An Arithmetical Demonstration of Inadequacy of Fixed Frets

[Alan Brookes]

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

Like every invention, there is dispute as to who invented/discovered it first. Usually, well-known figures are credited with what had previously been a long discussion. The problem of tuning was the latest news in Bach's time, and Bach, being the accepted spokesman of the time, made pronouncement which over time have become attributed to him.
The problem arises because no-one has come up with a better way of putting music down on paper. I think that the error is the naming of notes. The notes ABCDEFG were so named because they were the notes in the Amin. scale. They also happened to be the notes in the Cmaj. scale and in several other modes, which we don't use nowadays.
When we started writing in other keys the mistake was to refer back to C/Amin. What they should have said at the time, but they didn't have the technological knowledge, was to take any key, and from that extrapolate the notes. In other words, write everything in C and then assume you were putting some sort of capo' on to get to the other keys. The mistake was writing everything in C and then adding sharps and flats to the key signature, which never worked. You cannot write music in C (which is the way the lines are described) add a few sharps and flats and come up with another key. All the notes described in the key signature are in the key of C/Amin. In every other key they are inaccurate.

[Paul Arntson]

No need to import an excel spreadsheet for this formula.

In excel, just type the following in cell A1:
=10^(LOG(0.5)/12)

A1 should now read
0.9438743 give or take a few places on the right.

Then in cell B1 type :
1

Then in cell B2 type:
=$A$1*B1

Then copy cell B2 and paste down for as many frets as you want to go.
In most versions of excel you can copy by holding down the mouse button on a corner of the cell and dragging.
The dollar signs make the reference to A1 stick there, so you are always multiplying by 0.9438743...
The B1 reference in cell B2 will automatically refer to the cell directly above for each cell as you go down.

There it is.
For a scale length other than 1" just change the 1 in cell B1 to your scale length.

Presto.

If you want to turn the formulas into values after you are done,
highlight the whole column,
Copy
Click on a cell to the right where you want the numbers to be stored.
Paste Special and then select Values and they are transformed into actual numbers.

You can do this for a variety of scale lengths.

For frequencies instead of string length, just change the 1 to 55 or 110 or 22 or 440 or whatever
and change the formula in A1 to
=1/(10^(LOG(0.5)/12))
A1 should now read 1.05946309 and the column will get bigger as it goes down.

(You could also change the formula in A1 to be
=10^(log(2)/12) and it would do the same thing.

Twelve is the number of intervals you want in an octave. If you wanted to calculate quarter tones you could use 24 instead.
Fun to mess around with.

Sorry for slight thread drift.

[Alan Brookes]

No need to import an Excel spreadsheet for this formula...

That's okay if, like me, you've been using Excel day in day out as part of your job, ever since it came out, but a lot of people don't know how to write formulae in Excel, and that's what I created the spreadsheet. You just type in the string length from nut to bridge, in inches or centimetres, and it instantly shows the positions of all the frets, with distances from nut and bridge.

...Left my guitar like that over thirty ears ago (when I started playing PSG), and just checked... It still works fine, but my fingers are not prepared to play that way anymore, too soft skin on my left hand fingertips these days...

I recently had the same problem. I picked up my 12-string guitar and was rattling all over the place. Playing for so long with a tone bar has made the muscles in my left hand lazy and the skin on my left-hand fingers go soft. I've been playing the hell out of that 12-string ever since and the buzzing and rattling have all but gone.

[Paul Arntson]

Oops you're right, Alan. I took that part for granted but I shouldn't have.

I actually use it more when looking at frequencies.

I've had that exact same finger issue also. I now try to be careful to maintain at least a little callous, but sometimes I forget.

[Earnest Bovine]

=10^(LOG(0.5)/12)

Introducing the number 10 (the number of toes on a normal human) seems arbitrary and unnecessarily complicated. The ratio between adjacent frets is just
2^(1/12)
and between N frets is
2^(N/12)

[Earnest Bovine]

(nurd alert!)
There are several reasons why I think it's not useful to try to tune each note in your scale to a whole number ratio such as Alan cites

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

or similar versions of Just Temperament.

