Missing Fundamental

[LTig]

I have an electric bass guitar - which has strings - and picks up the string vibrations magnetically - so not dependent on the bridge or soundboard.

Typically, the fundamental, though strong, is weaker than the second harmonic.

I figure it is just how strings prefer to vibrate.

I think this happens because you pick the string close to its end. Do it in the middle and the fundamental should be stronger.

 

[RayDunzl]

I think this happens because you pick the string close to its end. Do it in the middle and the fundamental should be stronger.

Still difficult, when you apply an analyzer to the output signal.

---

There are some TV towers nearby.

Years ago, before it was more fenced and protected, we "played" some of the guy wires.

At the base of the guy wire, there were some dampers - chunk of metal on a stub of cable, that could wiggle,.

We wiggled them enough to slap/hammer the cable, very near the end.

The sound produced was a very extended EEEEEEEEEEEOOOOOOOOOOOOWWWWOOOOOOOHHHHHHHHH - sliding pitch from high to low, as more and more of the cable vibrated.

We got too rambunctious and eventually had the whole cable swinging a bit, and then some loud rattling was heard from the top of the tower, and the tech came out to see WTF was going on, and saw us, so we left.

The base is different now, the dampers are further up the cables, and "No Trespassing"...

https://www.google.com/maps/@27.840...CbePXp_eVBTnYA0JPQ!2e0!7i16384!8i8192!5m1!1e1



I tried to see if the guitar string exhibited the same effect (but much quicker) and the results were indeterminant. Maybe will try again.

 

[LeftCoastTim]

My comment was more along the lines of, after reading the thread and catching up with what was moved here vs. what was new, not only could I see that the discussion of the missing fundamental was talked across by people arguing different points (i.e. "pianos and acoustic instruments are in fact capable of generating deep bass" vs. "listen to this deep bass, oh wait it's only harmonics in this recording but whatever"), but that a new discussion about tuning and pitch perception had cropped up and that people were arguing that absolute pitch means you're out of luck if everything isn't perfectly tuned to your trained pitch. Every person I've ever met who had acquired "perfect pitch" had no problem adjusting to the relative pitch center of an ensemble if their instrument allowed for it. Tuning perception doesn't mean being bound by it even if you happen to know you're sharp or flat relative to A440.

Music and pitch are pretty interesting. Some other factoids that I know / have experienced.

Soloists (singers, solo violinists) playing with an orchestra will tend to play/sing slightly sharper than the pitch the orchestra is in. Why? To stand out more. I find it a bit annoying when they do too much of it. A related phenomenon is how orchestra kept raising their pitch amongst each other (pitch-inflation). There is some story that says 440Hz was standardized because the violinists started breaking their cat-gut strings too often, and a truce was called for. No problem for modern violins, of course.

Another related phenomenon is that musicians will notice a flat note much more strongly than a sharp note. In experiments, general population will have an even distribution of sensitivity around a pitch, but a musician will have a biased one. I am one of those with a biased sensitivity (measured with very precise instrumentation).

 

[MRC01]

Another one: the sensitivity of our pitch perception is greater at higher frequencies. We detect when frequencies are slightly off or out of tune by hearing beats, which is related to the intermodulation distortion of our hearing. Due to the fact that we hear frequencies logarithmically (octaves / notes are constant ratios), at higher frequencies, the same pitch/intonation error becomes more obvious and dissonant because it beats faster.

That means intonation is more forgiving for bass and low brass players, and it leads to jokes like, "Why does a symphony have only 1 piccolo? ... hahaha" ... and ... "How do you get 2 piccolo players in tune? ... Shoot one of them."

Here's a simple experiment I set up a while back demonstrating this.

 

[rdenney]

30 Hz is fine for me, too, though I do get something down into the upper-mid 20’s at least.

Rick “who can play tones from a tone generator” Denney

 

[infinitesymphony]

The "sounds" picked up by the mics were probably either evanescent waves and/or hydrodynamic waves, which decay exponentially with distance, and would not be detected (either by ear or mics) at the typical listening distance.

Those show the 28 Hz fundamental we discussed earlier. I've seen the same in some piano recordings in my collection. I suspect the reason it only appears in some recordings is (1) most obviously, not much piano music actually uses that note (though nearby notes may cause low level sympathetic vibration) and (2) it's a weak frequency only audible when miced at a certain position & distance.

