# What is a "pitch period"?

The term pitch period appears in the book Speech and Language Processing by Daniel Jurafsy:

As we just said, a pitch-synchronous algorithm is one in which we do something at each pitch period or epoch. For such algorithms it is important to have very accurate pitch markings: measurements of exactly where each pitch pulse or epoch occurs. An epoch can be deﬁned by the instant of maximum glottal pressure, or alternatively by the instant of glottal closure.

I have some difficulties to understand what a pitch period means here. Could someone please explain more about the concept of pitch period to me? Visualize an example of a pitch period in a graph like waveform or spectragram would be more helpful.

So based in @rbj answer, can i try to show you visually ...

Take a look in this pure tone signal, one sine at 100hz and sample rate at 44100hz. just by looking you can find where the pattern starts to repeat itself, I marked it with the naked eye as being in position 441(Period)

So we take exactly 100hz --> 44100/441=100hz

Another example: Another 100hz signal, it was sampled at 48000hz, one more time just looking the wavform signal i try mark where this start to be periodic, for my eyes i marked it at position 447(Period)

So it give me 48000/477=±100hz to be exact = 100.6289hz

• thanks！Is the term pitch period just the same as the term period ? Jul 13, 2021 at 14:50
• pitch and period can be equivalent terms, if you have the period you will be able to turn it into pitch and vice versa Jul 13, 2021 at 16:56

The pitch period of a perfectly periodic function, $$x(t)$$, is the smallest positive value $$P>0$$ such that

$$x(t+P) = x(t) \qquad \forall t \in \mathbb{R}$$

Now, simply because a function is periodic with period $$P$$, then it is also periodic with periods $$2P$$ or $$3P$$ or $$4P$$ or any integer multiple of $$P$$, but we don't pick $$2P$$ or $$3P$$ or $$4P$$ for the period of $$x(t)$$, we pick the smallest possible $$P$$ that is a positive value.

The reciprocal of that period $$P$$ is the fundamental frequency

$$f_0 \triangleq \frac{1}{P}$$

From that, it is possible to represent the period signal $$x(t)$$ as a Fourier Series

\begin{align} x(t) &= \sum\limits_{k=-\infty}^{\infty} c_k \ e^{j 2 \pi k f_0 t} \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} |c_k| \cos(2 \pi k f_0 t + \phi_k) \qquad \phi_k = \arg\{c_k\} \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} \big(\underbrace{|c_k|\cos(\phi_k)}_{a_k}\big) \cos(2 \pi k f_0 t) - \big(\underbrace{|c_k|\sin(\phi_k)}_{b_k}\big) \sin(2 \pi k f_0 t) \\ \\ &= c_0 + \sum\limits_{k=1}^{\infty} a_k \cos(2 \pi k f_0 t) - b_k \sin(2 \pi k f_0 t) \\ \end{align}

where $$c_k = a_k + j\,b_k = \int\limits_{t_0}^{t_0+P} x(t) e^{-j 2 \pi k f_0 t} \ \mathrm{d}t \qquad t_0 \in \mathbb{R}$$

Now most audio that sounds like a note or a tone, including voiced speech, is not exactly periodic, but is "quasi-periodic". So our defining condition of a quasiperiodic $$x(t)$$ is

$$x\big(t+P(t)\big) \approx x(t) \qquad \forall t \in \mathbb{R}$$

Now, instead of constant $$P$$, the period is a virtually constant (or slowly varying) function of time $$P(t)$$. The fundamental frequency then would also be a virtually constant function of time

$$f_0(t) \triangleq \frac{1}{P(t)}$$

Now the pitch of that tone or note is the logarithm of the fundamental frequency $$f_0$$. If that pitch is expressed in units of octaves, it's the base-2 logarithm. If the pitch is expressed in units of semitones, it's 12 times that base-2 logarithm. The pitch has to be relative to a standard pitch. For MIDI, the pitch is

$$p(t) = 12 \log_2\left( \frac{f_0(t)}{440 \, \mathrm{Hz}} \right) + 69$$

and is in units of semitones with Middle C having a pitch of $$60$$.

• thanks！Is the term pitch period just the same as the term period Jul 13, 2021 at 14:49
• As far as i can tell. It's also might not be a constant value like you would see in a perfectly periodic tone. "period" might imply a constant value. "pitch period" might imply a slowly varying value just as pitch is slowly varying. Jul 13, 2021 at 16:54