String theory! - spreadsheet -The mathematics of intonation

[JohnH]

I’ve been wanting to get my head around how best to set up guitars and understand how the various factors interrelate. So I’ve made this spreadsheet to work out the theoretical amount of compensation at the bridge, based on all the various parameters. This includes string gauge and type, tuning frequency, scale length, action height, neck relief and nut height.

To do this, I used a formula that relates string tension, frequency, length and mass, and combined that with what I know about stress and strain, and how when a string is depressed, the extra length required will stretch it slightly more, adding tension and so raising its pitch.

The spreadsheet is here:
Stringtheory1.1



Inputs are in yellow cells, results are green, showing string tension, the amount of compensation in terms of added length at the bridge, and a graph of how intonation will vary at each fret, once adjusted for correct open tuning, and compensation based on the 2nd harmonic and 12th fret.

Scale length is input in inches, but all the maths is in metric SI units (it’s the only way to make physics work properly!). The strings are entered as numbers 1 to 6. This calls up data from a table for diameter, mass, and the base frequency in standard tuning. Now it would have been nice if all strings were consistent for these parameters, in which case string properties could be calculated. But this is not the case, particularly with wound strings where the shape and diameter of the core, and the weight of the winding wire varies for each brand, and between strings. Even the plain strings are not entirely consistent, since they are both steel, plus coating. So the data here is from the only complete set of info I can find, which is a set of 10-46 strings from Dean Markley, with wound D, A and low E, as described here by Noyce Guitars (which is also a good article for reference):

www.noyceguitars.com/Technotes/Articles/T3.html

So, to operate the spreadsheet: Input the string number and scale length. Then the action you want at the 12th fret, and the neck relief. This is input in terms of what you will probably measure, being the relief measured beneath a straight edge from 1st to 17th fret. 0 is dead straight. This is then converted to an overall curvature of the neck. Finally, the height at the bottom of the neck slot. 0 means the slot is exactly at the depth of a 0 fret, following the profile set up by the other frets. This gives the best result, but if you want it a bit higher so help buzzes, enter it in mm.

Then, press the ‘Intonate’ button, and a goal-seek function will adjust compensation until intonation at the 12th fret is correct. You can now see how the result varies at each fret, with errors shown as cents of a semitone (50cents = ½ semitone – ew!). Im not sure what a perfectly acceptable result would need to be, but I suspect that 10 cents is too much, 5 cents may be OK and 2 cents is good. There is also a slider, to make a further manual adjustment of +/- 1mm. Macros need to be enabled to run these features.

The calculations are based on a perfectly light touch to depress the string down to the fret. Any heavier fretting action will go sharper, but is not captured here.

Observations

So what does this tell us? Here are some conclusions about what may make acceptable intonation easier or more difficult to achieve:

• Shorter scales are trickier than longer scales
• Low fundamental frequencies are harder than higher ones (eg G is more problematic than high E)
• High action and high neck relief makes the problem greater
• A nut that is not cut deep enough makes a big difference in the first few fretted positions, but about 0.1 or 0.2mm excess height is OK
• Wound strings work better than plain strings for intonation
• Drop tuning of a semitone or two doesn’t make too much difference
• High frets 18 and up seem to tend to go flat. Does anyone find this in practice? It may be compensated by fretting pressure and bending
• String weight doesn’t actually make much difference (eg changing from ‘10s’ to ‘11s’). For theoretical strings made of the same material and tuned to the same frequency and same set-up, the compensation is the same. In practice, irregularity between strings and sets makes a small difference.
  
  That’s all for now, I hope that is interesting!
  

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cheers
John

EDIT 2/9/13 Added to version 1.1: Nut compensation, and string action heioght at 12th based on setting treble side and bass side, with linear variation between. 

[JohnH]

I added another input box to this, to allow for compensation at the nut. Ie, the extent to which the nut is moved slightly towards the bridge. Its really quite interesting. With about 0.5mm of compensation there, and maybe leaving the intonation at the 12th about 1/2 cent sharp, one can achieve intonation within about a cent at all positions on all strings, with practical amounts of action and relief. I think nut compensation is a factor that would be well worth considering when replacing a nut.

See the link in the post above.

Here is a very comprehensive series of articles about intonation, from Dooling Guitars:

www.doolinguitars.com/intonation/intonation1.html

[4real]

All interesting stuff, I read the whole series of artiles that cover things well and fairly concisely as to what the issues are to a large degree. A lot of the intonation thing though will always be something of a compromise, that is the nature of the temperament.

Personally, tuning is critical but there are some ways to work around the limitations of the intonation problems inherient in the guitar.

I am intrested in the technique ways of dealing with such things, once one has an intonation as best it can be. To me, the worst thing are those 'sharp 3rds'

" But Those Thirds Sound So Sharp...

