Harpsichord & Piano Tuning

Introduction

We present an automatic tuning tool for string instruments, e.g. harpsichords and pianos.

The current application (exclusively realized in Python) collects a mono signal from the computer's input audio stream (USB connector), splits it into smaller, overlapping slices, and applies the Fourier transform to each. This practice is known as short time Fourier transform, i.e. STFT. The slices overlap by multiples of 1024 samples. They have sizes of L = 2^N samples, where N=15 or 16, resulting in sampling periods of 0.74 or 1.49 s for each slice. The sampling frequency is f_s = 44,100 s^-1

Each slice is apodized utilizing Gaussian windowing, where L = 7σ, resulting in a FWHM (@-6dB) = 2.62 bins and a full width of the main lobe of 10.46 bins. This translates to 3.53 or 1.76 Hz and 14.1 or 7.1 Hz for N = 15 and 16, respectively. The highest side lobe is -71 dB. Worth mentioning, the Gaussian L = 7σ is similar to a Blackman-Harris-Nuttall window.

Subsequently, in the frequency domain, Butterworth high-pass filtering is applied to suppress noise at the bottom side, before fundamental and overtone frequencies are sought. In order to achieve the highest accuracy in their positions, we fit a parabola - by three-node interpolation - to the logarithm of the Gaussian in Fourier space, for the strongest NMAX peaks found. The parabola centroids are utilized to derive the frequency f₀ and the inharmonicity factor B.

For an ideal string the frequencies of higher partials are just multiples of the fundamental frequency

$$f_{n} = n f_{1}\tag{1}$$

where n is the nth partial. The ear hears the fundamental frequency most prominently, but the overall sound is colored by the presence of various overtones, i.e. frequencies greater than the fundamental frequency, each partial at a diverse intensity. However, a real string behaves closer to a stiff bar according to a forth-order differential equation

$$\ddot{y} \propto {-y}''''\tag{2}$$

with a quadratic dispersion. Hence, its partials can be approximated by

$$f_{n} = n f_{0} \sqrt{1 + B n^{2}}\tag{3}$$

All peak positions are correlated to each other, such that they can be identifed as higher partials to one common base frequency f₀. By rewriting (3), we get for two frequencies that can be applied to all permutations of peaks

$$B = {{C - 1} \over j^{2} - C * i^{2}}\tag{4}$$

where

$$C = ({{f_{j} i} \over f_{i} j})^{2}$$

$$f_{0} = {f_{i} \over i \sqrt{1 + B i^{2}}}\tag{5}$$

The measured frequencies of the partials are denoted f_i < f_j and 1 ≤ i < j ≤ NPARTIAL. The maximum inharmonicity coefficient needs to be adjusted in parameters.py, depending on the instrument to be tuned, B < 0.001 and < 0.05 for harpsichords and pianos, respectively.

Finally, a synthetic spectrum is calculated from f₀ and B and compared to the measured one by minimizating the L1-norm of the coefficient vector, the coefficients being |f_i^measured - f_i^calculated|.

The L2-norm was tested, but behaved inferior. I employ the module scipy.optimize.slsqp, originally designed for least-square minimizations. Although it requires the Jacobian of the L1 to be computed as well - imposing additional CPU-power - it seemed to be the most reliable and fastest minimizer, when compared to brute-force or 'L-BFGS-B'.

Features

The frequency of the first partial f₁ is compared to a value derived from the pitch level and a tuning table tuningTable.py, currently comprising Werkmeister III, 1/4 Comma Meantone, and Equal Temperament - feel free to edit/enhance it for yourself. We have not yet considered enharmonic equivalency in meantone, hence, one would have to enable/disable certain keys, such as A♭ vs. G#. The key in the center of the pie shows what key was pressed and its deviation, in units of cent, for the specified tuning and pitch level: too low (in red) and too high (in green).

The orange vertical bars indicate the peaks identified by the peak finding routine. The red vertical bars show the partials up to n_max ≤ NPARTIAL as derived from the computed base frequency and inharmonicity coefficient when applying (3).

The hotkey 'ctrl-y' or 'ctrl-x' stops the program or toggles between halt and resume, respectively. 'Ctrl-j' and 'ctrl-k' shorten and lengthen the shift between the audio slices, whereas 'ctrl-n' ('alt-n') and 'ctrl-m' ('alt-m') diminish and increase the max (min) frequency displayed. 'ctrl-r' resets parameter to initial values.

The background noise can be measured through an integration and averaging over multiple slices by toggling hot-key'3'. This is an optional feature. Caveat: background noise must be kept low during that time. The hot-keys '1' and '2' increase or decrease sensitivity levels, respectively, if noise threshold equals 'No'.

Run the program with: python3 -m Tuning

Caveat

1. When tuning you may consider preventing the display from blanking, locking and the monitor's DPMS (on UNIX) energy saver from kicking in. Caffeine is a solution https://launchpad.net/caffeine

2. On certain Linux distributions, a package named python-tk (or similar) needs to be installed, when running in virtual environments.

3. Also note that the module pynput utilized here may encounter plattform limitations

Results

Owing to the lack of reliable data to date, I refer to results of ref. 4. The authors achieve a relative error of a Gaussian interpolation of 0.052% per bin for the center frequency of the parabola. This converts to 0.07% for N = 15, resulting in an error of 0.29 Hz @ 415 Hz (1.2 cent). Given the fact, though, that in the current application the frequency of the first partial is derived from a combination of all partials measured, its frequency is anticipated to be more precise. This is achieved through comparparing the measuremed spectrum which that calculated from f₀ and B by minimizing the L1-norm, such as in Compressed Sensing. The vector coeficients are normalized to the frequency, because higher partials impose a greater weight on the L1-norm than lower partials. In a preliminary test the max. relative error turned out to be around 1.4 * 10^-4 for frequencies and 7 * 10^-3 for B @415Hz.

References

1. HARVEY FLETCHER, THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 36, NUMBER 1 JANUARY 1964
2. HAYE HINRICHSEN, REVISTA BRASILEIRA DE ENSINA FISICA, VOLUME 34, NUMBER 2, 2301 (2012)
3. JOONAS TUOVINEN, SIGNAL PROCESSING IN A SEMI-AUTOMATIC PIANO TUNING SYSTEM (MA OF SCIENCE), AALTO UNIVERSITY, SCHOOL OF ELECTRICAL ENGINEERING (2019)
4. M. GASIOR, J.L. GONZALEZ, IMPROVING FFT FREQUENCY MEASUREMENT RESOLUTION BY PARABOLIC AND GAUSSIAN INTERPOLATION, EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH/ ORGANISATION EUROPEENNE POUR LA RECHERCHE NUCLEAIRE, AB-Note-2004-021 BDI (2004)

Contact

Ralf Antonius Timmermann, email: rtimmermann@astro.uni-bonn.de, Argelander Institute for Astronomy (AIfA), University Bonn, Germany.