Standard Staff Notation Issues

Standard Staff Notation Issues | Maths and Music | N J Wildberger

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1,693 views Aug 20, 2023 Maths and Music

There are two major issues in representing music on a written page: how to systematically specify the pitch of a given note, and how to systematically specify the duration of a given note. We introduce some of the standard features of our system by looking at a famous Mazurka Op 7 No. 2 of F. Chopin.

In the Western world, our solution to the first of these issues is firmly rooted in the particular architecture of the piano, in which the white notes are seen as first class citizens, and the black notes as second class citizens. I want to stress the arbitrariness of this current system, as well as its intrsinsic lop-sidedness. Unfortunately all musicians, whether or not they play keyboards, are forced to use this particular idiosyncratic system, and of course centuries of use and overfamiliarity renders it largely immune to critical examination.

The second issue of note duration is certainly somewhat more systematic, but still has some rather ad hoc elements, notably the use of the distinction between black noteheads and white noteheads to separate quarter notes from half notes (and whole notes).

However we will see that in fact interesting alternatives to both issues have been raised, and in our next video we will add our own contribution: a balanced "parity" system for music notation.

This is part of the Maths and Music Playlist: please check out the other videos in the series at

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Hello everyone I'm Norman Wildberger today I'd like to discuss the issue of Staff notation

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for our music that's commonly used in the western world where we have some story like

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this especially suited really for piano but pretty well also used for a wide variety of

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other instruments percussion is a little bit different and we'll have to talk about that

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in a quite separate way so there are really two major issues that I want to start moving towards

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and they're very important because I think it's pretty clear that the system that we're using

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now even though historically really well grounded is not really very optimal and presents a hurdle

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for young people and in fact just anybody trying to enter the musical world and it definitely is a

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hurdle and there's probably a lot of opportunities for improving our understanding of music by having

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better notation so this is especially sensitive issue for a mathematician because mathematicians

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are acutely aware care of the importance of notation in being able to guide you in fruitful

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directions of investigation and discovery Frederick Chopin Mazurka Opus 7 Number two

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[Music]

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[Music]

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[Music]

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So here is the first line or so of the Mazurka of Chopin that I played you and it's just a way

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for me to illustrate some of the features that we find with the modern staff notation so what

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do we see here well we have this this whole thing is called a staff this configuration we have five

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lines on the top and five lines on the bottom this is called the treble clef there's the bass

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clef typically this is the right hand and this is the left hand on piano and these funny looking

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things here are sort of sort of centering where this clef is sort of located in terms of pitch

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um it's like a G clef here okay this three over four is representing the time signature that's

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represented by these bars these segments that we have separated by these vertical lines are called

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bars and each bar has a certain number of beats in it okay and in this case there are three beats per

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bar that's what that numerator of that fraction kind of thing is telling you there are three beats

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in a bar and the denominator is telling you that a quarter note which is a special kind of note this

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is a chord note here okay that's what it looks like this is also a quarter note these are both

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different representations of a coordinates of a black filled in note with a single stem

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that note gets a single beat so there are three of these guys in each of these bars okay

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what else do we see well we see some Italian Viva ma non Tropo so some brief indication of the tempo

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and here is actually a metronome unit I'm not sure if Chopin put that in but perhaps the editors put

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that in suggested time speed 160 beats per minute perhaps and and what we have here is

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in the right hand typically sort of a melody and you can see various different variant notes okay

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quarter notes this is called a half half note these are eighth notes these are guys here these

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are 16th notes and there's actually a triple of them there we see some signs either going

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smaller like diminishing and in tone in in volume or increasing it's a crescendo there we see little

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dots the dots are to indicate staccato so the note is supposed to be just touched briefly and

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importantly especially for Chopin are these these phrase lines that tell you that this should all

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be in one phrase it's like when you're singing you sort of start here and you go there and then

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you take a breath okay so phrasing but sometimes these kind of bars also indicate that notes are

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to be played continuously that is you're not going to lift up uh what else there's a rest here this

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is a quarter note rest so there's three beats in this bar the first beat is sort of taken up by

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the rest meaning that you're not actually playing there you're playing on the second and the third

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now in terms of the actual notes themselves well this is a system of notation that that we are

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familiar with somewhat okay that we're going to have to talk a lot about and you see that

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um the their key signature which we talked about last time is is absent here or or put it another

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way this is suggesting that we're in the key of C well actually we're not in the key of C in this

