Notating the exact placement of noteheads

[stillofthenight]

I understand the basic notations , the simple divisions and subdivisions and why they are placed where they are. But sometimes I come across a part where I cannot understand the mathematical reason/method and preciseness of why the note head is placed where it is.

Look:



In the second measure, the second eighth note in the treble happens to start just a nudge after the 2nd note of the last triplet. The treble E note is sandwiched in. But how was it known that it is supposed to go EXACTLY there?

And then in the 3rd measure, the sixteenth note of the last beat just so happens to come a more slighter nudge AFTER the last note of the triplet...but I want to know how and why this is mathematically correct. This sixteenth is a "split" of an eighth note in reference to the 2nd measure...and this sixteenth happens to come in a peculiar spot.

I have tried thinking about it but cannot figure it out.

[lelle]

Lets look at the last beat of the 2nd measure mathematically

I'm sure you'll agree that eight notes are halves of a quarter note and triplet eight notes are one-thirds of a quarter note

So in the right hand we have two eight notes on the fourth beat, ie two halves

1/2 + 1/2 = 2/2 = 1 = 1 beat ( = 1 quarter note)

in the left hand we have an eight note triplet, so each note uses one-third of the beat

1/3 + 1/3 + 1/3 = 3/3 = 1 = 1 beat

So let's convert these fractions so they have a common denominator

1/2 = (1*3)/(2*3) = 3/6

1/3 = (1*2)/(3*2) = 2/6

So, if we divide the beat into six parts (1 = 6/6), each right hand eight note will use three sixths of the beat, and each left hand note will use two sixths of the beat.

So lets divide our beat into six parts and mark where each note is

Right hand

* * * * * *

The six stars represent the beat divided into six parts and each underlined star represents an eight note, and as you can see each eight note uses three out of the six parts of the beat

Left Hand

* * * * * *

Each note of the eight note triplet uses two-sixths of the beat

Put together

* * * * * *
* * * * * *

To figure out the sixteenth note, which is 1/4 of the beat, you find the common denominator for 1/3 and 1/4 and do the same thing.

[Bob]

Looks ok to me.  2 vs. 3.  and 3 vs. 4 (four 16ths).

[stillofthenight]

Aha so it seems that I was not using common denominators. It is just a matter of thinking of it as one beat and then comparing using common denominators. A 1/8 note essentially is a half of the quarter note beat and can be thought of as 1/2 in fraction form ,  a beat just cut in half. I was using the fraction 1/8 instead because that is what was written when instead should be 1/2. If I used 1/8 the error would be that I cut the beat into 8 pieces. So I guess that is where I was having a problem.

Thanks lelle.

[stillofthenight]

If a quarter note divided by 2 = 1/8 + 1/8
If an eighth note divided by 2 = 1/16 + 1/16
If a sixteenth divided by 2 = 1/32 + 1/32

Why 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.

[nyiregyhazi]

If a quarter note divided by 2 = 1/8 + 1/8
If an eighth note divided by 2 = 1/16 + 1/16
If a sixteenth divided by 2 = 1/32 + 1/32

Why 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.

Because you'd have to do even more stupid maths than this dumb system already demands in order to make any sense of anything.

Take a "quarter note" as being your base duration of 1 for most calculations and it's all very simple. The triplets are thirds then and nobody need start doing anything but the most basic calculations to make easy sense of it all. 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, it's a ridiculous place to assign a base value of one. Admittedly, seeing a triplet as a twelfth is no more stupid than calling a semi quaver a sixteenth, but both are totally impractical labels. Whoever started the terminology should be shot. Practical counting has nothing to do with those ridiculous pieces of terminology, so please don't make the mistake of adding up tiny fractions instead of seeing practical ratios between notes.

[stillofthenight]

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.

[lelle]

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.

It's correct. Your "1" in your calculation will usually be the beat, or whatever time period you need to play the tuplet. So if it was a bar in 4/4 time with a quarter note quintuplet your "1" would be the whole bar (a whole note). The only reason to even do this is to understand exactly where each note will end up so why make it more cumbersome than it already is? Especially since your notating software will place them correctly anyway

[mjedwards]

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.

    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.

The only relevant fractions are fractions of beats. Given that no piece has semibreve beats,[...]

    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.

[nyiregyhazi]

    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.

Ah, interesting. I've literally never heard of a single example of a one at the bottom before. Any others out there?

Regarding names, Americans don't like the British system merely because it needs to be learned by rote and they haven't done so. Everything is confusing when you don't know it. The problem with the American system is that it confuses when you DO know it. Personally I'm attached to neither. I teach names merely because people need to know terms that will be used by others (and actually, I'm a little slack on that even, as I've never seen much importance in the label but merely in what is meant rhythmically). In an ideal world, I'd use fractions like the American system but the crotchet would be the base of 1. It's simply how virtually all rhythmic thought actually operates. The positive of the British system is that it doesn't ever introduce a contradiction about having to count 1 while referring to a quarter. But I'd rather just both count 1 and call it a 1. Obviously we still adapt for 2/2 etc but it's a lot easier overall. Awareness of proportions matters, not external labelling. But crotchet is a much easier code word for 1 than a quarter is. Who wants to learn labels that must be multiplied by 4 in 95 percent or more of practical counting?