Exactly, the 27.5 Hz fundamental of A0 has a wavelength of around 40 feet. You'll need the right distance, mic, and recording gear to capture it. I know an engineer who uses an Earthworks QTC omni (3 Hz response) low-passed just to capture these low frequencies. It's difficult.

Add the fact that you're fighting this battle only for people with high-end systems and sacrificing bandwidth to do it. Most engineers conclude it's not worth the cost for the majority of music.

I have an electric bass guitar - which has strings - and picks up the string vibrations magnetically - so not dependent on the bridge or soundboard.

Typically, the fundamental, though strong, is weaker than the second harmonic.

I figure it is just how strings prefer to vibrate.

Bass has a smorgasboard of flavors and possibilities. The fundamental vs. overtone spectrum of a bass guitar is going to be highly dependent on a number of factors:

a. Pick or fingers. (bright to dark)
b. Plucking location from bridge to neck. (bright to dark)
c. Round wound, half wound, tape wound, flat wound strings. (bright to dark)
d. Type, magnet, and output level of pickups.
e. Tone pot setting.
f. Bass amp & cab settings and rig capability of playing the fundamental. (e.g. not sure how much 40 Hz is coming out of a 1x8" cab)

I like hearing the fundamental at the bottom, so I play a P-bass into a 15" woofer and roll the tone forward or back depending on how sine-wavey or cutting I want it to be for the song. J-bass players are going to have a lot more focused overtones / mids. I see some bass players around town with rigs that aren't even capable of reproducing the fundamental with any kind of volume.

Soloists (singers, solo violinists) playing with an orchestra will tend to play/sing slightly sharper than the pitch the orchestra is in. Why? To stand out more. I find it a bit annoying when they do too much of it.

Another observation: I've found that string orchestra members tend to gravitate toward playing in Pythagorean temperament, perhaps because they're so used to tuning their instruments and ears using fifths. Relative to equal temperament, the notes in Pythagorean temperament are more often sharp:

A related phenomenon is how orchestra kept raising their pitch amongst each other (pitch-inflation). There is some story that says 440Hz was standardized because the violinists started breaking their cat-gut strings too often, and a truce was called for. No problem for modern violins, of course.

And the corollary is true, early music ensembles with period-correct instruments and gut strings often tune lower (ex. A415) because the strings are more fragile and sensitive to tension.

Another related phenomenon is that musicians will notice a flat note much more strongly than a sharp note. In experiments, general population will have an even distribution of sensitivity around a pitch, but a musician will have a biased one. I am one of those with a biased sensitivity (measured with very precise instrumentation).

100%. Sharp is usually okay, flat is never good unless it's specified in the piece. I see it as "you overachieved" vs. "you didn't quite make it."

 

[rdenney]

With a brass instrument, though, it’s a lot easier to lift the pitch than it is to pull it, especially in the upper register. And I wonder to what extent this is a treble-clef thing. I often play with trumpet players who push the pitch so high I run out of tuning slide. It’s a contest everyone loses.

Rick “tubas have wacky intonation” Denney

 

[infinitesymphony]

With a brass instrument, though, it’s a lot easier to lift the pitch than it is to pull it, especially in the upper register. And I wonder to what extent this is a treble-clef thing. I often play with trumpet players who push the pitch so high I run out of tuning slide. It’s a contest everyone loses.

Rick “tubas have wacky intonation” Denney

Even at my best, high notes had a seat-of-the-pants kind of feel. Gotta blow pretty freely and with air column support not to pinch notes sharp. I found more success by avoiding shallow screamer mouthpieces and working with deeper cups and larger bores. Sincerely, an ex trumpet player.

 

[JustAnandaDourEyedDude]

I found some recordings of individual piano notes at U of Iowa. They are recorded with a Steinway B, with the left mic 8" above center bass strings, and the right mic 8" above center treble strings. Here are the spectra of a few notes (A0, B0, C1, D1, E1, G1).
http://theremin.music.uiowa.edu/MISpiano.html

View attachment 102745 View attachment 102746 View attachment 102747 View attachment 102748 View attachment 102749 View attachment 102750

It is interesting to see that the fundamentals mostly only showed up on 1 channel for A0, B0 and C1, with C1 switched to right channel from left. By about E1, the volume of the channels are comparable. I think it is because the Steinway B is not of sufficient size and/or rigid enough to radiate the fundamentals of the lower notes as proper acoustic waves. The "sounds" picked up by the mics were probably either evanescent waves and/or hydrodynamic waves, which decay exponentially with distance, and would not be detected (either by ear or mics) at the typical listening distance.

