What are the true frequencies of the piano keys?

Question

In theory it is easy, to get the frequency of the note other than A4=440Hz you just multiply/divide 440Hz by the proper number. For example, to get A2, you divide by 4 and get A2=110Hz.

I read for example here: http://en.wikipedia.org/wiki/Railsback_curve#The_Railsback_curve that the true pianos are not tunes in such ideal way, and the further you get from A4, the greater the abberration is. So I wanted to get the list of the true frequencies of keys, but I cannot find it. Can you point me to one? Or maybe there is the improved equation that takes into account the Railsback curve? Or the equation of Railsback curve itself? All tables I found on the Internet present this ideal frequencies, which is useless, because I can count them myself.

I expect that the true frequencies might differ from piano to piano, but maybe you can give me just general idea what the frequency range for each key can be?

Answer 1

Yes, you are correct, the "true" frequencies will differ from piano to piano.

In addition to the answers already given here, I would like to add more information regarding inharmonicity. The amount of offset or "stretched tuning" for the strings of an acoustic piano will vary with the size and type of the piano. It will be different for a spinet, upright, baby grand, grand, or concert grand. So there is not one strict formula for all acoustic pianos.

Discussion of stretched tuning and inharmonicity at Wikipedia.

In the article on Piano Tuning at Wikipedia, we find this quote:

The amount of stretching [in pitch] necessary to achieve this is a function of string scaling, a complex determination based on the string's tension, length, and diameter.

With the different sizes of pianos that I mentioned above, each different kind of piano will have a different set of strings with different sets of length, thickness of the core string, and thickness of windings on the bass strings. These will result in different amounts of tension for the strings used to produce a given pitch, between different makes, models and sizes of pianos. Again, this illustrates that there is no "one-size-fits-all" "list of true frequencies" for an acoustic piano.

A professional piano tuner knows that the exact tuning that is required will vary with every piano that he or she works with. Tuning each acoustic piano is a time-consuming and interactive process, and it does not depend entirely on a fixed set of exact pitches or frequencies.

Answer 2

The figure in the Wikipedia article tells you what you are asking, if you're willing to tabulate the deviations by reading the green line.

The vertical axis is the number of cents that the key is tuned away from equal temperament, e.g. the C two octaves above A440 (C7) is about 10 cents sharp, i.e. the frequency is a factor of 2^10/1200 sharp, or the actual freuquency is

   f = 440 * 4 * 23/12 * 210/1200 = 2105.13 Hz

(note: an equal tempered C7 is 2093 Hz)

The factor are:

1. the tuning reference 440 Hz,
2. go up two octaves, a factor of 4
3. go up 3 semitones, a factor of 2^3/12
4. apply the tuning correction, a factor of 2^10/12000

If the green line were flat at zero then all of the keys would be tuned to equal temperament.

Answer 3

Are you asking this question because you're writing a synthesizer? That kind of detail will help the answer that you get...

If you're working on a plain sample based subtractive synthesizer, the adjustments of the frequencies is already done for you by the sample set that you have. So unless you're trying to do physical modelling synthesis, you can ignore the offsetting.

Answer 4

The question is actually self-defeating. The problem is that the reason a piano gets stretched tuning in the first place is disharmonicity, meaning that the sine waves constituting the various harmonics travelling the string itself are not simple multiples of the fundamental.

As a result, the compound signal is not even periodic as a whole, so talking about its "frequency" is somewhat misleading. Of course, the strongest component will be the fundamental sinoid wave from the simplest string mode vibration, but if you repeat a sampled signal at that frequency, the disharmonicity will get replaced by border artifacts and the resulting harmonics will wobble.

In case this question was asked in the context of sound synthesis: you will need several oscillators per key at different frequencies in order to manage disharmonicity properly.

If you create a truly periodic signal with just a single frequency for synthesis, a stretched tuning will be pointless.

Answer 5

As the question itself admits, there is no "true" mathematical answer to the question since the best tuning varies from piano to piano. But one can use a formula that includes a quadratic term to give some approximation of the Railsback curve. It is therefore probably a closer fit for most pianos than equal temperament would be.

