier transform to have infinitey narrow peaks at frequency f = 1/T and the other harmonics 2f, 3f etc. When we sample such a signal and use a program to calculate the Fourier transform, we obtain peaks of finite width. (If we sample a signal with vibrato we might see even broader peaks, but let's restrict this to a strictly periodic signal.) In fact, we could get very narrow peaks if we did one of two things.
First, we could use a very long signal and a very long sample window for the transform. This would give narrow peaks, going to zero width as the signal and window length approached infinity.
Second, we could use a window whose length was exactly an integral number of periods nT. (Of course, without having performed a Fourier transform on a very long sample we don't know exactly what T is...)
In practice, we rarely do either of these: even if the software used for the transform may offers a choice of sample length, its maximum value may not be long enough to produce very narrow peaks. In general, the width of the peaks is of the order of 1/t, where t is the length of the sample window – except in the case where the window has a length nT.
This is a specific example of the impossibility of measuring time or frequency with infinite precision. I've devoted a whole page to this: it's called Heisenberg's uncertainty principle and the musician's uncertainty principle.
- Why are so many physicists and mathematicians (and engineers and ...) good at music?
I am often asked this, and one can make a few observations about the similarities between the two. For instance, in physics or maths we start with a relatively small number of definitions and laws and with these, we attempt to explain almost everything in the universe. We build elaborate and detailed patterns in a heirarchy of structures, starting with quite simple elements. In music, we start with a relatively small number of pitches and durations. Again, we build elaborate and detailed patterns in a heirarchy of structures, starting with quite simple elements. The physicist and the musician recognise
these heirarchies and underlying structures, and find the elegance and beauty in them. I've elaborated on this and other ideas in a paper called The creation and analysis of information in music.
However, on a more pragmatic level, I think that it is helpful to look at it from the other direction. A good musician knows about practice: that an hour's solid work yields only modest advancement and that regular practice is necessary. S/he is capable of abstraction at several levels. S/he is capable of processing information rapidly and precisely.
So, take someone who is good at abstractions, capable of processing information rapidly and precisely and has the temperament to work in order to progress. Is it surprising that some musicians have aptitude for physics? All that one would need to add to this list, I suspect, is curiosity and wonder about the world. It's not surprising that many intelligent, creative people are good at physics, maths and engineering. It's also not surprising that they are good at music.
- Why does a loud note still sound different from a soft note, even when you turn the volume down?
In many musical instruments, there is an oscillator (such as a reed, or the player's lips) that behaves in a non-linear way. For small vibrations, however, the behaviour is nearly linear. So louder playing means more non-linear behaviour, and more non-linearity means more higher harmonics. This is explained in more detail in, for example, how a reed works in a clarinet. There are sound files and spectrograms illustrating the effect in What is a sound spectrum?, whence this illustrtion:
- Why does rubbing your finger around the rim of a wine glass make a note?
Like a percussion instrument, the glass will vibrate with a range of frequencies when you tap it (lightly). However, when you rub your finger around the rim, you are continuously putting in energy and so you produce a sustained note.
It works like this: over a time of a millisecond or so, the fingertip 'sticks' briefly to the glass, then 'slips' a little and, if the conditions are just right, the glass will vibrate so that the finger 'sticks' again, one period of vibration later. The mechanism is rather like that of the violin bow on the string: see Bows and strings.
Wine in the glass impedes the vibration of the part that it occupies, so you can tune the note by drinking some of the wine. Glasses of different sizes also have different pitches.
Some glasses ring for longer when you tap them: there is less internal loss of energy in the glass. These seem to be easier to play with the fingertip, too.
Benjamin Franklin invented an instrument called the 'glass armonica', consisting of a set of glass bowls, of varying sizes, on a spinning axle. The player touched a bowl to make a note. One of Mozart's last compositions (K617) was for glass armonica, flute, oboe, viola and cello.
Questions related to string instruments
- What is the difference between artificial and natural harmonics?
This contrast is made, to my knowledge, only by string players. Natural harmonics are those that are played on an open string, whereas 'artificial harmonics' are the harmonics of a stopped string. The latter are of course more difficult to play, as you need one finger to stop the string and another to touch it at the desired node. See Strings, standing waves and harmonics and Standing waves.
- How important are the materials from which string instruments are made?