My interpretation/theory on these plots is that the strings themselves are vibrating at the fundamental, and when the mic is close enough to it, it can be picked up. But the vibration of the strings must pass through the bridge and to the soundboard in order to be heard by the audience. I'm willing to bet that the bridge and the soundboard is unable to transmit such low frequencies.

Very interesting! Thank you.

Very stimulating thread! I had never read anything other than "pistonic driver motion" about producing sound, and it is fascinating to learn how some musical instruments produce useful sound. The parts of the thread about perception of absolute pitch are hard for me to grasp, because I do not know the "ABCs" of music, have never played any musical instrument and can't carry a tune . But I find it enjoyable to follow along and analyze the logic of the arguments anyway.

Never heard of evanescent waves before either. Just googling the term shows a lot of hits to explanations of apparently well understood evanescent waves in the well known phenomenon of total internal reflection in the field of electromagnetism. The phenomenon in acoustics seems to be not as widely known, and the description @NTK linked to is very illuminating. The evanescent waves in this case are a type of hydrodynamic motion impressed on the fluid, and do not propagate to the farfield at acoustic speed, and attenuate exponentially quickly with distance from the surface producing them.

Googling basic facts about how a piano makes sound, and assuming that the strings of the Steinway B used to make those recordings have been designed and tensioned/tuned to produce the right notes, and that the strings are "fixed" at both ends of their speaking length,
speed of flexural wave in A0 string = 2 x speaking length x fundamental frequency = 2 x 1.5m x 27.5Hz = 82.5 m/s. This is much smaller than the speed of sound in air at room temperature of about 343 m/s. Making the sweeping assumption, with vigorous motions of my hands, that the string radiates sound in its plane of vibration like the bending plate example that @NTK linked to (and would have an oval-ish sound polar in the cross-sectional plane), it seems that the A0 string (and the B0 and C1 maybe; I do not know their frequencies) does not radiate acoustic waves but produces evanescent waves instead that would not be sensible at any appreciable distance as @NTK pointed out. As @LeftCoastTim pointed out, perhaps the string vibrations do not get through the bridge to the soundboard. Even if they did manage to set the soundboard in vibration, the same problem might occur with the soundboard as with the string. The speed of transverse flexural waves in the soundboard is unknown to me. Making yet more sweeping assumptions that the soundboard is flexible enough to work by flexural wave mechanism (rather than pistonic motion of a rigid soundboard with infinite wave speed), and that the soundboard length is just barely sufficient to support the A0 fundamental frequency, and keeping in mind that the soundboard at the bridge would be a "free" end (anti-node of the standing wave) so that the strings and bridge can move it transversely to its plane, the wave speed in the soundboard would be 4 x speaking length x frequency = 165 m/s. Still much less than the speed of sound in air. So in this case too, the soundboard would produce evanescent 27.5 Hz waves instead of acoustic waves, according to the mathematical analysis cited at the link @NTK provided. The soundboard is actually a plate, and so fits the evanescent wave example better than does the string. If the preceding speculation resembles the real physics at all, the first company to support and move the soundboard in pistonic motion for low frequencies would produce a grand piano that trumps the Bosendorfer Imperial. Maybe beryllium or carbon-fiber-reinforced composite soundboards on polymer suspensions?

 

[dasdoing]

I found some recordings of individual piano notes at U of Iowa. They are recorded with a Steinway B, with the left mic 8" above center bass strings, and the right mic 8" above center treble strings. Here are the spectra of a few notes (A0, B0, C1, D1, E1, G1).
http://theremin.music.uiowa.edu/MISpiano.html

View attachment 102745 View attachment 102746 View attachment 102747 View attachment 102748 View attachment 102749 View attachment 102750

It is interesting to see that the fundamentals mostly only showed up on 1 channel for A0, B0 and C1, with C1 switched to right channel from left. By about E1, the volume of the channels are comparable. I think it is because the Steinway B is not of sufficient size and/or rigid enough to radiate the fundamentals of the lower notes as proper acoustic waves. The "sounds" picked up by the mics were probably either evanescent waves and/or hydrodynamic waves, which decay exponentially with distance, and would not be detected (either by ear or mics) at the typical listening distance.

