Piano World Home Page Forums Our Most Popular Forums Piano Tuner-Technicians Forum FAQ - Tuner-Technicians Forum Logarithmic scale?

I've been hearing about logarithmic scales with regards to piano design. What does this mean? Logarithmic with regards to tension? String length? Something else? Why would this be an advantage or not?

Just curious from a layman's perspective, but I'm also a physicist, so can handle the real explanation

Does anybody have any references (books or online) to basic piano design? Curious subject.

Dan

It simply means that the length of each string is longer or shorter than its neighbor by some fixed logarithmic multiplier. That same multiplier will be used throughout the scale. At least the tenor scale. This is a concept that has long been talked about but has seldom actually been practiced. Indeed, even when the rare pianomaker of today claims to have a log scale it is generally only partially true. Nearly all pianos on the market today have bass bridges that “reverse” curved opposite to the direction they would curve if they sported log bass scales. The single exception among current production pianos is the Walter 190 grand.

Log scales are quite easy to develop using a simple spreadsheet. I can’t speak for others but I start at C-88 and work down. I know it’s backwards, but it works easier that way — at least for me. Depending on the desired tonal characteristics of the piano in question I’ll usually start with a speaking length of something between 50 mm to 54 mm at C-88. Then, knowing how many unisons I will have on the tenor bridge and knowing what lengths I want to start with and end with it is a simple matter to plug in the numbers and come up with a printout of string lengths.

Using a log scale enables the designer to use an even progression of wire sizes starting, usually, with #13 (or 0.031”) at C-88 and getting progressively larger on down the scale. Depending on the log multiplier used this progression will typically be something like 6 unisons of #13 (0.031”) wire, 4 unisons of #13 ½ (0.032”) wire, 6 unisons of #14 (0.033”) wire, 4 unisons of #14 ½ (0.034”) wire, etc. This progression may vary some depending on the log sweep of the bridge.

The main advantage of adhering to a log sweep to the bridges is the uniformity of string characteristics and bridge loading it makes possible. It is very helpful in balancing the tone quality of a piano across the full compass of the scale. It is also makes it possible to achieve a relatively uniform string inharmonicity curve which helps the tuner in setting a uniform stretch to the tuning.

Of course, coming up with lengths for the speaking portion of the strings is the easy part. Laying them all out into a workable form, balancing them against an appropriate soundboard assembly and coming up with a good looking and beautifully performing piano in the end — now that’s the tricky part. As the saying goes, the devil’s in the details.

Del

I don't think I quite understand log scale / wire sizes / whatever the relationship is... it's not like you take the speaking length of C8 at, for example, 50mm, and multiply by 2 to the 1/12 power, do you, to get the next note (and multiply each result by 2^(1/12)?

Hello Del!

Does the Walter grand (I understood that it is your own design, right?) have a logarithmic string length progression even in the bass?

Could you please elaborate on the different tonal characteristics you can obtain by starting at C88 with a length between 50 & 54 mm?

What is the factor of multiplication used to achieve the logarithimic scale? And why?

Any thoughts on a scale that uses double lengths for each octave? I have heard about this used in old harpsichords, but not in pianos.

Regards,

Calin

In my understanding of scale design the logarithmic scale is more of a starting point - a theoretical "ideal" that has rarely been put into practice.

Del, do you mean to say that the Walter is a log. scale?

Does anyone know of any pianos, built now or in the past, that use a fairly pure log. scale?

Thanks Del - great answer (of course I hoped you would reply). I can see how this would help the dreaded "fixing your fixes" problem. I work as an engineer, and I've noticed that if you start a design without a basically good solid framwork, you end up putting fixes in pretty quickly to patch up problems with the first draft.

Well, those fixes usually come with a cost, so then often you have to put in fixes, to smooth out those fixes. You can see that quickly you get to some kind patchwork of compromises, unless it gets so bad you have to start over.

I assume it's similiar to piano design, where starting out with some regularity in the scale design helps you from having to accept too many compromises in the end, or just happening to hit a design finally by accident.