OK, I'll admit that even I have a hard time listening to those Equal Tempered thirds sometimes. And sometimes, I'll cheat, just a little, to get the effect of Just Temperament out of an Equal Tempered guitar.

The way do this is to reserve a string for the third of the chord, tune that string down a bit, and play around it the rest of the time. The simplest example is to tune the open E chord to Just intervals, and play barre chord versions of that chord up and down the neck. As long as you only play that Major chord shape, it will be in Just Temperament everywhere on the neck. This may sound pretty limiting, and it is, but it's a trick that's been successfully used on thousands of studio recordings.

This idea can be expanded upon by refingering more complex pieces of music so that the third always falls on the same string, moving anywhere on the neck to make that happen.

Much beyond this though, and you'll start thinking about split frets, more than 12 frets per octave, B-Benders, pedal-steel style string stretchin' contraptions, hexaphonic pickups with hexaphonic harmonizers and MIDI presets to track your momentary temperament desires, DSP plug-ins and all-night digital editing sessions in the studio... beyond here lies madness. "

from 'intonation V' www.doolinguitars.com/intonation/intonation5.html

For a long time, I've tried hard to voice chords and even melodies in the centre of the neck octave.An open A major is typically where the problems are most apparent, that B string C# typically is sharp. To me, 'sharp = sour' and far better to be a little flat. but one can not reasonably put the whole guitar out all over, for the sake of that one chord for instance. Any time there are octavels, such as in a C chord, or an open E say, where there is an octave between the B and B string close to the nut are going to be very problematic. But what if one voices the same not combinations between the 5th and 7th fret, perhaps include some open strings for the 'jangle' if this is the effect required.

More recently, in arranging tunes, I trying to arrange it to have both the thirds, and open strings, often the added the 2nd, the effect of which is to kind of provide a blur the problematic thirds, even if the guitar is very well in tune and arranged as best one can, the temperament has always that quality.

But I've gone through all kinds of places 'where madness lies'...such as tuning everything in 4ths, or embracing quartal harmony or avoiding the thirds...going very clean to avoid the intermodulation of distortions...

But practicality must reign. Standard tuning is pretty darn special, but open and altered tunings do offer a lot and many people temper their tuning rather than to strictly allow a tuner to be 'accurate'. I tend to tune in various ways, but in particular the octaves between all strings, as if that is out, things are seriously bad to any ear.

I suspect that I did, perhaps a little conciously as I ahd it in mind, but to the slightest degree, moved my tele's LSR nut a little forward...as fitting this necessitated that some of the fretboard be removed and that it is held on by screws so adjustable and potentailly moved...but the tiniest amount. It does play beautifully in tune, but very anacdoatal, so dont take that as gospel. On my last 'jazz strat' thing I was considering adding a compensated nut and did a bit of reading then too, but I'm not at all sure about this kind of thing really.

But...those articles are interesting reading and the spreadsheet is kind of cool, though not sure if all the variables are accounted for and always a good ear and awareness of the compromises and finding the best fit for the music you want to play and your technique, the degree one is going to pursue this kind of thing, how you hear and how much courage one has to experiement with this kind of thing, such as moving the nut. I do know that a lot of custom lutiers have their own 'adjustments' to things like the nut distance to first fret and there are lots of 'systems' out there...but these days, I think that perhaps there is a good combination of factors to get it superior to most without radical structural things, and plenty one can do technically to avoid the problem aspects.

[JohnH]

Sept 2, 2013 2:52:16 GMT -5 4real said:

But...those articles are interesting reading and the spreadsheet is kind of cool, though not sure if all the variables are accounted for and always a good ear and awareness of the compromises and finding the best fit for the music you want to play and your technique, the degree one is going to pursue this kind of thing, how you hear and how much courage one has to experiement with this kind of thing, such as moving the nut. I do know that a lot of custom lutiers have their own 'adjustments' to things like the nut distance to first fret and there are lots of 'systems' out there...but these days, I think that perhaps there is a good combination of factors to get it superior to most without radical structural things, and plenty one can do technically to avoid the problem aspects.

Thanks for your thoughts. Let me know if you can think of variables that need to be added to make it more useful. I see the spreadsheet mainly as a tool to help understanding, rather than getting actual numbers of it to build with, but with the right data, its intended to be as correct as possible. It helps to counter misunderstanding of some rather non intuitive effects, such as the fact that if you bring the nut closer to the bridge, you might think everything goes sharp with a shorter string, but of course actually, it all gets flatter etc. also, I have not seen eleswhere, plots of how intonation varies at each fret. Its only addressing comparision to an even tempered scale, not the differences with the less common types specific to keys.

One thing Im aware of is that the actual shape of neck curvature to provide relief is quite complex and will not be a constant radius, nor the sinusoidal shape that I assumed so far in the spreadsheet. It will vary per guitar, but I might work out a more liley profile based on a structural analysis model.