For string instruments, in which vibrations of the material of the instrument is what radiates sound, the mechanical properties of the materials are of great importance. The stiffness, density, anisotropy and losses in the wood (or other material) are all important to the response and performance of the instrument. A complete answer would be very long. Briefly: the bridge is not quite stationary: it must move a little so that it takes a small quantity of the energy out of the string in each vibration cycle. This is used to vibrate the body and, particularly for low frequencies, a substantial area of the body must move in order to transmit energy effectively into the air. For a loud instrument and a good sustain of a plucked note, relatively little of this energy should be lost in the body. Further, the mechano-acoustical properties of the body should help make the sound interesting. The spectra of different notes should have overall shapes that are different (but not too different) so that the instrument has some character. For bowed string instruments especially, it is important that the body has properties that vary rapidly with frequency, so that a vibrato induces substantial changes in the spectrum. This is quite important to the characteristic warm sound of a bowed string vibrato. (See also How important are the materials from which wind instruments are made?)
- Why do you get odd tunings if you tune by harmonics?
If your instrument has four strings, tuned in fifths or fourths (violin family and bass) then it is likely that you will tune by harmonics, and you won't get very noticeably "odd" tunings. This question was asked by a guitarist.
Some background, and a method for tuning by harmonics, are given in Strings, standing waves and harmonics. But here is a specific guitar question. "I tune by first setting my E strings to a standard pitch, then using harmonics to match this string to adjacent strings. On the lowest E string, when one I hit the 4th fret harmonic to get G# (the third of E), it is slightly flattened. On the 7th fret harmonic to get B (the fifth), it is pretty much on an even division of E. When the chord is played after tuning this way, there are no beats, and this is what sounds in tune. Tuning the whole guitar this way yields some flat and some sharp strings, that all sound good together. I understand that this is the essence of temperement, well vs even or just. My question is, why is the 4th fret harmonic flat and the 5th and 7th frets on, and why does this eliminate beats? Is there a mathematical explanation that can be easily transmitted here?"
Your guitar fretboard is designed to produce pitches that approximate equal temperament - i.e. each of twelve semitones has the same frequency ratio (in that sense they are equal - our sense of pitch is close to logarithmic). An octave is 2:1 so that makes each semitone the twelfth root of two, 21/12, which is about 1.059. (Americans: semitone translates as halfstep.)
The ratio between the third and second harmonics of an exactly periodic sound is 3:2 which we call a perfect fifth in just intonation, or a pure fifth. (More on the "exactly" later, and see Sound spectrum for more explanation.)
The third harmonic, touched at the seventh fret: Seven equal-tempered semitones is 27/12 = 1.498, which is quite close to 3/2 (= 1.500). So an equal tempered fifth, plus an octave, almost equals the third harmonic, and so produces only very slow beats at ordinary pitches. Further, to play a fifth on the seventh fret, you've reduced your string length by about 1/3, so this is where you touch the string to get the third harmonic.
Your fourth fret observation: stopping the string here ought to give an equal tempered major third (four semitones). Touching the string here will give the fifth harmonic, which is two octaves and a major third above the fundamental. Four equal-tempered semitones is 24/12 = 21/3 = 1.260, which is not very close to a 5/4, which is a major third in just intonation. Here is the root of most of the problems which require temperament, which is described in another FAQ called temperament.
See also Tartini tones and temperament: a musician's introduction and the multimedia introduction at Tartini tones, consonance and temperament.
- What is the "secret of Stradivarius"?
Why this question is difficult. Although much has been written on this, it is a subject that attracts a lot of speculation and few facts.
The astonishing thing about Stradivarius is that virtually all of the instruments attributed to him are judged to be excellent, and most of them comparable in quality with the very best modern instruments. To a small extent, some argue that this may be a tautology: his instruments have so long been treated as the optimum, that is almost impossible to do better: an instrument sounding brighter than all (modified) Strads would be judged to be too bright; one sounding mellower would be judged too mellow.
To start, there are some complicating factors: One of the most obvious reasons why these instruments sound so good is that they are almost always played by superb violinists. It is also possible that much of the comment about the quality of Stradivarius' instruments may be complicated by the fact that the people who own or play them have a strong financial incentive to maintain their market price. Consequently, it is usually usually difficult to get them to agree to double blind tests. Non-owners who have the chance to play one very rarely do so blind, and often have expectations that may colour their opinions and/or their assessments. So one must be cautious. However, on a FAQ in music acoustics, we cannot get away from this question, so here is my contribution.
How do they compare? Note that I wrote 'comparable' above and not (necessarily) 'better'. In a recent study conducted at the 8th International Violin Competition of Indianapolis, 2 famous Strads, one Guanari and three excellent modern violins were played and compared by 21 experienced players in double blind conditions. One of the modern violins was rated best, and one of the Strads last. (The lead author, Claudia Fritz, was a PhD student in this lab several years ago.)
Differences among strads. However, it is important to note that Stradivarius' violins are judged to differ subtantially in character from one to another, just as modern instruments do. This is in part because wood samples differ substantially in mechanical properites, even if taken from the same tree. Makers can compensate for these differences to some extent.
