A new Staff Notation based on Parity
A new Staff Notation based on Parity | Maths and Music | N J Wildberger
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1,220 views Nov 23, 2023 Maths and Music
Many people over the years have lamented that our current musical notation, based as it is on the rather idiosyncratic design of the piano's white and black key system, is a hindrance to musical learning and understanding. I fully concur with this view. It is long past time to consider rationally what better alternatives there are.
In this video I bring a mathematical orientation to this question, and propose a vastly simplified system, in which the parity distinction between even and odd is reflected in the visual distinction between black and white notes, in which sharps and flats are entirely removed, where the treple and bass clefs are read exactly in the same way, and where transposition from one key to another is so much simpler. This will be a major help in understanding also the structures of chords in musical theory.
In keeping with this I also propose a modification of our notation for the length of notes.
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Hello everyone I'm Norman Wildberger and today we're going to be looking at new kind of ideas
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towards staff notation in music so this is a subject that's interested a lot of people
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for a long while for example in the earlier part of the 20th century Arnold Schoenberg the famous
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12-tone composer advocated a system to enable him to work more smoothly in his 12-tone system where
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the usual staff with five is not replaced with six but we're still going from a c here which
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is in the same relative position as usual up to this one here now so going a total of one
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two three lines up but still all 12 tones in this range and the way he does that is he creates three
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note positions between any two lines so I'm not actually on the lines but between the lines sort
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of on the bottom line in the middle and below the top line and further a little diagonal Strokes to
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help Orient you as to which one of these you're looking at so this actually has some similarities
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with the proposal that I'm going to make okay because I also want to have a system where we get
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all 12 tones sort of equally spaced in a nice way without having Sharps and flats which confuses the
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situation okay but since then there have been a lot of other attempts at convincing people that we
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should change our musical notation but of course there's a lot of tradition to to work against but
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a notable example of that is one due to Cornellus Pot a Dutch fellow who introduced what's called
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Klavar notation and it's quite innovative because the the notes that you're playing at a given time
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instead of being sort of vertical as they are with most systems are actually horizontal and his idea
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is to represent the actual physical locations on a piano with a horizontal strip of of notes
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so that even a beginner piano player could sort of read now on the page and see where they should put
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their fingers uh you know as as you go down so this is quite interesting that was in the 1930s
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but there have been lots of others for example George Skapski introduced what's called Panot
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notation in the 1950s J Beyreuther introduced a system where black and white notes were both used
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in 1959 and that's actually quite important for us because we're going to be doing that too okay
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so we're going to be following him actually in his footsteps also uh J Braunstein actually more
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recently introduced what's called chromatic lyre notation and also used black and white
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notes prominently and here is an example of something in his notation actually the
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first Prelude in Bach's Well Tempered Klavier in C major in this chromatic lyre notation
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Now to access a lot more ideas in this direction there is a book by Thomas Reed published in 1997
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called Directory of Music Notation Proposals so there's actually quite a large history of these
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things and it really points to the fact that our current system has a lot of weaknesses and
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and many people are looking for ways of improving that to make it easier for people to access music
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essentially and also for composers and players to to not be inconvenienced by the somewhat ad hoc
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nature of our current system which is largely piano based okay so a very quick review of the
Problems with standard staff notation
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difficulties or some of the difficulties with our standard staff notation here is the piano octave
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from C to C here are the representations of the white notes but unfortunately the black notes are
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much more complicated so there's a basic sort of asymmetry between between the white notes and the
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black notes furthermore that also manifests itself in in the sense that the interval sequence here is
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not uniform so there's some intervals between white notes which are just one steps while the
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others are two steps and that has the consequence that for example to go from the e to the f is
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actually even though it's just going up one step it looks a lot like the step from g day it's
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actually not quite different this is a two-step and this is a one step so there's actually three
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uh such places from E to F and from B to C and then also from E to F again three places just
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on this treble clef so that creates problems for translation or transposition it means that
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you can't just look at a configuration of nodes say like this Triad here in red and immediately
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see what kind of a chord it is in fact this chord here with e g and B is a Minor triad a
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minor chord well this one here which has the same kind of shape from G to B to D is a major Triad
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or major chord well this one here which again has the same kind of form from B to D to F that's a
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diminished chord or Triad so you know there's a a complete disconnect in terms of moving things up
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and down and things being the same so the natural uniformity that we have in a 12 tone system with
