Chords in Parity Staff Notation
Chords in Parity Staff Notation | Maths and Music | N J Wildberger
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672 views Mar 1, 2024 Maths and Music
Parity Staff Notation (PSN) is an alternate, much simplified system for annotating music, which avoids sharps and flats, and steps away from the dependence of our current system on the architecture of the keyboard. In this video we begin by getting a deeper understanding of chords and their interval sequences, comparing traditional and PSN notations, and focusing especially on inversions. For those with an interest, I now have dozens of TikTok musical improvisations and other informal pieces, of various levels of quality no doubt, but hopefully with something of a different flavour or sound from what you might be used to. Check it out at 
/ ucxl0zbk8_rvjylwar-xh9pqThis is part of the Maths and Music playlist, at:
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Hello everyone I'm Norman Wildberger and we're talking about mathematics and music
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and I'm talking about chords today in the context mostly of this new Parody Staff Notation that I'm
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proposing so we've talked about chords before in the context of the guitar so maybe a little bit
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of a review. A chord is for us a list of tones maybe that's a little bit different from what I
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said before it's good to have a list because we should allow ourselves some repetition okay but
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it's an ordered structure of tones which we're going to regard as notes in in ascending order
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okay so and we'll also make this convention that between any two tones we'll assume that there's
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at most of an octave separation so for example if we're going to go up on the keyboard we're
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never going to go more than an octave from one tone to the next. So for example here is a chord
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it's a list with tones four eight eleven two six so these are all always numbers between 0 and 11.
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interpreted as tones but we're going to think of this collection as being sort of an ascending
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sequence of tones creating a block: a block of notes sort of that you can play at the same time.
Interval sequence
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Now associated to such a thing is an interval sequence. The interval sequence is just the
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differences so four three three and four and here in the usual standard classical
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treble clef notation is that the chord displayed as a vertical slab there e that's
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note number four and then G sharp is eight, B is 11, D is two, and the F sharp is six
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and in the parity staff notation story we don't have a treble clef the two clefs are exactly the
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same as far as the notes are concerned the E this one here maybe I should write it down
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this one here is an E this next one is G sharp
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and then this one is the B and then we have D and then we have an F sharp.
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So this is going to be an opportunity for us to just slowly get a little bit more used to the PSN
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system and I'm going to be talking about different kinds of chords today today we'll probably stick
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to triads but importantly we're going to introduce the crucial idea of inversion and that's a way of
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having a richer story of chords by considering inversion also and it turns out that there's
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interesting mathematical aspects to that okay so the familiar triads chords with three notes are
Triads
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the major ones and the minor ones we've seen them before major chords or major triads have type 4 3
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and minor triads have type 3 4. an example of a major triad is five nine zero five is the F
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nine is A and zero is C in the usual convention and here is the corresponding picture in the PSN
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notation using the parity staff notation there's the F there's the A and there's the C and you
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can see the separation between here and here that's a four separation going from one line
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to another line the same color and this is a three separation going from this line to
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this line but changing from white to black over here we have a minor chord it's a E flat minor E
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flat G flat and B flat it's a 3 6 10 in terms of the actual notes and here are the corresponding
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parity staff notation signs for it the E flat there in white the G flat and the B flat
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now we want to consider also variants of these sort of standard forms by doing this inversion and
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inversion is okay in this case it's a relatively simple what we're just going to do is we're going
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to rotate the the three elements in the chord very much the way we rotated in interval sequence
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we were talking about modes okay but now we're actually rotating the the notes or the tones in
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the chord so if we invert 5 9 0 the 5 goes to the back everything slides forward we get nine
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zero five so that's a different chord which is however very closely related to this one so we
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say that these are inversions of each other and this is an important way of deriving some
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additional chords to sort of flush out the the space of chords in a nicer way so let's look at
Major Triads
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Major Triad inversions first we'll use the same example of our F chord five nine zero that's a
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four slash three chord if we inverted we get 905 and now the intervals are three and five so it
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becomes a three five chord and in the classical fashion here's what it looks like the F at the
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bottom here goes up to the top so there's an F there and becomes a F if we invert once more then
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the 9 goes to the back we get 0 5 9 and now the interval sequence is 5 4. so this is a 5 / 4 Chord
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obtained visually by taking that A and bringing it up an octave to that line above the stave so
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we get CFA okay now in the PSN system the thing is as follows so here is the original F A C chord
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inverting means that the the note in the bottom goes up an octave and in in this system going
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up an octave is always sort of going up from one line to the the line three above it or from one
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space to the space three above it and the color does not change okay so this F if we inverted
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it goes up here and stays white so we get A C F if we invert this one then this a goes up 3
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up to this line here and stays white the other will stay where they are so there's what the
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inversion looks like in the parity system correspondingly let's look at Minor triad
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inversions so we'll stick with an example the 3 6 10 Minor triad E flat G flat B flat
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which is a three / four chord we invert we get six ten three which now is a four / five
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chord and if we invert again we get 10 3 6 which is now a five / three chord and here are
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the various versions in the classical system and here are the various versions in the PSN
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system E flat G flat B flat then we move the E flat up to that space there which doesn't change
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color we get G flat B flat E flat and then we change this one here to invert it up there
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there we go B flat E flat G flat now another important kind of Triad is a diminished Triad
Diminished Triad
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that's of type three three for example one four seven which corresponds to C sharp E G so minor
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third minor third a three step followed by another three step when we invert that we get four seven
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one which becomes a three six chord when we invert again we get seven one four which becomes a six
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three chord so here are the various inversions of the diminished Triad in the classical formulation
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and here is the PSN formulation where there aren't any sharps or flats in the actual notes and I'm
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just putting the traditional notes here just to help us make this bridge between the classical
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story and the standard one in in fact it's better just to use these numbers so you know ultimately
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we want to get rid of the E's G's and C Sharps Etc and concentrate more on the numbers but
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okay this is a it's important to try to make this bridge to become a little bit more familiar with
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how this system connects with the standard one okay so here's C sharp note one and then Note 4
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okay that's the first line remember the lines are four eight zero four eight and here is G okay and
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then when we invert this C sharp goes up to there when we invert again this e here goes up to there
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so these are the shapes of the diminished Triad and its inversions and finally the last of our
Augmented Triad
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major examples of Triads are the augmented Triads which in some sense as you'll see are the simplest
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so for example here's an augmented Triad 2 6 10 it's a 4 / 4 Chord when we invert we get 6 10 2
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which is also a 4 / 4 Chord and we invert again we get another four / four chord 10 2 6 so this
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is the augmented Triad built on D so D F sharp A sharp there's the first inversion D goes up
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here and the second version F sharp goes up here and here are the corresponding PSN forms the D
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the F sharp the A sharp when we invert the D goes up there when we invert again the F sharp goes up
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there and you see that the the story visually for these augmented Triads and their inversions
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are all very simple in fact the inversion of any one of these is also directly an augmented
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Triad the form is not changing at all so that's particularly simple okay so that's an introduction
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to chords but of course the situation gets a lot more interesting and complicated when we
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go to the tetrads so how do tetrads look in this new PSN system and do we get some insight into
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the structure of these important constituents of music I'm Norman Wildberger, thanks for listening