Here are some:



1. There are always many other pairs of harmonics present besides the one pair that you are tuning to zero beats. (tho higher ones matter less; see #2)

2. You don't really hear much, if any, of the higher harmonics above about 6. And those that you do hear die away quickly. So there is not much reason to tune the 9th harmonic of C to match the 8th harmonic of D. (Well, actually, there may be a reason in that 8/9 makes a good just interval of 2/3 against the G.)

3. Actual real world strings have inharmonicity. An ideal string's harmonic frequencies are 2,3,4,5,6.. times the fundamental, but this sort of string exists only in theory. A real string's harmonics are a little higher than that because strings have stiffness.
Here is a table I copied from a piano site.
[tab]
Harmonic 1 2 3 4 5 6 7 8
Measured 440.0 880.9 1323.9 1769.8 2219.7 2674.6 3135.4 3603.1
Calculated 440 880 1320 1760 2200 2460 3080 3520
Diff (cents) 0.0 1.8 5.1 9.6 15.4 22.5 30.9 40.4
[/tab]
Guitar strings can have even more inharmonicity, because they are shorter. And the inharmonicity is more random on short strings like guitar, so a simple stretch tuning does not help as it does on the log, thin strings of the piano. When you fret up high on the neck, on guitar or steel, you hear it. You all know what your fat low string sounds like when you play at the 20th fret. You can hardly tell what note it is, because the harmonics are not even close to 2,3,4,5.
This would only matter if you tune with a meter; if you tune by listening to beats, you don't have to worry about it because you are taking the inharmonicity into account.

4. Of course, your guitar and pedals will keep going out of tune to some extent. If you tune an interval to the precise ideal width (such as 4/5 = 386 cents for major third), it may sound very bad if it gets a little wider or narrower (such as a little narrower than 4/5). There may be a range of interval widths that sounds acceptable (suc as 386 .. 400 cents for major third) and I think it is better to tune your intervals to the middle of this range (such as 393 cents for major third) so it still sounds OK when the inevitable happens and the tuning varies a little as you play.

[Tony Glassman]

I choose to ignore this thread & any similar ones.

Like "Bill Hankey postings", they tend to generate more heat rather than shed any light on a given topic.

[b0b]

It's worth noting, Tony, that this topic is not about tuning a steel guitar. It's about the mathematics of musical pitch. The only "heat" generated among the learned participants is esoteric and, I think, refreshingly civilized.

I found Earnest's comparison of real instrument strings to theoretical ones fascinating - it was something I'd never realized. It did shed quite a bit of light on the subject.

[Alan Brookes]

...There are several reasons why I think it's not useful to try to tune each note in your scale to a whole number ratio such as Alan cites ... or similar versions of Just Temperament....

It's worth noting, Tony, that this topic is not about tuning a steel guitar. It's about the mathematics of musical pitch...

Bobby Lee is right. This is not a discussion about how to tune a steel guitar, and the whole numbers given are the real mathematical relationships between string lengths and pitch.
I'm not advocating using Just Temperament or Equal Temperament to tune instruments. I'm just demonstrating the mathematical relationships and why, try as one may, one can never create a fretted instrument that plays in tune in more than one major and one minor key.

[Paul Arntson]

Alan, I think you've made an excellent case and I've enjoyed this discussion a lot. I hope I didn't sidetrack your original point too much. Bottom line (the way I think of it)is that
(1)making notes exactly in tune (rational number frequency ratios) is at odds with the fact that
(2) doubling frequency every time we halve the string length creates a sequence of irrational numbers for notes in between the octave points unless we have really wide variations in the intervals between some of the notes in the scale.

You're right, Ernest.
2^(1/12)=10^(log(2)/12).
Your description is easier on the eyes.
And yes I too have ten toes.

I posed the problem to myself by originally finding the solution to X^12=.5 and using logs is just my default approach. I never bothered to memorize the resulting number or formula, rather I only remember the approach to the problem and I generate the actual number fresh when I want to use it. I guess it's all in how different minds work. In my case I have just decided to live with the quirks of my own memory and go with how my brain is wired. I've made a good living over the last 40 some years looking at problems in weird ways, but sometimes it makes it harder for me to communicate.

