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Interesting: clavinet and e guitar have cyclical spectra too
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Author Interesting: clavinet and e guitar have cyclical spectra too
ricko
Learnt something new today: any instrument with multiple pickups selected (clavinet or twangy electric guitar) will have a comb-filtering effect, and this gives it the same kind cyclical spectrum as the "constant-breadth pulse" instruments (oboe, bassoon, cor anglais, harmonica, harmonium, to some extent trumput, etc.) Cyclical spectra from constant-breadth pulses were the basis of the old Variophon synthesizer.

When I made some sound samples for my Blip! waveshaper module, I intended to make oboe-ish sounds, but I gravitated to more funky plucked sounds and bass, and it never made "theoretical" sense why until now!

This comb filtering will produce fixed cyclical spectrums: where there are regular dips in the frequency band at a repeating frequency interval (e.g around every nth harmonics). The effect is because the pickups are physically separate, and so they sum the wave at different points as it travels on the string. The distance between the pickups and the speed of the waves in the string causes a kind of constant delay.

So this growing list of instruments with a fixed (or fairly fixed) cyclical spectrum re--inforces my feeling that these constant-breadth waves (pulses, in particular) are a fundamental waveshape that we have been largely missing out on. (These are pulses with a fixed duration in time, as distinct from standard VCO rectangular waves which have a duration relative to the note frequency.)

This fixed spectrum is mentioned in passing (fig 3b) in an academic paper A digital waveguide-based approach for Clavinet modeling and synthesis by Leonardo Gabrielli. People interested in the science behind the cyclical spectra of constant-breadth pulses can Google for papers on the Variophon. If you want less academic approach, see the backstory page on my Blip! waveshaper at fricko.home.blog, or hear those examples.
gtrmstr53
Interesting stuff! Thanks for the extra reading.
UltraViolet
Very interesting indeed! Another part of the puzzle for synthesizing the guitar sound.
Paranormal Patroler
That's fascinating!
cretaceousear
Delay line around 0.3 ms with one repeat should do similar I think?
fac
This is interesting. I'll check out the paper you mention. I've implemented feedback delays, phasers, reverbs and basic Karplus-Strong synthesis, but haven't really gone into waveguide synthesis and complex physical modeling.

But this leads me to a question.... is there some kind of physical electric instrument where you can manually vary the distance between pickups? Or better yet, is there a way to induce feedback between pickups? I'm thinking about phaser-like sounds.
colb
cretaceousear wrote:
Delay line around 0.3 ms with one repeat should do similar I think?


Yes. A short delay can transform basic waveforms. I hadn't really thought about why, but I guess it's this cyclical spectrum effect caused by the comb filtering.

One of my experiments a while ago (link) resulted in some sounds that are reminiscent of a bassoon or something similar (at least to me). This was using Doepfer basic VCO and a Disting as a delay. The delay time changes through the piece, but there are parts with a very short delay where there is a real deep woodwind kind of effect. I think there was maybe also some soft sync being used here. Also more than one delay repeat.
Keltie
fac wrote:
This is interesting. I'll check out the paper you mention. I've implemented feedback delays, phasers, reverbs and basic Karplus-Strong synthesis, but haven't really gone into waveguide synthesis and complex physical modeling.

But this leads me to a question.... is there some kind of physical electric instrument where you can manually vary the distance between pickups? Or better yet, is there a way to induce feedback between pickups? I'm thinking about phaser-like sounds.


Not physical, but the clavinet model in logic has moveable virtual pick ups, and moving them in real time always sounded kinda sorta phaser ish to me.

I’d be surprised if no one has ever built something physical with moveable pick ups, even as a one off. Not hard to do I guess...

Interesting thread, OP....
cornutt
fac wrote:

But this leads me to a question.... is there some kind of physical electric instrument where you can manually vary the distance between pickups? Or better yet, is there a way to induce feedback between pickups? I'm thinking about phaser-like sounds.


