Pinning a node makes higher modes act as fundamentals of shorter strings

Evidence

The nodal point at the center of the string belonging to the second mode might just as well be pinned down in that mode and indeed all the even modes, since it isn’t moving anyway. Once pinned, we see that the second mode is really the fundamental mode of a string of half the length. The frequency is doubled compared to the fundamental of the whole string, since f α 1/L, where L is the length between the pinned ends. Sauveur explained correctly that the motion of the second partial with its two parts could be going on while the lower partial with twice the wavelength was also activated. It is possible to check this idea by watching the motion of the nodal point of the second partial with the first partial also excited: that point should execute pure single-frequency sinusoidal motion, since the second partial is not causing any displacement there. — Eric J. Heller, Why You Hear What You Hear- An…, 874

Categorization

• DEFINE: Explains why harmonic modes behave like fundamentals of halved strings — a core mechanics concept for why harmonics have integer frequency ratios.
• ADDENDUM: Pairs naturally with the grafting/trees infrastructure piece’s recurring theme of structure-and-constraint: pinning a node is like grafting a branch at a fixed point to re-derive the fundamental.
• XREF: connects to: A string’s nth mode has n+1 nodes and n antinodes — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1081