The modular discriminant $\Delta (z) \in M_{12,0},$ the linear space of cusp forms of weight $12.$ Now dimension $M_{12,0} = 1$ (see, for example, T.M. Apostol: Moldular Functions and Dirichlet Series in Number Theory)
Every modular form for the full modular group is a polynomial in $E_4$ and $E_6,$ where $E_{2k}$ are the Eisenstein series defined by
$$E_{2k}(z) = \frac{1}{2\zeta(2k)} \sum_{(m,n) \ne (0,0) } \frac{1}{(m+nz)^{2k}} \quad \textrm{for } Im(z) >0,$$
where the summation extends over all integral $m$ and $n$ not both equal to $0$. So if we use the Eisenstein series to construct cusp forms of weight $12,$ because $\textrm{dim } M_{12,0} = 1$ these will necessarily be constant multiples of $\Delta (z).$
Now we can show that
$$E_{2k}(z) = 1 - \frac{4k}{B_{2k}} \sum_{n=1}^\infty \sigma_{2k-1}(n)e^{2 \pi i n z}
\quad \textrm{for } Im(z) > 0,$$
where $\sigma_k(n) = \sum_{d|n} d^k$ and the $B_{2k}$ are the Bernoulli numbers.
Therefore we can expect identities between the divisors functions $\sigma_{2k-1}(n)$ and $\tau(n)$ to exist. Ramanujan's paper "On Certain Arithmetical Functions" shows how to express
$$\phi_{r,s}(x) = \sum_{n=1}^\infty n^r \phi_{s-r}(n)x^n$$
for integer $r,s \ge 0$ and $ |x| < 1$ as polynomials in $E_2,E_4,$ and $E_6,$ from which many such identities follow.
EDIT: To cite a random example:
$$\tau(n) = n^2\sigma_7(n) - 540\sum_{k=1}^{n-1}k(n-k)\sigma_3(k)\sigma_3(n-k)$$
and so $\tau(n) \equiv n^2\sigma_7(n) \textrm{ mod } 540.$
EDIT2: $B_{12} = -691/2730.$
EDIT3: For completeness, using Ramanujan's notation
$$P = 1 - 24 \sum_{n=1}^\infty \frac{nx^n}{1-x^n}$$
$$Q = 1 + 240 \sum_{n=1}^\infty \frac{n^3x^n}{1-x^n}$$
$$R = 1 - 504 \sum_{n=1}^\infty \frac{n^5x^n}{1-x^n},$$
the relevant entries from his paper "On Certain Arithmetical Functions" are $(3)$ and $(6)$ from his Table I, given below. (Note that $Q$ is $E_4$ and $R$ is $E_6.$)
$$\begin{align}
1 - 24\phi_{0,1}(x) &= P \\
1 + 240\phi_{0,3}(x) &= Q \\
1 - 504\phi_{0,5}(x) &= R \quad (3) \\
1 + 480\phi_{0,7}(x) &= Q^2 \\
1 - 264\phi_{0,9}(x) &= QR \\
691 + 65520\phi_{0,11}(x) &= 441Q^3+250R^2 \quad (6) \\
1 - 24\phi_{0,13}(x) &= Q^2R \\
3617 + 16320\phi_{0,15}(x) &= 1617Q^4+2000QR^2 \\
43867 - 28728\phi_{0,17}(x) &= 38367Q^3R+5500R^3 \\
174611 + 13200\phi_{0,19}(x) &= 53361Q^5+121250Q^2R^2 \\
77683 - 552\phi_{0,21}(x) &= 57183Q^4R+20500QR^3 \\
236364091 + 131040\phi_{0,23}(x) &= 49679091Q^6+176400000Q^3R^2 + 10285000R^4 \\
657931 - 24\phi_{0,25}(x) &= 392931Q^5R+265000Q^2R^3 \\
3392780147 + 6960\phi_{0,27}(x) &= 489693897Q^7+2507636250Q^4R^2 + 395450000QR^4 \\
1723168255201 - 171864\phi_{0,29}(x) &= 815806500201Q^6R+88134070500Q^3R^3 + 26021050000R^5 \\
7709321041217 + 32640\phi_{0,31}(x) &= 764412173217Q^8+5323905468000Q^5R^2 \\
&+ 1621003400000Q^2R^4
\end{align} $$
Equations $(3)$ and $(6)$, along with $1728 \sum_{n=1}^\infty \tau(n)x^n = Q^3 - R^2,$ give
$$691+65520 \sum_{n=1}^\infty \sigma_{11}(n) x^n = 441 \times 1728 \sum_{n=1}^\infty \tau(n) x^n + 691R^2.$$
The result follows by taking this equation modulo $691$ and noting that $566$ and $691$ are coprime.