The Clock Diagram

let's say that we had a keyboard just a regular keyboard with all the notes labeled on it like so and me being the difficult one that I am decided to replace all of the letters with numbers so C is zero c-sharp is one D is two and so on until we get to B which would be 11 and we restart the numerical pattern at the octave now let's get rid of the keys and let's replace the numbers 10 and 11 with just the letters T and E that way we have every note represented by a single character and now instead of having this numerical line repeat over and over again what if we arranged it in a circle and voila we have something called a clock diagram which was first proposed by dr. Paul Metz right around the time I was born in his essay the clock diagram an effective visual tool in set theory pedagogy and this clock diagram is really useful when it comes to demonstrating certain musical practices which are otherwise really difficult to do with just a staff without having to throw some math at it for example if I showed you this augmented triad and fully diminished 7th chord could you tell me that they were symmetrical at a glance well what if we use a clock diagram we can pretend that 0 is C and plot the notes of these chords suddenly you can see how symmetrical they are they both have bilateral and rotational symmetry and this is really useful when it comes to demonstrating the stability of any given scale for example people like Debussy liked using a whole tone scale because it was really stable and sounded like it never resolved if we chart a whole tone scale on a clock diagram you can see how extraordinarily symmetrical this scale is compare that to like our diatonic scale or our major scale it really only has one axis of symmetry right here that asymmetry of the major scale gives it that pull towards certain notes which in essence created the foundation for tonal harmony but when you go back to that whole tone scale you can see how this scale can just play forever and never really feels like it needs to stop anywhere so if you saw my video on the music of Miyazaki films you might remember me saying the same thing about the pentatonic scale well if we chart it out you can almost see how it's like halfway between a major scale and the whole tone scale it's more stable than the major scale but not as completely stable as the whole tone scale but at the same time just by looking at the clock diagrams you can see how the major pentatonic scale is just a regular major scale missing two notes in technical terms this means that the major pentatonic scale is a subset of the major scale and the major scale is a superset of the major pentatonic scale and this is how I figure out how to get those traditional Japanese tetra chords to fit together I just started them on clock diagrams and use their corresponding shapes to construct a major pentatonic scale and this is where the clock diagram really begins to shine you can see how different collections of notes fit in together and relate to one another for example if you look really closely and you rotate the pentatonic scale in your mind you can fit the pentatonic scale into the gaps of the major scale at what you end up seeing here is a regular C major scale and the f-sharp major pentatonic scale or in other words the white notes and black notes of the keyboard when two sets come together to form the aggregate which is the technical term for all 12 notes of the octave then we call these two groups of notes complements of one another the pentatonic scale and the major scale are complementary in the case of the white notes and black notes then these two sets are what we call a literal complement because together they form the aggregate but in the case of C major in C major pentatonic even though they're complementary we call them a non-literal complement because you have to rotate one of these sets in order to get them to fit in with the other and form the aggregate and the same thing goes for literal and non-literal subsets and supersets which is all nice and complicated sounding but where things get really cool is when you start digging into the even more complicated stuff for example if I wanted a C minor chord and we're still using C on 0 just to keep things as simple as possible then this would be a C minor chord but I can invert that minor chord I can draw any one of these twelve lines through this clock diagram and use that line like a mirror and reflect all the notes from one site to the other and invert whatever collection of notes I have on the clock diagram so let's say I want to invert this C minor chord on the note C then the axis of bilateral symmetry would extend from C to F sharp or in this case 0 to 6 and that means that C will stay the same but the e flat will become an A and the G will become an F and what we have left is an F major chord so at this model whenever you rotate a collection of notes it functions like a transposition just like moving notes up and down a staff but if you look at the reflection of a collection of notes then you get the inversion of that collection of notes and now you can see that a minor chord is an inverted major chord and vice versa see if you look at how many steps you're moving in the other direction for a major chord after you've inverted the minor chord the distance between notes is all the same so you can express any major or minor chord as having the notes 0 3 and 7 this is called a set and it's the foundational building block of something called set theory and once you start exploring the world of set the nice clock diagram has become that much