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The benefits of a slotted headstock (left picture) over an unslotted one with vertically arranged tuning posts (right picture) seem to be largely subjective (The biggest advantage is problably that you have less wind resistance when spinning on the stage...).

Slotted headstock Normal headstock

I have always wondered though, why the slotted form is mainly used for classical (spanish) guitars while the normal one is used for, well, everything else. Is there a specific technical reason, maybe concerning the string's material?

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This is almost entirely down to tradition - early guitars used similar construction to equivalent instruments - lutes, violins etc all had that slot mechanism for the tuning pegs, where you have a slightly tapered tuning peg that is pushed into the hole to tighten.

Obviously, with more recent technology, tuning pegs with worm gear mechanism have removed the need for friction to hold the pegs from slipping, but tradition is strong enough that the shape has been retained.

The only other slight advantage for nylon strung guitars when using a slotted headstock is that there is no chance of a string slipping over the machine head (which could happen on the more recent flat style of headstock) as nylon strings require more string to be wound on. I call this a very slight advantage, because it really shouldn't be an issue unless you wind on far too much string.

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    Tuning a classical guitar requires winding a lot more string around the tuner than would be necessary on a steel guitar, but slotted tuners can accommodate a lot more string without the string having to overlap itself than typical upright-peg tuners can. On the flip side, tuning a steel strings generally requires making very fine length adjustments, and having smaller upright pegs helps with that. – supercat Jun 25 '16 at 13:59
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    you're quite right supercat, nylon strings need to stretch much more to get tuned, specially the lower (higher pitch) ones, requiring more turns of a gauge that's already thicker. Also nylon is more slippery than steel, so I think it would be really hard to properly wound nylons string on a (normal) vertical peg. Regarding using slotted pegs in non classical guitars, another (counter) factor is that the headstock with slotted tuning pegs is inherently more fragile and the risk of breaking with the higher tension steel strings would be higher. – José David Jun 25 '16 at 16:36
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    I'll word that in specifically. That's what I was meaning in my final paragraph. You can successfully use modern tuners with nylon strings, though. It's really not a problem – Doktor Mayhem Jun 25 '16 at 16:52
  • Upright pegs are common for ukuleles with nylon strings. It keeps the headstock lighter. – hpaulj Jun 26 '16 at 10:56
  • Yep. I'd just like to add that slotted heads, or the equivalent of the violin pegbox, have the advantage over a flat head that the tuning peg is supported on both ends. Again, this is an advantage that's not so important with machine heads with metal pegs and bearings. – Scott Wallace Jun 26 '16 at 20:30
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The main reason is probably tradition, though it may be a tradition for a reason:

It does help increase the angle of the string at the nut. Nylon strings tend to be more elastic than steel, so the sharper angle supposedly helps prevent the string from slipping or sliding at the nut.

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