Etymology
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gamut (n.)

1520s, "low G, lowest note in the medieval musical scale" (the system of notation devised by Guido d'Arezzo), a contraction of Medieval Latin gamma ut, from gamma, the Greek letter, used in medieval music notation to indicate the note below the A which began the classical scale, + ut (now do), the low note on the six-note musical scale that took names from syllables sung to those notes in a Latin sapphic hymn for St. John the Baptist's Day:

Ut queant laxisresonare fibris
Mira gestorum famuli tuorum,
Solve pollutis labiis reatum,
Sancte Iohannes.

The ut being the conjunction "that." Gamut also was used for "range of notes of a voice or instrument" (1630s), also "the whole musical scale," hence the figurative sense of "entire scale or range" of anything, first recorded 1620s. When the modern octave scale was set early 16c., si was added, changed to ti in Britain and U.S. to keep the syllables as different from each other as possible. Ut later was replaced by more sonorous do (n.). See also solmization.

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mi 

symbol-word for the third tone in a musical scale, 1520s; see gamut.

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solmization (n.)
"act of using certain syllables to name tones of a music scale," 1730, from French solmisation, from solmiser, from sol + mi, two of the syllables so used (see gamut).
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fa (n.)
fourth note in Guidonian scale; see gamut. Used from 13c. in Old French. It represents the first syllable in Latin famulus.
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la (1)
musical note (sixth note of the diatonic scale), early 14c., see gamut. It represents the initial syllable of Latin labii "of the lips." In French and Italian it became the name of the musical note A, which is the sixth of the natural scale (C major).
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ti 
seventh note of the musical scale, 1842, earlier te (1839), replacing si to avoid confusion with so, sol (see gamut).
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sol-fa (n.)
"syllables used in solmization taken collectively," 1540s, from Italian, from Medieval Latin sol and fa, two notes of the musical scale (see gamut). As a verb from 1560s; compare solfeggio "use the sol-fa system" (1774), from Italian solfeggiare.
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do (n.2)

first (and last) note of the diatonic scale, by 1754, from do, used as a substitution for ut (see gamut) for sonority's sake, first in Italy and Germany. U.S. slang do-re-mi "money" is from 1920s, probably a pun on dough in its slang sense of "cash."

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