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When we get picky about intonation, we have to nail down pitch. A digital tuner won't do that. A strobe tuner won't either.

Take this example: an open A string has the following partials: 110, 218, 329, 439, 550, 661, 773. Is this string sharp or flat relative to A=440? By how many cents? Would nut compensation improve things? would saddle compensation help? Does volume matter? Does musical context matter? Does age of the listener matter?

Does political affiliation matter? I hope not. Todd, if it does, please open another thread.

Well the open A will have a fundamental of 110(for A440) so it is perfectly in tune but some of the partials are a little off

When I get picky about intonation, I'm usually thinking about the pitch of the fundamental. When I'm looking at the pitch of the partials I'm usually thinking about (in)harmonicity.

Electronic tuners normally pick out the strongest (highest amplitude) partial and call that "A" (or B or C etc.). The strongest partial is not always the fundamental, because we tend to pick at 1/4 string length which can excite the second partial more than the first, and if you have a lot of inharmoncity that can be a problem. Sometimes the second partial will decay faster than the first, so the tuner will switch partials showing a jump in the tuning as it locks on to the new strongest partial. Some tuners (e.g. G-Tune) allow you to chose which partial you want to trigger on, so you can tune the partial of your choice. Other tuners (e.g. AP tuner) will graph all the partials in real time relative to harmonicity (it's scarey how they wander around) so you can see just how bad things can get.

Messing with nut and saddle compensation is usually done so as to get the fundamentals of the fretted notes as close to equal temperament as possible (at least, that's what I aim for) and harmonicity issues are generally left to look after themselves, as they are usually regarded as a string issue. However, that's not entirely true. The higher partials can be frequency shifted by coupling with body resonances, just as fundamentals can, so it is possible to have a single, particular partial seriously inharmonic when most of the others are fine. The fix, such as it is, is to keep the body resonances off scale tones.

Most of the seriously bad inharmonicity that shows up on a spectrogram (particularly of "nylon" strings) we don't hear, anyway. This is due to the psychoacoustic phenomenon known as masking, whereby a partial that is ~10dB lower in amplitude than another is masked by the louder one (that's the abridged version), so generally, we don't perceive too much of a problem. However, if you ever hear a guitar with very low inharmonicty (and good intonation), one tends to hear it as clarity and separation (knocking on into loudness), because all the harmonics align, making it easy for out ears/brain to discriminate pitch rapidly. It's not a bad thing to aim for, if that's what you want.

So, I'll answer your question:
with a question.

Does it sound in tune?

I suspect Todd is no longer of this world.

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

Trevor,
I think if I understood this as well as you, I would drive myself crazy with all the possibilities and corollaries. At least that's how I will convince myself to be happy in my ignorance. Now where is my glossary?

My blue Snark just exploded.

No stretching of the octaves? I thought 6 cents of stretching from the 6th string to the 1st would be a conservative minimum.

I notice that my digital tuner doesn't seem stable especially on the unwound strings. Based on Trevor's post above, I just tried tuning the open strings plucking at the 12th fret. Tuner was dead steady, and it produced a noticeably better tuning! I'm going to incorporate that into my tuning from now on.

I pluck my strings and turn the knob thingy until it sounds good.

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Old growth, shmold growth!

I know, that's what I wound up doing after the digital tuning, because it was never satisfactory! I like this new method though, because for me it is faster. Some people have better ears than others. Mine are slow to tell me what is right, but quicker to tell me what isn't!
Mike

You might think about getting your dog's opinion.

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It's not what you don't know that hurts you, it's what you do know that's wrong.

I thought I wouldn't complicate my previous post further by going into stretch tuning as well! But seeing as you ask....

When I tune I use an octave/double octave matching method. So I'll tune, say, the 1st string to an E reference (equal temperament) then tune the 4th partial of the 6th string ("harmonic" on the 5th fret) to match. That leaves the open 6th about 2 cents flat compared to the equal temperament E, so the stretch I use is only 2 cents (which is below the limit of resolution of a lot of the LCD type tuners). It does make an audible difference for me, at least. 6 cents would be a lot of stretch, in my experience. The 4th string, 2nd fret is then tuned to the 12th fret harmonic on the 6th string etc., so everything is tuned to E by matching open string "harmonics" with fretted notes.

Works for me, anyway...

_________________
Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

This post really came in handy for me. While working on the intonation for my current project the fretted 6th string was sharper than I thought it should be based on the saddle and where the other strings ended up. Before I pulled out the file I played a bit using some this information.

The suggestion I used was to tune the open string picking at the 12th fret. I was surprised at the difference. Also I recently picked up a Peterson tuner with a mechanical strobe light and used it instead. With it I could see some of the partials at slightly different frequencies. I was a bit intimidated at first with all of the swirling rings, but I finally got the concept of tuning the large ring (lowest frequency) that would tune. That way I was more likely tuning the fundamental frequency.

It may be the placebo effect, but I also noticed that tuning the guitar with the cheap tuner picking at the 12 fret or using the Peterson instead of the electronic tuner the guitar sounded better in tune as I played.

To close the story, with proper tuning the 6th string showed it was much closer to being properly intonated, I could have very well de-intonated the string, if I had completely trusted the electronic tuner.

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http://www.Harvestmoonguitars.com

In practice, I find that inharmonicity is an issue that has to be dealt with on autoharps and zithers, and tune them accordingly. On guitars, it makes more sense to me to simply tune the fundamentals. They are inherently kind of out of tune anyway. (yes, I know about nut and saddle compensation... )

I started this thread based on ideas provoked by Arthur Benade's "Fundamentals of Musical Acoustics". Mr. Benade offers several examples of musical notes that cannot be adequately defined by their fundamental frequency. He gets very technical, and provides many careful measurements of frequencies for the partials of various instruments. For him, the bottom line is how these measurements compare to the gold standard, which is: "I pluck my strings, and turn the knob thingy until it sounds good".

I've had good luck tuning my guitar by several methods. My idea is that every tuning method has its faults, and they all work as long as you cross-check with another method. Understanding the errors built in to the tempered scale helps me make sense of some of the discrepancies. Accounting for the inharmonicity due to string stiffness explains a bit more. Recognizing that even on the most carefully built guitars, there will be large displacements of several partials because of body resonances, keeps my ears open.

Tuning the lowest resonances of the guitar to avoid a strong wolf has to be a good idea. Trying to tune every resonance that might interact with a partial of any note is a Herleculean errand (The works of Tommy Smothers have also informed my inquiry.) Even stopping at the sixth partial, there are hundreds of partials for the notes of a guitar. They group themselves into about 50 areas. No one I'm aware of is tuning 50 natural resonances on their guitars.

Does it matter? Well, yes--the pitch errors of partials caused by body resonance ranged from +10.7 cents to -13.4 cents on the guitar Arthur Benade measured. Well, no--as long as you cross-check with other tuning methods, and as long as you don't equate pitch with fundamental, the errors will work themselves out.

We talk about cents, but we mostly hear beats. For example, on the guitar Benade measured, the second partial of the low E (6th string harmonic at the twelfth fret) was 10.7 cents sharp. That would be a 1Hz beat frequency. On the same guitar, he measured the 5th partial on the 4th string as 9.4 cents sharp. that would be a 4Hz beat.

Our ears try to average all of this out, and make sense of it. The conclusions they arrive at depend on volume (pitch v volume sensitivity), listening environment (masking noise), and listener sensitivity (most of us could hear to 30k when we were little, most of us can't hear beyond 20k now).