It would be intresting analysing a fortepiano (classical piano) since it was "purer". on a modern piano you hear the soundboard/wood, on a classic one you hear more of the strings

 

[SMc]

Is there any actual music that requires sounding the extended low keys on a Bösendorfer Imperial? I always thought those were just for extending the deep resonance of the instrument.

William Bolcom's New Etudes:
No. 4 Scène d’opéra uses the low notes.

I saw a piano studio with an Imperial (reputedly played by Dave Brubeck) that had a cover for those extra keys. Some pianists found the extended compass visually distracting.

 

[rdenney]

I’m trying to remember if the Bösendorfer Imperial only comes in a 12-foot length.

Grand pianos have body lengths commonly ranging from 4 feet to 12 feet. That makes a huge difference in the clarity of the low range, and in the loudness capability of the instrument. Small pianos are grainier, which is a word I believe means that they ask me to interpret low pitches by their higher harmonics to a greater extent than their lower harmonics. It also means that the pitch discrepancy caused by large string deflections is minimized on a bigger instrument—a short string tensioned to low pitches has to be pretty loose, and when played loudly flops around and is subject to to all sorts of spurious vibrations and variations compared to a long string tuned to the same pitch. I have a recording of Liszt two-piano tone poems, one of which ends its fanfare on a low note played very loudly with the sustain pedal in use. The pitch starts high—really it starts rather broad—and as the spurious frequencies attenuate settles onto pitch with clarity. I don’t know what instruments they used, but I’m sure they were concert grands tuned that day. The initial confusion of pitch would be greater with a smaller instrument.

Of course, pianos use strings in pairs or triples, adding complication because the strings don’t all behave identically. I suspect there are all sorts of nonlinearities that contribute to the characteristic sound.

It seems to me the soundboard is a broadband resonator, and acts as an amplifier more than a pitch clarifier. The soundboard on my piano, when thumped, magnifies the sound and adds reverberation but doesn’t ring much at identifiable frequencies.

Rick “always fascinated by pianos but can’t play with any skill at all” Denney

 

[SMc]

I’m trying to remember if the Bösendorfer Imperial only comes in a 12-foot length.

There's a smaller piano, the Bösendorfer 225 (named for its 225 cm length, about 7 feet), with four extra notes and black keys to distinguish them from the normal compass. The Bosie website claims "[c]ertain works by Busoni, Bartók, and Ravel, can only be performed accurately on these instruments." Here's a list of works that "sparkle" on an Imperial: http://www.company7.com/bosendorfer/specs.html

Another consideration for piano tone is straight strung vs cross strung. Virtually all pianos of the last century are cross-strung with exceptions such as Daniel Barenboim's custom instrument.

 

[LeftCoastTim]

Maybe beryllium or carbon-fiber-reinforced composite soundboards on polymer suspensions?

Carbon fiber soundboard on a piano is a real thing. I've never heard on in person, but even on recordings they sound very "different":

 

[dasdoing]

I can't hear it, and have a hard time believing that someone can.
do those 4 last pedals realy sound deeper then the above for someone?

I just FIR brickwall low passed this at 30Hz. there is very little audible stuff, and the 3 lowest notes have almost nothing

 

[paulraphael]

Out of curiosity, I plugged my in my bass and recorded a few plucks of the low E. Signal path was bass (G&L bolt-on neck 4-string) > Aguilar bass amp DI, with 2nd gain stage and eq section bypassed > Focusrite Clarett > Logic Pro X.

As I suspected, the fundamental is there, but is overpowered by the 2nd harmonic, especially as the note decays. This would be more the case when played through most bass speaker cabinets, which roll of the bottom octave quite a bit. But the fundamental is usually there ... you can both hear and feel a difference when you filter it out completely.

FWIW, this bass is strung with modern flatwounds, which make fewer upper harmonics than round-wounds, but more than old-timey, thuddy flatwounds.

 

[dasdoing]

Out of curiosity, I plugged my in my bass and recorded a few plucks of the low E. Signal path was bass (G&L bolt-on neck 4-string) > Aguilar bass amp DI, with 2nd gain stage and eq section bypassed > Focusrite Clarett > Logic Pro X.

As I suspected, the fundamental is there, but is overpowered by the 2nd harmonic, especially as the note decays. This would be more the case when played through most bass speaker cabinets, which roll of the bottom octave quite a bit. But the fundamental is usually there ... you can both hear and feel a difference when you filter it out completely.

