Frequency response, noise and harmonics (dedicated to MH)

[vdm]

Hello everyone,
It's been a good while since I've made any attempt to post here, but I just finished my first semester of medical school and have a few weeks off. I was dragged back into making effects after hearing Jeff Parker live with tortoise here in january and loving his distortion tone... so here I am searching again.

The real reason for this post is to reiterate and expand on an idea that Mark Hammer has been flogging for a good while - the need for frequencies allowed in a guitar effect to be controlled throughout in relation to guitar, and not hi-fi equipment.


NOTE: if you do not wish to read the scientific basis, simply skip to the end for the practical uses in guitar effects


Firstly the range of a standard tuned guitar is E to let's say E (24th fret). The respective frequencies of these notes are 82Hz and 1319Hz. If we tune our E down to D it becomes 73Hz, or C = 65Hz. This shows us that the fundamental (or first harmonic) of any tone produced by a guitar in standard tuning must lie in this range.

The lower extremity is useful here because we can see that even if we create a pedal planned for use with drop D we do not need to pass any signal below 73Hz (37Hz for bass) enabling us to filter any of these frequencies with high-pass. This is because no harmonic content can be added below the bottom frequency unless there is some kind of pedal trying to (octave divider/harmonizer).

The highest E on the guitar (1319Hz) is however not very useful. We know that harmonics are produced above the fundamental tone which we can hear, and like to hear. Following is a list of the harmonics of the highest (fretted) E on a guitar:

Harmonic  -  Relation to fundamental  -  Note  -  %volume of fundamental
1st  -  Fundamental  -  E  -  100%
2nd  -  Octave  -  E  -  25%
3rd  -  8ve+5th  -  B  -  11%
4th  -  Dbl 8ve  -  E  -  6%
5th  -  D.8ve+Maj3  -  G#  -  4%
6th  -  D.8ve+P5  -  B  -  2.8%
7th  -  D.8ve+min7  -  D  -  2%
8th  -  T.8ve  -  E  -  1.6%

This table shows how each successive harmonic is quieter than the previous. A harmonic above the 6th or 7th is very rarely audible amongst the other tones.

The following is a discription of the tonal quality of each harmonic

2nd - Adds clearness and brilliance
3rd - Thickens and adds hollow nasal sound (important in clarinet tone)
4th - Brilliance and shrillness
5th - Rich, horn-like quality
6th - Delicate shrillness with nasal quality

All odd numbered harmonics above the 5th introduce dischordant notes that are unpleasant to the ear and have an adverse effect on tone. We can also see that the 5th harmonic is also dischordant to a lesser degree as a major third (which is slightly sharp) is not always going to be concordant with the chords in a song.

With this knowledge, assuming we cannot, and do not want to, hear any tone above the 6th harmonic of the highest E on the guitar, we can safely say that any pitch above this is noise and not a product of the strings vibrating, but some external artifact. The 6th harmonic of E has a pitch of 7911Hz, making it obvious that as a guitarist we could not want any pitch above this passed in our effects pedals.

NOTE: for bassists the 6th harmonic of the highest G is 2353Hz, or for a 6-string, the 6th harm of highest C is 3140Hz.


---SUMMARY and PRACTICAL APPLICATIONS---


We have seen the highest and lowest pitches made by a guitar that are pleasant to our ears and as such have points to filter from:

Guitar: 82Hz - 7911Hz
DropD Guitar: 73Hz - 7911Hz
Bass: 41Hz - 2353Hz
Bass6: 31Hz - 3140Hz

Many effects use large electrolytic capacitors for output filtering while we can see this is unneccesary. The following is fantastic tool to see the response of different values in highpass and lowpass filters.

http://www.muzique.com/schem/filter.htm

This discussion shows us that there is a great deal of audio garbage that is not part of the guitar's music that we can filter out right at the beginning of the circuit to reduce noise to be amplified later on. We can also think about how high up the neck we practically use effects for, and how much harmonic content.

I know people who have a distortion pedal just for power chords... why not just cut the frequency response to 2kHz and you still have full harmonic content for the first 12 power chords, and the slight loss you might get for anything higher is most likely inaudible.



These are simply some musings I've had the last day or two and i'd like to recommend Sir James Jeans "Science & Music" to anyone that was interested in the above. I took a great deal of info from there to write this and it's helped me to understand this topic a lot more.

So now off to all my pedals to locate all the HP and LP filters, and then figure out how they actually work :wink:

Cheers guys,
Trent

[Mark Hammer]

Thanks for the nod, mate, and thanks for the math.

One of the more interesting properties of a lot of guitar speakers, most notably the 12" type, is that they rarely have much capacity beyond 5k-6khz.  Indeed, the sluggishness that a 12" speaker can possess often masks a whole lot of "audio sins" committed along the way to that speaker, and also forms the very basis for needing/wanting cab simulators for plugging straight into sound cards or mixing boards.

Though I have a couple of other amps, the amp I play through most often is a small Fender with a sweet 8" JBL speaker that yields a usable range from 60hz-10khz (if one believes the specs). I suspect one of the reasons why I tend to be far more attentive to lowpass filtering than many others might be is because I have the privilege of hearing all that useless harmonic crap where many others don't.

Yep, the message is clear.  Watch your bandwidth, folks!

[Joe Kramer]

Nice post Trent!  The harmonic order spectrum is an especially useful tool in sound synthesis.

However, David Blackmer (the "D B" of DBX and also the founder of Earthworks), has written a number of papers on the perceptual importance of frequency response far outside the traditional 20-20K limits.  His premise, which is born-out in experiment, is that while the human ear cannot properly "hear" above 20K, a person can most certainly "sense" frequencies far beyond this, and that reproduced music with an artificially limited bandwidth has a generally "irritating" effect on listeners.

