Pascal’s Triangle and the Mathematics of Reed Combinations – Originally published on my old blog in October, 2008. Republished here January, 2017
One of the things I like about piano accordions is their versatility. The mechanics of switch mechanisms permit you to open and close the valve holes for entire reed blocks at the touch of a button, allowing you a large range of sound out of an instrument that is relatively small.
As I write this I have in my hand an old Petosa brochure advertising, among other things, “over one quarter century of accordion eminence.” That dates the pamphlet to around 1950. Which makes it kind of an interesting glimpse into accordion history. One page of the pamphlet extols the “direct action switches” (which just means that when you press a switch, it opens and closes multiple slides at one time) feature on their accordions. Modern accordions of course all have this feature, but there was a time when a single switch mechanism did only one thing –opened the slides for a given reed block or, if they were already open, closed them.
In some respects, this type of manual set-up is more versatile, because every combination of reeds is available. The drawback is that you may have to open or close more than one switch to get the combination that you want. As long as there is a manual switch for each set of reeds, however, any combination of reeds is possible. This is worth noting because with modern direct action switches this may not always be the case. Frequently, for example, you will see student model accordions with three sets of reeds, but only three or only five switches. With three sets of reeds, the accordion can potentially have seven different combinations of reed blocks (actually eight, but the combination where every reed block is closed off isn’t very useful –unless you’re playing the avant-garde composition 4’33″ by John Cage.) The general formula is that for an accordion with n sets of reeds, there are 2^n – 1 possible combinations.
If the reader has an interest in number theory, the fancy explanation for this is that the sum of the elements in the nth row of Pascal’s Triangle is equal to 2^n. You then subtract the useless combination and the general formula becomes 2^n – 1.
The mathematics of this is actually relatively straightforward and intuitive. For each set of reeds there are two possibilities: the valve holes can be open (air passes through when a key is pressed,) or closed (air cannot pass through the reeds.) With only one reed block you have two possible combinations: all the reeds are open, or all the reeds are closed. And every time you add a reed block you are essentially doubling the number of possible combinations because you have every combination you already had, and for each of those you have two possibilities for the new reed block (either open or closed.) So that when you add a reed block, you just multiply the old number of combinations by two to get the new number of combinations. Two sets of reeds gives four combinations (three not counting the combination where everything is closed off,) three sets of reeds gives eight combinations (really seven,) and so on up to five sets of treble reeds which gives a potential 2^5 – 1 = 31 possible combinations. I’ve never seen an accordion with more than 5 sets of treble reeds. Five sets of reeds is frankly a little overkill.
Leaving the math, (fascinating thought it may be) aside for a moment, the point is that with direct action switches you are frequently shortchanged on reed combinations. And not just with student models either. A Petosa AM1100, for example, has four sets of reeds but only 12 different treble switches. With four sets of reeds there are potentially 2^4 – 1 = 15 combinations. To be fair, however, the three combinations that are left out are arguably redundant. The AM1100 is missing the bassoon/flute/piccolo, bassoon/flute, and flute/piccolo combinations, which isn’t such a big deal at all since they give you the bassoon/clarinet/piccolo, bassoon/clarinet, and clarinet/piccolo combinations.