There have been several guitars with sliding pickups. I seem to recall that Ibanez made a version of the Iceman that had one.
UltraViolet
I have seen people do this on basses by hollowing out a channel for the neck pickup to slide in.
GuyaGuy
fac wrote:
This is interesting. I'll check out the paper you mention. I've implemented feedback delays, phasers, reverbs and basic Karplus-Strong synthesis, but haven't really gone into waveguide synthesis and complex physical modeling.

But this leads me to a question.... is there some kind of physical electric instrument where you can manually vary the distance between pickups? Or better yet, is there a way to induce feedback between pickups? I'm thinking about phaser-like sounds.


DeArmond made pickups that mounted to a pole that could slide up and down. There’s also this:

https://www.kickstarter.com/projects/1165874210/pole-position-sliding- pickup-guitar

Of course with piezo you can place it just about anywhere.
ricko
cretaceousear wrote:
Delay line around 0.3 ms with one repeat should do similar I think?


Definitelly, and I like the word "similar" . That is the (static) flange effect, that I mention on my page. And certainly modulating the constant-breadth pulse breadths has a family resemblence to flange (without feedback) or chorus or PWM or some speech synths chips, or even, if you like, convolution filters, without being the same in practise as any. They can all make kinds of cyclical spectra independent of the notes fundamental frequency.

(Indeed, if you had a perfect delay, you could even simulate the pulse output of Blip! (not the Sharktooth out) by delaying and inverting a saw wave (though in reality you might also have to fiddle with the output level too.) That is just basic superpositioning. Not sure why you want to use an expendive delay and mixer etc for something a two-transistor monostable does so easily though... :-)

In plucked string instruments, you get waves traveling in both directions from the impulse point, and both being reflected at each end. So for guitar pickups at each different points will have a different waveshape for each note, because the reflections need at different phases at each point: this is why different pickup positions, even unmixed, gave different tones. But presumably clavinet, ehise strings are excuted from an endpoint, dont have dual relection (so no initial dual pulse) and therefore the initial impulse woukd gave sone simpker shape ?

So perhaps you would want to delay a slightly different waveshape than provides the dry waveshape, for sounds i n the same family as the electric guitar?

Actually, it is even more complex. Higher frequencies travel faster (in a slack string) than lower. So for strings, the start can be more like a constant double pulse from the impulse plus reflections, then it smears out as the higher frequencies move ahead in phase to the lower, then towards then decays into a more sine-like first harmonic wave ( for accoustic guitar) or everything but the first harmonic (in the case of clavinet.)

Getting this slight inharmonicity (which may be experienced as beating of different harmonics rather than than out-of-tuneness) is a bit more tricky: frequency shifter? all-pass filter? I expect reflection would increase the effect.

And finally, it is worth noting that where you gave pickups offset ftom each other or the strings, you are likey to get a slightly different cyclical spectrum repeat count for each string: the gaps in one spectrum may allow harmonics from a loud part of the cycle of another string to be more obvious: speculation, but seems logical.
UltraViolet
ricko wrote:

Actually, it is even more complex. Higher frequencies travel faster (in a slack string) than lower. So for strings, the start can be more like a constant double pulse from the impulse plus reflections, then it smears out as the higher frequencies move ahead in phase to the lower, then towards then decays into a more sine-like first harmonic wave ( for accoustic guitar) or everything but the first harmonic (in the case of clavinet.)


Complexity always seems to come in when you want to really duplicate the sound of a classic physical instrument. The higher frequencies traveling faster is an interesting twist. That makes for a really challenging problem.
ricko
Yikes. There is more. It is getting combier than a President!

Don Tillman has a good page http://www.till.com/articles/PickupResponse/ on pickups. It turns out that even a single pickup has a comb filtering effect.

So when you mix two pickups, I think you actually get three comb filters: one from each pickup that derives from the integer harmonic null positions on the string , and one from combining them that derives from the distance between the pickips independent of the strings. (Plus there can be a note-independent shape of the initial pluck, which would give a monentary fourth comb, to some extent.)
tim gueguen
Another example of an instrument with a movable pickup is Westone's The Rail headless guitars and basses. The pickup was mounted in a piece of wood with a pair of holes in it. The holes allowed it to slide along a pair of stainless steel tubes that connected the front section, to which the neck was mounted, with the rear section, on which the bridge and tuners were mounted. Numbers made were small, with the bass seemingly more common than the guitar.
ricko
Movable pickups do seem a goid idea. They provably don't alter the first 3 yatmics thst much, but above those definitely.

