Bits and bobs
Before I ramble about the Festival of History at the weekend, here’s a little round up of what’s been happening. Most important: after much dribbling, chomping on fingers (hers and anyone else’s in reach), and a bit of grumbling, A has her first tooth!
Katy has managed to buy us an upright piano off eBay for £5.50, although the removal costs will be quite a lot more! It’s currently in a pub about 30 miles away and they don’t need it any more - should be arriving tomorrow. J and K have been having lessons from a friend, and they’re now at the stage that lack of access to a piano is holding them back.
Last week a colleague of mine showed me the piano tuning kit and fabulous book (written in the early 1900s, with a foreword about finding one’s calling in life etc.) that he had just bought to give him something to do and an in-tune piano while on holiday with his parents in a rented cottage (with probably dodgy piano) somewhere. It was a very interesting set of tools - each one obviously just right for some specialist job - and I hope I can persuade him to practice on ours.
Emma, W and E invited Katy and everyone else to a local country park last week for a mini beasts thing run by the park rangers. The weather was great and there was catching insects in big nets, looking up on charts, looking under little logs and having a great time (photos on Flickr). A return fixture planned for this week is likely to be rained off.
August 14th, 2007 at 9:47 am
Would love to be able to tune my own piano - it’d be a wonderful sense of achievement and a not inconsiderable sum of money saved.
There’s intriguing maths involved too: if you go up an octave (say from A at the bottom of the keyboard to the next A) you should double the fundamental frequency. An octave plus a perfect fifth (eg from the bottom A to the next-to-bottom E) should treble the fundamental (because you get to the next harmonic). If you then go down an octave (to bottom E) you should halve the frequency, so you get to 3/2 times the number you first thought of.
So, going up an octave gives you 2 times the fundamental; going up a fifth gives you 3/2 times the fundamental.
So, going up n octaves gives you 2^n (ie 2 to the power n) of the original fundamental (let’s call it f): we have (2^n)f.
And, going up n fifths gives you (3^n/2^n)f.
Now let’s imagine we want to tune the A 7 octaves above the bottom A. This is usually the top A on the keyboard. What should its fundamental frequency be? Clearly, (2^7)f; that is, 128 times the original fundamental. Easy peasy.
Now let’s try the same trick with fifths. Count up the keyboard:
A-E-B (passing an A)-F#-C# (passing the second A)-G#-D# (passing the 3rd A)-A# (passing the 4th A)-F-C (passing the 5th A)-G-D (passing the 6th A-A (arriving at the 7th A above the original). This makes 12 leaps of a fifth. So the frequency of this high A should be (3^12/2^12)f, which is (531441/4096)f, which is about 129.75 times the original fundamental.
Oops. Despite the fact that we’ve defined both tuning techniques from the notion that harmonics are integer multiples of the fundamental frequency, they don’t seem to tally. Using a concert pitch for A above middle C (440 Hz) and working down in octaves to bottom A, the actual frequencies for this high A would turn out to be 3520 Hz by the octaves method, or 3568 Hz by the fifths method. This is certainly an audible difference (in fact it’s almost a semitone).
Solution: fiddle your tuning system so it doesn’t quite fit the harmonics. But it intrigues me that there should be such a problem in the first place.
August 14th, 2007 at 10:42 am
Thanks for that Beardy - most illuminating. Is the fiddling done a little bit at a time, or is it concentrated in a small number of intervals? I assume it’s the first one to sound less lumpy, but that would probably make the tuning system messy everywhere.
We have a similar problem with our code at work, where you’ve calculated that e.g. the customer’s due $15 discount, but it has to be divided equally back over 9 phone calls so that it can be taxed via a reduced price of the calls rather than as a thing in its own right. Do you have 8 calls getting $1 discount and one getting $7 (if so, which one?) or do you go 1.66, 1.67, 1.67, 1.66, 1.67, 1.67, 1.66, 1.67, 1.67 (which is fiddlier to calculate)?
(This is all because US tax is silly: don’t ask!)
August 14th, 2007 at 10:54 am
Equal temperament (common nowadays) defines semitones as having a ratio of the 12th root of 2, so 12 of them - an octave - add up to doubling the fundamental. This means that the fifth doesn’t have a ratio of 1.5 (ie 3/2, as a perfect fifth does): it has a ratio of about 1.498.
There are piles of alternative tuning systems that I’ve never bothered/had the time to try and understand but because of the maths, they all come unstuck somewhere. This is why before Bach’s day (when more modern kinds of fiddling with the tuning were developed), most western music was written in a very restricted set of keys. Otherwise it would have sounded horribly out of tune.
This is beginning to sound like rather a good excuse for my singing: I sing with pre-Baroque tuning and that song is in the wrong key for my voice!
By the way, I’m hardly an expert: I think I’ve now exhausted the sum of my knowledge. And the sum of my knowledge needs a long soak in the bath and a good sleep.