Tuning an Aluminum Tube and iphone apps to test overtones.

[Vic]

Hi,

I had some really good help here with the Radio Frequency project I did a sort time ago.
So, I thought I would seek some help/advise on a current project.

I have an aluminium tube it is 1m in length, 10cm Outer Diammeter and approx 4.6mm wall thickness.I have mounted it on a stand through the nodes (22.4% from each end) so as to not alter the tone when struck.

When struck with a rubber mallet it has a beautiful tone that site close to C#.

On friday I will spend the day in the workshop doing experiments to see how much I have to cut of to tune it as I need to make 12 of them of various tones.

Does anyone happen to have any calculations that might assist me before I start the experiments as a rough guide to lengths of this type of material and say for example how much needs to be cut off to shift one semi-tone?

I am happy to find it via trial and error but I am also interested in the physics behind it and what equations there might be.

Any help would be appreciated.

Also, has anyone had experience with Iphone/Ipad apps to test frequency response, harmonic overtones etc? How accurate have you found these?

Thanks

Vic

[Milo]

If you are using the well-tempered scale, then the frequency interval between each note is sq rt 2. Invert to calculate length (l = 1/f)

[Manning]

I began writing a quick response and it turned into an essay - sorry.

Here's the equal temperament scale maths. (The maths for other scales ranges from slightly different to VERY different.)

First you need to know your frequencies. Then you need to determine the pipe length ratio required to obtain those frequencies.

Frequency ratios:
The freq ratio between two octaves is 2^1 (that's "two to the power of 1", aka plain old two). This is easy enough to understand. (Why I'm being pedantic and saying "two to the power of one" will become apparent soon.)

It is thus obvious that the freq ratio between twelve semitones must also be 2 to the power of 1, because 12 semitones equals an octave.

Less obvious is the conclusion that the freq ratio between two semitones must therefore be 2 to the power of (1 divided by 12). This is also known as the 12th root of 2. (In Excel it is produced by the formula: =POWER(2,1/12) and roughly equals 1.059463).

How did we get to this apparently bizarre number?

Well, the question actually being asked here is: If I start at frequency "x", how do I consistently work my way to "2x" (the octave) in 12 identical steps?

The key here is that the process requires multiplication at each step (ie - it is "geometric").

Why is this process multiplicative? You can prove this by eliminating the alternative of addition. If the process was additive, to get to the next octave you'd just add 12 equal increments. (eg start at 120 Hz, add 12 steps of 10hz and bingo you get to the octave - 240Hz. However this adding system tells you that the next octave would occur at 360Hz, which your ears will quickly tell you it doesn't. Thus the only way of doubling any starting number in 12 steps must be to use a multiplicative process.

Hence we need to divide a doubling ratio into 12 equal multiplicative steps. A doubling ratio is 2 to the power of 1, so each individual step must simply be 2 to the power of 1/12. Here we use a branch of maths known as 'surds' (or Nth roots), which exploits the fact that any root can be expressed as a fractional power. Hence SQRT2 = 2^1/2 (square root of two = two raised to the power of 1/2). The twelfth root of 2 becomes 2^(1/12).

Multiplying numbers raised to a power is equivalent to adding their powers together. So (2^2) * (2^3) = 2^(2+3) = 2^5. And indeed 4 * 8 does equal 32 which is 2^5. Likewise mulitplying 2^(1/12) twelves times is equal to 2^(1/12 + 1/12 + 1/12 +.... etc) which is equal to 2^1, which is what we wanted all along.

Pipe Lengths
We'll assume your tube is perfectly tuned, meaning A above middle C = 440Hz. Dividing 440 by 1.059463, then dividing the result by 1.059463 etc, we eventually get to C# = 277.182635 Hz. We'll use this as our starting frequency.

Your pipe's thickness and diameter has a constant modifying influence to the pitch it produces, and we need to find that modifier so as to then calculate the lengths of the other pipes. We'll call this modifier "Pmod".

Frequency is proportionate to 1/(length)^2. Some manipulation of this shows that length is thus proportional to 1/(frequency)^1/2 or 1 divided by the square root of the frequency.

Your pipe modifier (Pmod) therefore equals 1000 mm (the length of the C# pipe) divided by 1/(sqroot(277.182635)) where 277.182635 is the freq of C#. A bit of calculation results in Pmod = 16648.80281

So from here, calculate your frequencies and apply the formula Length(mm) = Pmod * 1/(sqroot(freq)). This produces:

Code: Select all

Note     Freq            1/sqroot(freq)      pMod            Pipe Length (mm)
C#       277.182635      0.060064379         16648.80281     1000
D        293.6647722     0.058354463         16648.80281     971.5319412
D#       311.1269882     0.056693224         16648.80281     943.8743127
E        329.6275617     0.055079278         16648.80281     917.0040432
F        349.2282365     0.053511278         16648.80281     890.8987181
F#       369.9944281     0.051987916         16648.80281     865.536561
G        391.9954417     0.050507921         16648.80281     840.8964153
G#       415.3047036     0.049070058         16648.80281     816.9577266
A        440.0000064     0.047673129         16648.80281     793.700526
A#       466.1637683     0.046315968         16648.80281     771.1054127
B        493.8833084     0.044997442         16648.80281     749.1535384
C        523.2511382     0.043716452         16648.80281     727.8265914
C#       554.36527       0.04247193          16648.80281     707.1067812

Note that halving the length of a pipe does NOT raise its pitch by one octave.

