1580s, "allotted measure," a sense now obsolete, from French module (1540s) or directly from Latin modulus "small measure," diminutive of modus "measure, manner" (from PIE root *med- "take appropriate measures").
Sense of "a standard measure to regulate proportions" is from 1620s. Meaning "interchangeable part" is recorded by 1955, via the notion of "length chosen as the basis for the dimensions of parts of a building, etc." (1936); that of "separate section of a spacecraft" is from 1961.
It forms all or part of: amenorrhea; centimeter; commensurate; diameter; dimension; gematria; geometry; immense; isometric; meal (n.1) "food, time for eating;" measure; menarche; meniscus; menopause; menses; menstrual; menstruate; mensural; meter (n.1) "poetic measure;" meter (n.2) unit of length; meter (n.3) "device for measuring;" -meter; Metis; metric; metrical; metronome; -metry; Monday; month; moon; parameter; pentameter; perimeter; piecemeal; semester; symmetry; thermometer; trigonometry; trimester.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit mati "measures," matra "measure;" Avestan, Old Persian ma- "to measure;" Greek metron "measure," metra "lot, portion;" Latin metri "to measure."
1610s, in music, "vary or inflect the sound of," especially to give expressiveness, "vary the pitch of," back-formation from modulation, or else from Latin modulatus, past participle of modulari "regulate, measure off properly, measure rhythmically; play, play upon," from modulus "small measure," diminutive of modus "measure, manner" (from PIE root *med- "take appropriate measures").
General sense of "modify, adjust, adapt, regulate in measure or proportion" is from 1620s. The intransitive musical sense of "pass from one key to another, or between major and minor" is attested by 1721. In telecommunications from 1908. Meaning "exert a controlling influence on, regulate" is by 1964. Related: Modulated; modulating.
"science of versification," 1760, from Latinized form of Greek he metrikē "prosody," plural of metron "meter, a verse; that by which anything is measured; measure, length, size, limit, proportion" (from PIE root *me- (2) "to measure"). Middle English had metrik "the branch of music which deals with measure or time" (late 15c.), from Medieval Latin metricus.