Permalink Submitted by Anonymous on January 24, 2015

If we are to do this we then would need to consider not the number of different values a note could have, but the ratio of counts between two notes (ex. two counts to four counts, which simplifies to one count to two counts). By making a list of all possible rations and simplifying them, you can see the repeats and cross those out as possibilities. You are thusly left with twenty four unique ratios. Because these are ratios between two notes, we must raise twenty four (the number of unique ratios) to the n-1 then multiply. This leaves us with the following values
2 notes: 600 melodies
3 notes: 270,144
4 notes: 1.08e8
5 notes: 4.06e10
As you can see, it quickly escalates into much larger numbers of possible melodies than when we only considered the values of separate notes. It does, however, give us a much more accurate set of values for the possible numbers of melodies. One may intuitively think this cannot be correct on the basis of note values. For example, take a note with a value of one count to a note of a value of two counts, a one to two ratio (my apologies for not including the correct note names; I do not know the British convention for note values) then for the third note, a three to two ratio to the second note. Although this may seem unappealing at first, it is possible because these are absolute, unique ratios, i.e. simplified. therefore as long as the third note is twice as long as one third of the second note, it is allowed. In this case, the note has a value of that note is four thirds, one of the established notes that is being used.

## If we are to do this we then

If we are to do this we then would need to consider not the number of different values a note could have, but the ratio of counts between two notes (ex. two counts to four counts, which simplifies to one count to two counts). By making a list of all possible rations and simplifying them, you can see the repeats and cross those out as possibilities. You are thusly left with twenty four unique ratios. Because these are ratios between two notes, we must raise twenty four (the number of unique ratios) to the n-1 then multiply. This leaves us with the following values

2 notes: 600 melodies

3 notes: 270,144

4 notes: 1.08e8

5 notes: 4.06e10

As you can see, it quickly escalates into much larger numbers of possible melodies than when we only considered the values of separate notes. It does, however, give us a much more accurate set of values for the possible numbers of melodies. One may intuitively think this cannot be correct on the basis of note values. For example, take a note with a value of one count to a note of a value of two counts, a one to two ratio (my apologies for not including the correct note names; I do not know the British convention for note values) then for the third note, a three to two ratio to the second note. Although this may seem unappealing at first, it is possible because these are absolute, unique ratios, i.e. simplified. therefore as long as the third note is twice as long as one third of the second note, it is allowed. In this case, the note has a value of that note is four thirds, one of the established notes that is being used.