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Physics, Mathematics, and Linguistic Stack Exchange simply refuse to accept questions about the mathematics of tablature and guitar.

But what is tablature besides lines and numbers?

These sites often say I should ask these questions on Music Practice and Theory, but I have never seen a question about the mathematics of tuning.

Mathematics SE said this is not a mathematic question. I think they just don't understand how guitar works. What do you think?

Here's my question:

The algorithm for Standard tuning EADGBE is 055545. This means start with the lowest string E = 0, then tune the next higher string to the first string, fret 5. The third string is tuned to second string fret 5, fourth string to third string fret 5, fifth string is tuned to the fourth string fret 4, and finally the sixth string is tuned to the fifth string fret 5. So that is 055545.

Most guitarists agree that 055545 is the best, and then there is a series of alternative tunings that are very good but less common. And then there are a group of tunings which are not used generally but have been used on rare occasions. And then there are a lot of tunings that are useless.

So the probability of Standard tuning 055545 is very high and the probability of Lute tuning 055555 is almost zero.

Playing chords in 055555 is very difficult.

It seems like there a form of least action calculus on the guitar tunings in which Standard tuning is at a minimum value and no one can find a better tuning because it does not exist.

So my question is whether a computer could calculate that Standard tuning is the best given a guitar with 6 strings and no tuning?

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    Right away the biggest difficulty in your question is you have neither qualitatively nor quantitatively defined what a “better tuning” or “best tuning” is. With a qualitative definition, musicians could speak about the qualities of different tunings and how they fit with “better” or “best”. With a quantitative definition, mathematicians and computer scientists could speak to numerical relationships and algorithms that would help determine which tuning is “best”. With neither kind of definition, no one can give any meaningful answer. Commented Dec 7, 2023 at 18:18
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    Welcome! There are certainly plenty of math-y questions here, mostly about acoustics and temperaments. Quick note, talk about how the different Stack Exchanges behave or should behave should be on the "meta" section of each. Meanwhile, it seems to me the real difficulty here is in applying objective approaches to subjective phenomena. Which tuning is "the best"? By what standard? what makes it "best"? But when we talk about "most guitarists agree," and the "probability" of a tuning, then this is about collecting data on common usages. That's sociological research, a soft science. ... Commented Dec 7, 2023 at 18:18
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    Your line "playing chords in 055555 is very difficult" suggests a "goal-oriented" fallacy when viewing music history. Rarely are instruments "invented" to suit a musical need; more often, they evolve and then practices evolve around them. Saxophone is a good example of an exception, engineered by one guy and laid out in a logical way. But many instruments exhibit "QWERTY phenomenon": the QWERTY keyboard is less efficient, but we keep using it because everyone is used to it this way and it would be hard to make a different version "stick." Commented Dec 7, 2023 at 18:25
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    I’m confused. How do you know other stacks don’t like this question? You seem to have zero prior activity anywhere on the stack exchange network. Commented Dec 7, 2023 at 18:38
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    @ToddWilcox - unregistered account. Could have 6 all in the same name & no-one would ever find them, realistically. [Unless you go looking - math.stackexchange.com/questions/4817767/… ]
    – Tetsujin
    Commented Dec 7, 2023 at 19:18

4 Answers 4

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I assume that an algorithm or AI could come up with a different useful tuning, but I suspect it would be a slight variation of the currently popular tunings. After all, the tuning is very much limited by the practicalities of moving your fingers around the fretboard. That's why a violin is tuned differently (in fifths) than a guitar, because of the difference in scale length.

Let's look at this practically. How do we avoid the algorithm being influenced by the current guitar tuning status quo? We could e.g. look at piano music instead; it is probaby the instrument that's most similar in its use to guitar, a polyphonic instrument used both for accompaniment and melody, in many different genres. Analyze piano music in different genres and styles, and list the most commonly used chords, scales and arpeggios, and also the more advanced things that might be interesting to have.

