Perfect Tuning is a Mathematical Impossibility

[IamGroot]

In Western Music, we most commonly use the Equal Tempered System and the A 440 Hz standard. An octave is an interval where the frequency ratio is 2. An octave is divided into 12 semitones, with two sequential semitones differing by a fixed ratio equal to 2 raised to the 1/12 power.

Two raised to the 1/12 power is an irrational number since 1/12 is irrational . Therefore, any integer power of (2 ^1/12) other than a multiple of 12 is also irrational.

So A is the only note with a precisely defined value. All other notes are an approximation to whatever degree of accuracy is desired.

 

[JW56789]

I never quite grasped this until I saw a plot of the tuning of a piano, all 88 notes.

As you said, only the a below middle C was a spot-on 440hz. Looking at the plot, the lower you went from there it was progressively / fractionally flatter the lower you went, and going up from there was the same incremental curve, reversed, going sharp. Yet to play it upon completion, it sounded perfectly in tune. Tuned in perfect mathematical increments, and it would sound beyond bizarre, according to the tech that was giving me the USA TODAY short course in 'Piano Tuning for Dummies'. Some keyboard synths allow for 'unlatching' the standard tempered tuning and going to alternates tuning methodologies, if you want to experience this yourself. You won't want to do it twice . . .

 

[Mr_Moo]

I'm sorry, but 1/12 is rational. The 12th root of 2 is irrational, however, for the reason that 2 is prime.

That does not mean that the 12th root of 2 is in any way imprecise.

There are mathematical difficulties with all tuning systems, which is why there are so many of them. The problem you cite is not among them.

 

[SteveCS]

...

As you said, only the a below middle C was a spot-on 440hz. Looking at the plot, the lower you went from there it was progressively / fractionally flatter the lower you went, and going up from there was the same incremental curve, reversed, going sharp. Yet to play it upon completion, it sounded perfectly in tune.
...

It's also known as 'stretch' tuning...

 

[BassCliff]

Hi,

There's meantone tuning, equal tempered tuning, well-tempered tuning, etc. Each has their own color. All I know is that I can rarely get my clip-on tuner to match my pedal tuner. So yeah, tuning is a "best effort" situation.

I heard there would be no math today.


Thank you for your indulgence,

BassCliff

 

[IamGroot]

I'm sorry, but 1/12 is rational. The 12th root of 2 is irrational, however, for the reason that 2 is prime.

That does not mean that the 12th root of 2 is in any way imprecise.

There are mathematical difficulties with all tuning systems, which is why there are so many of them. The problem you cite is not among them.

The joke was since the 12 root of two was irrational, it does not terminate or repeat, it is always an approximation in a mathematical sense, even if you carry it out to a billion places.

 

[Mr_Moo]

The joke was since the 12 root of two was irrational, it does not terminate or repeat, it is always an approximation in a mathematical sense, even if you carry it out to a billion places.

That's fine, but incorrect, so not terribly successful as a mathematics joke.

 

[IamGroot]

That's fine, but incorrect, so not terribly successful as a mathematics joke.

So what is the precise value?

 

[Mr_Moo]

So what is the precise value?

The precise value of 2^1/12 is 2^1/12.

Look man, I am happy to yield to your superior knowledge of music theory, jazz standards, and bass playing in general. But if you want to talk mathematics, that's different.

 

[IamGroot]

Cheers

 

[vonbladet]

Irrational numbers aren't that terrible. The square root of two cannot be represented by a finite (or repeating) decimal expansion, but that isn't at all the same as being "a mathematical impossibility". If you draw a square with sides of length one, the length of the diagonal is the square root of two. (Either to any precision you can achieve or, if platonically, exactly.)

The twelfth root of two is also irrational (although not because a twelfth is irrational; obviously it isn't) and would need a fancier construction, but in practice it can easily be calculated to any precision you need.

The limits on the tolerances you can achieve in practice are a result of engineering, not mathematics.

 

[Gravedigger Dav]

Hi,

There's meantone tuning, equal tempered tuning, well-tempered tuning, etc. Each has their own color. All I know is that I can rarely get my clip-on tuner to match my pedal tuner. So yeah, tuning is a "best effort" situation.

I heard there would be no math today.


Thank you for your indulgence,

BassCliff

You misread it. It said no meth.
Maybe next week.

 

[Gravedigger Dav]

 

[Slevage]

That's why fretless instruments can be played perfectly in tune ( not by me although I have a couple of uprights).

Someone recently posted the difference between F# and Gb. There is a difference because there are 9 divisions between F and G. Where's the "half way point."?

 

[Robscott]

Irrational numbers aren't that terrible. The square root of two cannot be represented by a finite (or repeating) decimal expansion, but that isn't at all the same as being "a mathematical impossibility". If you draw a square with sides of length one, the length of the diagonal is the square root of two. (Either to any precision you can achieve or, if platonically, exactly.)

The twelfth root of two is also irrational (although not because a twelfth is irrational; obviously it isn't) and would need a fancier construction, but in practice it can easily be calculated to any precision you need.

The limits on the tolerances you can achieve in practice are a result of engineering, not mathematics.

Well most of the guys I play with are irrational (especially drummers) so that's ok then, right?

 

[CapnSev]

Talkmath.com

"Helping the low end get back to sleep since 2019"

 

[IamGroot]

I prefer the directors cut.

 

[rashrader]

I will worry about perfect tuning when I can play perfectly in time. Until then, I'll just be an imperfect player.

 

[IamGroot]

Imperfect future tense?

 

[two fingers]

The precise mathematical value of the tuning of my bass going through my Boss TU-3 tuning pedal on stage in a bar about to play rock and country to at least 100 adoring fans is......

(Takes deep breath)



.....plenty close enough.