I wonder how much of the characteristic sounds of each famous guitar design just comes from the particular pickup geometry?
commodorejohn
ricko wrote:
I wonder how much of the characteristic sounds of each famous guitar design just comes from the particular pickup geometry?

It's definitely a factor - there's a thread around here somewhere where one Wiggler hacked up a Jazz Bass to offer different pickup placements in an attempt to get it to sound like a Rickenbacker. The analysis and writeup were very enlightening.

Edit: ah, here we go.
ricko
Snooping around some academic papers on cyclical spectra, I think I figured out the difference between the comb effect and the formant effect: notch sharpness! ?

There was a swell of interest in cyclical spectra last decade in Vienna, from Drs Oehler and Reuter, who had a family connection with the cryptic Variophons, and their postgrad students.

I found a thesis paper Visualisierte Impulsform (Visualized Impulse Shape) by a Dr Baumgartner, which included nice graphs of the spectrum of different fixed impulse shapes (usually a 10Hz wave). Worth a look for anyone interested who is a visual learner like me (German literacy not required.)

The key insight is that the cycle of louder and softer harmonics stays the same regardless of the note frequency: fixed formants (or comb effect.) For fixed-breadth impulse, the cycle in the spectrum is a quadi-sinusoidal one, rather than the sharp notches we always see in diagrams explaining comb filters. I suspect comb filters would have some higher Q resonance/feedback causing this deeper notch effect.

(Comb filters must have deep narrow notches, because they were developed to get rid of power hum, eg 50 or 60 or 100 or 120 Hz and their early harmonics, and leave as much of the renaining signal as possible.)

Which brings up a conjecture about electric guitars. I would expect the notches in response from the pickup to be wide (sinusoidal) like fixed-breadth impulses. But when a guitarist has their amplifier loud enough to get good sustain, does this feeding-back also make the notches sharper (more like the comb shape)?


Cheers
Rick
fricko.home.blog
ricko
And more. Many piano designs (both upright and grand) have a fairly fixed distance of the hammers to the strings.

"A piano bass tone is an intriguing stimulus. Viewed in the frequency domain, it is veryrich. The spectra of the lowest bass tones may contain morethan 100 partials, and generally extend up to 4–5 kHz within a 60-dB amplitude drop. The magnitude spectrum has a char-acteristic formant-like envelope with groups of six to eight partials between spectral minima, determined by the striking position on the string.  partials are progressively stretched due to the inherent inharmonicity of the stiff strings, as well as a motion of the string terminations. The decay rate generally increases toward higher partials."

Effect of Relative Phases on Pitch and Timbre in the Bass Piano Range Cuddy Askenfelt, 2001.

(The groupings of 8 is corroborated by a spectrum if A0 in Bensa et al, Piano String Simulation. Looks like groups of 8 below 1kHz and groups of 9 above.)

Looking at pictures of piano frames, it seems the lowest 2 octaves have this relatively constant geometry (even on stretch pianos with more keys.)

(This cyclical spectrum of groups of 8 at A0 (27.5Hz) is like a 12.5% relative pulse at that frequency. Which is a pulse of about 8ms.

Using the standard BOM, the Blip! module's maximum pulse breadth is about 5ms, so a slightly larger timing cap would be needed. But this would give a 1/8 pulse at 27.5Hz and a 50% pulse at 110Hz, and that does not seem correct. )

It seems that piano strings get their tuning more from tension than from length (contrast with pipe instruments); so is there some effect of string tightness to push up the formants?

(I need to see more spectra, definitely. But what seems more plausible is that it goes from 12.5% at A0 to, say, 20% at A2 by which note other effects dominate over the cyclical spectrum. That gives the groupings of 6 to 8 mentioned in the article. And you could use vanilla PWM just as readily Blip! with modulation for it.)

