String "inharmonicity"?

[Alan Carruth]

Niel K Walk wrote:
"There are a lot of things that come into play, including:

- the saddle material
- the width of the slots in the nut
- if a capo is being used; and if so, if it's seated firmly but not too much so as to pull the strings sharp.
- whether or not a string's ball end is seated firmly against the underside of the bridge plate
- if the bridge is slotted
- the nature of the bridge material
- the stiffness and thickness of the plectrum material"

Some of these have an effect on the intonation, but none of them affect the inharmonicity of the string. That's all about the string. Sorry.

It's pretty simple, really: we're not dealing with 'ideal' strings. Physicists talk about ideal strings, which are not like real ones in some important respects. In particular, ideal strings have no stiffness, so the only force they feel to straighten them out when they're displaced comes from the tension. Real strings have stiffness.

Suppose the string is vibrating in it's 'fundamental mode'. That's the lowest resonant frequency of the string; it's vibrating 'up and down' with every part of the string going in the same direction at any given time. It only has to bend once. For those with the math chops it's fairly easy to come up with an equation that will tell you what that frequency is, but the real beauty of it is that you can use the same mathematics to figure out what the string does as you divide it into pieces. The whole string can be looked at as two strings half as long with the same diameter and tension, joined in the middle. As long as the two sections at the ends are moving in opposite directions the central point doesn't move. So as far as the string is concerned it's just like an end point, and each half works like a separate string that's half as long, with a pitch that's twice as high as the original string. You can divide it up into as many equal pieces as you want, and each one will work like a short version of the original.

The problem is that for those shorter sections stiffness does get to be an issue in real strings. Greater curvature means more bending, and the stiffness adds to the tension as a force trying to straighten the string out. This is especially a problem for fatter strings, since the stiffness of the string rises as the fourth power of the diameter. A .020" string is 16 times as stiff as a .010" string of the same material, and that's enough to shift the upper partials quite a bit sharp.

It really helps with this to have the string up as close to it's breaking tension as it can stand without actually breaking too often. For steel strings the rule of thumb is to run them at around 75%T. Octave G strings on 12s are closer to 85%-90% T, and we all know how often they break. They do sound good, though.

In theory, and very nearly in fact, any string of a given material and length can be tuned up to the same pitch before it breaks. Smaller diameters tend to go a little higher, but not all that much. Your B string could be tuned up to E without too much risk (to the string), but it would still be more inharmonic. For a given material the tension at pitch goes as the square of diameter, but the stiffness goes as the fourth power of the diameter, so the stiffness rises faster. Tune it to B and the stiffness is the same, but the tension is much less. Now think about a plain steel G (or the nylon G on most Classical guitars...).

A lot of inharmonicity may affect the perception of pitch. Our ears and brain seem to report a pitch that's something of a 'weighted average' of the fundamental of the strongest partials it can pick up. The fundamental doesn't even have to be there.

Another issue is that when you have a low pitched note that has strongly inharmonic partials those can clash with the fundamentals of higher notes. This is a real issue on small pianos. They commonly use 'stretched' octaves, so that the fundamentals of the upper notes agree reasonably well with the higher partials of the lowest notes, and don't clash too badly.

[JimCA]

Originally Posted by robj144

This is more or less true, but you're talking about the natural scale where the intervals are ratios of whole numbers. In the tempered scale, the major scale is based on intervals of 2^(1/12) = 1.059. So the fifth is 1.498 of the root note and if a string was perfectly harmonic, the third harmonic, or fifth, is 1.500 times the root, as an example.

You're right this is the natural scale versus the tempered scale. But the tempered scale 5th is actually 1.4983, which is within 0.11% of the natural scale 5th. I would say it's much more true than less true.

[Stomp]

Originally Posted by tadol

That is fascinating - If you have links to any videos, showing how this is achieved, I’d be most interested.

Multi-dimensional tuning indeed!

A few examples (we don't just make guitar tuners )

Here's a video made at one of our customer's premises, Whitechapel Bell Foundry:



Handpan making by PanART in Switzerland:



Steelpan tuning:



Marimba bar tuning:





A cool harp gong:

[rick-slo]

A visualization I just drew. The higher the harmonic of a vibrating string the greater percentage of it's (in theory) wavelength is reduced by string material stiffness.




An aside: as a plucked string gets towards losing most of its energy (gets quiet) the pitch tends to increase (due to the same deal of string stiffness) - most noticeably
on the bass strings. I first noticed this audibly in some of my recordings as I let the last notes in a recording fade out.

[joellidstrom]

I am in my sixth decade as an electric guitarist, was a performance major in classical guitar at the U of MN, and am a concert piano technician and a rebuilder specializing in Steinway grand pianos. I’ve been editing Wikipedia pages and came across some misleading information about guitar inharmonicity, and so have given it no small amount of consideration.

While there is inharmonicity present in guitar strings, there are many factors that contribute to out-of-tuneness to a vastly greater degree, rendering inharmonicity moot. (None of the following are new to most of you, but they are important to re-visit.)

1. Bridge saddle. The relationship between bridge saddle placement and fret spacing is critical. These go hand in hand to provide an accurate octave into which the frets define the chromatic scale. Even if frets are perfectly positioned, if the saddle is misaligned, every fret will be in the "wrong" place.

2. Fret accuracy. Frets must be precisely placed. To what, the nearest hundredth? Thousandth? Needless-to-say, any variability from the intended placement-algorithm narrows some intervals while widening others.