Strads don't sound like the violins Stradivarius made. The word 'modified' in the preceding paragraph is important. Few, if any, of Stradivarius' instruments today sound anything like the instruments he made, so no-one knows what a violin made by Stradivarius sounds like. Over the intervening years, virtually all have been subjected to the following changes:
To have an idea of the sound that Stradivarius would have heard from his instruments, go to a concert played by a group specialising in authentic performance. Some will play on old instruments, or more commonly on reproductions of old instruments, using gut strings and usually pitched about a semitone lower than the modern instrument. Many of them are in face imitations of violins by Stradivarius. The contrast with the modern instruments is striking. So the difference between how a Stradivarius violin sounded as he made it, and how it sounds now is great. John McLennan did his PhD in this lab, studying these changes.
His thesis and related research is on this link.
- a bass bar has been fitted, the plate and bar retuned, and the sound post has been replaced with a thicker one,
- the necks and fingerboards have been removed, discarded and replaced with longer, heavier necks and fingerboards at a greater angle to the body,
- new, taller bridges, with different shapes and acoustical properties, have been fitted,
- they have then been strung with more massive strings (steel instead of gut) at a much higher tension and tuned about a semitone higher,
- quite a few of them have been repaired in ways that would be expected to make substantial changes to the sound.
That said, the highly modified Stradivarius violins are still judged to sound good, so is there a secret?
Is it in the varnish? Some people talk of secrets in the varnish. Most makers and players agree that violins sound better 'in the white', i.e. before varnish is applied, than after. So one of the tricks in varnish (certainly not a secret) is to use only only enough to protect the wood, and not enough to change the sound much. For those who like the romantic idea of the "secret of the Stradivarius", it's very attractive, however, to imagine some secret ingredient, and the varnish is a convenient place to imagine putting it.
Is it in the wood? Was the wood special, or did it become special due to environmental or artificial treatments. Again, for those who like the romantic idea of the "secret", this would be a convenient place to imagine putting it. We list some reported additives below.
So how did he do it? How is it that Stradivarius made a lot of consistently good quality instruments? Although some 'secrets' are listed in the next paragraph, it seems that nobody knows for sure. So here are some observations and speculations of mine. First, he had good materials. Modern demand is high and modern makers compete for limited stocks of the best wood. (The demand by the aviation industry for spruce in the first half of the twentieth century did not help: a lot of potential violins were shot down in the first world war.) Second, he had good training: he was an apprentice of Nicolo Amati. The third and most important point is this: he was a virtuoso maker. Take someone really gifted with all the right talents, give him superb training, provide him with a large supply of excellent materials, then put him in a town where good instruments are bought for good prices, in competition with other good makers. Result: Excellent instruments.
The various 'secrets of Stradivarius'. It seems that, whenever someone claims to have discovered the 'secret of Stradivarius', the media become excited about it. It has been suggested that we should maintain a list of these 'secrets of Stradivarius'. Here are a few, to which readers may wish to add by emailing us:
So, a range of theories. But it is worth reading this recent study by Claudia Fritz.
- Chemical treatment of the wood - un-named components. Nagyvary et al, 2006.
- Chemical treatment of the wood - urine and dung. Nagyvary et al, 2000.
- Chemical treatment of the wood - Borax. Nagyvary et al.
- The 'little ice age' made the wood grow more slowly. Grissino-Mayer and Burckle, 2004.
- The 'little ice age' allowed fungus to grow on the wood. Michael Rhonheimer and Francis Schwarze, 2009.
- Stradivarius used the 'golden ratio' in the geometrical design. Anon., but widely mentioned, even by some violin makers.
- "Experience. Stradivarius made his best violins between 1710 and 1720 at age 60 (died at age 93). His contemporaries didn't live to be 60 years old, so lacked experience." (Contributed by Jan Woning.) This hypotheis could be tested by comparing instruments made in his forties or earlier with instruments made in his sixties or later.
- Shape and density. Sirr and Waddle, 2011. Measuring the shape and density of the wood using Xray scans could allow accurate replication of shape and density. But, given the variability in wood properties, would this produce the same acoustic properties? BBC report here.
- Imperfections in the geometry. Tiny modifications including the geometry of the f-holes can be decteced by synchrotron radiation. Zanini 2012.
- How important to violin quality is ageing, exposure
to climatic variation and the amount of playing?
We have begun a long
term study on this and related questions.
- How can you work out the appropriate size for a sound hole when designing an instrument?
Usually, the air resonance of a string instrument is set somewhere near the tuning of the second lowest string (a bit lower for guitars, a bit higher for bowed strings). I have included a discussion of the calculation on Helmholtz resonance, which gives a simple equation, and some warnings about the approximations used.
