The following essay was originally published in the July/August 2006 edition of Accordion World magazine (editor: David Keen) and is presented here by the permission of Accordion World.
(See our review of Accordion World Magazine).
When I began repairing accordions professionally many years ago, my main interest was in tuning. It is difficult to give an exact date, but at that time motorcycles were mostly made in Birmingham and it was possible to buy a loaf that tasted like bread. Late sixties, I would say.
I soon realised that just getting reeds to the correct pitch is not the whole story. For a tuning to be effective and stable there are many checks to be made and deficiencies put right. Every component contributing to the output of sound should be at its best. Those faults which do not influence the pitch of the reeds but can sabotage their performance should also be sorted out. For example, when an accordion has been in use for some time, dust, fluff, slivers of wood, the occasional digestive biscuit etc builds up behind the keys and buttons. Some of this collection can find its way inside the accordion and choke reeds, especially the smaller reeds at the knee end of the treble keyboard where gravity has taken most of the debris. Or it will become embedded in the leather faces of the key palettes, spoiling their airtightness. Airtightness, or compression, has an important bearing on the instrument's performance.
With the above in mind, and because I prefer to work on a clean instrument anyway, on all accordions reconditioned in my workshop for resale every moving part, treble and bass, is serviced before the reedwork is even started, tuning being the final task.
Tuning has its basis in mathematics, hence its academic appeal, but tuners do not need math's [sic] to exercise their craft. The earliest tuner 1 have heard of was the Greek, Pythagoras of right-angled triangle fame who, around 550 B.C. invented the Monochord while sitting in his bath, perhaps.
I mentioned Pythagoras and his Monochord. I have had many requests about this but decided to tell you about them anyway. The Monochord, then, consists of a taut string stretched between two fixed posts which are mounted on a sounding board or box. A movable bridge allows the effective length of the string to be varied and some important relationships can be demonstrated. Consider an example, if the open string (i.e. no bridge) is A440, then if the bridge is placed at the half-way point we find that A880 (an octave higher) is produced on each side. With the bridge at the two-thirds point, E660 is produced on the long side and E1320 on the short side. And at the three-quarters point D587 is on the long side and D1760 on the short side.
Guitar players will see this immediately, violin players even sooner. And if you can prise little Johnny away from his GSCE maths homework to assess the evidence he will tell you that the pitch of the note varies inversely as the length of the string. If you have a table of standard pitches to hand, (doesn't everyone?) you can see that the numbers quoted for E and D are not quite the same as those given by the Monochord. The reason? Your table will show the frequencies of the twelve-note scale in "equal temperament" which is the tuning system in general use in the Western world. It was perfected in 1691 by Andreas Werkmeister and adopted pdq by musicians because in this system all keys sound equally concordant. Johann Sebastian Bach celebrated the fact with his "Well Tempered Clavier" comprising 48 preludes and fugues using all 12 major and all 12 minor keys.
When a note is raised in pitch by an octave its frequency, measured in Hertz, is doubled, i.e. multiplied by 2. In the equal temperament system the octave is uniformly divided into 12 semitone intervals. Uniformity is achieved by using a multiplier, the twelfth root of 2 which is approximately 1.0594631.
To show how this works we can construct a short table of standard musical frequencies, starting at A440. You may be able to use your computer for this task, but a pocket or desk calculator is more than adequate. Start by feeding in 1.0594631 and make this a constant multiplier. On mine I need to press "multiply" twice for this feature. Next feed in 440 and press "equals" to display 466. 16376 (B flat). Press "equals" again to display 493.8833(B) and again for 523.25113(C). Continue until the octave is completed and beyond if you wish.
You ladies may prefer to dance backwards, in which case press "divide" twice at step 2 before inputting 440. This will give decreasing frequencies down to A220 and beyond.
Without really trying, you have also built a compound interest table. We deposited £440 in a savings account offering just under 6% interest per annum and we left it intact for 12 years. Then we built a discount table which showed that our £440 will be worth £220 in 12 years time if inflation is just under 6%p.a. each year. Next time - how these numbers are used to tune your piano.Copyright 2006