Dear etaoin: Badword anyone who says YART. It is rude and
unreasonable to expect new members to search before they
post. And if etaoin doesn't remember it, few others will
I don't see any likelihood of a close etymologic relation between harmonic and arithmetic, but I'll go search.

arithmetic

SYLLABICATION: a·rith·me·tic
PRONUNCIATION: -rthm-tk
NOUN: 1. The mathematics of integers, rational numbers, real numbers, or complex numbers under addition, subtraction, multiplication, and division. 2. Archaic A book on this kind of mathematics.
ADJECTIVE: ar·ith·met·ic (rth-mtk) also arith·meti·cal (rth-mt-kl) 1. Of or relating to arithmetic. 2. Changing according to an arithmetic progression: The increase in the food supply is arithmetic.
ETYMOLOGY: Middle English arsmetike, from Old French arismetique, from Medieval Latin arismetica, alteration of Latin arithmtica, from Greek arithmtik (tekhn), (art) of counting, feminine of arithmtikos, from arithmein, to count, from arithmos, number. See ar- in Appendix I.
OTHER FORMS: arith·meti·cal·ly —ADVERB
a·rithme·tician (-tshn) —NOUN

harmonic

SYLLABICATION: har·mon·ic
PRONUNCIATION: här-mnk
ADJECTIVE: 1a. Of or relating to harmony. b. Pleasing to the ear: harmonic orchestral effects. c. Characterized by harmony: a harmonic liturgical chant. 2. Of or relating to harmonics. 3. Integrated in nature.
NOUN: 1a. Any of a series of musical tones whose frequencies are integral multiples of the frequency of a fundamental tone. b. A tone produced on a stringed instrument by lightly touching an open or stopped vibrating string at a given fraction of its length so that both segments vibrate. Also called overtone, partial, partial tone. 2. harmonics (used with a sing. verb) The theory or study of the physical properties and characteristics of musical sound. 3. Physics A wave whose frequency is a whole-number multiple of that of another.
ETYMOLOGY: Latin harmonicus, from Greek harmonikos, from harmoni, harmony. See harmony.
OTHER FORMS: har·moni·cal·ly —ADVERB

I can remember vaguely something about Pythagorus seeing
arithmetic in musical scale. But the two word do not appear to be cognates.