- How can a resonating chamber amplify sounds? Where does the extra power (seemingly) come from?
Let's compare a string on immoveable mountings (an unplugged electric guitar approaches this) with a string on an acoustic guitar. In the former, the bridge (almost) doesn't move, so no work is done by the string. The string itself is inefficient at moving air because it is thin and slips through the air easily, making almost no sound. So nearly all the energy of the pluck remains in the string, where it is gradually lost by internal friction.
In contrast, the string on the acoustic guitar moves the belly of the instrument slightly. Even though the motion is slight, the belly is large enough to move air substantially and make a sound. So the string converts some of its energy to sound in the air. Consequently, its vibration decreases more rapidly than does that of a similar string on an electric guitar. (Internal losses in the string are still very important, however.)
So there is no extra energy: the energy for the sound comes from the string. Which raises an obvious question: if there is no amplification, how does such a little vibrat make such a lot of sound? The answer is that our ears are rather sensitive (see our page on decibels and hearing). Consequently, even a small energy (even less than a millijoule) over several seconds makes a reasonably loud sound.
- How much does one more violin add to the sound level of a section of n violins?
Let's make a few simplifying assumptions: that each violin radiates the same power P (not usually the case in amateur orchestras!), that the listener is equally distant from all of them (hard to arrange) and that there is no simple phase relationship among the sounds from the different violins (this one is safe).
Consider n violins, each with power P, that produce total power nP. Say that the intensity, at our listener's ear, due to one violin is I1. Thanks to our simplifying assumptions, the intensity due to n violins is then In = nI1. At this stage, you may need to look at our section on decibels, sound level and loudness. The sound level L1
(in decibels) for one violin, Ln for n violins and Ln+1 for n violins are all given by the definition
L1 = 10 log(I1/I0) ,
Ln = 10 log(In/I0) and
Ln+1 = 10 log(In+1/I0)
where I0 is an arbitrary, finite reference intensity (and is not the sound level due to zero violins!). Now to the question: the increase in sound level when you go from n to n+1 violins is
ΔL = 10 log(In+1/I0) − 10 log(In/I0)
Because of our simplifying assumptions, the intensity is proportional to the number of violins. The site on decibels etc shows you how to handle logs, and explains why log(a) - log(b) = log(a/b). This allows us to write:
ΔL = 10 log[(In+1/I0)/(In/I0)] = 10 log(In+1/In) = 10 log((n+1)/n) = 10 log(1+1/n).
Reaching for your calculator, you'll see that adding the second violin adds 3 dB to the sound level produced by the first, the third adds 2 dB to the level produced by the front desk, the fourth adds 1 dB, and so on, and adding the 15th violin gives you an extra 0.3 dB. The decibel page gives you sound file examples of how much changes of 3 dB, 1 dB and 0.3 dB in level sound like.
Of course, there is much more to it than that: multiple instruments give chorus effects that make the sound more complicated and give it a different quality. But if your orchestra has 15 firsts, the biggest difference will be your empty chair on stage.
- Why does a thicker string sound less bright (have weaker high harmonics) than a thin string?
All else equal, a thick string doesn't bend as easily as a thin one: it is harder to produce a sharp corner in a thick string. So, when you pluck or bow a thin string, you create a shape that has sharper corners. When you look at the harmonics needed to make up this shape (see What is a sound spectrum? for background), you'll see that more and stronger higher harmonics are required to make a sharp corner. So bowing or plucking a 'hard to bend' or stiffer string puts in fewer high harmonics.
Further, a substantial fraction of the energy you put into a string is not converted into sound, but is lost in bending and unbending the string. So the stiffer string usually loses its high harmonics more quickly.
Finally, at the same pitch, a thicker string is usually shorter than a thin one: to play E4 on the top string of a guitar, you use the whole length. To play it on the B string, you need 3/4 or the length. To play it on the lowest string, you need only 1/4 or the length. So, all else equal, the high harmonics require sharper corners on the lower strings.
I've referred to stiffer or 'hard to bend' strings rather than just thick strings. It is possible to make a thick string (or, more importantly, one with a high mass per unit length) that bends relatively easily by using a thin core and winding wire around it to increase its mass per unit length (and therefore stiffness). The low strings on guitars, pianos and usually violins are wound strings. This allows them to have stronger higher harmonics, and also improves their harmonicity. However, if you make the core too thin, the string is easy to break.
For some basics about string vibrations, see Strings.
- What causes sympathetic vibrations in a string? How is it different from the coupling between string and soundboard?
- Sympathetic vibrations: You pluck the A string of a guitar, then damp it. You notice that the (high) E string is now vibrating, although you didn't touch it. Or you play an A on the G string of your violin, and