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our equal temperament a well-tempered approach that that natural uniformity of translation is
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lost in our notational system and of course that's even accented more when we also think about the
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bass clef which is yet another system where again the the various idiosyncrasies are in different
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places so you have to learn the idiosyncrasies for the treble clef and for the bass clef separately
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so we want to try to think of a better way so my new proposal is that we should step away
New staff notation
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from the piano orientation rather we want to move to a clock view of things where we have
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our 12 tones and where we're flexible as to the interpretation of the zero so we may be playing
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different instruments guitar saxophone trumpet whatever they naturally are sort of based on
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different pitches than a piano so it's good to be flexible so for us zero is just some fixed note
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but we're not saying which one it is and we're going to have essentially one staff everything
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is describable just in terms of one step you can think about this as the treble clef or the bass
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clef so it turns out in this system that those two things look exactly the same okay the treble
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clef and the bass clef there's going to be two of them but they look exactly the same and the
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notes are going to be exactly the same just a couple of octaves down okay so what are we
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going to do so we want to represent the 12 tones say starting with this one here this is a black
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black note in the usual position of C if this was a usual treble clef and that's not a c for
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us that's node zero whatever zero is and then the subsequent notes one is the same level okay
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but it's a white note so we go from black note to White note it's going up one step we agree
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and then we'll say the next level is going from this line to the next space so that
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would be that one there we get a black note in that space that's going to be
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Note 2 and then changing it to a white one that's three and then we just keep on repeating that we
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just go up a level so from this space to this line then from this line to this space the
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space to this line Etc and at each stage there are two notes first a black one and then a white one
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so it's a little bit simpler than Schoenberg's notation for example because there's no sort
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of different positions there's just one node position in the space just like there is in
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the usual system that we currently use so it looks a lot like an ordinary system except
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that the notes are black and white that's a big departure let me say a little bit about that so
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the distinction between black and white notes on a page is a very noticeable one if you look
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at a page of music from a distance you may have trouble saying exactly you know whether that note
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is on the line or a space but you're not going to be confused as to whether it's a black note
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or a white note that's very visible so it makes sense for us to use that highly clear visual
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signal of the color of the note in a strong way currently our system uses that particular
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color difference only more or less to distinguish between a quarter note and a half note so
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it's reserved for saying something about the length or the extent or the duration of a note
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that's a relatively minor kind of application of this really powerful idea so I want to bring
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this idea center stage and I'm following again a Beyreuther and and Braunstein's ideas too this is
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exactly the same thinking that they would have had Okay so we're going up okay by one step you know
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one two three four five six seven eight nine ten eleven twelve or zero that's our 12 tone chromatic
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scale and we've started on this line here just below the the regular staff and we've gotten up
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to this middle line here so we've gone basically from here to here so that's an octave for us
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and that's very easy to remember because we're just going up three lines or if we started on
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a space we're going up three spaces that's an octave all of these are octaves so the entire
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pattern is repeated so this note here is the same as this one here so after we get to this
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node if we want to keep on going up we're going to go this way okay that's zero again one two
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three four five six seven eight nine ten eleven and then we get to like a high high C or a high
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zero okay so overall we've spanned two octaves from this note just one below the staff to this
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note just one line above the staff and we have all the the notes 12 tones plus 12 tones in this
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system and we have no Sharps and flats importantly okay so it's a quite interesting and you can see
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some interesting patterns so for example the black notes are all even numbers so they they correspond
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to a uniform two scale that I've showed you before remember there were two uniform two-step scales
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C D E F sharp G sharp a sharp B was one of them and then d flat E flat F G A B
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uh d flat again was the other one so those two things are represented by the black notes and
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the white notes respectively okay so the uniform scales are in there and if we go up by four well
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then we're going from a line to the next line so it's also a very important sort of Step from here
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to here that's like a major third an interval a four-step is a major third so the separation
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between these lines here is in each case a major third interval that's actually quite
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Pleasant because a major third is actually a very nice interval it's actually very common interval