You make a good point about real world string behavior. I agree and that is partly why this discussion travels into the impossible realm of "ideal" behavior. Compounding that issue is the effect of elasticity of the string, where a string initially is sharp and then the frequency drops as the magnitude of he vibration decreases. (More apparent with thicker strings.)

I just wish I could find a nice proof of either of our exponential solutions being incapable of being expressed as a whole number fraction (rational number). That would put my mind at ease.

[Earnest Bovine]

I just wish I could find a nice proof of either of our solutions being incapable of being expressed as a whole number fraction (rational number). That would put my mind at ease.

If I remember 7th grade algebra correctly, we proved that the square root of 2 is irrational by assuming that A and B are integers such that (A/B)^2 = 2, and that A and B have no common divisor. But that leads to a contradiction, since both A and B must be even numbers. There is probably a way to generalize this to other roots of 2.

[Earnest Bovine]

I think the same argument applies for the twelvth root.
Assume (A/B)^12 = 2
ergo A^12 = 2 B^12
ergo A is even
ergo B is even
ergo A/B is not in lowest terms
ergo A/B cannot be expressed in lowest terms
ergo A/B cannot exist

[Tony Glassman]

......and so this thread proceeds ad infinitum, ad nauseum.

[Earnest Bovine]

......and so this thread proceeds ad infinitum, ad nauseum.

Yes, I'm sure there is a less nauseating proof, but I'm too busy trying to tune my Sho-Bud. Give it a few minutes and let us know what you come with.

[Alan Brookes]

Luthiers, for hundreds of years, have derived the position of the first fret by multiplying the scale length by 16.834 and dividing by 17.835, which amounts to multiplying by 0.943874404261284. They then continue the same calculation for each fret, and the octave works out to 0.5.

In days of yore they would sell rulers with these calculations already done, for various string lengths.
Most of today's luthiers have already made templates for that formula, which they have tempered to allow for string pull-down; something which is unnecessary on a non-fretted instrument such as the steel guitar.

I have to say that a good proportion of people setting out to build a guitar will buy a ready-made fingerboard and set the bridge in the position which is double the length from the 0 fret to the 12th fret, or they just hold the blank wood against a guitar and mark off the string positions.

I choose to ignore this thread & any similar ones.
Like "Bill Hankey postings", they tend to generate more heat rather than shed any light on a given topic.

......and so this thread proceeds ad infinitum, ad nauseum.

So why do you read it? More to the point, why do you comment on it? Others find it interesting. I guess ignorance is bliss.

[Dave Mudgett]

IMO, the most important thing a steel player needs to learn is to play in tune. I'd rather hear one note in-tune than a flurry of out-of-tune stuff, and it is not exactly trivial to figure out how to play this contraption reasonably in-tune. So, again IMO, no amount of effort spent figuring out how to play in tune is wasted.

OK, I agree that mathematics is useful to perhaps understand the physical relationships, but no panacea for playing in tune. The ears always trump any mathematics, and without a good ear, all the theories and fancy electronic tuners won't make someone play in tune.

Still, I think it's damned useful to be able to make a mental model about how things work, and that is what mathematics is for. I think we shy away from these conversations too much, and do a disservice to new players who haven't been through this. Yes, they can look up old threads, but a lot of those old threads truly generated at least as much heat as light, and I find this discussion pretty civil indeed. I personally found that understanding the physical/mathematical issues helped push me in the right direction with ear-training.

Yes, Doug - that's a basic sketch of a basic proof-by-contradiction of the irrationality of something like 2^1/2 or 2^1/12. Some details missing, but that's it.

I also agree that robustness of the nominal tuning center is essential when faced with detuning and significant string nonidealities. But I think that having a mental model of the 'idealized' situation is useful, with nonidealities viewed as relatively small perturbations. Seems to have served physicists and engineers pretty well for the last century or two. I honestly doubt we'd be sitting here typing at these computers without that type of thinking.

