# How can I measure the harmonicity of an audio spectrum?

I would like to estimate the "harmonicity" (harmonics-to-noise ratio ?) of an audio signal from its spectrum. What kind of algorithm can I use?

• What's the mathematical formula that you use to define harmonicity ? It must have an explicit mathematical definition, otherwise you can neither estimate nor compute it. Please add that formula to the question. – Fat32 Jul 10 '17 at 11:41
• perhaps to add to Fat's question: what does it mean to have a totally harmonic signal? or a totally inharmonic signal? if "harmonic" means periodic and "inharmonic" means lacking any component of periodicity (like white noise), then i would measure hamonicity to be the maximum value of the autocorrelation (other than the peak at a lag of 0) normalized by the maximum value that is at a lag of 0. – robert bristow-johnson Jul 10 '17 at 20:24

Please take a look at this previous answer. Using the semantics and mathematical definitions from that answer...

Let $x[n]$ be the input signal from which you want to measure harmonicity or periodicity. Assume $x[n]$ has no DC component (like it is the output of a DC blocking filter). And $n_0$ is the sample index of the neighborhood of signal around where you want the harmonicity to be measured. $N$ is a large integer defining how wide your rectangular window is.

Average Squared Difference Function, ASDF:

$$Q_x[k, n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} \left(x[n+n_0-\left\lfloor \tfrac{N+k}{2}\right\rfloor] \ - \ x[n+n_0-\left\lfloor \tfrac{N+k}{2}\right\rfloor + k] \right)^2$$

$\left\lfloor \cdot \right\rfloor$ is the floor() function and, if $k$ is even then $\left\lfloor \frac{k}{2}\right\rfloor = \left\lfloor \frac{k+1}{2}\right\rfloor = \frac{k}{2}$.

From that define an alternative form of Autocorrelation:

$$R_x[k,n_0] = R_x[0,n_0] - \frac12 Q_x[k, n_0]$$

where

$$R_x[0, n_0] \triangleq \frac{1}{N} \sum\limits_{n=0}^{N-1} (x[n+n_0-\left\lfloor \tfrac{N}{2}\right\rfloor])^2$$

Since $Q_x[0, n_0] = 0$ and $Q_x[k, n_0] \ge 0$ for all lags $k$, that means that $R_x[k, n_0] \le R_x[0, n_0]$ for all lags $k$.

You will find a period $P$ such that $P > 0$ and $R_x[P, n_0] > R_x[k, n_0]$ for all lags $k>k_0$, other than the highest peak at $k=0$.

$k_0$ is the smallest value of $k>0$ such that $R_x[k, n_0]<0$.

Once that peak location $P$ is found, the harmonicity is

$$r_x(n_0) \triangleq \frac{R_x[P, n_0]}{R_x[0, n_0]}$$

If $r_x(n_0) = 1$, then $x[n]$ is perfectly periodic (or "harmonic") in the vicinity of sample index $n_0$. If $r_x(n_0) = 0$, it's noise, no periodicity to be found.

• Thanks for your answer. However, as I asked, I would like to compute the harmonicity of the spectrum and not of the signal. – sandoval31 Oct 19 '17 at 14:14
• Then inverse Fourier transform the spectrum into a signal. Or consider what implication these operations in the time domain have in the frequency domain. This will be about comb filtering. – robert bristow-johnson Jun 13 at 23:32

To answer the broad question "what algorithms you could use": There are different ways of computing an harmonics-to-noise ratio, in the time or in frequency, from basic interval cutting in the frequency domain to more involved cepstral/liftering techniques with local peak detection, using a limited number of partials, etc. Advantages and drawbacks of different methods are described for instance in Temporal and spectral estimations of harmonics-to-noise ratio in human voice signals, Qi and Hillman, 1997

Other potential sources :

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