Dan

Note that I said "some fixed logarithmic multiplier..." Not the 12th root of 2. But, yes, each succesive length is obtained by multiplying the previous length by the same number.

Del

Well, yes, it is a theoretical ideal but it’s much more than a “starting point.” Though you are correct in that it has rarely been put into practice. There are several reasons for this. Even back when Wolfenden wrote “A Treatise on the Art of Pianoforte Construction” he was lamenting the fact that most manufacturers, when bringing out a “new” piano exercised the false economy of simply copying something already in production rather than mathematically working out a proper scale. This observation was echoed in the collection of minutes of the Piano Technicians meetings of roughly 1914 to 1919. Even though the principles of good (tenor) scaling were generally understood at least by a few designers they were rarely utilized.

Sadly, not much has changed over the years.

Yes, the Walter grand is a true log scale through both the tenor and the bass (though, for obvious reasons, the bass uses a different multiplier). As is now the tenor section of the Walter vertical.

I’ve not kept a record of them, but there are a few around. Sometimes in a surprising package. Several years ago I was asked to do some redesign work on a 4’ 7” Howard grand (no, don’t ask why) and in evaluating the stringing scale found it to be laid out to a nearly perfect log progression. As for current production, the new M&H AA almost certainly has a true log scale along its tenor bridge. As probably does the new Seiler grand.

Del

Yes, it is. Much of what now passes for piano design is really design patching. There are some fundamental design flaws built into the tradition design formula. A great deal of innovative tweaking goes into ameliorating the limitations imposed by these flaws. It is a credit to some of today’s piano builders that they do as well as they do. For example, the Shiguru concert grand. This instrument is relatively similar to the Steinway D’s fundamental design. Still, it is exceptionally smooth and dynamic. The bass/tenor break is barely, if at all, discernable. There is little, if any, drop-off in killer octave region. The limitations of the fundamental design have been masked over to a remarkable degree through a combination of design tweaks and superb workmanship.

Now, while I admire the effort made by the Shiguru’s designers and builders, I would prefer to see the industry moving on to a cleaner design base as a starting point. I think the industry has at least one more evolutionary step left. Perhaps more, but that is the limit of my vision just now.

Del

Hi Del,
Could you share what Charles Walter asked for on the CW190? I'm curious to see how it compares to my perception of the piano (which I really love).

Dan

The thing that I have notice since delving into some of these mysteries is that there are pianos with very good scale designs which don't necessarily have outstanding sound, and pianos with outstanding sound that don't have good scale designs. The latter group can often be improved with careful string selection. (There are some models I would like to get my mitts on, to see what the results would be.) It leads me to believe that not everything can be reduced to math calculations, though.

I find that some of the conclusions that Frank Hubbard came to in his book on harpsichord construction hold true for piano design as well, such as the fact that shorter scales seem to work better on lighter instruments, like Steinways, and longer scales work better on heavier instruments, like M & Hs. But I haven't done nearly as much research as Del.

Remarkably little at first. In part because we share many of the concepts of piano performance. He wanted a very smooth inharmonicity curve and he wanted a smooth and balanced scale. Beyond that he specified the length of the piano and expressed his general desire for tone quality. Then he monitored the progress of the design all the way through and made me justify every design decision I made. In so doing he ended up with the piano design he wanted and left me happy with it as well.

Del

Nice, sounds like you both got what you wanted, something that doesn't always happen when consulting.

I'll admit it, I really admire the CW thus far. I was trying to describe it's tone to my wife last night, but couldn't do it. It's too elusive, which I believe is a positive. Best I could do was say "Well, on a scale of Bosendorfer to M&H, I'd say it goes

Bosie - Steinway - CW - M&H

With the bosie being the most mellow and melodic (low harmonics), and the MH being the most individualistic with high harmonics."

I also have difficulty describing the tone of a Steinway. Other than it (and the CW) both have a powerful bass and singing treble.

Dan

Hi Del and thanks for taking the time to answer!

Here are a few more questions:

What is the difference between a small and great sweep in the bass? Which one is preferable? I guess the greater sweep, as it should have a smaller increase in inharmonicity and would mean you get fundamental frequencies in the lowest bass closer to the correct pitch?

