An Arithmetical Demonstration of Inadequacy of Fixed Frets

[Alan Brookes]

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 lets 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 or 531,441 = 524,288

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. Mediaeval 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

[Bill Moran]

Heck, I knew that ?

[Doug Beaumier]

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

I am constantly aware of that.

[Richard Sinkler]

[Dickie Whitley]

Richard, your collection of "smileys" sometimes just make my day....of course I agree with the take on the math....got lost in the first paragraph somewhere....

[Earnest Bovine]

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

hmmm; I would write those fractions inverted. In other words, frequency of Pythagorean D is 9/8 the frequency of Pythagorean C.

I don't think anybody would use this tuning to play chords. Look at the width of the "major third" from C to E: 81/64 is 1.265625, or about 408 cents! Icky poo! Anything wider than 400 sounds like crap. Most steel players use about 390.

[Barry Blackwood]

Alan, will this have any negligible effect when playing 'A Way to Survive?'

[Alan Brookes]

...I don't think anybody would use this tuning to play chords. Look at the width of the "major third" from C to E: 81/64 is 1.265625, or about 408 cents! Icky poo! Anything wider than 400 sounds like crap. Most steel players use about 390.

Which just goes to show how attuned we've become to Equal Temperament in modern society. I wonder how this affects other creatures which still hear the same notes as we do, yet are not educated in human music.
http://bb.steelguitarforum.com/viewtopic.php?t=265014
Maybe the reason dogs and cats howl at certain notes is that they sound dissonent to them.

[Donny Hinson]

"An Arithmetical Demonstration of Inadequacy of Fixed Frets"

True, fixed frets are inadequate for mathemeticians.

However...they are perfectly adequate for good musicians.

Not everything that can be counted counts, and not everything that counts can be counted.

[Alan Brookes]

...However...they are perfectly adequate for good musicians...

It's the good musicians who are the most concerned. Luthiers all over the world spend their lives trying to build guitars that will play in tune in all keys and no-one has ever succeeded. It's not just for mathematicians. What I quoted here is from an address that I gave many years ago to the Northern California Association of Luthiers about why they would never succeed in the task. It's like the apothacaries' dream about turning copper into gold. Many have spent their lives trying to do it, but it's impossible.

[Alan Brookes]

Georg, you've plotted the reciprocals of the way I was presenting the figures. I was using string length but you're using frequencies.

Whichever way you look at them, the differences don't work out that much, bearing in mind that when you add vibrato you're moving the bar by about 1/8" each way.

The problem goes away when you're playing without using the frets, either with a tone bar or when playing an instrument such as a 'cello or violin, except that you have to fit in with the keyboard player who is stuck in Equal Temperament, and the guitarist who won't retune when he changes key.

I notice you've shewn C# and Db as two different ratios, which, of course they should be, and would be different still depending on which key one is playing in. Unfortunately, guitars and *piano's don't have additional frets or keys.

*I know that plurals don't take an apostraphe before the 's', but "piano'" is an abbreviation of "pianoforte", just as "'cello" is an abbreviation of "violoncello", so the apostraphe denotes the missing letters, and the plural of "piano'" is, therefore, "piano's".

[b0b]

I notice you've shewn C# and Db as two different ratios, which, of course they should be, and would be different still depending on which key one is playing in. Unfortunately, guitars and *piano's don't have additional frets or keys.

Actually, he's showing the two different D notes. The 10/9 is in tune with A and the 9/8 is in tune with G.

When you tune the E9th pedal steel by harmonics, this phenomenon shows up on the F# string. If you tune the F# to B, it will be sharp when compared to the (pedaled) C#. I cover this in my article Just Intonation of the E9th Pedal Steel.

[Alan Brookes]

...Actually, he's showing the two different D notes. The 10/9 is in tune with A and the 9/8 is in tune with G...

Absolutely right. In addition to the keys of D, A and G, the note D also appears in the keys of D#, F, A# and C, and is a different value in each case.

[Andy Volk]

This thread makes me feel like Charlie Brown!

[Alan Brookes]

Unfortunately, on fretted instruments without pedals you can't do that, which reinforces my original point, that it's impossible to tune an instrument with fixed frets.
In days of yore, when guitars had gut strings, they also had frets made from tied gut, which you could move around to suit the key you were playing in. The lute still has tied frets.

[John RJ Wilson]

The equal temperament problem was tackled by the company below a few years if you buy into this system, as adopted by Steve Vai (who else ) and others,
it produces wiggly frets.

http://www.truetemperament.com/site/ind ... go=0&sgo=0

[Paul Arntson]

Alan, have you ever checked out Harry Partch?
He was really deep into this.
https://en.wikipedia.org/wiki/Harry_partch

Yours is a great discussion of why an even tempered scale can never be truly in tune.

