How does the relationship between carbonate, pH, and dissolved carbon dioxide work in water?
$^{\circ}KH$ or $dKH$ is called (degree of )carbonate hardness, but it is usually about bicarbonates $\ce{HCO3-}$, not carbonates $\ce{CO3^2-}$ . Carbonates are present in water just in traces, unless it is e.g. highly alkaline water from soda lakes of East Africa rift (like where cichlidae from lake Malawi live).
There is the equilibrium for carbon dioxide solution acidity, keeping the expression below constant:
$$K_\mathrm{a1,H2CO3^{*}} = \  \frac{\ce{[H+][HCO3-]}}{[\ce{CO2(aq,total}]}=10^{-\mathrm{pH}}\cdot \frac{\ce{[HCO3-]}}{[\ce{CO2(aq,total}]}=const \cdot 10^{(7-\mathrm{pH})}\cdot \frac{KH}{\mathrm{ppm}  \ce{CO2}},$$
where [] denotes the molar concentration.
Therefore $$\mathrm{ppm}  \ce{CO2} = \frac{\mathrm{const}}{K_\mathrm{a1,H2CO3^{*}}} \cdot 10^{(7-\mathrm{pH})}\cdot KH \approx 3 \cdot 10^{(7-\mathrm{pH})}\cdot KH$$

Pure water in equilibrium with air has theoretical $\mathrm{pH=5.6}$, given by dissolved $\ce{CO2}$ from air. There is established equilibrium
$\ce{CO2(g) &lt;=&gt;[H2O] CO2(aq) &lt;=&gt;[H2O] H2CO3(aq) &lt;=&gt; H+(aq) + HCO3-(aq)}$, but all components are present just in traces.

Addressing feedback:
Carbonate hardness implies there are dissolved bicarbonates $\ce{Ca(HCO3)2}$ resp. $\ce{Mg(HCO3)2}$. These are determined indirectly by titration, that determines bicarbonate content.
This of course fails, if there is sodium bicarbonate or carbonate presence, or if all bicarbonate comes from $\ce{CO2}$. Both cases have carbonate hardness lower or zero, compared to what would say its measurement.
In fact, measuring carbonate hardness ( dKH ) is measuring alkality:
$$\ce{HCO3-(aq) + H3O+(aq) -&gt; 2 H2O + CO2(aq,g)},$$
assuming it is equivalent to content of $\ce{Ca(HCO3)2}$ or $\ce{Mg(HCO3)2}$. But it can be aslo sodium carbonate/bicarbonate, if added to water to increase alkality.

pH of water is determined by this equation:
$$\mathrm{pH} = 6.35 + \log {\frac{[\ce{HCO3-}]}{[\ce{CO2}]}}$$
both in molar concentration.
Calcium carbonate has solubility about 15 mg/L, what makes roughly 1 dKH.
$$\ce{CaCO3(s) &lt;=&gt; Ca^2+(aq) + CO3^2-}$$
$$\ce{CO3^2-(aq) + H2O &lt;=&gt;HCO3-(aq) + OH-(aq)}$$
This solubility increases in presence of dissolved carbon dioxide,which decreases concentration of carbonate:
$$\ce{CO3^2-(aq) + CO2(aq)  + H2O &lt;=&gt; 2 HCO3-(aq)}$$
There is direct relation between pH and relative ratio carbon dioxide : bicarbonate : carbonate.
pH ( in usual aquarium range ) is directly determined carbon dioxide / bicarbonate ratio. If pH change is forced  by other chemicals, the new ratio is established.
BTW, there are just 2 degrees of freedom. You choose 2 of 3 parameters. The third one is determined by the former 2.