I'm working on some computer code to detect musical notes and have just found out about in-harmonics (up until now I was under the impression that the overtones were always integer multiples of the fundamental frequency).
So now I am wondering if the inharmonic frequencies in a piano (and other instruments) are actually shifted/replacing the theoretical harmonic frequencies in the spectrum or are they in addition to the harmonic frequencies?
A solid "it depends". The overtones are all sinoid signals, and sinoids stay sinoids given linear shift-invariant systems. Which most elements of sound transmission are. So if there is no significantly non-linear element after the sound-generating disharmonic element, the "unnatural" overtones are all there are.
With a piano, this will be mostly the case. If there is any snaring part in there, the overtones of the snare will be harmonic overtones clashing with the disharmonic ones.
As an extreme example, take a free reed instrument like accordion, harmonium, bandonion. The metal reed generating the sound is vibrating by bending, and there are several higher modes of bending and torsion that the reed can also exhibit apart from the fundamental. Those higher bending modes have frequencies wildly different from harmonics but due to the overall shape of the reed and the (destructive) manners of tuning it, sometimes they might come close to a harmonic in which case the reed tends to sound unclean and become more liable to breaking.
Now here the trick is that the mechanical oscillator has disharmonic modes of oscillation, but the actual sound is not generated by listening to the vibrations it may transfer to the instrument (which is rather less than more desired) but because the vibrating reed "punches holes into the air stream" which is quite a nonlinear operation and generates rich overtones. Those air stream overtones, in contrast to the vibration overtones, are a function of the "form of the holes" punched into the air stream and thus are strictly harmonical as they are part of a periodical signal with the period of the fundamental.
Another example is an electric guitar: both the magnetic pickup process as well as any additional distortions added by effects and amplifier are nonlinear and will tend to generate harmonic overtones while the vibration of the thicker strings might be responsible for disharmonic overtones. An acoustical guitar in good shape, in contrast, will almost only deliver the (slightly disharmonic) overtones from the strings without adding competing harmonic ones of its own.
Actually both, but the "shifted" inharmonic overtones usually dominate and thus mask the harmonic ones, making them hard to detect with an FFT.
Due to stiffness and non-zero diameter, the resonance modes of the big strings on a big instrument will cause the higher vibration modes (overtones) to be measurably sharp. But there are usually other tiny non-linear responses (due to stiffness and material inhomogeneity, etc., in other parts of the instrument) to the fundamental vibration mode that can create some pure harmonic overtones as well.
So I found an answer to my question on this page - at least for the case of a piano the overtones are actually shifted (so there is no peek in the spectrum at the theoretical frequencies), specifically the measured frequencies of the partials become higher than the naively calculated frequency as you travel up the keys.
The page linked provides a table and curve showing how the A4 note partials on a Steinway B piano deviate (up to 40 cents by the 8th partial). Also included is a formula for calculating the deviation but it relies on a coefficient that needs to be measured for each piano.
Another instrument that uses an anharmonic series is the tubular bells. The theoretical frequencies of oscillation of a tubular bell struck at one end are 1:9:25:49... (all squares of odd numbers), rather than the standard 1:2:3:4... of the regular harmonic series. However, the frequencies at 81f0, 121f0, and 169f0 are close enough to being in a 2:3:4 ratio that the brain interprets the sound as having a pitch around 40f0.
This "missing fundamental" occasionally occurs with other instruments as well, most notably the pedal tones of brass instruments. If you do the signal analysis on a trombone or trumpet playing a "pedal" note (A1 or A2, respectively), there will not be a component of the waveform at the the "fundamental" frequency (55 Hz or 110 Hz, respectively.) Your ear does the work of filling in the fundamental frequency and interpreting it as a low note.
What an awesome conversation. I don't have any technical references for the main question. It seems to me as an instrument player that there is no "shift" of the naturally existing harmonics, rather instruments try to latch on to the existing frequencies occurring in nature and amplify them through the instrument. The instrument vibrates at certain nodes producing a sound. A brass instrument such as a trumpet gets its sound because along its perfectly measured metallic tube it is vibrating according to a frequency at those nodes.
The sinoid signals do in fact work as was mentioned by user23472, I would like to add that especially for lower frequencies that are "missing" the fundamental that there are other waves that can sound those pitches. Take a look at this link http://onlinetonegenerator.com/ and enter frequencies of pedal tones http://www.phy.mtu.edu/~suits/notefreqs.html (this link has a chart to frequencies based on Western musical notation), you will see that you can't really hear the sine waves, but try the square or sawtooth waves and you will hear better results. Even better, try mapping out these lower frequencies in multiple windows and you can REALLY hear how the pedal frequencies work with high frequencies in the same chord, you can also test it on intervals like octaves where you can really hear the overtones using a sawtooth wave. You'll usually hear these lower notes at an orchestra concert or low instrument ensemble recitals. In fact, those especially awesome moments in classical music usually involve the use of these lower frequencies which you can't hear using a pure sine wave .
Here's also a link to a Cymatics video which shows on a vibrating plate that things modify themselves along certain frequencies.