FWIW, this bass is strung with modern flatwounds, which make fewer upper harmonics than round-wounds, but more than old-timey, thuddy flatwounds.

video has no sound

 

[paulraphael]

video has no sound

It's just a screen grab, but when I have a minute I can add the sound as a track. I can also do a version with a low-pass filter.

 

[JustAnandaDourEyedDude]

Grand pianos have body lengths commonly ranging from 4 feet to 12 feet. That makes a huge difference in the clarity of the low range, and in the loudness capability of the instrument. Small pianos are grainier, which is a word I believe means that they ask me to interpret low pitches by their higher harmonics to a greater extent than their lower harmonics. It also means that the pitch discrepancy caused by large string deflections is minimized on a bigger instrument—a short string tensioned to low pitches has to be pretty loose, and when played loudly flops around and is subject to to all sorts of spurious vibrations and variations compared to a long string tuned to the same pitch. I have a recording of Liszt two-piano tone poems, one of which ends its fanfare on a low note played very loudly with the sustain pedal in use. The pitch starts high—really it starts rather broad—and as the spurious frequencies attenuate settles onto pitch with clarity. I don’t know what instruments they used, but I’m sure they were concert grands tuned that day. The initial confusion of pitch would be greater with a smaller instrument.

Yes, a longer, tighter string will produce more of the fundamental relative to the harmonics, and will also produce relatively less of what I saw referred to (in my brief googling on piano construction) as anharmonicity. I think the anharmonicity would be introduced by the initial deflection of the string by the felt-head hammer striking it, a deflection which does not match the needed sine deflection and cosine transverse velocity profiles needed to start with pure fundamental and harmonic frequencies. Yes, the initial anharmonicity will rapidly attenuate as its energy gets transferred into the resonant standing waves of the fundamental and its harmonics, clarifying the tone.

It seems to me the soundboard is a broadband resonator, and acts as an amplifier more than a pitch clarifier. The soundboard on my piano, when thumped, magnifies the sound and adds reverberation but doesn’t ring much at identifiable frequencies.

Agreed, the soundboard is designed to non-selectively amplify the sound of the vibrations it receives from all of the strings, and so is unable to reject the anharmonicity of each string. The soundboard collects all the vibration frequencies it receives via the bridge and uses its greater surface area to amplify the sound of all of them relative to the direct sound from the strings.

Of course, pianos use strings in pairs or triples, adding complication because the strings don’t all behave identically.

Good observation. Yes, there will be small phase, amplitude and anharmonicity differences among the members of each pair or triple. I imagine each piano stringing design and the imperfections of each of its realizations will help give the piano its characteristic sound.

 

[Francis Vaughan]

as anharmonicity.

Anharmonicity is an intrinsic part of any vibrating string that does not have zero diameter. It isn't an issue with the way the string is energised (struck, plucked, whatever) but is a problem with the harmonic structure of the vibrating string. Basically the problem is that the apparent length of the string is different for the different harmonics. Since the string has width, there isn't a simple defined end point where there is no vibration and the ideal string pivots from. Rather the point at the end of the string where it vibrates from is somewhere inside the string at a point that can move depending upon how much the string bends. This means that the fundamental can see a string that is effectively of a different length to a harmonic. This means that each harmonic is not an exact integer multiple of the fundamental, but usually a bit higher in pitch. Thus anharmonic. The very heavy wound stings in a piano are particularly subject to this issue, but it affects all strings. Shorter pianos need heavier strings, and are worse affected. The problem is that a piano is intrinsically out of tune with itself.
Managing anhamonicity is part of why pianos are tuned with a stretched tuning, and why smaller pianos need more stretch than large ones. In order to make the piano sound in tune with itself octaves are stretched so that anharmonics between octaves more closley match the next octave's fundamentals.

I have often considered that it is possible the one reason why pianos were once considered as a difficult instrument to reproduce is that excessive harmonic distortion in the reproduction chain lead to audible harmonics that were integer multiples, an these did not sound the same as the anharmonic harmonics inherent in a piano, so the instrument never sounded quite right.

Multiple stings on a note in a piano is a really interesting part of its construction. The strings are not tuned identically. They are very slightly split in tuning. When struck they act as a set of coupled oscillators and they pass energy between one another, with each string in turn oscillating more than the others in a continuous dance. The coupled system acts as a whole in generating the note.