Tabhair dom do Lámh

"Give Me Your Hand" (Tabhair dom do Lámh in Irish) is a tune from the early 17th century by Ruaidri Dáll Ó Catháin (c.1570-c.1650), perhaps in honour of a lady. It is one of the most widely recorded pieces of Irish traditional music.

"Give Me Your Hand" is also the title of many other songs; e.g. Dorothy Stewart's unrelated song is well known.

According to Edward Bunting, in The Ancient Music of Ireland, this harp tune was written in about 1603 by Ruaidri Dall Ó Catháin.

This tune, revived by Seán Ó Riada, was originally a composition of the blind Derry harpist Ruaidri Dáll Ó Catháin. He wrote it while (in) Scotland, where he had a disagreement with a Lady Eglington. He composed the tune for her when she apologized.

from the Wolfetones.

Captain Francis O'Neill suggests

Proud and spirited, he resented anything in the nature of trespass on his dignity. Among his visits to the houses of Scottish nobility, he is said to have called at Eglinton Castle, Ayrshire. Knowing he was a harper, but being unaware of his rank, Lady Eglinton commanded him to play a tune. Taking oftence at her peremptory manner, Ó Catháin refused and left the castle. When she found out who her guest was her ladyship sought and effected a speedy reconciliation. This incident furnished a theme for one of the harper’s best compositions. “Tabhair Damh do Lámh,” or “Give Me Your Hand!” The name has been latinized into “Da Mihi Manum.” The fame of the composition and the occasion which gave birth to it reaching the ear of King James the Sixth, induced him to send for the composer. Ó Catháin accordingly attended at the Scottish court, and created a sensation.

There are other stories and legends, but no authoritative evidence or references are known.

The tune is sometimes claimed to be written by the famous harpist [O'Carolan], who lived some years later. However, there is no reference in the Bunting collection of O Carolan’s music. Nor should Ruaidhrí 'Dall' Ó Catháin be confused with another blind poet at around the same time, Rory 'Dall' Morrison.

A number of apocryphal stories have circulated about the circumstances of the tune's composition; further details can be seen at Andrew Kuntz's The Fiddler's Companion.

The Fiddler's Companion says

The Latin title first appears in the Wemyss manuscript of 1644 and in the Balcarres manuscript of 1692

and then

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Wikipedia