If a quarter note divided by 2 = 1/8 + 1/8If an eighth note divided by 2 = 1/16 + 1/16If a sixteenth divided by 2 = 1/32 + 1/32Why can't for eighth note triplet you do a quarter note divided by 3 = 1/12 + 1/12 + 1/12 Why does it have to be 1/4 = 2/6 + 2/6 + 2/6 which is like thinking of the quarter note as 1 and dividing 1 by 3 to get 1/3's and then converting the 1/3's to equivalent 2/6's.
Ok so if I understand correctly I just think of the quarter note as "1" instead of 1/4 for thinking of where the exact spacing of notes go.So say I want to mathematically understand how to play a sixteenth note quintuplet in 4/4 time. Now most people I assume would say "just play 5 evenly spaced notes over one beat." That would be 5 evenly spaced notes played over a quarter note, or equivalently 5 evenly spaced notes over 4 sixteenth notes. So if you were to analyze a 16th note quintuplet over 4 sixteenth notes...I thought of each note of the quintuplet as lasting 4/20 and then each note of the four 16th note grouping last 5/20. From what I drew out looks pretty similar to what my notating software produced.
The concept of a quarter as the normal way to perceive a crotchet really is lunacy. Anyone with an ounce of practicality counts whole beats, not needlessly complex fractions. The given names are extremely silly and the less attention paid to the terms (instead of simpler ratios) the better. Take them as labels, not a basis for having to do intricate maths.
The only relevant fractions are fractions of beats. Given that no piece has semibreve beats,[...]
There might be a case for adopting the British (and Australian) system of calling the notes minims, quavers, crotchets, and so on. This divorces the names of the notes from the supposed fractions associated with them. Since I am Australian, I use this system myself. I do understand the American system, but I find it a bit distracting too, and sometimes have to think a moment to determine just what note an eighth or sixteenth or thirty-second, etc., is. But I have heard American people criticize what they regard as the confusing and illogical mess of names that the British system uses. So maybe it comes down to what you're used to. Well, there are a few exceptions - such as the coda to Saint-Saens' Symphony no. 3 in C minor (the "Organ" symphony), which is in 3/1 time. But certainly, such beats are rare now.Regards, Michael.