The Targum Psalms provides some clues. The Targums were amplified translations of the Hebrew Scriptures into Aramaic. That is, they provide some nuance as to how the Hebrew Scriptures were understood by Jewish scholars who translated the texts from Biblical Hebrew into Aramaic. The Targum Psalms is Late Jewish Literary Aramaic, and therefore appeared as early as the Fifth Century (Flesher & Chilton, 2011).
The passage at hand appears as follows in the Targum Psalms. Please click on the image to enlarge.
The nuance here is that the tent cord is a LINE of utterances. In other words, the idea is that the very heavens declare the glory of the Lord and his existence through the "lines" of the utterances of creation, which speak through silent words. These "lines" are statements (as the "lines" of these statements represent my inaudible "voice"). In other words, these lines of statements, or utterances, are the voice emanating from heaven, which communicate to man.
For example, in the Masoretic Text of the Psalm the Hebrew word here is קָו (H6957), which means cord or measuring line. This same Hebrew word appears in the following verses, where the word is translated as "line" of words, which are not understood.
Isaiah 28:10-13 (NASB)
10 “For He says,
‘Order on order, order on order,
Line on line, line on line,
A little here, a little there.’”
11 Indeed, He will speak to this people
Through stammering lips and a foreign tongue,
12 He who said to them, “Here is rest, give rest to the weary,”
And, “Here is repose,” but they would not listen.
13 So the word of the Lord to them will be,
“Order on order, order on order,
Line on line, line on line,
A little here, a little there,”
That they may go and stumble backward, be broken, snared and taken captive.
The context is the inability to understand what is communicated, which are "lines" and concepts. That is, just like Psalm 19, there are the "lines" of declarations of the glory of the Lord throughout creation, which many do not find intelligible (e.g., Psalm 14:1 speaks to one who denies the existence of God).
In conclusion, the Targum Psalms provides one provocative explanation as to why the LXX uses "voice" in its translation: that is, the Hebrew word קָו (H6957) represents not only a cord, but an actual "line" (of utterances). Such imagery appears resonant in Isaiah 28:10-13, where the same Hebrew word refers to "lines" of unintelligible statements to those who reject the rest and repose of the Lord. That is, in Psalm 19, the same rest and repose of the Lord are evident through the creation (i.e., "the tent of the sun" in Ps 19:4-5, which the Targum Psalm calls "the resting place").
In summary, the creation is an erected tent, whose "lines" are the inaudible utterances of glory from heaven. The LXX here uses the word "voice," which declares this glory. For those who reject the rest of the Lord found in the tent of creation (where even the sun finds repose), such "lines" -- or voice -- become unintelligible and/or meaningless.
Flesher, P. V. M., & Chilton, B. (2011). The Targums: A Critical Introduction. Waco: Baylor University Press, 235.