(The paper mentioned is also interesting for its actual topic too, btw.)
Joe_D
ricko wrote:
And more. Many piano designs (both upright and grand) have a fairly fixed distance of the hammers to the strings.

I'm a piano rebuilder. The relatively fixed distance between the hammers and the strings has nothing to do with the harmonic spectrum. It simply limits the period of time and travel during which the hammer can acquire inertia before letting off and flying freely to the string and rebounding--or to put it differently, it limits the potential dynamic range. The maximum is always constrained by other mechanical considerations, but we generally strive for the widest possible dynamic range.
ricko wrote:

"A piano bass tone is an intriguing stimulus. Viewed in the frequency domain, it is veryrich. The spectra of the lowest bass tones may contain morethan 100 partials, and generally extend up to 4–5 kHz within a 60-dB amplitude drop. The magnitude spectrum has a char-acteristic formant-like envelope with groups of six to eight partials between spectral minima, determined by the striking position on the string.  partials are progressively stretched due to the inherent inharmonicity of the stiff strings, as well as a motion of the string terminations. The decay rate generally increases toward higher partials."

This is fairly accurate and widely agreed upon.
ricko wrote:
(The groupings of 8 is corroborated by a spectrum if A0 in Bensa et al, Piano String Simulation. Looks like groups of 8 below 1kHz and groups of 9 above.)

Looking at pictures of piano frames, it seems the lowest 2 octaves have this relatively constant geometry (even on stretch pianos with more keys.)

This cyclical spectrum of groups of 8 at A0 (27.5Hz) is like a 12.5% relative pulse at that frequency. Which is a pulse of about 8ms.

The harmonic spectrum of piano tone is primarily influenced by several factors, none of which to my mind are the result of a fixed pulse width (though I'm always open to learning more). They are: the strike position or point (that is, the position on the string where the hammer strikes the string, usually indicated as a fraction of the speaking length, meaning the regularly vibrating portion of the string), the hammer-string contact time (determined by mechanical properties of the hammer and string and strike point), the makeup of the hammer (determined by the hammer's mass, springiness, surface, and the degree of mating of the hammer's striking surface to the string), the hammer's velocity (determined by the player), the soundboard/bridge impedance (determined by the piano designer and builder/rebuilder), and the effect of other resonating strings and aliquot string portions. I'm not seeing any of those as resulting in a fixed pulse length. The most fixed of them is usually the strike point; in fact, the hammers' strike line (the strike points of all 88 hammers on the strings) is usually the first mark drawn when designing a new piano model. It is responsible for the characteristic gap in the partial series in pianos. You can look up how the excitation point of a string cancels partials for more info. Or you can just pluck a guitar or bass string at different points and hear how the tone changes. So, there is a similar effect in that there is a characteristic partial gap, but to my mind, for different reasons.

ricko wrote:
It seems that piano strings get their tuning more from tension than from length (contrast with pipe instruments); so is there some effect of string tightness to push up the formants?
Inharmonicity, string stiffness, string mass, core to winding ratio, and all are probably too big a topic to get into here. Glad you're interested, though! If you want to spend a fair chunk of money, you can get the Pianoteq physically modeled piano plugin and play around with design parameters and hear/see how they affect tone. You'd have to get one of the more expensive versions in order to be able to muck around with all of the design parameters, though.
ricko
First off, I did not mean the distance from the hammer to the string. I meant the distance along the string: the "fraction length" being less interesting (which is not to say less important) than the absolute distance.

Second, I did not mean to imply that the lowest piano note(s?) actually have a pulse shape. Merely that (on some pianos at some dynamics and mic placement etc) it has a cyclical spectra, which pulse waves also have. (Note the post said "like a 12.5% pulse", not that it was such a pulse.)

The theme of this thread is the realization that there are more instruments that produce tones with cyclic spectra than I had realized: before today I woul never have thought any notes on any piano would have clear cycles in their spectrum with "formant like" consistency (ie not tracking the note frequency), especially bass notes...far too much going on!

