Phase Coherence as a measure of Acoustic Quality

[Mike Sulzer]

Mike thanks for the explanation. That 3000Hz filter shows the amplitude leading way out in front of the original signal's trace. What's the explanation for that?
What did you use to excite the string (and where) and what is the pickup's location relative to the string's nodes?

Good question; I should have said something about this.

amplitude leading: In these digital filters, the relationship of the response of the filter to an impulse is such that the peak of the impulse response is coincident with the input impulse. That is, the response begins before the impulse. A filter made from Rs, Ls, and Cs would have to have a delay, of course, since it needs to be causal. Using causal filters here would meant that the delay would be different for each filter since the bandwidth of each is different. That would make it somewhat harder to see the frequency shifts of the harmonics; so I just did the most natural thing, which is a no delay filter.

You can see the lack of delay on all of the filter outputs; the lower frequency filters show more of an effect (earlier ringing) since they are narrower filters. The "beat repetition rate" increases with the filter frequencies, showing that the harmonics are going sharp.

The pickup placement is standard single coil strat bridge pickup location. The string was picked with a plastic pick, flexible rather than stiff, about midway between the neck and bridge pickups. (There is no middle pickup on this guitar.)

[David Schwab]

Slightly aside, but not much:

Mike just very astutely mentioned what is called "stretching" the tuning of a piano. Bass string fundmentals below a certain point, usually middle C, are tuned successively flatter, and treble strings tuned successively sharper in order to get the upper partials of the lower strings to be "in tune" with the fundmentals of the higher strings. The art of piano tuning is in the stretch, and different piano tuners have different approaches, and different piano players prefer one or another or another depending on their particular touch and also just how their brains put all these slightly out of tune notes and harmonics together. There is no correct way to do this as pianos are never in tune with themselves because of the compromise of tempered tuning AND because they use real world strings whose harmonics drift ever more sharp the higher up in the harmonic series they go. Then there's the issue of a pianists touch...how hard they hit also has an effect on the harmonic content of each note.

But these very imperfections can make for "signature sounds" for individual pianos as well as individual tuners and players.

Right. That's because the bass strings on a piano are very stiff. A stiff string vibrates like a rod, and the harmonics are out of tune.

But you knew that already.

[Rick Turner]

Yes, I knew that already!

About 40 years ago, there was a really good article in Scientific American on the physics of pianos, and it got deep into all of that. Harmonic theory is based on an infinitely flexible one dimensional string. In the real world, the harmonic nodes are far from perfect, and they are actually quite fuzzy in terms of location. Hence the transition of harmonic series from kind of string-like over to clamped bar-like. This is the fundamental reason (pun intended) for the need to compensate string length at the bridge (or elsewhere), and that is a separate issue from temperament. Not my own snarky temperament...pun intended...

[Mark Hammer]

....in addition to the sonic difference between roundwound and flatwound, as well as wound vs unwound strings.

There are many different ways to reach the same diameter.

[Mike Sulzer]

Rick and David,

That is not quite how it works. See: Inharmonicity of Plain Wire Piano Strings.

Simple theory assumes an infinitely thin wire. The stiffness referred to for the generation of inharmonicity is the same stiffness that says how much a string lengthens with a given applied force, and thus is involved with the string vibration. The string must have stiffness.

A brief summary of the very good but approximate theory:

delta = b(n)^2

delta is a measure of the relative frequency shift due to inharmonicity. It is usually measured in cent., that is, 100th of a semitone. The spacing between semitones is the twelfth root of two, about 1.0595.
b is a constant depending various things.
n is harmonic number.
So inharmonicity increases with the square the harmonic number. That is pretty fast.

b depends on:
-- the square of the diameter of the string
-- inversely with the square of the simple fundamental frequency
-- inversely with the fourth power of the length
-- linearly with the ratio of Young's modulus and the density

So it goes down very rapidly with increasing length, everything else the same.
Young's modulus is the measure of the stiffness, that is, how hard it is to stretch or compress it by small amounts that it can spring back from.