Also, as you stated, your pipe is not exactly in tune to C#. The above lengths will produce frequencies correctly related to your starting note - in other words they'll all be equivalently out of perfect tune.

Also my apologies for not cleaning up the decimal expansions. If you do manage to cut your pipe to within one ten-millionth of a millimetre, please upload pics.

[mylesgm]

Excellent post Manning.

[Alastair Reynolds]

Ditto That

That's one hell of an explanation Manning, but perfect.
I dips me lid to you

[Wiz]

I tried to understand it...I honestly did...8)...I read it five times...but then I passed out...when I woke up...I couldn't remember my name or how to tie my shoes....which is weird...cause I am wearing thongs atm.....8).

I have always wondered about the mental arithmetic to discern notes vs frequency.

I recently started using a test tone, within my daw to help me more accurately ascertain what frequency I don't like about a sound, say a ring on a snare, and then cut that frequency, rather than sweep the EQ.

Eg listen to sound. Hum note I don't like in the drum sound that I hear. Un mute oscillator...adjust tone of oscillator to get to that note....repeat till I found it...then readout on the oscillator what the frequency is, and then start EQing.

Its been interesting.

I understand that an octave, is a doubling of frequency, and that a 31 band graphic, splits the octave into three points between...third octave EQ three sliders per octave...but I always was confused about the not linearity between numbers..

But I have never been able to work out the frequencies between the octave in my head.

Now I know why, I am the mathematical equivalent of a besser block. I do however make a very nice BBQ roast lamb in a greek style, so all is not lost.

If at all possible, could you explain, simpler...the method of dividing up the notes, between the octave points....or am I hopelessly unteachable...

cheers

Peter

[Djembe]

Wow! That's some answer, Manning.
When I had to tune aluminium pipes a few years ago for a sound sculpture project, I just kept cutting bits off until I had the note.
I had to get mine anodized, so made sure I did that first because it affects tuning.

[Manning]

But I have never been able to work out the frequencies between the octave in my head.

Dude, cut yourself some slack.

Anyone who genuinely IS capable of calculating the 12th root of a number in their head probably lives in one of those "special homes" and has BBC documentaries made about them every couple of years.

[Wiz]

But I have never been able to work out the frequencies between the octave in my head.

Dude, cut yourself some slack.

Anyone who genuinely IS capable of calculating the 12th root of a number in their head probably lives in one of those "special homes" and has BBC documentaries made about them every couple of years.

thanks, I feel a little better now...8).

Peter

[Manning]

Wow! That's some answer, Manning.
When I had to tune aluminium pipes a few years ago for a sound sculpture project, I just kept cutting bits off until I had the note.
I had to get mine anodized, so made sure I did that first because it affects tuning.

Blair - are you the guy who recorded in our studio on the weekend? (I wasn't there, but Marc was.)

If this is you - I heard the tracks last night, great,great stuff mate. For quite a while we muted everything except your parts and sat back just enjoying them.

[Manning]

Here's an appendix to my ridiculous essay above, which demonstrates why "perfect scales" go horribly wrong.

Pythagoras (allegedly) discovered that the ratio between a note and its perfect 5th is 3/2.

I say "allegedly" because some people believe he really did discover it, some people believe he only falsely claimed credit for discovering it, some people believe that he never actually claimed credit for discovering it, and finally some people believe that he never existed in the first place.

So here's a purely mathematical exercise which demonstrates that you cannot create a 12 tone scale from this "perfect" Pythagorean 5th ratio of 3/2. It also uses the (undisputed) fact that if you halve a frequency you get a note exactly one octave lower.

The process is:
1 - take a starting frequency (we'll use A = 440Hz)
2 - Multiply it by 3/2 (or 1.5) to get the frequency of the 5th.
3 - Multiply that result by 3/2 to get the next 5th.
4 - If the resulting frequency is ever greater than 880Hz, halve it to get the same note an octave lower.

Here are the results:

Code: Select all

Note       Freq (adj)
A          440.00
E          660.00   
B          495.00 (down an octave)
F#         742.50
C#         556.88 (down an octave)
G#         835.31
D#         626.48 (down an octave)
A#         469.86 (down an octave)
F          704.79
C          528.60 (down an octave)
G          792.89
D          594.67 (down an octave)
A          446.00 (down an octave)

So as you can see, if you apply the 3/2 ratio through an entire 12 tone scale (adjusting the octave down as needed), you end up 6Hz away from your starting point.

[Drumstruck]

Manning you are the man!! Thankyou for those excellent posts. Succinct but of sufficient depth and accessibly written.

[Djembe]

Blair - are you the guy who recorded in our studio on the weekend? (I wasn't there, but Marc was.)