Source: Original research. (I am an amateur musician with a math degree.) Corrections are welcome.

• Assumption #1: The octave around middle C (or C4) is roughly equal-tempered.
• Assumption #2: Due to the inharmonicity of the strings, each octave above or below middle C should be "stretched" incrementally wider than the preceding octave.

Define a "stretch factor" s in semitones per octave. Then each note n, in semitones above middle C, should be tuned (s/2)(n/12)^2 semitones sharper than equal temperament (and each note below middle C flatter by the same amount).

For my piano, a stretch factor of s = 0.05 semitones (or 5 cents) seems to work well. In other words, the octave around C5 will be tuned 5 cents wider than equal temperament, the octave around C6 will be tuned 10 cents wider, and so on. Using the above formula, we find that each C above and below middle C should be tuned as follows:

C5: 2.5 cents sharp (5/2)
C6: 10 cents sharp (5 + 10/2)
C7: 22.5 cents sharp (5 + 10 + 15/2)
C8: 40 cents sharp (5 + 10 + 15 + 20/2)

Now how do we get actual frequencies from this?

In an equal-tempered tuning, the frequency of a note n is x = C4 * 2^(n/12) Hz. Adding in our adjustment term, we get x = C4 * 2^((n + (s/2)(n/12)^2) / 12) Hz. (Notes below middle C should be flat rather than sharp, so subtract the adjustment term rather than adding it.)

For concert pitch (A = 440 Hz), the correct frequency for middle C depends on the choice of stretch factor s. Substitute x = 440 Hz and n = 9 semitones in the above formula, then solve for C4. For s = 0.05, the correct frequency is 261.41 Hz.

Plugging that value for C4 back into the formula, we can then compute:

C1: 32.25 Hz (flatter than E.T. @ 32.70 Hz)
C2: 64.98 Hz (flatter than E.T. @ 65.41 Hz)
C3: 130.52 Hz (flatter than E.T. @ 130.81 Hz)
C4: 261.41 Hz (flatter than E.T. @ 261.63 Hz)
C5: 523.58 Hz (sharper than E.T. @ 523.25 Hz)
C6: 1051.71 Hz (sharper than E.T. @ 1046.50 Hz)
C7: 2118.66 Hz (sharper than E.T. @ 2093.00 Hz)

And so on.

Answer 6

Just to expand on a point I think is important, and the reason I think learning frequencies is semi-useless:

As supercat said in his comment @Dave's answer, each string will vibrate at multiple frequencies.
Physicist mode on:
In theory, if the string was made from a perfectly elastic material (like they assume in physics class), it would vibrate at infinitely many frequencies, but the strength of each frequency rapidly becomes very small, such that only a finite amount of energy is involved.
Physicist mode off.

Now comes the important bit: these additional frequencies are essential to the sound of musical instruments. They're basically the reason you hear a difference between a violin and a piano both playing an A4.

I would recommend that you listen to a pure sine wave at 440 Hz, e.g. here:

If you've ever had your hearing tested at a hospital or somewhere, that's what it sounds like. (Make sure to ignore the pseudoscientific babble "related" videos that appear on the right-hand side on youtube.)

Answer 7

This chart displays the frequency of all notes:

As a reference, 60 midi number = middle C on the piano.

A piano has seperate strings for each note. So the fact that the A is different wouldn't necessarily mean the others are not in tune with 440. Depending on what the tuner did.

Are you then asking: if I want to tune the entire piano to say 420 (A), how do the other frequencies change? In that case, you can use the formula, as given here:

fn = f0 * (a)n
where
f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz. But in this case it could be 420.
n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away.
a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...

The wavelength of the sound for the notes is found from
Wn = c/fn
where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature." 

For more details and understanding about music and frequencies I suggest you take a look at the excellent (open source) book of David J. Benson, available here.

Regarding:

maybe you can give me just general idea what the frequency range for each key can be?

I hope this is clear. Take a certain range for A (say 420-460) and you will be able to calculate all ranges from that. maybe you can give me just general idea what the frequency range for each key can be?