Here's my take on the conundrum of an even tempered scale:
The ideal ratio of adjacent string lengths (ignoring string stretch and how hard the string is picked) on two adjacent frets is:

10 raised to the power of (log(.5)/12). (Where log is really log base 10)
That equals 0.943874313...


I believe this to be an irrational number, i.e. one that can't be perfectly expressed by any fraction. (I was never sure how to prove this mathematically.)
Therefore there will always be dissonant sum and difference frequencies when we play two notes at once because the waves can never line up.

The reason the 5th sounds so in tune is that the ratio of the G length to the C length is 0.667419927085017… and that number is really close to 2/3.

Other notes seem to sound out of tune by the amounts that differ from a nearby rational (defined as the ratio of two integers) number.

When I lay out fretboards for my own hobby projects, I use whatever scale length is convenient and then create frets at the following proportions:
(numbers are shortened to 9 decimal places because they seem to go on forever without repeating).
1 (open unfretted scale length)
0.943874313
0.890898718
0.840896415
0.793700526
0.749153538
0.707106781
0.667419927
0.629960525
0.594603558
0.561231024
0.529731547
0.5 (octave or 12th fret)

You can use the formula to keep going as far as you want. I just showed the first
octave because the second octave is 1/2 of that, and so forth.
I think we play slightly different notes when we play an unfretted instrument, just so we sound more in tune for what's happening at the time.
I'm pretty sure I unconsciously use different fret pressures on fretted guitar depending on what is going on around me.

[Alan Brookes]

John, yes I've seen the wiggly fretboard idea before. It can't possibly allow you to play in tune in every key because, as mentioned earlier in the discussion, notes vary in frequency with the key they're in. If, for instance, D is a different frequency in the key of C than is the D in the key of F, which D appears on the wiggly keyboard? If it were that easy, everyone would be doing it. Certainly, all the luthiers in the western states are familiar with the wiggly fretboard but none of them uses it. I've even seen fanned fretboards, which are made for ease of playing rather than accuracy of tuning, and adjustable nuts, which help in some keys but don't solve the problem, because the problem is insolvable. Making a fretboard that is in tune in every key is like building a perpetual motion machine. I'll go further. It's impossible to make a wind instrument with fixed holes that will play in every key.

Paul, the formula that you point out is the same one that Bach came up with when he "invented" Equal Temperament tuning. I put it into an Excel spreadsheet about fifteen years ago to enable me to work out fret spacings on the instruments I build. I've been using that formula since 1963, but it was a lot more tedious in the days before computers when we had to work it out by long multiplication for each fret. To save myself the trouble, in those days I had created rulers marked up with fret positions for different string lengths and I used to scribe the positions on the fretboard from the rulers. If anyone would like a copy of my Excel spreadsheet, just email me at afbrookes@aol.com and I'll sent it to you. Unfortunately, spreadsheets cannot be sent through the Forum.

[b0b]

Bach didn't invent equal temperament, Alan, nor was The Well-Tempered Clavier written for it. http://en.wikipedia.org/wiki/Equal_temp ... ly_history

[Paul Arntson]

Thank you, Alan, I'll check that out further.
According to good old wikipedia, "The method of logarithms was publicly propounded by John Napier in 1614", so it's not inconceivable that it could have been applied to the problem. However, it does not seem to be addressed in the wiki article
https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier
The most interesting thing to me is discerning a proof that the ratio I propose is indeed an irrational number. Does anybody know how to prove or disprove this?

If my idea is provable that the ratio is an irrational number, then we are forever condemned to either playing in even temperament slightly dissonant or else restricted to certain keys with minimal dissonance.

b0b, you and I posted at the same time. I'll let you and Alan hash that one out.
Alan, I think your original inequality can be traced to the difference between
2/3=0.666666666666666… and the number I came up with which is approximately 0.667419927085017...

[John RJ Wilson]

ooph my brain hurts,
I admire you guys that can get your head round the maths of this.

And I am proud that none of you have mentioned that dreaded A=440 is the work of the devil, and we should all go back to A=432 so that all music will become in harmony with the universe due to the fact that notes will no longer be trailing around large numbers of significant figures after the point, and thus there will be more space around the music.

[b0b]

Actually, A=440 Hz is the key to world peace. Half of the world uses AC electricity at 50 Hz, while the other half (including US) uses 60 Hz. This creates a standing electromagnetic wave of 10 Hz encompassing the globe.

A=440 Hz divides down in octaves to A=220 Hz, A=110 Hz, and lastly to the bass note A=55 Hz. This compromise mediates the two conflicting systems, dispersing the tension and bringing harmony to the entire planet.

A=440 Hz can save the world. Turn it up!