[2] Why do long scales induce backscale problems? Isn'y there a limit for backscale length beyond which there is no gain in flexibility?
What is the minimum feasible backscale for A0, that doesn't restrict the free movement of the bridge?

[4] I don't mean to have all the octaves doubling. Just for the plain wire. I made a small calculation that shows that, for instance, with a C88 of 50 mm, one could make a scale that doubles at each octave until let's say E20 which would be 2540mm - a length that would probably fit in a concert grand. The bass, of course, must be made with another multiplication factor. Would such a scale work? It should have much less inharmonicity than normal ones. But how would this influence other factors?

Another issue: when you want to design a grand of a specified length, how do you decide where the bass break should be?
Just see what the longest plain wire string is (from a logarithmic progression), that fits in the case?
That would of course mean that the smaller the piano, the more bass strings it should have, which doesn't always happen in practice.

Regards,

Calin

It’s not so much that one sweep is more or less desirable in the bass, it’s more a matter of which is possible within a given overall length. Things all have to fit within a given length and shape.

[2] Again, within a given piano length, everything has to fit. That means if the speaking length is made longer the backscale is going to be shorter unless the piano is made longer. Making the piano longer is usually not an option.

I don’t know the limits. It usually doesn’t become an issue because the range is usually restricted by other factors. I do know that 50 mm is so short as to essentially prevent any bridge motion in the 27.5 to 55 cycles per second range. Anything shorter than that (and there are some) begins also to restrict meaningful motion through the 2nd harmonic range as well. I don’t know how long is too long.

[4] I’ve not done any work with strings this long. It would take building a monochord and measuring the tonal characteristics to get a general idea of what was there. But, in the long run, you’re going to have to build the piano.

F-21 is approximately 1830 mm to 1850 mm in the typical 275 cm concert grand. And it is already a difficult fit. Making any string on the tenor bridge another 600 or 700 mm longer would mean substantially lengthening the piano. Again, these decisions usually boil down to what is practical within a given piano length.

Yes, the decision of where the bass/tenor break should be is primarily based on what will physically fit. And, yes, in the smaller piano more wrapped strings should be used. I am aware that this doesn’t always happen in practice, but it should.

Del

Del:

I hope this hasn't already been covered in one of your other replies(which are beyond my understanding) already. I think you said in one of your first posts that in a logarithmic scale each string length is the same multiple of the string length adjacent to it. Being a math teacher, I first assumed that the multiplier had something to do with logaritms. Is that true or is the multiplier some number unrelated to logs(and if so, why the name *logarithmic* scale?)?

Thank you!

Tradition. And when it's plotted on a log scale it makes a nice, straight line.

Del

That was a simplistic answer.

Basically, the process is this:

Let's assume you want to start with a C-88 having a 50 mm long string and you want to end up one octave lower (C-76) with a length of 93 mm.

93/50 = 1.86. This is the octave sweep.
So 50 mm * 1.86 = 93 mm.

There are 12 notes to the octave so, taking the 12th root of 1.86, or 1.0531 gives you the note to note multiplier. Multiplying 50 mm by 1.0531 = 52.7 mm, the speaking length of B-87.

Or, put another way, 50 mm * 1.0531^12 = 93 mm, (the speaking length of C-76), 50 mm * 1.0531^1 = 52.7 mm (the speaking length of B-87), and if you want to know the length of A-85 (3 notes down) you can multiply 50 mm * 1.0531^3 and come up with 58.4 mm. Etc.

There is a much cleaner way to write this with an equation editor, but you get the picture.

Del

Hi Del,

Thanks so much for explaining in detail why the term "logarithmic scale" is used, per PianoLoverus's query.

One more question though ... It's not crystal clear to me why the octaves aren't in the ratio 2:1 (in your example the ratio is 93/50 or 1.86).

Going all the way back to Pythagoras and his school, I had learned that octaves were 2:1, and thus consecutive semitones would have the ratio of the 12th root of 2; apparently not, though, according to your figures.

Care to clarify?