Tempered intervals differ slightly from just ratios: a tempered fifth is ~2 cents narrower than the natural 3:2 fifth

In 12-tone equal temperament, a fifth spans 7 tempered semitones. In Max/MSP, computing a fifth by adding 7 to a MIDI note and converting with mtof produces a tempered fifth; computing it by multiplying the fundamental frequency by 3/2 produces a just (natural) fifth. The natural fifth is slightly wider — the difference is about 2 cents (1/100 of a tempered semitone). Though subtle, this distinction is audible in sustained tones and affects tuning decisions in additive synthesis, physical modeling, and microtonality work.

Examples

MIDI note 60 (C4) → mtof → 261.63 Hz. Adding 7 → MIDI 67 (G4) → mtof → 392.00 Hz (tempered). 261.63 × 3/2 = 392.44 Hz (natural). The natural fifth is the higher of the two.

Assessment

Why does patching mtof → *~ 1.5 in Max produce a slightly different pitch than note + 7 → mtof? Name the interval difference in cents.