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arguably it's maybe the most important interval it's arguably okay so it's it's Pleasant for us
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to have the separation between one line and the next line being a a four step or similarly between
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a space and the next space above that's also a four step so from here here is a four step
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um and then the nature of these even intervals like a four step or a six step let's say look
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at a six step okay so we're going from here to here that's a sixth step so we're in terms
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of levels we're going sort of up one two three level and the colors are the same
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so similarly a six step from this one here going up three levels one two three color is the same
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so it's that to that so the the even steps are really easy to to visualize and they're the same
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essentially whether we're starting on a black note or a white note however the odd step intervals are
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a little bit more complicated so for example just the one step the one step from zero to one you see
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what that looks like it just looks like from a black note to the white note on the same level
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that's a one step however there's another kind of one step going from a white note to a black
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note now we have to go up a level okay so white note at that level to the black note on the next
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level that's also one step so there are these two different kinds of one steps depending on whether
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you're starting on a black note like from here to here or on a white note from here to here but
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once you've understood those well then the pattern is the same all throughout okay so yes there's a
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little bit of work in in looking at these various intervals and and sort of seeing what does you
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know a five steps that's like perfect fourth you know from here to here that's one kind of five
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step but also from here to here is a different kind of five steps so you have to sort of stare at
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it and get familiar with this is a five step and that's also a five step but once you've understood
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that well then it's the same thing here let's go from here to here that's a five step go from here
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to here it's a five step it's the same pattern everywhere okay so the the odd uh intervals
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the odd step intervals are a little bit more complicated require a bit more work to get your
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head around but it's all I would say a lot simpler than our current system a lot simpler because of
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this built-in uniformity and the fact that what I'm saying here applies equally to the bass clef
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and to the treble clef so if there was a bass clef down here there would also be two octaves from the
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the zero down here to the zero to the zero here and then another two octaves from this zero to
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that zero so four octaves all together spanned in a nice way by the two uh the bass clef and
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the treble clef together so a little bit more experience getting your heads around this new
Major chords
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proposed system which we might call sort of like a parity system because the black and the white
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stories are pretty well in the same footing here so here are major chords so starting on any tone
Major chord
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there's a major chord it's like a four three chord consisting of a four step followed by a three-step
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and it's important for students of mathematics to know what the major chords on any tone are and
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here's sort of the classical version as the major court or try it on c d flat D E flat I don't know
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why I mixed these up e and then F and then F sharp and then G and I'll do the next on a separate
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slide so you can see that in the usual system it's not clear what the pattern is at all you
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know some serious memorization has to be put in to get your head around where all these things are
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here in distinct contrast the situation is pretty simple there are basically only two shapes that
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you have to remember and then you just repeat them you just translate them into most obvious
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fashion so starting with node zero so if you like think of zero as being C then this is the same as
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this so there's a node zero there's node four okay and there is node seven because if we're
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counting so it'd be zero one the white note and then the the space two three and then this is four
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and then five and then the the space black and then the space uh White so that's seven so that's
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um like 0 4 7 note in our system now if we start on Node 1 instead it's a little bit
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different the thing has a little bit different shape the fourth step is still from this line to
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this line four steps are always that way but now the three step which is an odd interval
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step okay has a different shape than this one you have to go up a little bit further so you
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have to go from this one to this one because if we were counting the next note would be
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the the black note this one here okay then the next one after that would be this one and then
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the next one after that would be this one so from here it's one two three to get to that one
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okay so there's a four step here and a three step here and it looks more like a a regular Triad
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like this but there's two white notes here and a black note over here there were two black notes
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and a white note so remembering the colors is an important part of remembering this overall shape
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but once you've gotten your head around these two examples of a major chord well
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then the other ones are pretty simple you just translate them up so a chord built on two it's the
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same pattern we start with Note two we do a four step do the corresponding next space and then we
