Sinusoid extraction based on beating pattern

Oct 2017
I would like to ask if someone might have a tip how to approach a problem of sinusoidal signal separation in a musical recording sample. There is a recording at 44100Hz of 1,3sec (56855 samples) length and we have the knowledge that up to approx. smp 17100 there is just one sinusoid (1-st piano harmonic) – let us call it SIN-A. Starting from smp 17100 up to ca. 49600 there is another sinuosoid SIN-B overlaid with half a tone difference that causes amplitude beating. There is also another component apparently mixed up between 23000 and 30000 but that we can leave out for now. I am most interested in ideas how to best separate the two main components that cause cisible amplitude beating between 17100 and 49600. The first graph from the top – let’s call it Time1 - shows the input time points with all the sinusoids combined. What I am also given is the frequency and phase points of the first sinusoid SIN-A throughout the 17100 – 49600 region. This is output from a DSP filterbank sinusoid tracking software that follows SIN-A using frequency and phase continuity information. However this DSP software cannot cope with frequency beating caused by SIN-A + SIN-B. To separate SIN-A and SIN-B a gave it a first simple try that is I linearly approximated SIN-A amplitude linking the region boundary points. The result is shown on the second graph from above – Time2. Then I did a subtraction Time1 – Time2 hoping that magically I get the SIN-B without beating. But it did not happen – the result is shown on the third graph from the top – it has very irregular amplitude envelope which means that my approach failed knowing that SIN-B is another piano key stroke so it should have a more regular Attack-Decay-Sustain-Release and generally monotonic amplitude envelope. It seems that maybe independent modelling of SIN-B based on the beating pattern may bring better results ? How would you approach such a modelling – that is what would be the equation for the amplitude, frequency and starting phase of SIN-B if I know the combined time points and SIN-A frequency and phase values but not the amplitude in the 17100 – 49600 region ?