My friend and I have been arguing about this. Neither of us has a clue as to the formula. He says the answer is over five hundred thousand, I say it's much less. Can you help?

Given that there are 12 notes in a musical octave, what is the maximum number of musical scales possible within that octave, if each scale has a minimum of 5 notes and a maximum of 9 and we start all the scales from the same note?

In case you don't know anything about music, a scale is a progression of notes where you start on a specific note and end on that same note an octave higher. There are twelve different notes between these two similar notes. Which notes you choose to play determine the sound of the scale. Anything less than five notes would not make for a very interesting scale. Anything more than nine and you would be playing almost 'every' note in the scale, not leaving much room for distinction in how you organize these notes.

I assume you first have to figure out the maximum number of variations possible in a 5-note scale (with 12 notes at your disposal). Then do the same for a 6-note scale, then a 7-note, then an 8-note, and so on. Then add up the results. How to find this maximum number of variations for each scale size though is what I don't know.

Thanks,

Terence



Hi Terence,

You are on the right track regarding counting all of the 5-note scales, the 6-note scales, etc. and then adding them up. What I will clarify for you is how to count those scales.

I will assume that the starting note is fixed. According to your description, that also means the ending note is also fixed, one octave higher. Then there are 10 notes between the starting and ending notes, not counting the starting and ending notes.

I will also assume that the notes are a progression where each successive note is higher than the one previous so that no matter which notes we choose to play, they can only be played in one order.

For a 5-note scale, where the starting note is fixed and the ending note is fixed, you have 3 notes in between. There are 10 notes from which you can choose these three and mathematically the formula for the number of ways to choose 3 things from a group of 10 things is given by the formula 10C3 (in words, 10 choose 3) which is:

 10!/3!(10-3)! where n!is the product of the numbers from 1 to n
i.e. n! = n(n-1)(n-2)(n-3)...(3)(2)(1) = 10!/3!7!

For a 6-note scale, there are 4 notes in between to choose from the 10 and so there are 10C4 6-note scales:

 10!/4!(10-4)! = 10!/4!6!

For a 7-note scale, there are 5 notes in between to choose from the 10 and so there are 10C5 7-note scales:

 10!/5!(10-5)! = 10!/5!5!

For an 8-note scale, there are 6 notes in between to choose from the 10 and so there are 10C6 8-note scales:

 10!/6!(10-6)! = 10!/6!4!

Note that this is the same as the number of 6-note scales, 10C4.

Similarly, the number of 9-note scales, 10C7, is the same as the number of 5-note scales, 10C3.

Thus, to calculte the number of 5-note scales:

10C3 = 10!/3!(10-3)! = 10!/3!7! = 10(9)(8)(7!)/3!7!

Simplifying we get 10(9)(8)/3(2)(1) = 120 5-note scales

Since 10C3 = 10C7, the number of 9-note scales is also 120.

By similar calculations we find that 10C4 = 10C6 = 210 so the number of 6-note scales is 210 and the number of 8-note scales is 210.

10C5 = 252 so the number of 7-note scales is 252.

Adding all of these up we get a total of 912 different scales containing from 5 to 9 notes. Changing where you begin/end your octave will result in another set of 912 different scales.

I hope this helps,
Leeanne



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