Rí, or very commonly ríg (genitive), is an ancient Gaelic word meaning "king". It is used in historical texts referring to the Irish and Scottish kings, and those of similar rank. While the Modern Irish word is exactly the same, in modern Scottish Gaelic it is rìgh, apparently derived from the genitive. Cognates include Gaulish Rix, Latin rex/regis, Sanskrit raja, and German Reich.
Three traditional grades
There were three traditional grades of rí in Gaelic Ireland, whose grading was largely symbolic. As time went on, the real power of many lesser kings could equal or even eclipse those of higher grade.
A rí benn (king of peaks), or ri tuaithe (king of a single tribe) was most commonly a local petty king of a single túath, although "one" túath might be many times the size of another. There are generally estimated to have been between 100 and 150 in Ireland, depending on who really qualified.
Importantly, in theory every king of a superior grade was also a ri benn himself, and exercised no direct compulsory legal authority outside his own ancestral túath. Kings were bound to others by military allegiance and the payment of tribute.
A ri buiden (king of bands), also ri tuath (king of [many] tribes) or ruiri (overking), was a regional king to whom several rí benn were subordinate, and often other territories. He was in some sense still a petty king, but could also achieve provincial-level prominence, including, although rarely, the provincial kingship, and was often fully sovereign in any case. Depending on who was counted, there may have been as many as 20 genuine ruiri in Ireland at any time.
A "king of over-kings", a rí ruirech was often a provincial (rí cóicid) or semi-provincial king to whom several ruiri were subordinate. They were also referred to as ri bunaid cach cinn ("ultimate king of every individual"). Several kingdoms belonging to the 1st and 2nd millennia are listed below, but do not all belong to the same periods. No more than six genuine rí ruirech were ever contemporary, with the average being three or four. Originally, there were only five provinces, at least according to legend (see the Táin Bó Cúailnge, the actual text thereof).
Scotland appears to have had a variety of rí as well. In addition to the monarch or "high king" there were others, although these are conventionally styled only "lords" in the English language.
There were also a number of Kings of Moray, who are commonly styled mormaers in later Scottish tradition, but properly styled rí in contemporary Irish sources. The famous Macbeth of Scotland is argued to have begun his career as Ruiri of Moray.
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