[Paul Arntson]

Dave and Earnest, thanks! I like that. I'll think on it a bit but I bet you've hit it. I should have paid more attention in jr high...

[Alan Brookes]

IMO, the most important thing a steel player needs to learn is to play in tune...

I agree entirely Dave, but this is a discussion about fretted instruments, and why it's impossible to build one with the frets in the right place to play in more than one major and one minor key.

[Paul Arntson]

OK. A proof so straightforward I almost stubbed one of my ten toes on it.

Assume 2^1/12 is rational.
If this is true, then 2^1/12= a/b where a and b are both integers.

So if 2^1/12 is rational, that means a^6/b^6 is also rational.
However a^6/b^6 = 2^1/2 so it cannot be rational, since
2^1/2 is well proven to be irrational.
Contradiction. Ergo 2^12 is an irrational number.
QED.

Alan, great thread. It made me think over some other stuff I've had in the back of my mind for a while. I won't belabor this any more in this thread. When I gel my ideas I'll start my own thread.
Thanks to all who tolerated my thread drift.

[Earnest Bovine]

it's impossible to build one with the frets in the right place to play in more than one major and one minor key.

One of the points I was trying to make is that there is no way to tune "perfectly" even in just one key. Even in just a single interval, any pair of notes, no matter how you tune them, there will be audible "imperfections", which of course you can define in different ways.

[Paul Arntson]

How close is close enough?

Alan and Earnest, am I correct on this?
C 1
D 8/9 4c flat relative to even tempering
E 64/81 7c flat "
F 3/4 2c sharp "
G 2/3 2c flat "
A 16/27 5c flat "
B 128/243 10c flat "
C’ 1/2

[b0b]

Your fractions look upside down to me, Paul. When you multiply a frequency to get a higher note, the fraction has to be greater than 1/1.

I've never seen the major 3rd listed as 81/64 before. Most of us use 5/4, which is about 14 cents flat of ET. Isn't 81/64 actually sharp of ET? Any major third that's sharp of ET sounds out of tune to me. 81/64 isn't a very usable musical interval in our 12 note system.

[Dave Mudgett]

b0b - this discussion started in terms of string length, not frequencies. That is why the fractions are inverted. I also generally prefer to think in terms of frequency ratios, but string length ratios are also valid, at least in an idealized string model such as being considered here.

And yes - a major 3rd ratio of 81/64 = 1.265625 is indeed sharp of the ET version of a major 3rd, which is 2^(1/3) = 1.25992105, as compared to the pure harmonic 5/4 = 1.25. Yuck.

I agree entirely Dave, but this is a discussion about fretted instruments, and why it's impossible to build one with the frets in the right place to play in more than one major and one minor key.

Yes, I understand, and understood, the point of the initial post. My point was, and is, that wrapping one's head around all this is especially relevant for those of us that came to steel from guitar and/or keyboard. I had to convince myself that what I actually had to work at pretty hard for guitar (e.g., tuning equal temperament by ear and/or counting beat frequencies and really understanding the compromises being made) for pushing 40 years wasn't gonna cut it for steel before I would force myself to spend the time really focusing on a different way to tune and play the steel. I imagine I'm not alone, and why I think we should emphatically not try to shut down 'tuning' discussions. It's not just about frets or tuning, but also about being able to hear what's "in tune".

I also again agree with Doug that 'perfection' in tuning can never be obtained, even in one key. Math is math, physics is physics, and they are not exactly the same. Math of this type is useful for modeling physical phenomena, but it never captures them exactly.

[b0b]

One of the points I was trying to make is that there is no way to tune "perfectly" even in just one key. Even in just a single interval, any pair of notes, no matter how you tune them, there will be audible "imperfections", which of course you can define in different ways.

Even within the confines of the C major scale, you need two separate D notes - one in tune with the G and the other in tune with the A. These are the problems that meantone tunings attempt to overcome.

Is there such a thing as a meantone guitar fretboard? I've seen staggered frets on JI guitars...