A Visual Guide for Formula Pattern Scales (Free Download)

Contrary motion scales are awesome. Not only are they fun to play and sound cool, but they’re a wonderful way to teach scale fingerings – especially when students are first learning to play scales.

A step up from a simple contrary motion scale is playing scales using what’s called a “formula pattern.”

P.S. I’ve always wondered why it’s called a “formula pattern” so if you know, please educate me! I find it to be a boring name for such a fun scale! Ha!

Actually, I think we should call them zig-zag scales instead! What do you think?! LOL


What is the Formula Pattern?

If you’re unfamiliar with this scale pattern, it is basically a 2 (or 4) octave scale with a bump in the road.

  1. Begin by playing the scale ascending in parallel motion.
  2. At the halfway point, play a contrary motion scale, returning back to the middle.
  3. Finish the top half of the ascending scale in parallel motion.
  4. Once again, after descending halfway back down the parallel motion scale, throw in another contrary motion scale (out and back in).
  5. Finally, finish the pattern by descending the final half of the scale in parallel motion.


Why the Visual Works

The first time I tried to teach a student the formula pattern, it was a struggle. I try to avoid using formal “scale books” for students to have to read every note and fingering, so I needed to find an easy way to explain the pattern.

Since I’m a visual person, I came up with this simple visual for my students. Every student I’ve used this with has found it very helpful – I hope that perhaps it will help your students as well!


Formula Scale Progressions

Here is the leveling based on the Royal Conservatory of Music program’s technical skill requirements to give you a rough idea of a good progression of this particular technical skill.

Level 1 = C Major (2 octaves)
Level 2 = C, G Major (2 octaves)
Level 3 = D Major (2 octaves)
Level 4 = C harmonic minor (2 octaves)
Level 5 = A Major, A harmonic minor (2 octaves)
Level 6 = E Major, E harmonic minor (2 octaves)
Level 7 = D Major, D harmonic minor (2 octaves)
Level 8 = Eb Major, Eb harmonic minor (4 octaves)
Level 9 = Db Major, F Major, C# harmonic minor, F harmonic minor (4 octaves)

P.S. Joy Morin has a great free downloadable PDF of the Technical Requirements for the 2015 RCM Program.




  • Now that Iā€™m retired I have time to explore interesting ideas from other teachers. Thanks so much!

  • I call the “Formula Pattern” the “Rollercoaster Scales”! I agree, it does need a new name. šŸ™‚ My students and I put the “rollercoasters’ in our arms. We trace both arms going up the first “rollercoaster”, then make our arms go out to the sides and come back in. On another week, we add the “second rollercoaster”. They go the second hill, down, go out and in, and then down the long roller coaster track (descending scale). In short the arms trace the visual example you have above! We also draw out that same visual for students to remember the “rollercoasters”.

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