Now if you wanted to synthesize such cyclical spectra, various kinds of pulse-ish waves would be a reasonable place to start. The problem is than most VCOs do not provide any tracking mechanism that would allow the pulse width to respond to keyboard CV in a way that gives a more or less constant breadth pulse (eg 5ms or whatever) regardless of the note. On the other hand, my Blip! module does allow constant pulse but I don't know whether its modulation capabilities fit the bill either: getting a better idea of the design choices that evolved for pianos might help formulate what kinds of spectra and modulation are missing. (The point not being to make horrible cheezy imitations of pianos, but rather to develop synth modules that can play the same tricks as nature and physical instruments.)
Joe_D
Hi Ricko,

Good thread, thanks for starting it. It'll give me some things to think about in regards to some of the instruments with a fixed pulse length.
ricko wrote:
First off, I did not mean the distance from the hammer to the string.

Sorry if I misunderstood you.
ricko wrote:
I meant the distance along the string: the "fraction length" being less interesting (which is not to say less important) than the absolute distance.

Maybe I misunderstand you again. The fraction length is the most important thing. Earlier, you said:
ricko wrote:
Many piano designs (both upright and grand) have a fairly fixed distance of the hammers to the strings....Looking at pictures of piano frames, it seems the lowest 2 octaves have this relatively constant geometry (even on stretch pianos with more keys.)

As I understand it (I'm open to learning more), fraction length is the more constant thing, and as I said earlier, usually the very first design consideration. The fractional position of the strike point is decided, the strike line (point of hammer-string contact for 88 keys) is drawn, then all of the terminations (beginning and ending points) of the strings are drawn so that the fractional position of the strike point remains fairly constant.

It is this fractional position that determines which partials will be damped. Pluck a guitar string exactly in the middle and notice that the tone has a hollow quality--it's along the lines of a (filtered) square or triangle wave tone, because the even partials are somewhat damped by plucking in the middle.

So by keeping the fraction constant (rather than the distance from the end of the string to the strike point), we can make the various notes have highly related spectra and thus sound like one cohesive instrument rather than a collection of percussion instruments (well, really, isn't a piano 88 separate percussion instruments? We work as hard as possible to meld them into the illusion of being one instrument, and that starts with the uniformly fractional strike point). If the seventh partial of one bass note on a particular piano is weak because the strike point is 1/7 of the string length, we want the seventh partial of all notes in that piano's bass section to have a weak seventh partial. That gives them a similar tone. The result is that the weak partial is at the same place in the harmonic series from note to note, but it is at a different frequency from note to note, that frequency moving proportionally to the note's fundamental.

Here's a picture of a large Steinway grand. The hammers are the gray-ish shapes below the strings, between the black dampers and the gold colored strut that says "repetition action." You can see the agraffes (starting point of the vibrating portion of the string; they look like little bars in circles near the bottom of the picture) getting farther and farther from the strike point as the notes get lower. The same thing happens at the far end of the strings, to a much greater degree since the "bulk of the fraction" is on the far or tail side of the strike point.

ricko wrote:
Second, I did not mean to imply that the lowest piano note(s?) actually have a pulse shape. Merely that (on some pianos at some dynamics and mic placement etc) it has a cyclical spectra, which pulse waves also have. (Note the post said "like a 12.5% pulse", not that it was such a pulse.)

Sorry; I read too hastily.
ricko wrote:
The theme of this thread is the realization that there are more instruments that produce tones with cyclic spectra than I had realized: before today I woul never have thought any notes on any piano would have clear cycles in their spectrum with "formant like" consistency (ie not tracking the note frequency), especially bass notes...far too much going on!