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piece so it turns out that for a major scale if you were playing in a major scale the absence of

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Sharps and flats here at the beginning would alert you to the fact that you're in C major scale but

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this piece is actually in a minor scale reflective of the somewhat melancholic nature of the piece

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it's in a minor key and it's in a minor key which is associated to the c major scale sort of closely

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connected to it that's the a minor scale closely connected in the same sense that the c major scale

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and the G Major scale are closely connected namely they share a lot of common notes okay

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so you can't actually just look at the key signature and immediately tell what key you're in

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you could be in a major key or the corresponding minor key but nevertheless even in this little

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segment here we see there's some additional Sharps going on there's no Flats in this case but there's

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a sharp here okay so when we get to this note here it's or nearly a c which is a C sharp in

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front of it and here there's a d but that's a D sharp and down here there's in the bass clef

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that would be an ordinarily a g but there's a G sharp and this sharp here also applies to all the

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other notes similar notes in the bar okay so that note there's also a G sharp this thing here is a

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different kind of symbol this is a natural sign which undoes the sharp sign or the flat sign so we

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just had a D sharp here and this natural here is telling us that we should revert back to a normal

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D now strictly speaking legally speaking this flat is not actually required because the understood

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rule is that these sort of occasional sharps and flats that you find in bars only only go

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up to the end of the bar and then there then their effect is is off okay but this is just a kind of a

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reminder to the the player that okay you just did play a D sharp but now we're back to ordinary D

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okay so um that's a brief introduction to you know some of the the features that we have

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um and you see it's quite a rich story and centered all on this way

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of representing notes on the piano or more generally just musical notes on a staff

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okay and it's this correspondence that we want to have a very good look at okay so the main

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issue here is how do you actually represent notes different pitches different frequencies on a page

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how do you put a piece of music on a page so that somebody can just play it without having heard

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it before in the Middle Ages they had some rather primitive system where you just had

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um sort of some marks on a page sort of going up and down very roughly so of corresponding to

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whether the melody was going up or down but when you're playing a keyboard and there's lots of

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notes that you could be playing at any given time and the relationship can get quite complicated

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you need a rather more systematic procedure and so what we've adopted in today's music in the

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Western World at least is the staff notation which is firmly rooted on the on the keyboard

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okay you can think of the piano and we have to understand it in that context so here is a piece

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of a piano there's C there's a C there so this is like the Standard 12 tone scale 0 1 2 3 4 5 6

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7 8 9 10 11 then back to C with the white notes being named C, D, E, F, G, A, B and C again and

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the standard convention is that the treble clef so the top part of the staff thing that has this

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funny thing in front of it that's a place where these notes from middle C to the next C up are

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represented in a logical and systematic fashion starting with middle C being actually not on the

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staff but on this auxiliary line that you have to draw just below the the staff so there's another

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line here sort of meant to be like an extension of this one but just one down and there's the note C

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on on that and then what we do is we go up a one either going to essentially a space between lines

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or on a line itself so space line space line space line space Etc and we're just marching up the uh

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the white notes here so that's representing C this thing here is representing that note

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this e representing that there's F there's G there's a there's B and there's C and we

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could carry on so the next one up here would be D which would be the following one Etc and

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there's a corresponding story in the bass clef so that's somewhat logical at least if you're a piano

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player because it follows exactly what's happening with the white notes but the complexity has to do

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with the black notes because the black notes are also part of our 12 tone system and in principle

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they are they're playing an equal role but they definitely do not get an equal role when it comes

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to their representation in our staff system this is a crucial thing that we have to appreciate even

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though it's maybe a little bit uncomfortable for us so the black notes get quite different

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treatment so here they are just in this single octave range and each one of these black notes

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has two names because they're each associated to two white keys one on the left and one on

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the right so this black note here is to the right of C and to the left of D so it actually has two

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different names and two different representations one as C sharp where we take this C and we put

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a sharp raising it by a step so that's C sharp that's one way of describing this another way is

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to think of it as being D going down a step with a flat symbol soDd flat and the same then occurs

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with all these other ones this next one here is either D sharp represented this way so if we're

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taking that D and putting a sharp in front of it or we're taking this e and putting a flat in front

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of it D sharp or E flat to represent this one then this one here is either F sharp or G flat

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represented this way or this way this one here is either G sharp or G sharp or A flat pardon me

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oh

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there we go G sharp or A flat that one there and finally this last one here A sharp or B flat

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and you know here's the source of a lot of difficulty and and challenge for playing

