-4

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?

Terence Allen
  • 23
  • 2
  • You'd have to ask a computer. We're just musicians. … but why would you want to start on E, A & end up on C, F? That way madness lies. – Tetsujin Dec 07 '23 at 17:48
  • 7
    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. – Todd Wilcox Dec 07 '23 at 18:18
  • 1
    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. ... – Andy Bonner Dec 07 '23 at 18:18
  • ... It comes along with the questions that always attend such research, like "what were the study methods, what was the pool of who was surveyed, how well can we extrapolate from that data to make assertions about entire cohorts," etc. We certainly can't talk about "what all guitarists do" because they're not all surveyed; it would be anecdotal conjecture. And even with the most rigorous research, we wouldn't have "best," we'd have "most common." – Andy Bonner Dec 07 '23 at 18:21
  • 1
    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." – Andy Bonner Dec 07 '23 at 18:25
  • Why don’t you just ask an AI: “What is the best way to tune a 6 string fretted instrument for playing melody and chords”? See what it comes up with. – John Belzaguy Dec 07 '23 at 18:30
  • 1
    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. – Todd Wilcox Dec 07 '23 at 18:38
  • 1
    @ToddWilcox - unregistered account. Could have 6 all in the same name & no-one would ever find them, realistically. [Unless you go looking - https://math.stackexchange.com/questions/4817767/how-do-i-calculate-the-euler-characteristic-betti-number-and-genus-for-musical ] – Tetsujin Dec 07 '23 at 19:18
  • What is wrong with Drop D tuning - 075545? Power chords are so much easier with this tuning, and even classical guitar music uses this tuning at times (see Fernando Sor for more details)! – Dekkadeci Dec 07 '23 at 20:16
  • Wow, that math.se question is something. I don't understand any of it even though I'm familiar with some of the math it refers and I suspect it got closed because nobody else does either. – ojs Dec 07 '23 at 20:24
  • 1
    This question seems to me like "Why is standard tuning objectively the best" but asked in a very roundabout way. And that isn't a good question. Unless I'm misunderstanding something. – Edward Dec 08 '23 at 02:51
  • "the probability of Lute tuning 055555 is almost zero": note that so-called lute tuning isn't how lutes are tuned. The reasons behind standard guitar tuning are primarily historical, not mathematical, and based on principles established on instruments other than the guitar, before the guitar even existed in its present form. – phoog Dec 08 '23 at 08:33
  • I read this question not as "What is the best guitar tuning?", but as "Is there some mathematical description of guitar tunings that, when asked for the optimal one, will return the current standard tuning?" Is this a correct interpretation? – EdvinW Dec 08 '23 at 09:02

4 Answers4

9

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.

stimmung
  • 91
  • 2
  • 1
    Welcome to Music: Practice & Theory. Thanks for posting a great first answer! – Elements In Space Dec 08 '23 at 03:16
  • 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 Dec 08 '23 at 08:59
  • 1
    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 Dec 08 '23 at 12:02
2

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.

Aaron
  • 87,951
  • 13
  • 114
  • 294
1

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...)

Emil
  • 719
  • 1
  • 6
  • 16
  • 1
    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. – piiperi Reinstate Monica Dec 08 '23 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 Dec 08 '23 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 Dec 08 '23 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 Dec 08 '23 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 Dec 08 '23 at 23:01
  • "amount of music that can be played" ... where did that come from? Do you mean that it's your "purpose X" and maybe even the OP's? Or everyone else's, self-evidently, as a matter of course? My point was that the OP didn't explicate his idea of what the purpose of a guitar tuning system should be, in this supposed optimization problem. It may have been, ease of playing some kinds of chords. Is that also the reason of existence for guitars, or all instruments in general, everywhere ever. Dunno! Maybe it can be proven with mathematics or AI, like the meaning of life. My precalc table says 42. – piiperi Reinstate Monica Dec 09 '23 at 01:29
  • In my example I say "if you will only ever play one song with one chord, tune according to that chord". The expansion of that in my mind is all the music that he wants to play should be partitioned into subsets where each subset has a different optimal tuning. I have taken a course in AI and read multiple papers about multiple subfields and is reasonably well versed in torch, tensorflow, and openvino. I just don't get your arbitrary deliniation of the AI field. To me machine learning just means tweaking a couple of million of weights. – Emil Dec 09 '23 at 09:14
  • And I honestly don't think traversal search over all possible music would work. It just sounds like you are viewing a different problem. And I also reject the number of tunings being different. There is a huge difference between 440Hz and 442Hz. Maybe somewhere like 20000 different possibilities per string. And then the factorial for all strings together. And then check against all songs you want to play. It grows huge. And I can promise you some songs are easier in other tunings, so there is no optimum for all taken together. – Emil Dec 09 '23 at 09:21
  • So then you get the powerset of all songs as well. Maybe even retuning in the middle of a song! Now it grows truly unweildly. – Emil Dec 09 '23 at 09:25
  • Ok fair, I assumed the optimum would be for playing the guitar. I doubt the resale value or design of the guitar would be affected. Although tuning too hard is not good because then the guitar could loose its tuning in the middle of the night, so that could be what the objective function was. But honestly I think you know s/he wanted to know which tuning would allow me to play most music in an easy way, that feels like the default problem to solve. – Emil Dec 09 '23 at 09:31
  • (I guess in your case, the "tweaking of weights" was done by hand by you, and that is why you did not consider it machine learning? I guess it would be a stretch to call that learning, that is more like just evaluating, but you would probably have to change your objective function so the learning happens inside you instead) – Emil Dec 09 '23 at 09:50
  • (In an even longer stretch, you could consider the characters of the source code of your program being the weights) – Emil Dec 09 '23 at 10:13
  • The problem is that you call everything weights, the word loses its meaning – ojs Dec 09 '23 at 14:30
  • There are people who disagree that 440Hz and 442Hz is a huge difference, but it would be a separate discussion. – ojs Dec 09 '23 at 14:31
  • No it doesn't, is is just independent variables. They do not lose their meaning. The only thing that makes stuff harder is that you cannot do vanilla calculus since they are not nice scalar values, but that is just some tweaking on the algorithm to fix. Stop making baseless questioning of my statements please. It is a measurable difference, anyone who says it is not a difference has either not a trained ear or not a tuning app. – Emil Dec 10 '23 at 07:36
  • Seems I messed up, I don't know what the model variables are called. But it is them I mean anyway. I don't think anyone restricts you to use scalar fields, and you can probably make any kind of learning method you like, genetic algorithm, backpropagation, A*, I don't know, it was a decade ago I took the class. – Emil Dec 10 '23 at 07:53
-2

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

Michael Curtis
  • 56,724
  • 2
  • 49
  • 154
  • 1
    The question isn’t about AI. It’s about whether there’s a mathematical calculation that would predict the ideal tuning. – Aaron Dec 07 '23 at 20:54
  • 2
    "could AI...?" Right in the title. So, it seems about AI, mixed up with a "mathematical" hope. – Michael Curtis Dec 07 '23 at 20:57
  • 1
    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 Dec 07 '23 at 21:03
  • 1
    AI encompasses much more than language models, even though language models have had the most attention recently – ojs Dec 07 '23 at 22:08