Greetings in the Name of our Lord.
The basic answer to this begins with an appeal to the very important distinction between the elements and the circumstances of worship. The elements are the parts of worship: the part of the people to God (i.e. praying/singing, offerings, confession of faith), and the part from God to the people (i.e. call to worship, reading and preaching of the Word, the sacraments).
The circumstances of worship are those things our Confession defines as “… concerning the worship of God, and government of the church, common to human actions and societies, which are to be ordered by the light of nature, and Christian prudence, according to the general rules of the Word, which are always to be observed” (WCF 1.6). This would include such things as the time of worship, the number of hymns to be sung, how the offering is received, etc.
Given this, I would suggest that musical accompaniment through instruments belongs to the circumstances that aid the congregation in singing the praises of God. In churches that are non-instrumental in their worship, it is common for a pitch-pipe to be used to aid the singing. It is of course not essential to have musical instruments of any kind—we could imagine, for instance, the Scottish Covenanters in the 17th century in the highlands singing outdoors and without instrumental accompaniment. But so long as the purpose of the instruments in worship is to support the congregational singing, and does not eclipse it by becoming its own independent entity, I believe that instruments have a place in the public worship of the church while still honoring the regulative principle.
I hope this is helpful.
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