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music is that if you're not just always playing in the key of C dealing with these black notes

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uh is a lot more cumbersome because first of all they had this duplicity of representation

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and there's a an auxiliary aspect to them they don't fit so easily into our our staff notation

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we have to have these special symbols the sharps and the flats and we end up thinking of them as

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sort of secondary to the white notes so here is uh an example in that Mazurka that I just played

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the second part of what I played was a little bit more complicated in terms of the melodic

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structure there's a lot more sharps and flats so have a look at the second part okay you'll see

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that it's it's quite a lot richer and there's all these sharps and flats and Naturals and so on that

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have to be put in and so it definitely ups the complexity of the music and you know reinforces

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the idea that there's really a very strong ad hoc aspect to all of this because we basically

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just taking what's happening with the piano and sort of imposing it on the entire musical world

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now there's a second issue that has to do with how we represent the length or the extent of notes how

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long a note is going to be played so that might be in terms of Beats typically right if we have a set

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beat system in our piece then we might want to know well I give a note is going to be one

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beat long or two beats long or maybe a half a beat long Etc so here is the standard convention that

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we use and probably the the best one to start with is this quarter note which sort of plays

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a central role so the quarter note is represented by a solid note head so in terms of notation this

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part is called the note head this is called the stem and then if there's some things like this

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like little flags they're called tails okay so the quarter note has no tails there's a

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note head in black and a stem and the stem can be either up to the right or down to the left

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and okay typically the quarter note gets one beat which is a little bit idiosyncratic notation we

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have to admit and in terms of that if you want to have a note that's twice as fast

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that's the eighth note that's represented as the quarter note like this with a single tail

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so two eighth notes make a quarter note so if a quarter note let's say the beat is like this

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three four one two three four and if I was playing quarter notes I would play

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just with the beat that's the chord if I'm playing eighth notes it would be

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two to a beat if I'm playing four to a beat

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faster that would be a 16th note with two tails and much faster I won't attempt it uh 30 second

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note so we're dividing by two each time now uh when we have a lot of these together then

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typically we do the following kind of thing so here are some eighth notes okay so the tail here

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becomes part of this sort of extension uh that links these two so these are two eighth notes that

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would uh have the same time total as a quarter note and here are four sixteenth notes for these

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guys as you can see by these sort of this double bar there that's telling us that we're basing

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having two tails so these are 16th notes and four of them would be the same as one chord note

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now in terms of longer notes there's a half note and the whole note the half note as the notation

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suggests is worth two beats so here are Beats and I'm seeing half notes it'd be like this

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are half notes and whole notes well that would be

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four beats for a hold on now there's a little bit of variation sometimes

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we use dots to go halfway between these two things but that's a little bit more

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secondary and I also want to point out that the stems can either be going up to the right

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like this or down to the left like in this case here and there's another kind of situation that

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we actually saw in the in the Mazurka where we had three things together with a little three

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under there that's telling us that we should play three of these in the time of two um so if these

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are uh like uh eighth notes okay so if here is our beat ordinary eighth notes would be like this

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if we have something like this they look like eighth notes but there's three of them in the

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place of two so instead of we would play three three to a beat instead of two to a beat that's

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an eighth note triple three in the time of two now you can see that there's a sort of a difference

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here right there's some logic to what we're happening when we're making the notes shorter but

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when we are going to longer notes we're changing our logic okay instead of doing something to the

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stem what we're doing is we're changing the note head itself from a black note to a white note

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and then one more going from the half note to the whole note we're getting rid of the stem

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so this is definitely a little bit obscure or not entirely logical you know why we're doing this

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so whenever we have a system where we are doing illogical things there's room

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for us to think okay what would be a sensible way of doing this

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what would be a better more logical approach to both of the issues that I've talked about today

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and this is a hugely interesting question which I'm certainly not the first to have

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have thought about okay this this issue has come up over and over again in the history

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of Music people have looked at our notational system and said you know actually we probably

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could do better than this you know in the 20th century Arnold Schoenberg who's a sort of famous

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12 tone composer a very avant-garde influential 20th century composer spent a lot of time thinking

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about this but actually quite a lot of people over the years have devised different notational

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systems both from the the the first issue in terms of actually representing the pitches on

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on the staff and in terms of this second issue of how we represent the length or the extent of notes

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so both of these are are well worth thinking about especially from a mathematical point of view like

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what would an AI do we're going to think about that I'm Norman Wildberger thanks for listening

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