The Impossibility of Tuning Fretted Instruments

Alan Brookes · From: Brummy living in Southern California

Here is an extract from a lecture which I gave to the Northern California Association of Luthiers a few years ago. Steel guitarists should be grateful that they have no frets to worry about. Unfortunately they're playing along with other instruments which are stuck with the inadequacy, and keyboard instruments that are fixed with equal temperament.

An Arithmetical Demonstration of the Inadequacy of Fixed Frets
by Alan Brookes

Let us take a string tuned to C. We shall call the string length 1 unit. It does not matter what the length. In what follows, we shall keep the tension constant and vary the sounding length in proportions. The one unit could be inches, feet, centimetres or even miles !

C = 1
If C =1 then taking 2/3 of this length will push us up to G.
G = 2/3
Let us do this again. 2/3 of this new length will take us up to D’.
D’ = 2/3 x 2/3 = 22/32
Let us now get back into the original octave, by doubling the string length.
D = 2 x 2/3 x 2/3 = 23 /32
Okay, now let's apply the 2/3 again to take us up to A’.
A’ = 2 x 2/3 x 2/3 x 2/3 = 24 /33
Again, let us get back into the original octave by doubling the string length.
A = 2 x 2 x 2/3 x 2/3 x 2/3 = 25 /33
We shall continue the same steps until we have been around all twelve notes and come back to C.
E = 26 /34
B’ = 27 /35
B = 28 /35
F# = 29 /36
C#’ = 210 /37
C# = 211 /37
G# = 212 /38
D#’ = 213 /39
D# = 214 /39
A#’ = 215 /310
A# = 216 /310
F = 217 /311
C’ = 218 /312
C = 219 /312 = 524,288/531,441

But we know that C = 1

So 1 = 524,288/531,441
which means that 531,441 = 524,288
which is obviously untrue.

Next time you try to tune your guitar to play in any key, bear this in mind. None of the above steps is in the slightest way contestable. If C = 1 then G must equal 2/3. Likewise, each of the remaining steps is incontestible. So, why does this produce such an obviously incorrect conclusion ?

The answer is in what we call the notes. If C is 1, then G is 2/3 in the key of C. But D’ is only 2/3 of G in the key of G: it is not 2/3 x 2/3 in the key of C. That is because the D’ which occurs in the key of C is not the same D’ which occurs in the key of G. So a note can vary in pitch depending on what key we play it in. There is no way we can tune a string to D and expect it to be in tune in every key. Mediæval musicians knew this, so they had moveable (tied) frets on their instruments. Keyboard instruments were made to play in two or three major keys, and some of them had more than one key for each note. The start of the modern era came when Bach formulated what he called “equal temperament”. He averaged out the notes so that they sounded almost right in every key, but not completely right in any key. To celebrate this he wrote the works “The Well-Tempered Clavier”, which contains all 12 major and 12 minor keys in one piece.

Nowadays all orchestral work is written for equal temperament. The problem comes when you try to play something folk-based which was intended for natural temperament, usually in a diatonic scale. Most folk music is diatonic, so you will have to get used to retuning your instrument every time you change key. Folk singers are not alone here. Experiments performed on classical singers have shown that if you take away the orchestra they usually revert to natural temperament without realising it.

By the way, that fraction, 524,288/531,441, was arrived at by Pythagoras in his experiments with the monochord, and became known as Pythagoras’s Constant.

Alan F Brookes
6th April, 2002.
(not finished yet, scroll down !)

THE DIATONIC SCALE
For the sake of completeness, here is the diatonic scale in the key of C major:

C 1
D 8/9
E 64/81
F 3/4
G 2/3
A 16/27
B 128/243
C’ 1/2

chas smith R.I.P. · From: Encino, CA, USA

I've always liked the "traditional" Just scale:

William Lake · From: Ontario, Canada

Thank you for simplifying tuning. Sure a lot easier than using this confusing chromatic tuner. Laughing

_________________
Bill

Alan Brookes · From: Brummy living in Southern California

Mike Perlowin · From: Los Angeles CA

These Earvana intonated nuts make a guitar sound a lot more in tune.

_________________
Please visit my web site and Soundcloud page and listen to the music posted there.
http://www.mikeperlowin.com http://soundcloud.com/mike-perlowin

Alan Brookes · From: Brummy living in Southern California

There are lots of ways of making a guitar sound more in tune, but none that make a guitar sound completely in tune. Sad

Charles Davidson · From: Phenix City Alabama, USA

Damn,I been playing out of tune since I got that 13 dollar Stella in 1949 . Mad

Hope all those folks that have been paying me to pick for the last 60 years don't want their money back. Rolling Eyes

YOU BETCHA,DYK?BC.
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Hard headed, opinionated old geezer. BAMA CHARLIE. GOD BLESS AMERICA. ANIMAL RIGHTS ACTIVIST. SUPPORT LIVE MUSIC !

Mike Perlowin · From: Los Angeles CA

Alan, you're right about guitars being mathematically out of tune, even with the intimated nuts. But on a practical level, I tell you based on personal experience that the guitars really do sound better with these nuts. You can really hear a difference.