The theory in that paper is for a plain (not wrapped) string. Wrapping a string reduces the inharmonicity since the density is effectively increased without making it stiffer. But it is really more complicated than that.

[David King]

D'Addario actually make bass strings that are wound with tungsten wire to get the mass up at a smaller overall diameter, (W is it's symbol -it's a long story) is significantly heavier than steel.

[Mike Sulzer]

D'Addario actually make bass strings that are wound with tungsten wire to get the mass up at a smaller overall diameter, (W is it's symbol -it's a long story) is significantly heavier than steel.

As incandescent light bulbs are phased out, the price of tungsten might drop.

[David Schwab]

Rick and David,

That is not quite how it works. See: Inharmonicity of Plain Wire Piano Strings.

Simple theory assumes an infinitely thin wire. The stiffness referred to for the generation of inharmonicity is the same stiffness that says how much a string lengthens with a given applied force, and thus is involved with the string vibration. The string must have stiffness.

A brief summary of the very good but approximate theory:

delta = b(n)^2

delta is a measure of the relative frequency shift due to inharmonicity. It is usually measured in cent., that is, 100th of a semitone. The spacing between semitones is the twelfth root of two, about 1.0595.
b is a constant depending various things.
n is harmonic number.
So inharmonicity increases with the square the harmonic number. That is pretty fast.

b depends on:
-- the square of the diameter of the string
-- inversely with the square of the simple fundamental frequency
-- inversely with the fourth power of the length
-- linearly with the ratio of Young's modulus and the density

So it goes down very rapidly with increasing length, everything else the same.
Young's modulus is the measure of the stiffness, that is, how hard it is to stretch or compress it by small amounts that it can spring back from.

The theory in that paper is for a plain (not wrapped) string. Wrapping a string reduces the inharmonicity since the density is effectively increased without making it stiffer. But it is really more complicated than that.

Right, but stiffness is still a factor in pianos:

For instance, a stiff string under low tension (such as those found in the bass notes of small upright pianos) exhibits a high degree of inharmonicity, while a thinner string under higher tension (such as a treble string in a piano) or a more flexible string (such as a gut or nylon string used on a guitar or harp) will exhibit less inharmonicity. A wrapped string generally exhibits less inharmonicity than the equivalent solid string, and for that reason wrapped strings are often preferred.

Inharmonicity - Wikipedia, the free encyclopedia

So the harmonics in the bass strings out of tune, and the treble strings are stretched to match. You can also hear this in a short scale bass with flat wound strings. The notes on the low E often sound a bit out of tune.

But equal temperament is a compromise anyway.

[Mike Sulzer]

Right, but stiffness is still a factor in pianos:

Yep. The primary reason for using a very stiff material such as steel is to get powerful vibrations for high volume. The rest is side effects.

I do not think that the analogy to vibrations in a bar is a good one. A bar is a very different kind of vibrating system.

[bbsailor]

Yep. The primary reason for using a very stiff material such as steel is to get powerful vibrations for high volume. The rest is side effects.

I do not think that the analogy to vibrations in a bar is a good one. A bar is a very different kind of vibrating system.

Mike and David,

Strings have two lengths based on the stiffness of the string.
1. The physical length that defines the nut to bridge length that is used for the fret spacing based on the 12th root of 2 formula.
2. The speaking length. Here the string is restricted by the stiffness of the string such that it does not actually vibrate exactly at the nut or bridge but just slightly forward of these locations.

This must be why Earvana (http://www.earvana.com/) has developed a compensated nut to attempt to offset the differences in these string lengths to attempt to restore more accuracy in guitar tuning. Note that most of the compensation is in the back-set location of the nut string groves to attempt to better align the physical string length with the speaking string length. Note that each string is offset by a different amount with the stiffer strings being offset more.
The Earvana instructions also require the fingerboard to be shortened by a few mm (for some nuts on new guitars) to properly install this compensating nut but each string then has a slightly different length. The bottom line is that it still compensates for string stiffness.

