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If not then what kind of guitars/fretted string instruments can?

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    What reasons do you have to believe that fanned frets and Pythagorean tuning have any effect on each other? It would be easier to answer if we knew why you're asking. – Bradd Szonye Jan 6 '15 at 9:02
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    This question should remain open. We have taken questions on guitar intonation and historical temperaments and microtonal tuning before. – user1044 Jan 6 '15 at 16:27
  • Agreed with Wheat here, this is an interesting question with implications for musical performance practice. – jjmusicnotes Jan 7 '15 at 0:04
  • Please don't vote to close; to anyone who understands intonation and knows what fan fretting is, this question is quite clear. – neilfein Jan 7 '15 at 0:29
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    @neilfein I just don’t see what guitar intonation (fanning frets) has to do with just intonation. The two things are only superficially related, and I’m not sure where the question is trying to go with it. Clarification would be helpful, as the answers so far are sort of guessing at what the question means. – Bradd Szonye Jan 7 '15 at 4:05
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Along any single string, one could (in principle) achieve any desired intonation by appropriately placing the frets; fanning implies some desired relationship between the intonation on the different strings -- but it's unclear from your question where you are going with that.

Playing a fretless guitar (not very common, but fretless basses and all orchestral strings are fretless) or using a slide allows one to achieve arbitrary intonation (although achieving a given intonation on chords can often be difficult). In addtion, you may want to search out Tolgahan Çoğulu who performs using microtonal guitars, and the "true temperament fretting system". Neither of these cases involve frets that I'd consider "fanned".

If you really mean Pythagorean Tuning where you have 12 tones all generated by stacking fifths from a base tone, then the frets would not need to be fanned, rather, between any pair of adjacent strings separated by a fourth (let's ignore the major third between the g and b strings for now) there would be one additional fret that needs to be adjusted. So there is not an overall "fanning" of the frets.

Let's take G as our starting note string, then our 12 pythagorean notes are

      Notes:  E♭  B♭  F    C     G   D    A   E  B  F#  C#  G#
      Fret#:  8   3   10   5     O   7    2   9  4  11  6   1
      String 
      Length:  ...    9/16 3/4   1   2/3  8/9 ...

where I've indicated the note name, the fret number, and where the fret would need to be located to produce pythagorean intonation in terms of the "distance from the bridge to the fret" relative to the overall string length; this is the length of string that is vibrating when a note is fretted there.

If we go to the D string and make a similar table -- using the same frets as the g string we'd end up with

      Notes:  B♭  F   C    G     D   A    E   B  F#  C#   G#  D#
      Fret#:  8   3   10   5     O   7    2   9  4   11  6    1
      String 
      Length:  ...    9/16 3/4   1   2/3  8/9 ...

Note that we get a "D#" at the first fret which is a different note (has a different pitch) from the enharmonic E♭ on the G string. This is the Pyhagorean comma. To keep the same 12 tones as on the G string, we need to make this note at the first fret be the same tone as the E♭ on the G string (it's an octave lower), i.e. this fret's location will need to be adjusted. When you look at the A string, it will need to have the first and sixth frets adjusted, relative to the G string frets (but the first fret will be at the same location as the first fret on the D string), in order to produce the same 12 tones, and so on. Between the G and the B string 4 frets would need to in different place, since the interval between these strings is a major third, which is a stack of 4 fifths in Pythagorean tuning.

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Fanning frets have nothing to do with temperament. It's simply about ergonomics. The close ends of the frets have a smaller guitar scale than the spread ends (the bridge is further up). Fanned frets and standard parallel frets have an equal capacity for alternative temperaments, which is to say none unless you relocate them.

There are three means of establishing temperament on a fretted string instrument, the relative tuning between strings, the relative placement of frets down the neck, and if you wish to play chords, the relative placement of frets not only down the neck, but also in reference to frets on adjacent strings. Unfortunately simple fanning, even with a straight bridge and single scale length for the instrument will not achieve that. The patterns are bizarrely scattered, and even then only suitable for a particular key.

Sitars, and according to another answer here, viola da gambas, viols, and lutes have adjustable frets, which will at least afford you the first two methods of temperament for melodic playing. You could set up your open tuning to accommodate particular chords if you play slide style (different positions) instead of across strings, but there is no simple design solution for general chords in temperaments other than equal temperament, and it's worse if you want more than one key.

If you want to play chords with Pythagorean intervals, the only practical solution is to go fretless. There are alternative note choices even for a Pythagorean based scale, so fretless is more practical even for melodic playing. It's not unlikely that fretless players with sensitive ears are unconsciously employing some degree of Pythagorean intervals naturally already. With every note played, depending on the structure of a composition, the question becomes harmonious notes 'relative to what other notes?'. This question is perhaps answered more easily with natural intuition than theoretical analysis. There are choices, but each choice is an artistic compromise as well. You can harmonize intervals with a key, a chord root, the prior note, or a drone, but not all four at once, and even then you likewise compromise the step pattern of subsequent chord components.

  • I had always thought fanned frets were about transitioning fluidly between a shorter and longer scale length so that you could have a larger range from the top to bottom string. – Dan D Apr 8 '15 at 13:42
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    They are. That's what I was just trying to say. – Kristal McKinstry Apr 8 '15 at 21:28
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I can't address fan-fretted guitars at all, but as far as other fretted string instruments go, the Viola da Gamba, and other Viols (not to be confused with the modern viola), as well as the Lute were some Rennaissance/Baroque instruments that used tied-on frets, made from loops of gut string tied around the neck. Because these frets were only tied, their position could be adjusted to alter the temperament of the instrument, or could even be double looped, to provide two slightly different tunings for enharmonic notes (e.g. G♯ vs. A♭ could be slightly different). Based on the music of the time, it seems likely that they may have often used some form of Meantone temperament (which is neither 12-TET nor Pythagorean).

This website has a handy calculator that shows where you would need to place extra frets in order to get notes from a variety of temperaments, including Meantone and Pythagorean.

And though it doesn't address the temperament question, here is a video showing how to tie a viol fret:

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