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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 Also not exactly. Railsback has nothing to do with equal temperament. It's 100% a result of inharmonicity.
Keith Akins, RPT
Piano Technologist
USA Distributor for Isaac Cadenza hammers and Profundo Bass Strings
Supporting Piano Owners D-I-Y piano tuning and repair
editor emeritus of Piano Technicians Journal
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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. Also not exactly. Railsback has nothing to do with equal temperament. It's 100% a result of inharmonicity. Railsback is unquestionably the result of inharmonicity, but it is measured from equal temperament. Here's the graph of the Railsback curve that's used on Wikipedia. ![[Linked Image]](https://upload.wikimedia.org/wikipedia/commons/a/ae/Railsback2.png) Read the title of the left axis. It's plotting the tuning WRT [with respect to] equal temperament. If it were being plotted with respect to something else (for instance, raw frequency) it would have a different shape.
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It could just as well be measured from just intonation, or any other theoretic temperament.
Semipro Tech
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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. An ordinary frequency counter would likely count the frequency of the loudest part of the signal, and be oblivious to the various partials. Either that, or it would just be confused, and report nonsense--not the right tool for the job, in any case.
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Whatever the "loudest part of the signal" might mean. I cannot imagine that any meaningful definition would allow for being "oblivious to the various partials."
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Whatever the "loudest part of the signal" might mean. I cannot imagine that any meaningful definition would allow for being "oblivious to the various partials." Frequency counters are designed to look at simple waveforms, with some regular frequency. They will trigger on some part of the waveform and count up how many trigger events there are in a second, for example. They are not designed or capable of picking out the various frequency components of a complex waveform and reporting them. For example, a square wave of sorts has many harmonics, but a frequency counter will report on the first one. If you google "frequency counter," you'll see a bunch of models.
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Whatever the "loudest part of the signal" might mean. I cannot imagine that any meaningful definition would allow for being "oblivious to the various partials." Frequency counters are designed to look at simple waveforms, with some regular frequency. They will trigger on some part of the waveform and count up how many trigger events there are in a second, for example. They are not designed or capable of picking out the various frequency components of a complex waveform and reporting them. For example, a square wave of sorts has many harmonics, but a frequency counter will report on the first one. If you google "frequency counter," you'll see a bunch of models. A piano sound audio waveform is very complex because the frequency of the various partials do not align with the fundamental, or to each other. The waveform is ever changing with no uniform periodicity. A frequency counter may pick up on parts with more with more intensity (the "loudest part of the signal"), but those parts may be mixed and confused with ever changing and overlaying partials. Processing an audio waveform to get a frequency requires special techniques such as transformations to the frequency domain (e.g. FFT), or autocorrelation methods.
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I just used "frequency counter" as a general term for some device that associates a number to a given pitch, because people use some sort of device like that to assign frequencies to pitches. If someone wants to call it something else, that is fine. I still do not know what it does to gets those numbers, and more than that, I do not know what the numbers mean, and what they have to do with whether notes are in tune or not, especially when the numbers are different from what one would expect from theory.
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Today piano tuning apps can detect frequencies of partials realtime with an accuracy of less than 1/10 cent. No problem with that part.
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You have just introduced another unit, the cent, which you have not defined, which is detected by some process that is vague, that supposedly means something to do with tuning, which you have not explained. People use this sort of reasoning to show that they cannot tune pianos. What does all this mean to someone who wants to know how to tune pianos?
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Another unit used in tuning apps is the "beat". The accuracy within which beats can be measured by tuning apps is out of reach of human ear.
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One can make up any number of arbitrary units that have no relevance to the physical world.
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What unit is made up to measure downweight in the physical world ?
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And generally what other units are made up to design a piano scale in the physical world ?
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What unit is made up to measure downweight in the physical world ? Weight is just the combinatorial effect of the personal magnetism of the highly evolved turtles under your feet that hold up the surface of the earth which is flat. Kees
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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/hxMj1YSnP2If you tune octave above middle C (C4-C5) on "normal" instruments the fundamental frequency of C5 is just 2X the fundamental of C4. If you do this aurally you're listening for interference of C5 fundamental with C4 second partial. On piano problem is inharmonicity; second partial of C4 is not 2X fundamental but a little sharp because of string stiffness. So if you tune C5 fundamental to C4 partial the note will be a bit sharper than "theory". Deviation from this "theory" is this Railsback curve. E-pianos try to imitate this. And that's all there is to this, unless you believe the earth is flat, partials don't exist, cents were invented to corrupt "physical" piano tuning, and what not. Kees
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When I tune octaves, I listen for consonance* between the fundamental and the octave and two octaves. I check it so the relationship between other intervals agree† as closely as possible.
It is much simpler than worrying about frequencies and partials, because when you think about them, frequencies and partials are not defined in such a way as to give you enough information to tune a piano. In this example, if an octave is defined as twice the frequency of the fundamental, then either the definition of frequency of a piano note is wrong, or the definition of tuning is wrong, or both. I do not know to make sense of that, so I practice the above.
*A consonant interval sounds beatless and reinforces both notes.
†Intervals agree when they sound similar to one another, in beats and to the extent that they reinforce both notes, but are not necessarily beatless.
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