It's Sunday afternoon, so it must be time for some #Acoustics #DataVisualization and this time it's all about musical instrument strings...
You might remember my 'Acoustic Ashby chart' where I plotted bulk modulus versus density for various fluids. I've redrawn it with plotly; in due course there'll be a web version where you can hover over points to get the substance name etc, but for now here's a static plot. 
Bulk modulus (inverse of compressibility) provides the restoring force that makes the fluid return to its equiliibrium state when disturbed; density provides the inertia that makes them take time to do so. Liquids are stiff and heavy, gases are floppy and light.
The blue diagonals are lines of constant sound-speed and the the black diagonals of constant (characteristic specific acoustic) impedance. One take-home message is that liquids and gases can have similar sound-speeds but their impedances are *way* different.
You can do the same thing with strings. The restoring force comes from their tension and the inertia versus mass per unit length. Here they are for various sets of electric guitar strings in standard tuning from @DaddarioandCo (who helpfully provide tension data - thanks!).
Each dot is a string. Because all the strings are the same length all the notes tuned to a particular pitch must have the same wave-speed, which we can confirm by joining up matching notes like this:
The tension isn't completely constant across the strings for any set, and in all but one case the top E-string is slightly tighter than its neighbour, while the bottom E-string is slightly slacker than its neighbour (assuming I haven't introduced errors when transcribing data).
This chart adds three sets of classical guitar strings, described as normal, hard, and extra hard tension. The physical differences are surprisingly small, though the subjective effects might not be.
That's a bit crowded so here's the same things with thee electric guitar strings removed, but the wave-speed contours retained so you can keep your bearings.
Next up: ukuleles. Here, on the same axes, are concert, tenor and baritone nylon-strung ukulele strings. Concert and tenor are tuned the same but have different string lengths; they also have re-entrant tunings - I've joined the dots in pitch order rather than string order.
The top string of the Baritone uke (with these strings) has the same wave-speed as the second string of any guitar, but it doesn't play the same note, because it's a different length.
Now I've added another set of ukulele strings, these are 'Nyltech' strings, which simulate the properties of gut strings. This set includes a bass ukulele but the bottom of the legend got cut off - it's the greenish one with the highest mass.
It would make sense for tension to scale with length, so that the finger-force needed to displace the string when plucking it stays about the same. Let's test that hypothesis - here are the classical guitars and the ukes with tension scaled to length.
They all lie reasonably close apart from the nylon-strung tenor uke.
There are lots more instruments to add including banjo, mandolin, lute, orchestral strings, erhu, pianos (the liquids to the plucked string's gases) and so on.
There are lots more instruments to add including banjo, mandolin, lute, orchestral strings, erhu, pianos (the liquids to the plucked string's gases) and so on.
I don't yet have data for theorbo strings (sorry @Liz67Kenny); once I do I'll be able to take the fun out of all these theorbo vs ukulele jokes by analysing them scientifically.
