= Gear base pitch
{title2=$p_b=\pi m\cos\alpha$}
<Gear base pitch> is the spacing of successive <involute> tooth profiles referred to the <gear base circle>. For base diameter $d_b$, tooth count $N$, <gear module> $m$ and reference <gear pressure angle> $\alpha$,
$$
p_b=\frac{\pi d_b}{N}=\pi m\cos\alpha.
$$
It also gives the spacing between successive corresponding tooth contacts measured along the <gear line of action>. The <gear circular pitch> uses the reference circle instead, giving $p=\pi m$ and $p_b=p\cos\alpha$.<KHK Gear Nomenclature>[Technical reference, “Meshing of Involute Gear”, base-pitch relation; Basic Guide B §3, base-circle geometry]{ref}
For $m=2\,\mathrm{mm}$ and $\alpha=20\,{}^{\circ}$, $p=6.2832\,\mathrm{mm}$ but $p_b\approx5.9043\,\mathrm{mm}$. A second gear with the same module but $25\,{}^{\circ}$ reference angle has $p_b\approx5.6945\,\mathrm{mm}$. Matching reference tooth spacing alone therefore does not make these the same specified mating geometry.
Changing the mounting separation of a compatible <involute> pair does not change its cut profiles, base circles or <gear base pitch>. In operating-circle notation, $p_w=2\pi r_w/N$ changes with mounted radius, while
$$
p_w\cos\alpha_w=\frac{2\pi r_w\cos\alpha_w}{N}=p_b
$$
stays fixed. This links the mounted <operating pressure angle> to an invariant tooth-profile spacing.
For a tooth rotating at <angular speed> $\omega$, base-circle tangential speed is $r_b|\omega|$. Dividing by <gear base pitch> gives the <gear mesh frequency>,
$$
f_m=\frac{r_b|\omega|}{p_b}=\frac{N|\omega|}{2\pi}.
$$
A 30-tooth gear at $600$ revolutions per minute gives $f_m=300\,\mathrm{Hz}$. Finally, the <gear contact ratio> divides usable <gear path of contact> length by this base pitch, so profile spacing and available contact length jointly determine overlap between successive tooth engagements.
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