After installing one on one of my guitars, I plugged it into a tuner and checked every fret on all 6 strings, and every note was dead on. After that I've installed them on all my guitars. (I wish they would make one for a 12 string.)

Regardless of the math and the theory, my ears tell me that the guitar makes a more pleasing sound with these nuts than without them.
_________________
Please visit my web site and Soundcloud page and listen to the music posted there.
http://www.mikeperlowin.com http://soundcloud.com/mike-perlowin

Alan Brookes · From: Brummy living in Southern California

I'll try one out.

Isn't your electronic tuner programmed with equal temperament? If so, it would actually show natural temperament as out-of-tune. We're so used to hearing equal temperament in the western world that we don't notice the disharmony most of the time.

Concerning the problem of pull-down, luthiers go to a lot of time and effort to compensate for this by moving the nut slightly back and building a fingerboard which has a complex curve. Some luthiers build complex nuts with different contact points for each string. On fretted instruments that I build I always have a zero fret. The fact that open strings vibrate against ivory, whereas fingered strings are held down against brass or nickel silver means that open strings have different tonal qualities. This is completely eliminated with a zero fret, and I can't see why the zero fret is not standard on all fretted instruments. It can only be tradition ...the sort of tradition that lumbers the violin family with unreliable friction pegs instead of machine tuners, but that's another story.

David Mason · From: Cambridge, MD, USA

Regarding zero frets being rare, I think the reason is that it's a lot easier to replace a nut that's worn too low than it is to replace a single fret, over and over. If I cut a bone nut for the lowest possible action - fret height - I have to replace it at least every two years on a guitar that gets a lot of playing time. Also, a lot of buyers do like to play their cowboy chords really hard, and a higher nut can help them not buzz out on every single string. And then there's slide... ever since Sonny Landreth re-infected my inner slide monster, my main playing guitars have two nuts to chose from.

When I got really serious about playing in tune on pedal steel, I put in a lot of time playing against drone notes, an hour+ a day. It didn't take long at all before my "hearing in tune" surpassed the difference between JI and ET tuning, effectively ruining 1,200 or so CDs.... It took me a while to get back to being able to appreciate all the "out-of-tune" music that I grew up on. Laughing

Mike Perlowin · From: Los Angeles CA

Alan Brookes · From: Brummy living in Southern California

Gary C. Dygert · From: Frankfort, NY, USA

In my experience, the mandolin is impossible to tune.

Alan, I always wondered how musicians got their instruments anywhere near correct tuning hundreds of years ago when strings, many of them double, were gut, and frets were also lengths of gut tied to the neck. Were lutes ever in tune? How about those theorbos, huh folks?
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Alan Brookes · From: Brummy living in Southern California

All fretted instruments are impossible to tune. The smaller the instrument, the more difficult it is to tune, and so the mandolin and ukulele are very frustrating instruments to play.

In olden times strings were of variable quality. Steel strings were not very uniform in their gauge, and were prone to breakage, and gut strings, by their very nature, have never been very accurate: nature doesn't work that way. One of the reasons for the double courses is so that you can keep playing after one of your pair has broken. The other is to even out the inconsistencies of the strings. With two strings vibrating in and out of the same pitch, beats will occur. In a way, this added to the tone of the instrument.

Forget riding along on a horse strumming a lute. The lute is difficult enough to hold as it is. The round back makes it slide around in your lap, and on horseback it would be too quiet to hear. Of course, the lute was a courtly instrument. The peasants played folk instruments such as the cittern. Were lutes and theorbos, chittarones, etc. ever in tune ? Most of the time they were just playing the continuo to other instruments. The lute pieces were only played by skilled lutenists, who would do their best to keep their instruments in tune. Without being able to go back in time it's impossible to tell how in-tune the average lutenist would be, but I can say that most of the professional lutenists that I've known use gut strings and play in tune. My guess is that in the old days most lutenists didn't play in tune.

One thing that most people don't realise is that the lute and theorbo are very light instruments. You have to slacken the strings off before you put the instrument away, or you risk the bridge being pulled off during the night when the temperature goes down and the strings shrink, hence tightening and adding taut to the bridge. With as many as 20 gut strings that means that you spend more time tuning the lute than playing it. There's a well-known saying from centuries back that says that he who plays the lute for 60 years will spend 40 years tuning it. Whoa!

I thought I would slip this in here. It's a sketch of my youngest daughter, Josephine, playing the lute. The sketch was made by her elder sister, Katherine, who's an artist.

chas smith R.I.P. · From: Encino, CA, USA

Daniel Morris · From: Westlake, Ohio, USA

Hey, Mike, those Earvana nuts look fascinating.
If they did indeed make ones for pedal steel, do you feel they would solve open tuning issues, all PSG tuning issues, or maybe none? Just curious.
_________________
1979 MSA U12 Pedal Steel
1982 Kline U12 Pedal steel
2019 Sierra U12 Pedal Steel
2011 Bear Creek MK Weissenborn
Milkman 40W Mini amp w/Telonics 15" speaker.
Dr. Z Surgical Steel w/TT 15" speaker.
Frenzel MB-50 head.
Spaceman, Empress, Eventide, Pigtronix.

