Method for calculating musical note-based magnitude spectrum My specific question is: is there a standard way to do the kind of analysis I describe below? If not, is there a useful method that I have missed?

I'm exploring ways to produce a spectral histogram of audio signal magnitude based on musical notes. I wish to calculate a spectrum spanning 10 octaves, or 120 notes. This covers a frequency range from ~20Hz to ~21,000Hz. The sample rate will be at least 44,100Hz. 

The final desired result is a numerical value per note, that I can then plot on an axis of note bins. This is essentially a logarithmic x axis, with non-linear note bins. I wish to do this in realtime, on a desktop computer, and display visually at ~10Hz update rate or better.

I initially looked at the DFT (via FFT), however this has some drawbacks - the size of the DFT needs to be large enough to provide sufficient resolution at the lowest frequencies. E.g. to get ~1Hz resolution at 44100 Hz I will need to use an FFT of size 65536, which is longer than 1/10th of a second of signal (for 10Hz update). Secondly, the frequency bins for the DFT are linearly spaced in frequency, so there needs to be a warped mapping from these linear frequency bins to the non-linear note bins. In some cases many frequency bins will map to a single note bin. I understand that the shape of the frequency bins is related to the window function, so I imagine I would use something like sinc interpolation to collect frequency bins into corresponding note bins. 

I've also come across some other options:

Someone claimed to have found a way to tweak the DFT to use note frequencies as the basis functions, however I wasn't able to understand how they did this and there was no math provided to back it up. Is this possible? Can the set of basis functions for the DFT be changed to match note frequencies, and if so what are the implications for scaling, phase (not so important in my application). I'm a layman but my feeling is that if the basis functions are not orthogonal then there's going to be overlap which could result in some terms counting energy more than once.

Another suggestion involves applying the Goertzel algorithm 120 times. However based on the rule-of-thumb on Wikipedia, this would be significantly more expensive than using the FFT above.

Another idea involved applying a much smaller FFT to a single octave, with filtering and downsampling to shift each octave down. This would result in 12 smaller FFTs and 11 frequency-shift/filter operations. This may be more efficient than the large FFT above.

One final point - the system can be fine-tuned in realtime, so "A440" may not actually be exactly 440Hz and may shift slightly by up to half a semitone in either direction. Therefore I need a method that can be "tuned" accordingly, or is precise enough to correctly capture energy within +/- half-semitone of a note.
Have you looked into a Chroma/pitch-class analysis? That sounds more like what you want overall (expect perhaps for the octave-folding/collapsing). 

Check out librosa's CQT-based chromogram. Code to generate plot below on github:


You can use the tuning parameter to tune the musical scale, but study the full paramter set.

tuning : float
    Deviation (in cents) from A440 tuning


Finally, checkout these examples showing "enhanced chroma", to say isolate harmonic content:


https://librosa.github.io/librosa_gallery/auto_examples/plot_chroma.html#sphx-glr-auto-examples-plot-chroma-py




Regarding performance of the CQT computation, the FFT + a kernel can be used to make the computation much faster. See Judith Brown &amp; Miller "Super" Puckette's An efficient algorithm for the calculation of a constant Q transform.