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My two granddaughters (6 and 9) starting taking lessons in Sept. They have a spinet that was shipped cross-country several years ago. It hasn't been tuned since moved and who knows how long before that and is clearly out of tune. I decided to record the key frequencies before and after tuning.

The first thing I noticed was that trying to get a good measurement was difficult because the multiple strings on each key are mismatched so the measurement jumps all over the place. After plotting the results I got another surprise - the frequency does not match the ideal (especially from A5 and up). I used the equation 440 x 2 ^ ((key # - 49)/12) to compute what I called the "ideal" frequencies and the values matched the tables I found online.

Then I decided to measure my CA98. It's digital so it must match the ideal frequencies. I tried two different iPad apps and two different piano options on the CA98. The values were quite stable (no multiple string mismatch like the acoustic).

SURPRISE! All were very similar - a big rise starting at A5.

Why don't pianos match the calculated frequencies?

The GREEN and BLUE are the CA98 and the ORANGE is the acoustic.

https://link.shutterfly.com/hxMj1YSnP2

Variability in Wire stiffness due to the imperfect design of the string scale.

Piano wire is stiff enough that some of the restoring forces are due to the spring nature of the wire itself. There is the restoring force of string tension and a restoring force of stiffness.

As the string subdivides into higher "harmonics", the proportion of restoring force due to stiffness rises whereas the amount imparted from tension stays the same. Thus the harmonics become partials, and deviate sharp from the whole number multiples of simple harmonics.

This is called inharmoniocity.

The other key phrase you might search for is "stretched octaves".

Frequency counters do not necessarily reflect what intervals sound like.

Or piano tuning temperament and marvel at your piano tuner’s skill.

All of the wire stiffness comments make sense for the acoustic piano. But the CA98 is a DP and the deviation curve has the same characteristics as that of the acoustic. I would expect the DP to exactly match the calculated "ideal" curve.

I tried two different apps to verify that the apps were accurate and the results were very similar. The measurements were done on an iPad but I don't think that the microphone would be an issue. It might change volume due to frequency response, but I expected the frequency to be correct.

BTW - the graph that I provided is the deviation from the calculated values in percent. For example, A4 = 440 Hz. If the measurement was 442 Hz the deviation would be:

(442-440)/440 = -0.45%

Here are the frequencies for A1-A6 on the CA98 DP:

Calculated Measured Deviation
A1 55.0 54.9 -0.2%
A2 110.0 110.0 0.0%
A3 220.0 220.0 0.0%
A4 440.0 440.3 0.1%
A5 880.0 881.0 0.1%
A6 1760.0 1770.0 0.6%
A7 3520.0 3564.0 1.3%

Clearly as the frequency goes up the error increases. Maybe at the higher frequencies the errors are inaudable?

ee375:

This not an error. The piano should sound that way (digital or acoustic, dp are mostly sampled from acoustic pianos or try to mimic the sound of an acoustic). Since the inharmonicity of the string increases towards the treble, the more the string has to be stretched in order to match the strings below it.

When you play a note on an acoustic piano, you perceive the fundamental pitch, but all the other partials are also there.
The piano is tuned trying to match the partials of notes with each other according to the selected temperament, so that the piano sounds in harmony.

Most of the pianos are tuned to EQUAL temperament. But there are many historical temperaments as well.

Piano tuning is a broad subject and also an art.
Just find a good technician and let him/her tune your spinet.

It would be possible to make a digital keyboard with notes that are pure sine waves of specified frequencies in pure geometric progression, with a specified decay rate, and specified 'arithmetically perfect' harmonics for each note.

It would sound nothing like a piano.

ee375 if you are interested in this topic, you need to read a book on it. Though the explanations already given here are an excellent starting point.

Imagine two fence posts ten yards apart, with thick fence wire running between them. You could pull on that wire, five yards along, and get it to move a little. You might be able to "twang" it.

Now imagine a six-inch length of that same wire, secured at each end. You wouldn't be able to "twang" the six-inch length, even though it's made of the very same wire.

ee375:

Here is an article on the subject:

https://asa.scitation.org/doi/10.1121/1.4931439

I reviewed the links from the replies above (and thank those who provided them) and also found other links that explained "stretch", "equal temperament", "railsback curve", etc. So now I know enough to be dangerous.

If I understand correctly, most pianos are tuned to a railsback curve. While the general shape is the same for all pianos, the actual frequencies (octave stretch) are not due to the physical properties of the strings and the cabinet resonances. I also understand that most keys have multiple strings and that each string for a given key must be tuned to the same frequency.

If this is correct and assuming that I get the "perfect" tuning, can I measure the frequency of each string immediately after tuning and then periodically check it over time to look for changes indicating that it is out of tune?

If I find anomalies in the plot for a certain key after tuning does that mean the string is not ideally tuned? In other words, should the curve be smooth (i.e. no keys spiking above or below the curve)?

Can this data be used for future tunings? It seems like it would be much easier the second time - simply tune each string to the original tuned frequency.

It is all very well to talk of "simply tune each string to the original tuned frequency". It is not easy to tune any string to any frequency! A great proportion of the skill in tuning is the ability to manipulate the tuning lever in such a way as to get the string to go where you want, and stay there. It ain't easy!

