Notes on the Bible, by Albert Barnes, [1834], at sacred-texts.com
The captains of the host - Rather, "the princes" of Ch1 23:2; Ch1 24:6.
Under the hands of Asaph ... - That is to say, "under the direction of Asaph" - who himself "prophesied," or performed the sacred services, "under the direction of the king."
To lift up the horn - Some take this literally, and consider that Heman and his sons played on the born in the musical services; but there is no other evidence that the horn was so employed. Perhaps the most probable explanation is that it has been transferred from the next clause, where (as here) it followed the word "God," with the sense that "God, to exalt Heman's horn (or, increase his dignity), gave him fourteen sons and three daughters."
With their brethren - i. e., "with others of the tribe of Levi." Each son of Asaph, Jeduthun, and Heman, was at the head of a band of twelve skilled musicians, consisting partly of his own sons, partly of Levites belonging to other families 1 Chr. 25:9-31. The 24 band-leaders, together with their bands, formed a body of 288 persons (24 x 12 =288) Besides these, we hear of there being above 3,700 singers, who were probably divided, like the trained musicians, into 24 courses, which must have contained about 155 each Ch1 23:5.
As well the small as the great - Compare Ch1 24:31. The lot was not applied indiscriminately to all the 24 courses, but was only used to settle which course of Asaph, which of Jeduthun, and which of Heman, should on each occasion be taken. Asaph was given the precedence over his brethren, and his four courses were assigned the first, and then each alternate place. Jeduthun took rank next, and received alternate places, first with Asaph, and then with Heman, until his courses were exhausted. After this, all the later places fell necessarily to Heman, whose courses continue without interruption from the 15th.