[4real]

A lot of this stuff really is anti-intuitive, but all so important...well done for providing a tool to explore it without hacking away at a guitar...

However, there are some DIY things one can do rather than move the nut itself. One could for instance add a 'sliver' to the front of the nut to effectively move the nut forward a touch, it is after all a very tiny amount....

eNut offer a kit for such things... www.monteallums.com/enut_tuning.html  ...it is certainly an easier less invasive approach and cheaper too...DIY versions could be devised.

There are short samples on the site there as well...

   

Of course there are 'products' too such as the earvana...

www.earvana.com/



The buzz feiten thing too makes similar claims with a nut slightly forward and tuned in increments...

www.buzzfeiten.com/howitworks/faq.htm

There are others as mentioned in the article link...here is a video of the phil hartley method...



The above video again uses a shelf nut addition...the accurate tuner and before and after might help some get the idea about the problems of intonation and nut compensation effects...

This video expalins some basics about intonation...



This video demonstaates fairly well why picking strength makes such a difference. If you look at the tuner you will see that the initial attack is sharp and then returns to 'pitch' as was tuned 'softly' with a tuner. This is a major factor in tuning problems and especially now that people are so acoustomed to relying on electronic tuners and shows the practical limitations of strategies such as nut compensation, being 'technically' correct means very little if your playing technique is such that you are pulling everything 'sharp'...

Around 21mins in the video...you can also see some of these 'problems' with say the G string and advising tuning it a little flat, as much as 15 cents, and how the A two frets higher is now perfectly in tune. As I've mentioned, everything 'sharp' sounds "sour". If you play 'fast' with a heavey technique and picks, it will pull every note 'sharp' and there wont be time for the note to settle back to tune. Slower and it might settle back but it can cause that cats 'yowl' kind of effect as the attack,the most important part for percieving the pitch. All effects that don't sound good to my ear.Now a little flat might seem as bad, but the effect of a falt note tends to be much more pleasent, even desirable, certainly over anything 'sharp'.

So there are a few things there one might consider technique wise to avoid sounding 'bad', at least in regards to tuning. It's also important when you are playing with others...a heavy strummer will often sound sharp and some chords pulled really sharp, compared to a lighter player with good intonation due to playing technique.

These are some of the effects that one needs to keep in mind from the playing perspective as no matter how 'finely tuned' your guitar might be, the player ahs a huge factor in how 'in tune' they will sound. I feel strongly that this 'player tuning' makes a big difference to how another will percieve your ability as a guitar player and the enjoyment of your music. Even more so if you are playing harmony and not just needling away out of contest with a bad or other players or yourself as is the case with fingerpicking.

As a result, my playing has become increasingly 'light' and the effects of this player intonation is far less if noticable at all on a good set up. It perhaps explains my desire to explore amplification on the acoustic, though potentially my mini-jumbo is quite loud and it has compensated bridge and very good intonation wise, as that at least with the electric guitar, one can always increase volume with the amplification and not by hitting or plucking harder. Nylon string guitars such as classicals, onemight also notice, have far less fo these problems...but perhaps that is the way I play the things.

The mods to my guitar to locking 18:1 tuners has made it a lot easier to get those kinds of tiny adjustments and for those strings to remain stable...

Although I even have an onboard chromatic tuner, I still tune largely by ear as well, not just one string to another but in octaves between strings and to get the guitar "play in tune" not just 'technically in tune or what a tuner might say. As the video ends "tune it how you play it"...for most, attention to set up and putting the effort into this is going to make huge differences to the way you sound and are percieved byt the lay person who is I think, more likely to be impressed with the sound of the instrument than how fast one plays, how complicated the chords...or how many pickup combinations one has wired into the things.

If one likes to tune down, the effect is even more pronounced...as he discusses in relation to the way a piano is tuned and may be of interest to contemplate, say if you are playing with a bass player and seeking to get a great 'band' sound between the gutiar and bass as if a single tuned instrument. Being in tune yourself is only so good if the backing track or fellow band members are tuned differently to you.

In fact, it is common for notes of good vocalists or wind players to 'tail off' flat and so the effect of say, playing hard with a slightly flat tuned guitar that played 'sharp' will give this kind of 'sweet' sound compard to the 'yowl' effect that is so common, espcially if you are playing really light strings that deflect easily with the pick on the attack and always sounds a bit 'sour', most notably when you play with others or to a perfectly pitched backing track as you so often see or hear. Of course, if you just noodle by yourself lead lines, one might easily be fooled as you are not playing the harmony or against anything that provides that 'context' and be more spell bound by the technique or speed or 'tone' or what have you.

[gitpiddler]

Replacing a zero fret with another nut produced perfect intonation on my #1, with string height @ the nut very critical to that first fret bump on the graph. Excellent work John.