If this is you - I heard the tracks last night, great,great stuff mate. For quite a while we muted everything except your parts and sat back just enjoying them.

Yeah, that was me. It was great to record in your studio and a pleasure to work with Marc. Glad you enjoyed the tracks.

[Milo]

Ah, yes, but if you read further into the history of Greek and Indian modal music, you eventually find out what the Just tuning scale was for: its purpose was to describe the structure of the elements (any good quantum scientist will tell you that subatomic particles vibrate in ratios of 2/1 and 3/2, and that the smallest action - a quantum action - is measured by a multiple of Planck's constant, h, in ratios of 0, 1h, 3/2h and 2h - many other examples), so the Greeks (and the Chinese- same measuring) weren't saying that perfect fifths formed a workable scale. They derived workable modes from these elementary (naturally occurring) ratios, using these and the many dieses (microtones) such as the Pythagorean comma.
It was also used to explain time (circular) differences between the Perfect Year of 360 days and the solar year of 364.25 days .....etc, etc...yadda, yadda, yadda.
Fascinating when you really get into it....such as octave and fifth ratios can actually define the finite limits of the universe...stop, Milo, stop there!

[Chris H]

Cool, as a casual observer of this thread, I never imagined it would take us to the end of the universe........

[Alastair Reynolds]

I hope you've all been taking notes 'cause there'll be a quiz at the end of the day

Tomorrow, quantum mechanics and string theory

[Manning]

Whoa Alastair, don't go crazy.

Tomorrow we'll only cover axiomatic quantum field theory, with special attention to the Yang-Mills problem and imaginary time.

[rob]

strangely, some 25 years ago I recieved an Australia Council grant to develop a marimba pipe tuner. The pipes under the bars of a marimba are passive resonators that increase the volume and tone of the struck bar. Since the pipes themselves don't "make" a sound it its difficult to tune them other than by hitting the bar and adjusting the pipe length for the strongest resonance. This is tendious and time consuming. To the professional concert orchestra marimba player the pipe tuning is important and significantly it changes with changes of atmospheric conditions, making a rapid means of tuning before a gig desirable.
So we developed a computer controller audio generator and measurement system that within a short period of squirting out tones would tell you what the pipe resonant note was.

The maths for determining a pipe's pitch from dimensional information is difficult as it depends on all sorts of factors including how the pipe is "driven" .. ie struck or blown. I'm of little use here as my notes have long disappeared into the past and my memory has diminished.

good luck!

[Vic]

Wow, I turn my back for a second then there's 19 in depth replies.

Thanks for all this great information.

I am going to now print it out, sit with a coffee and read it all slowly and repeatedly.

Will get back to you shortly.

Thanks

Vic

[Vic]

Hi,

It took me a while to get back here to this thread, caught up on other things. But next Wednesday I have to tune these twelve pipes, So I have been having a good read of all this GREAT information.

Manning, thanks so much, that's such great information and a real help to me. Really appreciate it.

And yes, I have had to read it several times and will sit down and read it several more.

Here is a link to a sort video of 5 of them just cut to different lengths just as a test to see what different lengths mean. These ones are 1020mm, 1000mm, 980mm, 960mm. And two of them are the same.

(video link on the way - internet being crap, just uploading it to youtube, will post in another post shortly)

My understanding that thickness affected Volume, not pitch. Is that not the case?

On Wednesday we are to sit and start the fine tuning process now we have conducted some experiments.

Wagga Wagga Stainless Steel (although we are using Aluminum, the Stainless Steel tubes sounded crap) have been helping me heaps, giving me space and time in their workshop.

Vic

[stosostu]

Someone should submit this thread as an outline for an episode of the Big Bang Theory, we just have to find a way to include Penny in this........

[Manning]

Hi Vic

Thickness, density and diameter all influence the pitch as well, though in complicated ways which I won't even begin to attempt to explain (I don't think I fully understand them myself). Temperature and air pressure factor in as well.

Fortunately if you are using the same type of pipe those variables are usually constant. This is why you can use the pitch (determined by trial and error) of one length of pipe to calculate what lengths (of the same pipe) produce other pitches.

If you wanted to use different types of pipe together then your only option would be trial and error, as there is no simple method (and possibly not even a complicated method) of correlating all the different variables that would be in play.

I'm looking forward to the video

[Vic]

Can't upload to Youtube or Vimeo just now for some reason, so will post video whenever I'm able.

Luckily all the pipes are the same (we have 3 x 6m lengths of 4.6mm thick 100mm diameter Aluminium Tubes (The type they use to make Truck Bull Bars.

We are flooded in right now, So I'm not sure if we will get into town by wednesday to tune them. But if not, it means I will have more time to study.

Vic

[Vic]

Weird uploading problems, but I managed to make the video go to Vimeo.

So, as mentioned, this is 5 lengths cut to 1020mm, 1000mm, 980mm, 960mm (there are two here at 960). These aren't fine tuned, just made them these lengths to see what it would give us.

http://vimeo.com/37888017

Video from an Phone. And they sound much nicer with a rubber mallet.

Vic