Then explain to the algorithm what kind of fingerings are possible, and which are easy and which are hard. You could use the existing guitar corpus for this. Then, let the algorithm try every possible tuning, and see what percentage of basic and more advanced chords, scales and arpeggios are possible. The tuning which covers all the basics and has the most of the advanced features, wins.

Why do I think that the result will be close to EADGBE? Because there isn't actually that much wiggle room.
Choose larger intervals, like fifths, and the stretches become too large for the average player's hands. If you tune in diminished fifths, like FBFBFB, you have too many of the same note, and too many notes that only fit in diminished or dominant chords. Even just one diminished fifth may make some shapes a literal stretch.
If you tune in fourths, which some people do, you have the advantage of easy transposition of shapes across the strings and frets, but you lose the same-note top and bottom string, which will make most chords more complicated (although e.g. the thumb-over technique might help here).
Choose smaller intervals, and it becomes difficult to play common major or minor chords, with their succession of 3, 4 and 5 semitone intervals. And playing common scales in a fixed position will be replaced by playing diagonally across the fretboard, and thus a lot more hand movement.

If the algorithm finds other useful tunings, they will probably be standard tuning with the major third between other strings, like EADGCE or EADF#BE, or maybe with two major thirds, like EAC#F#BD#. Maybe you could throw in a minor third, but I doubt that would make anything easier to play. Of course there are a couple of alternate tunings already in use, like all-fourths and DADGAD and drop-D, which are clearly useful in some styles, but they are not generally used across genres.

It goes without saying that in order for a new tuning to actually gain traction, it would have to be both demonstrably superior to current standard tuning, and have advantages for all genres of music and styles of playing. Otherwise, it will just be another niche thing that some players use sometimes.

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    Welcome to Music: Practice & Theory. Thanks for posting a great first answer! Commented Dec 8, 2023 at 3:16
  • This is a good answer. I would personally put just somewhat more emphasis on the second-to-last paragraph, about the possible variations. Moving the third around and transposing leaves you with many tunings. 4 fourths and one major third thus gives you a few dozen tunings that are possible without changing the pitch or timbre too much compared to the current standard, and among these there are many possible choices that are not obviously inferior to the standard tuning. If you include 2 major thirds the total number about doubles, and and a major and a minor third the number doubles again.
    – EdvinW
    Commented Dec 8, 2023 at 8:59
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    With that said, the proportion of tunings with more than one third among their fourths is much smaller. On the other hand, including a fifth somewhere might be possible as well, and maybe even a diminished fifth. I agree that, if you want to use the guitar for roughly the same things as today, the optimal tuning is likely rather close to the current one, in the sense that most intervals would be fourths.
    – EdvinW
    Commented Dec 8, 2023 at 12:02
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I posit that the answer is no, an AI/algorithm could not determine the ideal tuning. Or, rather, it would agree that the standard tuning is ideal, but not on any objective, mathematical basis, but because the calculation would have to be based on biased information.

The question presents the computer with an unresolvable chicken and egg problem.

Guitar spec, sans strings

Suppose we give a computer complete specifications of all six-string guitars — dimensions, properties of the building materials — and ask for an ideal tuning that would, on average, be best across all such guitars.

Guitar string design

The computer would then ask, "what are the properties of the strings?" This information, however, is biased toward a specific tuning. Guitar strings are designed with the standard tuning in mind. In fact, this is already true of the rest of the guitar: the instrument is built with the answer to the tuning question already in mind.

But, let's ignore these problems and give our computer complete specifications for all strings designed for six-string guitars (including, of course, the type of guitar they're intended for).

Analysis of guitar music

But now our computer asks, "ideal for what kind of music?" Here again, there's a built-in bias toward standard tuning, since the guitar is designed with a certain musical repertoire in mind, and the music is written for an instrument designed with certain properties — such as a presumed tuning.

We give our computer access to all of the music every written or arranged for guitar. But this presents still another problem: who is going to be playing this music?

Mechanics of the human hand

Now we have to give the computer a complete specification of the human hand — or sufficiently broad access to human hand information to come up with a working range of parameters like length and diameter of each finger, dexterity, speed, span between fingers, ....