While this holds true for high-fidelity reproduction of music, it may be more-or-less debatable for particular instruments like guitar.  Nevertheless, I would not summarily discard frequencies that lie outside the practical response of the instrument, any more than I would summarily disregard everything that exceeds the reach of my eyesight.  Just because you can't see it (or hear it), doesn't mean it isn't there.

Regards,
Joe

[R.G.]

I think anyone who's beyond copying other pedal circuits should read "The Psychology of Music" by Carl Seahore.

It's old, but very pertinent.

[davebungo]

I would rather maintain a wide bandwidth thoughout my signal chain.  It sounds great so why change it?  I would only deliberately limit the bandwidth to achieve a certain effect.  A guitar is a non-linear system and so are most guitar effects.  If you apply mixed frequencies to any non-linear system what you get are complex intermodulation products including sum and difference frequencies which may cover a range of frequencies from very low to very high.  For instance, if you play through a distortion pedal and play two notes fairly close together you get a strong low frequency beat frequency (which sounds nice in solos etc.).  What about all the bumps and scratches which occur as you pluck or scrape a string?  Also, as some pedals may add high frequencies for effect, would you really want to deliberately filter these out?

This discussion shows us that there is a great deal of audio garbage that is not part of the guitar's music that we can filter out right at the beginning

It's a sweeping assumption that signals outside this range are garbage.  It all adds to the sound and people tend to like it.

A quick word about bandwidth also as many people assume that if a speaker has a usable bandwidth of say 6KHz that nothing much above can pass.  This is not strictly so.  Speakers like most other systems do not have a brick wall cut-off so there is always some significant response at frequencies way in excess of the stated bandwidth and in my view, this all adds to the complexity of the end result.  This is not to say that this is always desirable but in the main I think it is better to let it all rip at full bandwidth and tune out offending noise and stuff as an exception rather than the rule.  Don't forget the tone controls on your guitar and on your amp as well as those in graphic EQs etc.  These can all be used to help you achieve the sound you want.

[Mark Hammer]

Don't confuse bandwidth across the signal chain with bandwidth within specific elements.  You can bet your last dollar that if I had a Parker Fly or a Godin with a piezo bridge I would do everything in my power to retain bandwidth from one end of the signal path to the other.  At the same time, would I want more than 4khz going into a fuzz?  Not bloody likely.

And yes, you are absolutely correct that a rolloff at 6khz is merely that, a rolloff.  That being said, when you see a 20db dropoff just above that, clearly the ultra high end is not going to be *easy* to hear.

[davebungo]

Many effects use large electrolytic capacitors for output filtering while we can see this is unneccesary

I'm not confusing anything but I had worrying pictures in my head of vdm tearing out caps from his effects based on what I believe to be an if'y conclusion to a thoughtful piece of analysis (with respect vdm).  Nothing wrong with experimenting though - best way to learn.

[vdm]

Sorry i've been away for a while. too busy playing music to talk about it :wink:

Thank you all for the discussion and it is very useful to me and everyone around to hear both sides of each story.

Just a few points I'd like to clear up though.

1) I wasn't trying to suggest the use of these techniques everywhere in your signal chain, but simply in places where noise can be an issue, particularly with high-gain effects, or pseudo-noise-amplifying effects like compressors.

2) The comment about ripping electros out of my pedals: I'm not about to go doing things like this, but just for future reference, were I to build said pedals again, I would use smaller caps still retaining the low frequencies a guitar can give. I'm not the hugest fan of using intermodulation in my music, so i really have no method of creating frequencies any lower than a low E.

3) With regard to "all the bumps and scratches which occur as you pluck or scrape a string", I'm not sure how much of this noise would have frequencies above harmonics of the highest note on the guitar, and obviously they don't have resonant harmonics due to the nature of the noise (I think :roll: )

Nonetheless, thank you all for your comments, and hopefully we can all be more mindful of what is going on in our effects, amps and everything else, which was, funnily enough, the original purpose of the post

Trent

[davebungo]

A step change or fairly instantaneous change in any signal generates a very wide range of frequencies.  So when you pluck a string, the guitar generates a short wide frequency burst of signal before settling into the actual note being played along with any harmonics etc.  It's nothing to do with what is the highest "note" on the guitar.

Actually, that's a good question: What is the highest note on a guitar and what is the highest frequency the guitar produces when played?  I've never thought of trying to measure this.  Maybe there's a post somewhere showing this.  I'll hook up my pico scope anyway and see if it will perform a spectral analysis for me just out of interest.

Has anyone else already done this?

[Paul Perry (Frostwave)]

There simply isn't anything in the ear to respond to a stimulus above 20 KHz.
It is possible, though, that a low pass filter around 20KHz would produce phase artifacts that would be noticeable in an A/B test. (depending on the filter design).

[seanm]

How did I miss this post? I really liked the description of the different harmonics.

I think you have to take the volume levels at the different harmonics with a grain of salt. I play bass, and for the low notes the 1st and 2nd harmonic produce almost all of the sound. It has been shown that for low notes if you completely take away the fundamental, people can still tell the pitch of the note just from the harmonics.

In reality this happens all the time. Many sound systems are not really very good at reproducing a low E, but if I play a walking bass line starting on the low E, the notes will be distinguishable.

This is getting better. The old rock bassists rule was: "If you are lost or don't know the song, just pound away on the E string."  This worked because the E was so muddy it fit with everything Think old Fender Bassmans, a bass amp designed to kill the low end since the speakers of the time couldn't handle it!