I have an old digital synthesizer and I would like to understand how its pitch control was implemented. A perfect representation of a given pitch in an equal temperament scale with the concert pitch of $440$ is given by $$f(x)=2^{x/1200}\cdot440,$$
where different values of $x$ output different pitches and in this case we consider the values $x=300-1200, 310-1200, 320-1200,...,1500-1200$. In musical terms the output is an octave of pitches starting from middle C, in ten cent intervals.
The synthesizer uses $24$ bit arithmetic for representing pitches, and a sampling rate of $88200$ samples per second, this means representing only pitch values of the form $y\cdot \frac{88200}{2^{24}}$ is possible, for integer $y$. After recording, we obtain the following values $y$, and would like to find a function that goes through all of the points.
49767, 50052, 50344, 50635, 50927, 51225, 51519, 51820, 52118, 52421, 52725, 53029, 53336, 53645, 53955, 54268, 54584, 54900, 55219, 55538, 55860, 56182, 56510, 56835, 57166, 57497, 57819,
58156, 58490, 58831, 59171, 59514, 59860, 60207, 60556, 60905, 61258, 61613, 61971, 62330, 62691, 63053, 63417, 63785, 64155, 64526, 64900, 65276, 65656, 66036, 66418, 66804, 67181, 67570,
67961, 68356, 68751, 69149, 69550, 69954, 70358, 70765, 71175, 71588, 72004, 72420, 72840, 73262, 73687, 74112, 74541, 74975, 75409, 75847, 76284, 76728, 77171, 77627, 78076, 78532, 78988,
79443, 79905, 80367, 80831, 81299, 81770, 82244, 82721, 83201, 83684, 84170, 84656, 85145, 85637, 86135, 86633, 87137, 87639, 88149, 88659, 89172, 89689, 90220, 90743, 91268, 91797, 92328,
92863, 93401, 93941, 94485, 95035, 95585, 96138, 96693, 97255, 97820, 98385, 98956
Contrasting the values of $y$ with the perfect estimates $y_{est}=\text{round}(f(x) \cdot \frac{2^{24}}{88200})$ gives the following plot for $y_{est}-y$
Trying out the "obvious" polynomial approximations such as min-max polynomial for estimating $2^x$ and deriving a suitable approximation does not appear to work. As can be seen the above error function is very jagged and that is essentially why. I would like to ask for possible ideas on what methods could have been used to compute the output pitch values on the synthesizer?