more useful like in the grand scheme of things major and minor chords aren't that difficult but let's step it up a bit what if we start with the C major nine and we pick a random access that we can use to invert it like this one right here well the B would become B flat the C becomes an A the D would become G but the G would become D so that's a wash and E would become F and we end up with a G minor nine and that's the benefit of a clock diagram just at a glance could you assure me that these two monstrosity x' are actually the same set well look at them on a clock diagram first we might have to invert it or reflect it on an axis then we might have to transpose it or rotate it around the face of the clock diagram but this shows you how these two collections of notes are actually the same set and then if you want to find out what this set is or the prime form of this set then you can just maneuver the set around until it starts on zero and you have all of the biggest the apps between the notes toward the end of the set and there you go you're doing set theory now I want to clarify you can just stare at sets on a staff and throw math at it until you can prove that these are actually the same collections of notes or in technical terms the same set but that's really really difficult and clock diagrams don't just help us understand the arrangement and relationship between groups of notes but it can also show us something about the spaces in between the notes themselves the gap between any two notes is called an interval and chances are you've heard about these before perfect fifths major thirds minor seconds major sevenths tritones and the truth is that all of those are wrong because I'm bright and there's actually only ever been six intervals and it can prove that with a clock diagram so right here this is the smallest gap between two notes and that's going to be our first interval then there's two gaps between these two notes that's our second interval and here we have our third interval our fourth interval our fifth interval and finally our sixth interval now I know what you might be tempted to say you might say hey what about that seventh interval but the seventh interval is just an inverted fifth interval the closest distance between these two notes isn't seven spaces it's five and we can express the interval content of any given set with something called an interval vector so if we have the set zero one two then there are two instances of one gap in between notes and there's one instance of two gaps in between notes but there are no three four five or six note gaps in this set because it's so small and if we look at our set zero three seven which is any major or minor chord we can see that there are no one note intervals there are no two note intervals but there is one three note interval one four note interval one five note interval and zero six note intervals this really helps composers when they want to know how a group of notes is going to sound like now we know that a major or minor chord will have the same kinds of intervals and sound a certain way because they're both the same set so just at a glance we can look at the interval vector for a whole tone scale and see that it has six two note gaps six four note gaps and three six note gaps and this helps in more ways than you might think for example if I show you these two chords on a staff they might not seem that special but if I show you their sets on a clock diagram well they still might not seem all that special but if I show you their interval vector you can see that not only do they share an interval vector but these two chords have exactly one instance of every interval and when two different sets have the same interval vector we call them Z related and once you get these clocked diagrams down you can start seeing some really cool patterns and older music like those four tetra chords I use to analyze the music in Miyazaki films well once you look at them on a clock diagram you can see that there are actually only two different sets that have each been inverted and if you like the idea of symmetry and the sound of the whole tone scale then you can go out and find some other symmetrical and stable shapes and clock diagrams and if you look at their corresponding interval vectors you can get a sense of how they might sound and if you ever get bored you can start creating your own scales that don't have to have any total bases just by drawing different shapes on clock diagrams and figuring out what kind of intervals they have by drying out their interval vector so those are clock diagrams and they've been my secret weapon in the never-ending quest of exploring new music and hopefully they'll help you to and the next time you come across a piece of music that doesn't quite make sense chances are that a clock diagram is gonna offer a new perspective thanks for watching I'd like to thank my patrons for making these videos possible with a very special thank you to Anna Burch F and Matt Ben leaves fogger Ethan Rooney and Florian H Carr if you liked what you saw here be sure to subscribe and check out my other videos follow me on Twitter and twitch to have your musical questions answered live and if you really like what I'm doing consider supporting the channel on patreon but that's gonna be it for me for now thanks for watching [Music]

Channel: Sideways

Views: 121,294

Rating: 4.9825587 out of 5

Keywords: Music, Theory, Music Theory, The Clock Diagram, Clocks, Inversion, Transposition, Set, Set Theory, Metz, Clock

Id: oGeBem72R3Y

Length: 8min 41sec (521 seconds)

Published: Tue Feb 27 2018