Musician's Problem

Probability Level 2

In music, "scale" is a set of intervals between notes. Intervals can be whole tone and semitone. The scale is called "diatonic", if there are only 2 intervals of semitone, they are not adjacent and the total number of intervals is 7.

For example,

1.Tone Semitone Tone Tone Semitone Tone Tone - is diatonic.

2.Tone Semitone Semitone Tone Tone Tone Tone - isn't diatonic.

3.Tone Semitone Tone Semione Semitone Tone Tone Semitone - isn't diatonic.

So, how many diatonic scales can we make?

The answer is 15.

1 solution

Sardor Yakupov
Aug 15, 2017

We can simply count different options. If first semitone will be in the 1st interval, second semitone can be in the 3, 4, 5, 6 and 7th interval. (it can't be in second interval, because 2 semitones will be adjacent)

If first semitone will be in the 2nd interval, second semitone can be in the 4, 5, 6 and 7th interval.

3rd interval - 1, 5, 6 and 7th, but there already was a combination (1;3). SO, only 5, 6 and 7 count.

4th interval - 6 and 7th.

5th interval - 7th.

So, total number $N=5+4+3+2+1=15$ .

$\boxed { 15 }$

Note: Technically speaking, the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale

Calvin Lin Staff - 3 years, 9 months ago

I tried to give less technical information, for not making problem too musical.

Sardor Yakupov - 3 years, 9 months ago

If the first semitone is in the 1st interval and the second one in the 7th interval then they are actually adjacent.

Imagine building this scale with C as the root. Then, it would contain

C, Db, Eb, F, G, A, B, C

You see that B, C and Db are all consecutive semitones, which wasn't allowed.

Henry U - 1 year, 7 months ago

Musician's Problem

The answer is 15.

1 solution

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