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I found two papers from which two different formulae for the tetrasilicate (tetrasilicic acid) are found:
1.https://www.sciencedirect.com/science/article/pii/S0022309302015491.
2.https://www.ncbi.nlm.nih.gov/pubmed/26482329.
Of course, the usual formula for polymerization of oxacids suggest the first one is the right one, that is, the tetrasilicate anion being $\ce,{Si_4O_9^{2-}}$, however, how to understand the second formula? Is something missing?
"Tetrasilicate" is one of special guys. For example, sulfate is $\ce{SO_4^2-}$, while $\ce{SO_3^2-}$ is sulfite. But tetrasilicate is $\ce{Si_4O_n^{(2n-16)-}}$ - thus, it simply has four $\ce{Si}$ atoms (without connotation that they have special number of oxygens near 'em): check for
http://www.biomedsearch.com/nih/Novel-silicate-anion-Si8O2212-Hydrothermal/11487344.html, which is also tetrasilicate (here tetrasilicate is $\ce{Si_4O_11^6-}$).
https://www.researchgate.net/publication/50934696_Preparation_structural_and_luminescent_properties_of_Ba2Gd2Si4O13Eu3_for_white_LEDs (tetrasilicate is $\ce{Si_4O_13^10-}$), etc.
[edited] I've found for you even tetrasilicate with 15 oxygens: https://link.springer.com/article/10.1007/s00710-016-0447-1
Also check for amicite (zeolite): it has approximation $\ce{K2Na2Al4Si4O16·5(H2O)}$. It seems for me like 16 oxygens is maximum for tetrasilicates, but maybe someone would disagree - interesting to know that other tetrasilicates exist.
