Violin non-linearity

[Enthalpy]

Hi dear music lovers!

 

The violin and all bowed instruments has an important non-linear behaviour. It is noticed when playing two notes simultaneously on one instrument: the nonlinearity produces an additional beat, absent if two instruments produce the notes separately. When the frequencies are far from a ratio of integers, for instance in an equal-tempered small third interval, the beat is unpleasantly fast and seems out of tune, so violonists must train to play equal-tempered. The equal-tempered octave, fourth and fifth in contrast are nearly ratios of integers, the beat can be unnoticeable for being slow; violonists use it to tune the instrument.

No source of non-linearity is expected in a violin, because all vibration strains and stresses are small. I claim (others may have done it) that the bow-string contact is the source. In favour of this: we don't hear the beat if playing pizzicato, that is, by plucking the strings. Also: the beat is still audible if playing piano, while a non-linearity in the violin would decrease the relative strength of the beat when the notes weaken.

The bow-string contact operates in a well-understood fashion. Rosin makes the bow sticky; the moving bow pulls the string to the side and gives it energy until the string's stiffness wins, then the sticky contact breaks, the string relaxes back, until one-half to one period later, the speeds again similar permit the string to stick again to the bow.

By the way, this stick-slip process is somewhat random as sticking uses to be. As a consequence, sticking ceases within a period at a moment that fluctuates, and the sound isn't really periodic. It is known meanwhile (but not enough) that for a listener to recognize a violin, the sound must be non-periodic. Attempts to synthesize it from a harmonic spectrum, hence producing a periodic sound, consistently failed over decades - but a primitive synthesis with a simple sawtooth signal shape, whose transition had intentional jitter, was immediately "violin-like". This experiment (in a French university I believe) was among the first proofs that musical sounds are, and must be, non-periodic.

Relying on a threshold, the stick-slip is as nonlinear as possible. Several processes can couple the strings so that the instant phase at one string influences the moment when the other string tears off the bow:
- The strings' movement is knowingly elliptic, not just parallel to the bow's speed. As one string moves perpendicularly to the bow, it changes the bow's pressure on the other, hence the maximum sticking force, and influences the tear-off instant.
- The horsehairs at the bow have some elasticity. When a first string tears off the bow, the horsehairs reduce their strain, pulling suddenly stronger the other string, which tears off if it was about to do so.
- The violin's bridge is compliant. Less so than a string, to make there a displacement node, but enough to move the sounding boards. The movement of one string displaces the other a bit, which again influences the instant when the other string tears off.

These processes keep a beat amplitude essentially constant relative to both notes, which a nonlinearity downstream the oscillator can't.

Marc Schaefer, aka Enthalpy

[hoola]

as to the harmonic being heard when playing two notes on one violin and not hearing it (as much) when the same two notes are played on the two separate violins, the physical coupling between the note generators on a single instrument is tighter, especially in lower overtones. enhancing audibility. The overtone is then echoed from within the instrument, a small chamber acting as an amplifier. In two instruments, the separation dis-allows close coupling. The overtone is (almost completely) expressed in the air and so gets diffused. If the violins were to be physically connected (double-neck violin), or if the room that the two separate instruments played in were extremely small, that would somewhat replicate the one violin harmonic sound....I think this coupling is almost all through the bridge, to the body, and back to the bridge, with some resonant energy of the overtone coming out of the f holes.. some small coupling is through the bow......ed

a good test of this is to get two bows and play the separate notes on the one violin and mic the output. Compare with a single bow play of the same two notes as a control.......edd

Edited December 23, 2013 by hoola

[Enthalpy]

Musical synthesizers presently sample a real instrument and reproduce the record when the keyboard demands a note. This works well for a piano, less so for a violin, whose expression capabilities exceed a keyboard, and because two violin notes reproduced from samples sound like made from two instruments, due to the lack of beat.

Sound synthesis could imitate a violin's beat better than samples do. If using a jittered signal, say the sawtooth plus some frequency filters, the synthesis could just displace a bit the transition of one sound waveform according to the time interval to the transition of the other sound.

Marc Schaefer, aka Enthalpy

A good test of this is to get two bows and play the separate notes on the one violin and mic the output. Compare with a single bow play of the same two notes as a control.

I like this idea. It will distinguish the two first couplings I propose from the third.

 

Two bows on one violin will be difficult for lack of room, but violonists are always available for such stupid challenges.

[Strange]

When the frequencies are far from a ratio of integers, for instance in an equal-tempered small third interval, the beat is unpleasantly fast and seems out of tune, so violonists must train to play equal-tempered.

 

Isn't the fact that it is being played equal-tempered part of the problem? (I assume that is done to keep in tune with fixed tuning instruments like the piano?) I thought that if you tuned to a specific key, then the notes could have exact integer relationships and the beating would disappear.

[Enthalpy]

The beat persists at integer ratio relationship when the ratio isn't very simple, say 9/8. Though, it does get unpleasant when the equal-tempered interval differs from an integer ratio, like the small third:

(1.0595)³ = 1.1892 <> 6/5 = 1.2000

 

Violonists train to follow the equal-tempered scale in order to play with other instruments, yes. But not only.

 

The equal-tempered scale is also necessary for a lone violonist. If he plays successively several intervals according to simple ratios, he quickly gets badly out of tune.

 

Imagine four small thirds played as simple ratios:

(6/5)⁴ = 2.0736 <> 2.000

that's 4% away from the expected octave, more than a quarter tone.

 

While no violonist would be off after four notes, some pieces are feared for long series of such intervals without a reference like an empty string, and surprises do happen - before training that piece.

[hoola]

when a fellow came to tune my piano, he told me that the ear hears pitches inaccurately as the frequency goes up....so the maths being perfect in relationship would sound off to the ear, this according to Mr. Crowell. I can't remember which way he said that the ear perceives the higher frequencies, either as flat or sharp, but he said that he had to compensate for this to get a proper tune....and that the frequency counter he had hooked up did not do the whole job of establishing the final tuning.....does this "sound" right to you?......edd

[Enthalpy]

It's something all piano tuners claim. They allege that they have to stretch out the intervals at the higher notes of a piano so they sound in tune to human ear. I have big doubts about that, but did not experiment it in detail. Anyway, the highest octave of a piano makes such short sounds that one doesn't perceive properly the pitch. These notes are just too high for strings, because efficient radiation dampens them too quickly; a glockenspiel or a celesta play them properly.

[hoola]

I have noticed that the top key of most pianos sounds dead.....even on brand new ones. Seems odd that they can't make that sound better. I bought a kimball baby grand as it sounded good, plus the top key seemed playable....don't those high priced steinways sound good on the top key? I suppose it is a matter of getting a good hammer alignment, mechanism strike/release, and proper felt density......how are the two bow experiments going?

[Enthalpy]

Two bows experiment: I don't have my violin here - I left nearly everything in my country. Other people shall experiment that.

 

There is no good way to improve the highest notes of a piano. Strings just don't fit the task. They can't be longer or they'd break, can't be thicker to be strings, so they store little energy; and high frequency couples them well into the sounding board, plus increases the air drag, so the sound is extremely brief. Hammer alignment and material won't improve that. Piano makers have just exaggerated beyond the capabilities of strings.

[Enthalpy]

As it looks, a non-linearity isn't necessary to let sounds beat
https://www.scienceforums.net/topic/113243-sound-perception/?do=findComment&comment=1042336

The two bows experiment remains interesting to check if it changes the sound quality.