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in a piano's design?

Is it mostly the length of the piano? Do most pianos of about the same length have about the same number of bass strings or can this vary greatly even for pianios with the same length?

My impression is that as pianos get longer they tend to get fewer and fewer bass strings? Is this correct and is it an advantage?

The scale designer determines the number of bass strings. Cost constraints probably come into the equation.

Broadly speaking it will be the length of the piano.

Although older scales can be quite erratic most modern piano scales start at C-88 with a length of 50mm to 56 mm (1.97” to 2.2”). From there each successive note down will be determined by multiplying each length by some fixed number. The value of this number will be determined by the “length” of the desired scale which, in turn, will be determined by the length of the piano and by the desired acoustical characteristics of the scale.

At some point the scale lengths calculated this way will become too long to fit into the chosen height or length of the piano being designed or built. At this point one of two things has to happen: the scale has to be “foreshortened”—each successively lower string length made shorter than the mathematically optimum length—or a transition to wrapped strings has to be made. Most traditionally drawn scales use what I call a “reverse hook” at the lower end of the tenor bridge at which point the speaking lengths of the strings is simply made shorter than a strict mathematical multiplier would dictate. The tension of these strings will be lower than optimum unless the wire size is increased significantly. In either case the tone quality of the low tenor suffers.

Rather than foreshorten the strings a better practice—and that followed by most designers of ”modern” pianos—is to simply transition to wrapped strings further up the scale and avoid this foreshortening; this is what you will see in most modern short pianos. Due to the constraints of length smaller pianos simply do not have room to extend the lengths of the steel strings so designers transition to wrapped strings further up the scale. It has long been accepted practice to avoid placing wrapped strings any further up the scale than E-32. This keeps wrapped strings out of the traditional “temperament octave” which starts at F-33. If the wrapped strings are not properly designed it can be difficult to aurally set a smooth temperament if this octave includes wrapped strings. However, with good scaling this problem can be overcome and I have designed scales in very short pianos (≈ 150 cm, or 4’ 11” in length) with wrapped strings extending up to F#-34 that do not present tuning difficulties.

Longer pianos do not present these problems so the plain steel strings will extend further down the scale. The typical transition in concert grands falls between E-20 and F-21.

ddf

Del,
Can I ask perhaps a remotely related curiosity question:
were there any efforts to design non-uniform distribution of added mass (discrete or continuous) on wrapped strings to enhance or suppress desired harmonics?

Yes. The earliest that I am aware of were those of Harold Conklin while he was head of R&D at Baldwin. He developed and patented a method of scaling that controlled the pitch of longitudinal harmonics in wrapped strings and made it possible to tune them to specific pitches that blended with the fundamental pitch of the strings. This scaling was used by Baldwin for a time under the trade name “SynchroTone.”

The scheme worked but, like most everything to do with piano design, there were tradeoffs. In this case the ratio between the core wire and the mass of the wrap had to be very tightly controlled. Mr Conklin also developed what I believe was one of the first NC controlled string wrapping machines to wind these strings. It had been abandoned by the time I arrived at the company because, I was told, it was too slow to maintain the required production speeds. The strings were then wrapped by Mapes but, while the Mapes strings were of good quality they were unable to maintain the precise balance between core and wrap that was required to precisely tune the longitudinal harmonics to their desired frequencies.

Another problem with this scaling was the uneven inharmonicity sequences and the uneven tensions—and, hence, acoustical power and timbre—from note to note. As well, in some scales there were problems with premature string breakage.

Still, it was a remarkable development. I don’t know the full history of the development of this technology but I was told that to demonstrate the his ability to control the pitch of the longitudinal harmonics Mr Conklin produced a set of strings having the same length and fundamental pitch but on which he was able to play “Mary Had a Little Lamb” using just the longitudinal mode harmonics.

ddf

Yes, he did as part of his patent application. I have a CD that includes that recording (among other things).

Del,
Thank you for the great answer!
Controlling the longitudinal harmonics makes all the sense.
The examples of the longitudinal control and its effects in a online piano primer by Horald Conklin (that I've looked up after your answer) are impressive.
Perhaps more of a curiosity question: can transverse modes be affected. For instance, by using the beads or copper cylinders placed intentionally to stiffen the nodes of the higher harmonics redistributing their energy and affecting the timbre (Sorry if it may sound as gibberish practically - the closest related mentioning I've seen is the reference 5 in this online paper.)

I’ve not tried any actual experiments on this but, yes, I don’t see why this couldn’t be done. If you wanted to reduce the amplitude of a certain partial you could place mass loads at the antinode points of that partial or stiffen the node areas. Of course this would also alter the mass and stiffness of the entire string so the vibrating characteristic of the whole would also change.

I’m not sure just what you would be wanting to accomplish but placing any mass or stiffener on a vibrating string will change how it vibrates. This, for example, is what is done when a copper wrap is applied to a steel core wire; mass is added to all possible antinodes and all possible nodes are stiffened.

ddf

I was just thinking (not out of any experience) about the intrinsic limitation in length of the bass strings and their perceived tubbiness (due to higher harmonics being a significant contributor to an overall response, if I understand it correctly).
Now, the copper wire is wound uniformly (traditionally and, naturally, for simplicity) and thus all the higher harmonics are suppressed uniformly (as You've mentioned) and thus overall not very efficiently (given the perceived sound tubbiness).
Perhaps viewing it simplistically, it may be possible by using the added mass distribution in a form of some beads and cylinders to suppress most selectively and efficiently the nearest and/or most prominent higher harmonics (ideally, channeling the energy to the fundamental mode) in an effort to make the sound "more pure" and desirable in smaller instruments.