A science of the mouth, not of the page
Tajwid (تجويد) comes from the root ج-و-د, to make something excellent. Linguistically it means to do a thing well. As a technical discipline it means giving every letter its due: pronouncing it from its correct articulation point (مخرج, makhraj) with all of the qualities (صفات, sifat) that properly belong to it, and observing the rulings that arise when letters meet one another in a chain of recitation.
Notice what that definition excludes. Tajwid is not a system of decoration, and it is not a style of melody. It is a conservation science. The Qur’an was received orally, transmitted orally, and is still learned mouth-to-mouth today; tajwid is the body of rules that describes that oral transmission precisely enough for it to survive intact across centuries and continents.
وَرَتِّلِ ٱلْقُرْءَانَ تَرْتِيلًا
wa-rattili l-Qur’āna tartīlā
— and recite the Qur’an with measured recitation.
Classical scholars read this verse as the Qur’anic root of the discipline. ʿAlī ibn Abī Ṭālib is reported to have glossed tartīl here as tajwīd al-ḥurūf wa-maʿrifat al-wuqūf — perfecting the letters and knowing where to pause. Those two halves are, in effect, the syllabus of this course.