As I said above, I don't think that this is correct. The prominent gaps in the harmonic series due to the excitation point do generally track from note to note. There are other secondary resonances related to the soundboard that may have some formant-like constancy, but we work to minimize them as much as possible. In a violin-family instrument (especially basses and cellos), we call such a fixed resonance a "wolf," and it is a problem to be dealt with, not an advantage. And I should add that there are always at least some very inconsistent partial profiles from note to note due to the impossibility of manufacturing wound bass strings that scale exactly proportianally.

ricko wrote:
Now if you wanted to synthesize such cyclical spectra, various kinds of pulse-ish waves would be a reasonable place to start. The problem is than most VCOs do not provide any tracking mechanism that would allow the pulse width to respond to keyboard CV in a way that gives a more or less constant breadth pulse (eg 5ms or whatever) regardless of the note. On the other hand, my Blip! module does allow constant pulse but I don't know whether its modulation capabilities fit the bill either: getting a better idea of the design choices that evolved for pianos might help formulate what kinds of spectra and modulation are missing. (The point not being to make horrible cheezy imitations of pianos, but rather to develop synth modules that can play the same tricks as nature and physical instruments.)


A pulse wave is a good choice to experiment with synthesizing cyclical spectra. To get even in the ballpark of piano tone (just for inspiration and exploration, not to convincingly synthesize a piano with old-school synthesis techniques, which I don't think can happen), one needs to be able to produce inharmonicity and characteristic partial decay envelopes. A modern physical modeling module (maybe Elements?) would be a good place to start. Additive synthesis could maybe work, but you really would need more than 64 partials, inharmonic "stretch," and lots of envelope work.

As I said, good thread, thanks.
commodorejohn
I'm absolutely digging getting to read professional explanation/confirmation of some things I've long suspected about pianos smile Great read all around!
ricko
That is too kind. And thanks to Joe D, from me too.

Perhaps I can put what I am thinking about another way. (Lets ignore resonances, velocity, dampening, string tension, and other complications.)

If we had a piano where every hammer hit at exactly the same ratio ("fraction length”), say 1/2 then we would have an instrument with very "consistent" tone, but not cohesive. The harmonic coefficients would be the same for every note if you like: spanish guitarists use this effect when plucking at 50% along the string. (So this is like typical analog polyphonic synths, with the same wavesape for every note.) However, while the peaks of one note and its octaves would all have similar formants in effect (and so be difficult to distinguish) yet notes in between could have peaks where our first note had troughs (and so be clearly distinguishable: it is the basic principle of classical orchestration that instruments with the same formants are perceived as a unity.)

At the other extreme, if we made a piano where every hammer was a fixed distance along the string, say 4 inches, we would have an instrument where every string's peaks and troughs aligned: the spectrum if each note would have peaks and troughs at the same frequencies. It would be like a strong EQ. This is more like what a constant-breadth pulse shaper provides. Very cohesive but more difficult to distinguish individual note played at the same time.

But real pianos, of course, are at neither extreme. The designers make a tradeoff (among lots of trades-off) between clarity and cohesion.

However, there are several grand piano designs where the lowest octave or two looks much more like the fixed length design than the fixed ratio design. (Presumable because there is no musical need for those notes to be distinguishable when played at the same time., because it doesnt really get called for.) So for these, thinking in terms of formant (cohesive, similar tones) rather than gaps (clarity) may be useful.
Joe_D
ricko wrote:
If we had a piano where every hammer hit at exactly the same ratio ("fraction length”), say 1/2 then we would have an instrument with very "consistent" tone, but not cohesive. The harmonic coefficients would be the same for every note if you like: spanish guitarists use this effect when plucking at 50% along the string. (So this is like typical analog polyphonic synths, with the same wavesape for every note.) However, while the peaks of one note and its octaves would all have similar formants in effect (and so be difficult to distinguish) yet notes in between could have peaks where our first note had troughs (and so be clearly distinguishable: it is the basic principle of classical orchestration that instruments with the same formants are perceived as a unity.)

At the other extreme, if we made a piano where every hammer was a fixed distance along the string, say 4 inches, we would have an instrument where every string's peaks and troughs aligned: the spectrum if each note would have peaks and troughs at the same frequencies. It would be like a strong EQ. This is more like what a constant-breadth pulse shaper provides. Very cohesive but more difficult to distinguish individual note played at the same time.

