Shea Marshall

How many saxophone fingerings are theoretically possible? Recently, I searched the intertubes to see if I could find an answer to this question and couldn't, so I've decided to hit it with my limited knowledge of combinatorics.

Before we begin, I must make a few assumptions. I will not be addressing half-open keys, as that's beyond the scope of my analysis. Because of this, I'm also disregarding the front-F (fork-F) key, as it closes the B key and vents the high F and the effect can and will be reproduced and counted in my set of fingerings. I'm also using my Selmer Mark VI tenor for key layout/dependency reference.

I'll start by separating the core group of fingerings from the "independent keys" like octave, side E, etc. and will reintegrate them once we have our core total. I also separate the core fingerings into "Upper stack only" and "Both hands" to get around the problem of G#, which is deactivated by any of the 3 keys of the lower stack.

By upper stack, I am referring to the 5 pads which are controlled by the left index, middle, and ring finger using the B, bis, C, and G pearls. There are 11 possible distinct configurations of these 4 keys, allowing for the bis key's dependence on B. (figure 1)

Each of these fingerings can be used with 5 possible left pinky configurations: No keys, B, Bb, C#, or G#, giving us a total of 11*5=55 potential "core" configurations of the left hand stack.

Now we address fingerings with both hands. There are 7 possible configurations of the 4 pads controlled by the right index, middle, and ring finger using the F, E, and D pearls. The eighth (no keys pressed) was already addressed in the previous section. (figure 2)

I understand that on some older horns the "fork" 1 and 3 configuration of the right hand opens an Eb vent, this won't change the calculation at all, as it is still a distinct fingering.

There are 4 possible left pinky states when using the lower stack (G# is disabled): No keys, B, Bb, C#. The 7 possible right hand configurations can be multiplied by the 11 upper stack configurations and again by 4 left pinky states to count 11*4*7=308 fingerings involving the right hand.

Adding these 2 sets together gives us 363 Core Fingerings but we aren't done yet..

Lastly, I define the remainder as "Independent" keys which can be opened or closed without affecting or being limited by any other keys on the instrument. These are:

High D, High Eb, High F, Side E, Side C (B#), Side A#(Bb), Chromatic (fork) F#, Low C, Low Eb, and the Octave key. (Figure 3)

Although the function of the octave key is dependent on whether upper stack G is depressed, I still consider it an independent key because it can be added to every other fingering in exactly one way to effectively change it. This gives us 10 independent keys, with 2¹⁰=1024 possible configurations.

Multiplying the core fingerings by the independent key combinations gives us 363*1028=
373164 theoretically possible fingerings!

And for everyone who has been playing long enough to have played every possible fingering on the saxophone, if you wanted to try all 373164 possible fingerings for 5 seconds each to search for altissimo, multiphonics, etc. it would take (373164*5)/(3600*365)=1 hour 25 minutes every day for a year.

Again, it hasn't all been done. Proof that it's not over.

Some concluding calculations for different horns:

My Mark VI has no high F#. For a sax with a high F# key, we have 373164*2= 746328 fingerings

For Selmer Sopranos with a high G key, the high G activates the high F# so there are only 3 possible states for this structure, giving us 373164*3= 1119492 fingerings.

My Low-A Buecher 400 (stencil selmer USA) bari activates the low Bb when the low A key is pressed, so this Low A key is added to the left pinky term rather than the independent key term. This gives us 451 "core fingerings" and 451*1028 = 463628 fingerings.

Mr. Shea Marshall is a multi-instrumentalist in Phoenix, AZ -- and this is his website.