The actual curve will have spikes and other anomalies because iH does not progress smoothly from string to string. Also, iH can appear to vary from partial to partial on a single string.

This is a perfect example of "theoretical" vs "real world".

Example: Race cars use tires with no tread. Why? Because more contact with the surface translates into better traction. If this is so, then all cars should have better traction to improve control, right? They shoukd all have racing slicks because they have more traction. Yes, except for the fact that the average car is required to perform in adverse weather conditions in which racing slicks would fail miserably. Therefore a compromise is made for the family car by removing some of the traction and leaving grooves and channels for rain, snow, and mud to go into and still leave rubber on the road.

In reality, NONE of the intervals (including unisons) are PERFECTLY pure (i.e. "correct" in the nth degree). They are all compromised from the theoretical to one degree or another, to satisfy real world conditions found in the structure and capability of the piano. The more research you do the more you will see this to be true. In the end, these slight differences contribute to the unique sound of the piano.

Relative quality in design and manufacture is another issue entirely though.

Pwg

Well, we all have to start somewhere. Welcome to a potentially lifetime journey of understanding -- or at least coming to terms with -- piano tuning.

A few comments:

1) No, pianos are not "tuned to" a railsback curve. That's backwards. It happens that when pianos are tuned -- originally only by ear and now with the aid of devices that can do quite good jobs of copying the result of aural tuning -- they are tuned to adjacent or connected notes (called intervals) for a particular sound. With the advent of pitch measuring devices, it was discovered that when pianos were tuned "normally" in that manner that the pitches would match a certain plot on a graph which was first published by Railsback.

This is also an example of the aphorism "figures don't lie but liars figure". Not that anyone involved with plotting pitch curves are deliberately attempting to deceive but rather that the plot is illustrating a machine-centric prejudice about "correctness". One would get an exactly opposite (or inverse) plot if electrodes could be attached to a tuner's brain and the results displayed on a screen: the pitches plotted as the Railsback curve would be a flat line and pitches on a flat line would plot out as a reverse Railsback. It is often not recognized that we program or design-in our assumptions into our measuring devices -- or how we record the data -- so that the information is presented in a biased format.

2) Regarding results from a digital piano... The digital is attempted to be programmed to have the same output as a real piano. Obviously, if it had a different output, it would fail at synthesizing piano sound. (All digitals fail at some level. Expensive ones are pretty good ... others not so much. )

One of the first rules of measurement is to use a measuring device appropriate to the task at hand.

I am pretty sure I know what a frequency counter would be counting for a steady tone. But pianos do not put out steady tones, and what they count could be several things, and likely none of them are exact.

Great, you've got the basics. Other terms that might be helpful to understand might be "inharmonicity" and what's meant by terms like "4:2 octaves" which will help you understand why "stretch" is necessary.


Not exactly. I would restate that as, "Measuring a tuned piano and plotting its deviation from equal temperament gives a curve with the general shape of the Railsback curve. The shape of the curve is determined by the piano's scale, including how long the strings are (shape of the bridge), and which wire types/diameters are used."

Unrelated, but typically when you're plotting a Railsback curve you do it as a "Cents" deviation from ET, where 100 cents (roughly a half step) equals a frequency ratio of 2^(1/12) See also http://www.sengpielaudio.com/calculator-centsratio.htm


There is no such thing as a "perfect" tuning. It's always a compromise. But yes, you could in theory record a good tuning and then look for changes. But it could give you false positives. If a piano were uniformly a few cents sharp or flat of your good tuning, it would still sound perfectly fine to any listener. The thing that requires tuning is when it goes out of tune unevenly causing octaves or unisons to sound bad. (This applies generally to home-use situations, obviously not concert pianos, which aren't allowed to go out of tune in any way.)


It will never be perfectly smooth, and it probably shouldn't be, especially for a spinet. There are things that can cause bumps and spikes...things that affect the placement of harmonics like wire transitions in the piano's scale, soundboard resonances, etc. But there will probably be other small bumps and spikes from the tuner not being able to get each string exactly where s/he wants it, and from other factors outside the tuner's control (like when the heating/AC kicks on in the middle of a tuning).


In theory, yes. In practice, I suspect it would be tricky. Half the problem is knowing what pitch you're aiming for. The other half is getting the string to be stable there.

By the way, here is a graph showing a Railsback-like curve and, in my mind, illustrating why there is no such thing as a "perfect" piano tuning.

The dark black line is a "Railsback" tuning curve calculated by piano tuning software. The scale for this line is on the right, going from -50 to 50 cents, with 0 being equal temperament.
The blue dots represent the actual tuning of the piano. Even though the dots don't fall exactly on the black line, they're close enough for the piano to be considered in tune.
The 5 extra blue lines are a bit complicated. They show the "widths" of 5 different types of octaves (2:1, 4:2, 6:3, etc.) The scale for these is on the left axis, also in cents. Here, zero means that the octave is tuned pure or beatless. The point here is that it is impossible to tune all the octave types pure for any given note. In the middle of the piano the 4:2 octaves are fairly pure. In the tenor the 6:3 octaves are pure and the 4:2 octaves are stretched wide. 8:4 and 10:5 octaves are always narrow. In aural tunings the type of octaves tuned pure is a matter of judgement and taste.