▁re eds ▁differ ▁from ▁strings ▁in ▁that ▁they ▁are ▁held ▁at ▁one ▁end ▁and ▁free ▁to ▁v ibr ate ▁at ▁the ▁other . ▁The ▁frequencies ▁of ▁their ▁fundamental ▁and ▁harm onic ▁v ibr ations ▁are ▁subject ▁to ▁the ▁same ▁in harm onic ity ▁as ▁strings . ▁However , ▁because ▁of ▁the ▁compar ative ▁thick ness ▁of ▁the ▁bars ▁that ▁terminate ▁the ▁t ines ▁in ▁an ▁electric ▁piano , ▁the ▁larger ▁( and ▁stronger ) ▁v ibr ations ▁tend ▁to ▁" see " ▁term ination ▁points ▁slightly ▁deeper ▁in ▁the ▁bar ▁than ▁do ▁smaller , ▁we aker ▁v ibr ations . ▁This ▁enh ances ▁in harm onic ity ▁in ▁t ines . ▁ ▁Effect s ▁on ▁tun ing ▁▁ ▁In harm onic ity ▁" st ret ches " ▁harm on ics ▁beyond ▁their ▁theoretical ▁frequencies , ▁and ▁higher ▁harm on ics ▁are ▁stretch ed ▁proportion ally ▁more ▁than ▁lower . ▁Thus , ▁in ▁our ▁example ▁of ▁an ▁oct ave , ▁exactly ▁matching ▁the ▁lowest ▁common ▁harm onic ▁causes ▁a ▁slight ▁amount ▁of ▁stretch , ▁matching ▁the ▁next ▁higher ▁common ▁harm onic ▁causes ▁a ▁greater ▁amount ▁of ▁stretch , ▁and ▁so ▁on . ▁If ▁the ▁interval ▁is ▁a ▁double ▁oct ave , ▁exactly ▁matching ▁the ▁upper ▁note ▁to ▁the ▁fourth ▁harm onic ▁of ▁the ▁lower ▁comp licates ▁the ▁tun ing ▁of ▁that ▁upper ▁note ▁with ▁the ▁one ▁an ▁oct ave ▁below ▁it . ▁ ▁Sol ving ▁such ▁d ile mm as ▁is ▁at ▁the ▁heart ▁of ▁precise ▁tun ing ▁by ▁ear , ▁and ▁all ▁solutions