Piano pitch detection

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Is there an algorithm for finding a frequency without DFT or FFT?

FFT isn't a particular efficient way of building a tuner. Better (and cheaper) methods include auto-correlation, phased locked loops and delay locked loops, etc..

One example is to use tracking of local maxima and minima to roughly hone in on the fundamental frequency and then use a local oscillator and phase locked loop to track this frequency precisely. This can track a moving fundamental during tuning quickly, continuously, and with great accuracy even if the frequency is low and if the fundamental is weak.

A search for 'piano tuning software' or similar items will yield a large number of hits – some good, some not so good.

Every type of musical instrument has unique acoustic/physical/environmental characteristics that affect its sound. And it can get complicated, as thousands of books and research papers would suggest (eg: tonality, attack/decay characteristics, inharmonicity, etc.).

Pitch detection is itself a wide-ranging field. The following is but a tiny fraction of what's available: overview article 1 and stack exchange post and overview article 2

As for your specific questions: 1) your sample size seems like overkill – depending on SNR and waveform stability, you can get high frequency accuracy using other methods using fewer cycles. (some methods are FFT-based). And you may be capturing attack/decay with a long sample time, 2) any window other than rectangular will broaden the beam-width in the frequency domain, but that doesn't mean you shouldn't use one – Hann seems common with HPS, from what I've seen, 3) as noted in the first link above, HPS doesn't work very well at low frequency, and inharmonicity will affect you on the lower strings. As for your overall method, without having to write a lot of pages, I can only say that I would do it differently, depending on the frequency range and harmonics I was dealing with.

Another answer suggests PLL. I think you should stay away from PLL: most literature on pitch tracking focuses on auto-correlation (search for "YIN Pitch Tracking" -- YIN is a modern pitch tracking algorithm based on auto-correlation) and FFT. I believe PLL is more suited to tracking tiny fluctuations in frequency, like with radio.

Auto-correlation is a good place to start. It is fast, efficient and accurate. However, there are tricks for making FFT very accurate and fast (most techniques that use the FFT just look at the magnitude, but you can also use the phase information), so if you are familiar with FFTs you can use that technique as well.

If you do use either of these techniques, I suggest prefiltering with a lowpass to reduce harmonics and focus on the fundamental. With the FFT, you can, instead, or in addition, use tricks like looking at the first local maximum.

This might be a good starting point for filtering and so on. It will also give you some tips about avoiding doing too much work, and it links to source code: http://blog.bjornroche.com/2012/07/frequency-detection-using-fft-aka-pitch.html

This book has sections that explain both YIN and FFT using phase information: http://www.amazon.com/DAFX-Digital-Udo-ouml-lzer/dp/0470665998

Finally, you'll have to understand the specifics of pianos. I'm not sure if the tuner itself needs to do anything special WRT, eg, stretched tuning, or if that's left up to the person tuning the piano, but you'll need to at least understand that stuff. Another poster suggested looking at out of tune harmonics, but the main issue is to identify and tune the fundamental, so the harmonics being out of tune should not matter as long as you properly identify the fundamental.

The wide peaks you see may be the result of physical phenomena, not a signal processing artifact. In general, narrow peaks in an FFT result represents an unmodulated sinusoid exactly periodic in the window in the time domain. But piano string vibrations are not that stationary. They evolve over time, creating a noticeable modulation.

Several effects: multiple piano strings per note will exchange energy thru the soundboard; the total vibration energy will decay over time; the vibration modes may be slightly inharmonic to start with; the exact frequency of vibration of each mode (harmonic) may change with decay in amplitude due to non-zero string stiffness and diameter; and each harmonic may decay at a different rate, etc.

You may have to decide which of these multiple modulations you want to call "the pitch" (books on audiology may help), and find a method to better track it inside the FFT's "wide peak".

With the lower notes of pianos, especially for uprights, the spectrum tends to be stretched out (the distance between the fundamental and first overtone is a little more than an octave, etc). This is what gives pianos their percussive sound, the lowest note on older uprights will often sound more like a thud than a note, and as I understand it this is why cheep uprights have their particular percussive honky tonk sound. Because of this good piano tuners (the people not algorithms) will tune pianos by the lower overtones than the fundamental for lower notes, the human ear tends to focus on the interaction of the lower overtones for these notes. The stretching of the harmonic series could also be the cause of the wider than expected peaks in the spectrum.