This is just my understanding of the string stiffness issue. I'm sure others will have a different opinion.

Joseph Rogowski

[David King]

Luthiers have been compensating nuts since the beginning of time. I can't see anything patentable in that unless it's a way to sweeten some chords while inevitably souring others. The concept that harmonics go sharper than the fundamental is novel to me but I've since verified that the 7 foot Mason and Hamlin in my living room has the low notes tuned flat and the high notes tuned sharp by a very small amount, less than 1 cent either way. It's been 6 months since I had it tuned though. I recently acquired an old Conn tube strobe tuner which will presumably show me the fundamental and the harmonics moving in opposite directions.

[Joe Gwinn]

After digesting many of Griesinger's articles on the importance of phase coherence to engagement, it occurs to me that variations in phase coherence may be the reason that some pickups sound clear and immediate, and yet others are not, with little or no difference in their spectral response curves.

I'm still thinking about this, but in the context of a transducer (versus a concert hall), the way one achieves coherence is by ensuring that the transducer (as installed and used) exhibits a linear phase response, which is a fancy way to say that all frequencies suffer the same time delay (in seconds) when passing through the system.

Linear phase response also preserves the waveshape of attack transients, so long as one does not run out of headroom somewhere.

[Rick Turner]

Joe, you're speaking of "group delay", and that's the factor that cable guru George Cardas said was improved in my experimental Litz wire pickups and was responsible for my perception of "tighter", more immediate low end.

Mike, I'll have to dig up the old article in Scientific American on piano strings. It says exactly what bbsailor said here. Unfortunately, it came out a good 40 years ago.

bbsailor...that's it in a nutshell. The effective length of the string lessens for upper partials because of the stiffness at the nodes of the note, and that throws the harmonics ever sharpward.

If we could have one dimensional strings...strings with tension but no stiffness...then we'd have perfect harmonics and no need for intonation compensation.

David, yes, the tuning of your piano was stretched. BTW, an ancestor of my stepfather was Emmons Hamlin, co-founder of Mason and Hamlin. At their peak, they had a factory with 500 employees.

[Rick Turner]

Here you go:

c:\windows\TEMP\afx7072.TMP

It's not tension that makes partials go sharp, it stiffness due to the aspect ratio of the diameter of the string vs. the length and the strings natural stiffness.

I must have seen this article in 1970 or so, and it opened up my ears.

[Mike Sulzer]

Here you go:

c:\windows\TEMP\afx7072.TMP

It's not tension that makes partials go sharp, it stiffness due to the aspect ratio of the diameter of the string vs. the length and the strings natural stiffness.

I must have seen this article in 1970 or so, and it opened up my ears.

Well, to start with, it is completely incorrect to describe the piano as exceptional in having inharmonicity. One expects all string instruments to be so at least to some degree, and even with sustained-tone instruments, the behavior might not be so simple. Musical instruments are quite complicated, as hinted at here (I do not have the whole paper yet):
http://iopscience.iop.org/0034-4885/62/5/202:

Musical instruments are often thought of as linear harmonic systems, and a first-order description of their operation can indeed be given on this basis, once we recognise a few inharmonic exceptions such as drums and bells. A closer examination, however, shows that the reality is very different from this. Sustained-tone instruments, such as violins, flutes and trumpets, have resonators that are only approximately harmonic, and their operation and harmonic sound spectrum both rely upon the extreme nonlinearity of their driving mechanisms. Such instruments might be described as `essentially nonlinear'. In impulsively excited instruments, such as pianos, guitars, gongs and cymbals, however, the nonlinearity is `incidental', although it may produce striking aural results, including transitions to chaotic behaviour. This paper reviews the basic physics of a wide variety of musical instruments and investigates the role of nonlinearity in their operation.

The author of your article seems unaware of the very fine work done by Young, a scientist at NRL, about 15 years earlier. This is the article I referred to above. It explains the physics much better and presents many more measurements.

Remember, it is the stiffness of a string that allows it to support tension. It is not correct to treat the tension and stiffness as independent.