Conclusion

Finally, given all of these factors, it seems inevitable that the computer would spit out the standard tuning, because the standard standard tuning was derived from these factors and subsequently reinforced by them.

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Depending on what objective function you give a constraint optimization problem solving algorithm you will probably find different optimums. (Solutions usually vary as well depending on start point and other particulars unless the problem is very simple). If you will only ever play one song always and that song only has one chord - guess what? Tuning the guitar for that chord would be optimal. Unless you are optimizing for unplayability/difficulty... So it depends very heavily on what you will play/optimize for. I would also reject your proposal that all fourths tuning make chords harder, on the contrary it makes all (sane) ≤4 string chords easier, it is only 5 and 6 string chords that get slightly trickier. (And I think if I ever get 6 fingers in addition to the thumb it would be the go to choice for me always...)

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    Does it have to be a machine learning algorithm? If you can write a fitness for purpose X (which remains to be clearly explicated by the OP) evaluator function, it's quite trivial to even brute-force try all possible combinations. Commented Dec 8, 2023 at 13:37
  • How is a 'fitness for purpose X evaluator function' any different from an objective function in an optimization problem? Sounds like a synonym to me? I just assumed he had a corpus of music he wanted to optimize over, that is why I said machine learning. I guess maybe he could just write an AI that sends an email asking every person in the world what their favorite tuning is as well, then it would definitely be a different thing than machine learning.
    – Emil
    Commented Dec 8, 2023 at 21:32
  • To me machine learning is just a synonym for a good old optimization problem with constraints like in calculus... not sure what you mean with it. Changed so it doesn't say machine learning now.
    – Emil
    Commented Dec 8, 2023 at 21:41
  • But I disagree that it is a finite set, I would say the amount of music that can be played is at least a countably infinite set, because you can always tweak it a little to get a new song.
    – Emil
    Commented Dec 8, 2023 at 21:50
  • The finite set is the set of possible tunings. "Machine learning is just a synonym to optimization problem" is so technically true it must come from a person who doesn't have much experience with it.
    – ojs
    Commented Dec 8, 2023 at 23:01
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ChatGPT 3.5 answers the same way as humans on this exchange, because AI just rehashes existing human generated text.

The reason isn't specifically mathematical, it's practical for accommodating the shape of a hand, and learning time tested techniques.

ChatGPT transcript:

What is the best way to tune a guitar?

ChatGPT To tune a guitar, you can use a tuner device, a tuning app, or tune by ear. Start with the low E string and adjust each string until it matches the correct pitch: E, A, D, G, B, E. Alternatively, you can use the fifth fret method, where you match the pitch of the open string with the fifth fret of the next higher string.

User Why is that the most popular tuning?

ChatGPT The standard tuning (EADGBE) is widely used because it provides a balanced arrangement of notes across the fretboard, making it versatile for various playing styles and music genres. This tuning allows for easy chord shapes, scale patterns, and a good balance of tension and resonance on the guitar. While alternate tunings exist for specific styles or effects, the standard tuning remains popular due to its versatility and familiarity.

Message ChatGPT…

ChatGPT can make mistakes. Consider checking important

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    The question isn’t about AI. It’s about whether there’s a mathematical calculation that would predict the ideal tuning.
    – Aaron
    Commented Dec 7, 2023 at 20:54
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    "could AI...?" Right in the title. So, it seems about AI, mixed up with a "mathematical" hope. Commented Dec 7, 2023 at 20:57
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    Yes, AI is mentioned in the title, but based on the question itself, and the use of "mathematical" and "calculate", clearly the interest is not in a lookup-based AI but rather an AI (or, rather, an algorithm) that, given appropriate parameters, could independently determine the ideal tuning. It's a poorly articulated question, but obviously not one written with ChatGPT in mind, since the answer in that case is trivial.
    – Aaron
    Commented Dec 7, 2023 at 21:03
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    AI encompasses much more than language models, even though language models have had the most attention recently
    – ojs
    Commented Dec 7, 2023 at 22:08

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