Alan Brookes · From: Brummy living in Southern California

Those Earvana nuts are made to compensate for the pull-down of the strings by the fingers. Since the steel guitar has no frets and you don't press the strings against frets with your fingers, they would serve no purpose on a steel guitar.

Georg Sørtun · From: Mandal, Agder, Norway

Modified nuts for improved intonation on my 6 string guitars back in the -70s - somewhat like the Earvana nuts as far as I can see, and although they don't solve the natural vs equal tuning problems mathematically, they sure made it easier to play in tune - natural or equal - all the way up the neck by bringing the strings in the ballpark where finger-pressure adjustment does the job.

Tested out a simple intonation nut on my PSG around 1990 - while also modifying the bridge/changer to be at a slight angle for improved intonation. Has nothing to do with "natural vs equal tuning", but as strings on my keyed PSG comes in from the keyhead at very different angles, and strings of varying thickness/type don't vibrate at the exact same points lengthwise over nut rollers and bridge, lengthwise adjustable nut rollers did improve things a lot - as did the angled bridge.

Bridge stayed angled - approx 2 millimeter longer lowest string compared to highest for improved intonation. Went back to regular "in-line" nut roller arrangement - found a good intonation nut roller arrangement to be too complex to build at the time and on that instrument, and simply accepted that PSGs are best tuned by ear with the bar on the strings.

Donny Hinson · From: Glen Burnie, Md. U.S.A.

The mathematics that you have done is impressive, and it likely scientifically proves that the tuning method, whichever one you choose, winds up being "imperfect". But on the practical side, does it really matter? It's well established that seasoned players (the late Chet Atkins, for instance) can play and sound in-tune well enough to please even the most discriminating ears, even with all these mathematical eccentricities and mechanical foibles we're forced to deal with. Therefore, while all this science and metrology may make for interesting discussion, both in the classroom and in a pub over a pint of grog, we must sooner or later come to the realization that what we're dealing with here is an art form, one whose essence is subjective beauty, and not scientific exactness. Smile

Alan Brookes · From: Brummy living in Southern California

You have to bear in mind, Donny, that we were all brought up listening to music played on tempered scales, so our minds have come to tolerate the differences. You also have to bear in mind that the music that we listen to is actually out of tune most of the time. To explain that, imagine that you've slid the bar from C to D. C was in tune, D was in tune, but between that you've sounded out an infinite number of fractional notes which are all out of tune. Again, imagine a singer slurring his way up the scales, or a trombone player bending his notes. Between notes they're producing a lot of out-of-tune notes. When you add together all the music played by the entire band, it's full of dischords and disharmony. Why don't we notice it? The brain knows the meter of the music, and expects the music to be in tune at specific intervals. If all the instruments come together at each beat the brain forgets the disharmony between the beats.

Our brains are wonderful things. They compensate for all sorts of irregularities. Look at an illuminated clock in the dark, then close your eyes. You will still be able to read the time for several seconds. It's known as Persistence of Vision, and it's why we can see a series of 24 frames per second in a movie and put them together as one continuous motion. The brain does a similar thing with music. Cool

Let's face it, we enjoy music, but explaining what music is can be a problem. The closest explanation seems to be that music is a series of notes at regular intervals that are conjoined by mathematical relationships of vibration rates which are pleasant to the brain. Oh Well

Pete Burak · From: Portland, OR USA

Have you guys checked out the "True Temperment" fret method for playing guitar "in-tune"?
I would like to try one of these guitars:
http://www.truetemperament.com/site/index.php

Darrell Criswell · From: Maryland, USA

Alan Brookes · From: Brummy living in Southern California

Darrell Criswell · From: Maryland, USA

Alan Brookes · From: Brummy living in Southern California

From my participation in the organization, over the last twenty years, of the Northern California Association of Luthiers, I've gotten to meet and know most of the luthiers on the West Coast of the United States, and have had the pleasure of visiting many of their workshops. There's a lot of amazing talent in the area, and I can't imagine this area is any different from any other.
Yes, there were a lot of talented luthiers around Cremona in the 15th century, and there probably still are, but whether they were any better than other luthiers is all a matter of by what criteria you judge them. Most of the luthiers that I've met build fretted instruments. For some reason, violin builders seem to be a society unto themselves and rarely come to luthiers' meetings.
I just find it hard to believe that there's any less talent around nowadays than there has ever been. It's all a matter of what you're looking for. If you want a violin in the Cremona style, in the style of the 1400s, then one would expect Cremona in the 1400s to be the place to look. Most modern violinists wouldn't want a renaissance-shaped instrument, which would be more difficult to play. I myself listen to and play a lot of Early Music, so it's the type of violin I would play, but I prefer the rebec, of which I've built several, being more of a folk music fan than classical.

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