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do one of these adjacent uh White notes and then three looks just like this one except everything
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is sort of up one level from this line to this space and then this space and then this space
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two white ones followed by a black one and then it just carries on there's number four five six seven
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Etc in both base and treble clefs so you can see that
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the possibilities for understanding chords is much simpler because there's
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a uniformity here a natural uniformity which is reflecting the uniformity that
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we've built into our musical system by adopting this well tempered 12 tone system
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where all the notes are equally spaced as far as frequency ratios go so just for completeness
Major Triads
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here's sort of the other ones so again seven is where we ended up with the last slide and then so
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this bottom here is Note 7 and then going up node eight node 9 10 11 and then back to zero okay and
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here are the major Triads built on all of those the same kind of pattern um in terms of stems so
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stem UPS typically the the notes are sort of in the lower half of the staff then the stems go up
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and if they're in the upper half then the stems go down but it's not an absolute rule but so
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um you know this is the same kind of picture it's basically these two shapes uh repeated
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um two white notes and then a black note all sort of equally spaced or two black notes and
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then a a closer white note like here here and here and in contrast here are the other Triads
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in the usual system there's G and then there's a flat and a and B-flat and b and c and d flat and
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and d and you can see you know there's you know you sort of have to memorize these it's hard to
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have a really sort of good systematic understanding of these
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and yeah there's just a world of difference in my view between trying to understand music
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and Harmony from from this kind of point of view rather than with this one it's much more
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logical and sensible and and Accords more with the actual music you know the the mathematical
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symmetry that we have naturally in our music because it's a translationally and variant system
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so the final thing to talk about is the second issue having to do with Note length or duration
Note Length
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or extent and remember that we've taken the white black distinction which was the difference between
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a quarter note and a half note and we've utilized that in a different fashion so we're obliged to
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find some other way of distinguishing chord notes with half notes now there's all kinds of possible
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more fancy systems that we could use here but I think the simplest thing is pretty well to use the
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existing system basically starting with a quarter note as sort of the basic thing just a single note
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head with a stem either going up to the right or down to the left but now of course that note
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head could be a black note or a white note both are allowed so that's different from previously
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and then to get these other ones we just put little Tails each time we put a tail we're
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dividing the duration by two we're making it half as as long so this would ordinarily be called an
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eighth note but I prefer to actually have somewhat more sensible names for these things so this note
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here the quarter note is really unfortunate terminology um in 95 of all cases the quarter
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note is getting the beat okay not in all cases but most cases the the quarter note is getting
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the Beats so it's really the note length which corresponds to a beat so it's much more accurate
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and descriptive and I think useful to call it a one beat note so let's do that all right so that's
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a 1B note well that's going to be a half beat note and that's going to be a quarter beaten and that's
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going to be a 1 8 beat note and those are a little bit easier to deal with than 8th and 16th and 30
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seconds okay but what about the step from here to here from what's usually called a quarter note to
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a half note but we're going to call a one beat note to a two beat note so we have to make up
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something here because we can't just use the the white we've already used the black to white note
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already so what I propose is that somewhat in line with this that we use something like this where
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we take say a black note there and we have two little stems maybe one going up to the right and
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one going down to the left a little bit shorter so the thing doesn't take up quite so much room so
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maybe half the size but there are two stems here okay there's one stem here there's two stems here
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maybe suggesting that this thing is twice as long okay and whether the two stems go in this fashion
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or maybe they both go down or maybe they both go up well something in this Direction all right and
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then the four beat or hold whole note well that's we can do that by actually using a symbol which
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is already around I think it's called a breve so it's like a note either a black or white note with
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uh sort of two little auxiliary lines extending in both directions so in some sense there's four
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different little extensions two up and two down so it's somewhat consistent with with this thing
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here okay so I'm not saying this is too serious and no doubt we could consider all kinds of other
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uh things and we should okay we should think carefully about a sensible system like this but
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at least this is a an initial foray that matches up and so now we could in principle you know write
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music in this new parity notation uh for for music I'm Norm Wildberger thanks for listening