3. Fret width. Given precise placement, each fret has a width. Is the string always vibrating from the exact middle of the fret? What about as the fret wears, or as some do and others don't? Do they still vibrate from the middle?

4. Fret height. The pitch varies according to how tall the fret is, and how hard you push the string down. (Remember Gibson’s “Fretless Wonder”? I didn’t like them. I couldn’t get my calloused fingers under them enough to comfortably bend pitches. But there was such little space between the top of the fret and the fingerboard that you essentially couldn’t alter the pitch by over-pushing.)

5. Action height. The further the strings are from the fretboard, the greater the increased tension resulting from the greater deflection. This deflection varies greatly from nut to the last fret.

6. Guitarist's technique. Players with exceptional technique do not "squeeze" the strings, or dislodge them from a vertical push to the fret. Those that do force some pitches sharp. Further, with nylon strings, pulling strings back toward the nut sharpens the pitch, and pushing toward the bridge flattens the pitch. It is how a classical guitarist creates vibrato (as opposed to, say, a blues guitarist who displaces the string across the neck).

6. Equal temperament. While fret positioning theoretically solves the 12th root of 2 problem, simply by placing the frets spaced that very way, having done so assumes much. Open strings must be tuned as appropriately wide fourths, and the major third between the open 'g' and 'b' strings must be appropriately wide. Slight variations have large implications.

7. Fast beating intervals, and the complexity of sounding notes together. Major thirds, and to a lesser degree major sixths, are highly volatile intervals in the equal temperament system. If they are too wide, they sound "sour". And if they are narrow? They might sound lovely, but then other intervals, including other major thirds, become correspondingly too wide. Any pitch error creates not just one error, but a multitude. That is why expert piano tuning, vastly more complex than guitar tuning, often produces more satisfying results. Each note exists independently, and is tuned using a sophisticated series of crosschecks. With the piano you tune 230 strings; with the guitar you can tune no more than six, despite the instrument's ability to play 120 notes. Tune five of those six perfectly and you have, not 20 notes wrong, but those 20 interacting with thousands of other combinations of pitches.

8. String scale. The shorter the scale, the more it exacerbates all the above errors.

I’m sure others will have thought of even more things. It all adds up to inharmonicity in a guitar being more a theoretical consideration than it is a practical one.

[musicman1951]

Originally Posted by Stomp

We make tuners for instrument manufacturers who need to see the fundamental and partials separately displayed.

When manufacturing instruments such as steelpan, handpan, chimes, bowls, bells, cymbals, gongs, blocks, marimba & xylophone bars, they tune the fundamental and several overtones separately by removing material from specific areas.

With reed instruments like harmonica and accordion, you can also do this, although to a lesser extent.

I think of it as "multi-dimensional tuning".

I've experienced this. One of my percussion students had an internship in our summer music program. He was working on some instrument maintenance and came in with a carpenter's hammer and asked if he could try out some concepts from his reading - on our gong. I thought that hammer was an odd choice, but the gong actually sounded slightly less beautiful than hitting a metal garbage can lid, so I figured I had nothing to lose (as you might imagine, we used that gong almost never). About 45 minutes later everyone in the office jumped at the incredibly beautiful gong tone coming from the rehearsal room.

[PineMarten]

I tend to notice inharmonicity at the lower end of pitch that's viable on a given scale length, like the low C on my cittern, the low B on my 5 string bass or the low E on my very short scale parlour guitar. And I'd agree that sometimes a lighter gauge of string sounds sweeter and more in tune in these cases. There seems to be a juggling act between tension at pitch, the innate stiffness of the string and the setup and response of the instrument that's hard to anticipate. On my parlour guitar, so far Martin flexible core strings in 0.052 gauge are the most pleasant low E that I've found, and increasing the gauge with standard strings makes things worse. I hear it as a muddy, confused sound with pitch that's hard to hear clearly even when it reads in tune on a tuner.

[marciero]

Originally Posted by JimCA

You're right this is the natural scale versus the tempered scale. But the tempered scale 5th is actually 1.4983, which is within 0.11% of the natural scale 5th. I would say it's much more true than less true.

Its worth noting though that in terms of cents, the difference in tempered scale fifth to the "natural" scale fifth- what Ive always referred to as the Pythagorean fifth- is
1200log2(1.5) - 700 approx 1.955- about equal to the correction due to young's modulus at the first overtone given in the white paper/notes cited above (interestingly from an electrical engineering course "Technologies in Music" at my alma mater), which is listed in the table at 2 cents. The differences do get larger at the higher overtones.

Interesting to bring in the inherent dissonance of tempered tuning, but this is apples and oranges as inharmonicity is about the overtones of a single vibrating string, and results from the physical properties of a given string, while tempered tuning dissonance is more about tuning of the fundamentals of notes sounded together, independent of physical properties of whatever produced the notes

[Tunesalot]

Just popping in to add that I learned so much about all this by studying a bit on the sitar. Makes me feel like a neophyte studying an instrument with so many strings/tunings/uses. Lots of fun to be had with the sitar.

[rick-slo]

Hopefully the guitar was built with compensations addressed at inharmonicity. In steel string guitars the saddle is angled to help deal with it. Of course the best angle would be in part
related to the string gauges used and the tuning (pitch) of the strings. Nylon string guitars with much less inharmonicty don't use an angled bridge.