This is a creative and somewhat intriguing thought experiment in the abstract. If one were to somehow construct the fixed distance instrument (which couldn't exist in a piano as we know it, as I'll explain below), the effect would be like playing music through a resonator. Were you do so, you could get some very fun sounds and could make a piece out of it it, but it would be extremely limiting.
ricko wrote:
But real pianos, of course, are at neither extreme. The designers make a tradeoff (among lots of trades-off) between clarity and cohesion.
It's definitely true that designers make a million tradeoffs, but they aren't related to fixed hammer/string contact distance, or any desire for the resulting resonator/EQ effect, which would be so limiting in an instrument that no one would buy the instruments.

I didn't state explicitly yesterday the reasons why this can't work in the real world. There are at least two of them:

One, consistently attenuated partials sharing the same frequency from piano note to note can only happen for very closely related harmonic intervals like octaves and fifths. If the hammer/string contact distance were (for a round number example) 10cm and a certain string length were 20cm, the second partial would be attenuated. The note one octave below would be 40cm, and the hammer would strike the string at 1/4 of its length, cancelling out the fourth partial, which would be the same frequency/"note" as the attenuated partial for the first hypothetical string. That's how you concept works, and it's kinda cool.

But now consider the note one half step below our first hypothetical note--what partial will be attenuated here? Frankly, it's too late for me to do the math, but maybe the 17th partial? And descending chromatically, the other notes will have attenuated partials in different places. It's not until you get to the note a perfect fifth below the first note that your idea will come into play.

And the second reason is that for this effect to work, all of the strings would have to be made of the same (presumable mild drawn steel) wire, and their length would have to double every octave (and it's approximately true that they do in the upper treble).

For reasons I won't get into here, the highest string pretty much has to be very close to 2 inches (say a bit over 50mm). For such an exponential string length increase to continue, the periodically vibrating portion of the lowest string would have to be over 25 feet long! (close to 8 meters) There was at least one such upright piano built into the basement of a building and extending through the first floor and into the second floor. That's the only one I know of in the world (someone may have built one that I don't know about) in which your idea could be tried.

The way piano designers compensate for the obvious impracticality of building such pianos is to increase the mass of the strings by wrapping or double wrapping the bass strings with copper and drastically (really, really drastically) departing from the 2/1 string length ratio per octave. Once they do that, your idea no longer creates attenuated partials that line up even for fifths and octaves.

ricko wrote:
However, there are several grand piano designs where the lowest octave or two looks much more like the fixed length design than the fixed ratio design. (Presumable because there is no musical need for those notes to be distinguishable when played at the same time., because it doesnt really get called for.) So for these, thinking in terms of formant (cohesive, similar tones) rather than gaps (clarity) may be useful.


They look closer to a fixed length design but they aren't. What you're noticing is that in short pianos (and the vast majority of pianos are short), the bass strings get only a tiny bit longer as you descend from one note to then next. Since most of the increase (say, 6/7ths or 7/8ths of it) is at the end away from the hammers, the remaining 1/7 or 1/8 of string length increase is indeed tiny, and it starts to look a bit more like it might be a fixed length design. But it isn't. Here's one I serviced today:

It looks at first like a fairly uniform hammer/string contact length. But I put one end of my ruler against the overhanging lid (which is for our purposes about parallel to the hammer line) right above the highest bass string, and slid the "T" on the ruler right up to the string's upper termination (the very small pin where the string bends, not the tuning pin). I then positioned the ruler in the same manner at the lowest bass string. Here's the pic:

As you can see, the upper termination of the lowest bass string is about an inch (say 25mm) closer to the overhanging lid, and thus, the upper termination is for about an inch further from the hammer line. While an inch isn't nearly as much difference as the large Steinway displayed, it is still proportional, because in this medium-short piano, the bass strings don't get very much longer as you go down the scale. Here's the bottom of those bass strings:


I placed my screwdriver parallel to the bottom of the piano (sorry for the crooked pic) so you could get an idea of how much increase in the "below the hammer" string length there is for the lowest note. It looks to me like it's loosely in the ballpark of 6-7 inches, just what you would expect with a 1/7 or 1/8 fractional strike point.

OK, enough pianos; it's time to get back to our regularly scheduled modular programming!
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