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Comment by VBprogrammer | Hacker News Reader

I believe it's also one of the reasons the G-string is hardest to tune by ear because in equal temperament it's actually the furthest away from the true note.

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Yes, between most guitar strings is a perfect 4th interval (where equal temperament ratio [0] and the integer ratio it approximates [1] are quite close to each other) but between the G string and B string is a major 3rd interval (where equal temperament ratio [3] and the integer ratio it approximates [4] are a bit farther from each other).

When using a tuner to tune a guitar, you get the equal temperament normally desired for it's ability to play equally in any key. When tuning by ear, if you are just minimizing the beating you hear (instead of targeting the beating that exists in equal temperament, like good piano technicians are trained to do) you'll end up with integer ratios. Accidentally ending up at the integer ratio of a major third is "farther off" than doing so for a perfect fourth. Typically this manifests as a difficult time tuning the B string though, not the G string, if you go low to high. Solution: tune low to high for the E,A,D,G and tune high to low for the high E,B (doing a perfect octave for the E to E where the equal temperament ratio [4] perfectly matches the integer ratio [5]).

[0] 1.334840:1 or 2^(5/12):1

[1] 1.333333:1 or 4:3

[2] 1.259921:1 or 2^(4/12):1

[3] 1.25:1 or 5:4

[4] 2:1 or 2^(12/12):1

[5] 2:1

Does this mean there's actually a scientific reason to tune by ear instead of with an electronic tuner? I feel like I naturally do some of these things, like tuning low to high but then (seemingly arbitrarily) tune the high e against the low E.

Yes, integer ratios sound better than equal temperament ratios. The downside is that you end up with an instrument that has some integer ratios and some disgusting wolf intervals because it's mathematically impossible to have all integer ratios between every possible interval in every possible key you might decide to play in. That problem is what equal temperament solves for, and is a good reason to use an electronic tuner unless you know your songs will sound better without.

When tuning by ear, if you are just minimizing the beating you hear...

I assume you are talking about fretless instruments. With a guitar you just play a unison using the frets and the thicker string.

No, my entire comment is about tuning a typical fretted guitar. Playing a unison as you say, and turning the peg until beating completely disappears, yields integer ratios instead of equal temperament ratios. The result of that can sound nicer than equal temperament for some keys/chords but terrible for other keys/chords.

Unison means same sound. There's no ratio, unless you mean 1:1.

Ah shoot. I screwed up (only in the comment you're questioning, not my earlier comment per se). Of course the frets are positioned according to equal temperament, so when fretting a note for tuning the next string, indeed you end up at equal temperament when getting the beating down to nothing as you play the unison.

Rather, I'm talking from a point of view of playing two adjacent open strings. Or harmonics thereof (as some people like to tune with 7th fret harmonics and so forth). If you aren't fretting and you minimize beating down to nothing, then you end up with integer ratios.

IIRC harmonics ignore frets. You use the fret to approach the point where it happens, but its exact position is a function of the whole string lenght and it's therefore not equal temperament related.

I think you're right about the "beating"... in general this post has been extremely instructive for me. My teachers were piano players. Either they didn't know better (because they didn't need it) or they didn't bother to explain the fine details to me. Even the books I consulted later don't tell the whole story. That's the magic of the Internet :-)

Correct! Tuning open strings (listening to an interval) and tuning by harmonics (listening to a unison without using frets) both result in a lack of equal temperament if you eliminate the beating entirely. Only good piano tuners are really experienced in landing at the correct beat frequencies without electronic aid.

Yes, in this case that is correct. It is common to play the same note on two different guitar strings in order to tune them together.

I played classical guitar for years, and am a physicist, and feel a bit sheepish to say that I never realized this.

Please let me know if I'm understanding. This technique is for tuning open strings and listening, back-and-forth, to the perfect fourth (or major third) interval?

In contrast to matching the same note on the next string?

Yeah, I should have clarified, but at the time of writing, I didn't even realize it (because I'm more of a piano tuner than a guitar tuner so I don't have the luxury of frets that automatically impose equal temperament!) -- yes, this issue occurs when comparing open strings or unfretted harmonics.

Thank you. You also made me realize that it's better to tune open strings for avoiding intonation shifts by the pressing force.

That's actually the crux of the issue: if you fret the note to tune the next string, then you achieve equal temperament at the expense of any inaccuracy that comes from all sorts of things like imperfect action height, imperfect bridge tuners, excessive pressure beyond just contacting the fret, sideways motion, etc. which is all avoided by using open strings instead. But then doing open strings by ear doesn't get you equal temperament, it gets you integer ratios instead. Open strings with an electronic tuner gets you the best of both worlds (assuming you want equal temperament).

I find it easier not to tune either high to low nor low to high, but use lots of cross-string intervals and a variety of chords. Since my tuning fork is a standard A, I start on the A string, and use that to tune both E strings and the D, then B relative to high E and D relative to A. It’s good to tune G using both octaves from the low E and A string. This way you never tune a string more than 2 steps away from the reference, as opposed to tuning E,A,D,G, which gives you a G string 3 steps away from the reference.

But recently I discovered the most helpful trick, use a fifth from G down to the A string, and also use a fifth from the D to the low E string. Those last two intervals can be the most surprisingly out of tune intervals when using either ear tuning or an electronic tuner.

That's interesting. I start with "A" on the G-string, and then use its "B" to tune the B-string and so on.

I've always found the B string hardest to tune by ear. Generally I need to fine tune it in the context of a few chords.

Whether G is hard or B is hard depends on whether you're going high-to-low or low-to-high, as the issue is the interval between them, so it's just a matter of which one you're using as a known-good reference and which one you're trying to tune. There's an easier solution than taking multiple passes as you play different chords: see my other comment.

Thanks. I understand your approach but what method do you use to tune the high E? I assume you mean that the low E is used to tune the high E but perhaps you pick an E on the G string for this purpose. Is there any "best" E on the low side to use as a reference?

You always need some point of reference to start with, or if you don't have an external reference, you can just go with anything at all. I would typically choose either of the E strings, just to be able to immediately match the other E string to it, and then go inward (up to G and down to B).

I tried it and it works pretty well. One thing that I noticed is I subconsciously have been tuning GBE open because it’s a pain to reach around to the tuning pegs. How does this impact things?

When you play G and B open and tune them to eliminate beating, you're in pretty bad shape because you're making a "just" (integer ratio) major 3rd which is quite far from an equal temperament major 3rd.

When you play B and E open and tune them to eliminate beating, you're in okay shape because you're making a "just" (integer ratio) perfect 4th which is not terribly far from an equal temperament perfect 4th.

See my earlier comment [0] with all the numbers if you want to quantity "pretty bad" vs "okay". In that comment, my initial references to footnotes 3 and 4 should actually be 2 and 3, but it's too late to correct it.

[0] https://news.ycombinator.com/item?id=31145233

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