Rethinking musical note naming from a mathematical point of view

Let's step back a bit from our piano focused notation for naming musical notes, and adopt a more mathematical, logical terminology. This will have major advantages to our understanding of the structure of intervals, scales, modes and chords going forward. We use this notation to discuss the intervals that compose the most familiar triads, the major, minor, diminished and augmented triangles.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?

list=PL8403C2F0C89B1333

Year 9 Maths: https://www.youtube.com/playlist?list...

Ancient Mathematics: https://www.youtube.com/playlist?list...

Wild West Banking: https://www.youtube.com/playlist?list...

Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...

Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?

list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov

Math History: https://www.youtube.com/playlist?list...

Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...

MathFoundations: https://www.youtube.com/playlist?list...

Wild Linear Algebra: https://www.youtube.com/playlist?list...

Famous Math Problems: https://www.youtube.com/playlist?list...

Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...

Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...

Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...

Differential Geometry: https://www.youtube.com/playlist?list...

Algebraic Topology: https://www.youtube.com/playlist?list...

Math Seminars: https://www.youtube.com/playlist?list...

And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: https://www.youtube.com/playlist?list...

Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...

Triangle Centres: https://www.youtube.com/watch?v=iLBGX...

Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...

Algebraic Calculus One: https://www.youtube.com/playlist?list...

Algebraic Calculus Two: https://www.youtube.com/playlist?list...

0:00

hello everyone i'm Norman Wildberger and today we're going to talk about the structure of music

0:06

from a mathematical point of view so i want to introduce some thinking which allows us to zoom

0:12

out from the usual cultural historical approach to writing music naming music naming notes naming

0:21

chords and think a little bit more about the large-scale mathematical structure and how we

0:27

can use that perhaps to to clarify some things and maybe actually even make it easier to think about

0:34

music in a in a broader way so we're going to be basing things on this 12 tone chromatic scale

0:41

that we've talked about and i remind you that that's mathematically resting on these equal

0:49

ratios between these adjacent notes each of these intervals okay represents a ratio of

0:56

frequencies and that ratio of frequencies is the same for each one of these steps okay and um that

1:04

that ratio has the property that if we raise it to the 12th power then we go from we get to 2 because

1:11

that's corresponding to the octave which is the relationship between these two notes now this

1:18

actually is not exactly the way the pythagoreans thought about things it's a more sophisticated

1:24

away in some sense but it does contain some some irrationality in that there actually is

1:31

no exact number which has the property that when you raise it to the twelfth power you get two so

1:37

there is no twelfth root of two that's just the way the world is and we can accept that or we can

1:44

perhaps try to deny it but it's best if we accept it and just admit that that's an approximate

1:50

construction right so this well-tempered scale is necessarily something approximate there's no

1:57

no instrument on the on the planet which has an exact well-tempered scale

2:04

okay so having said that we do want to transcend the usual orientation in western music to the

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say the major scale the c major scale and we also want to step back from this naming convention

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that currently is very piano focused okay so the current naming convention is c d e f g a

2:28

b c and the black notes get names that depend on the adjacent white notes and there's two of them

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right so this one here is just a little bit above c so that's c sharp where it's a little

2:39

bit below the d so it's d flat so each of these black notes actually has two names

2:45

sort of different contexts that makes the whole thing rather a little bit cumbersome although the

2:52

architecture of the piano is brilliant okay it's really great because the the placing of the black

2:58

keys makes it obvious where the interval is so i mean you can easily see that's a c because it's to

3:03

the left of the two black notes and so this note here which is also to the left of the two black

3:07

notes is also c as is this one etcetera okay so it makes navigation and hence playing on the piano

3:13

relatively easy but it does come with this rather cumbersome notation which doesn't

3:19

really uh you know logically uh translate to a guitar or or a stringed instrument or or a wood

3:26

instrument uh you know very very naturally so what we want to do today is think about

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a purely mathematical way of doing that and it's kind of obvious i mean anybody who sort of

3:38

asks themselves how we're going to do this is likely going to come up with the same answer

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okay so here's the answer that what we're going to do is we're going to describe these

3:48

notes going from here to here which are in this sequence of 12 steps with the numbers 0 to 11.

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we could do 1 to 12 but in many counting situations as computer scientists well know it's

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better to start with zero so we're going to start with zero here okay that's going to be note 0.

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note 1 no 2 no three node four node five node six node seven node eight node nine node ten note 11.

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and then back up here well that's note 12 but if we're only sort of interested in the

4:28

the tone if we're sort of equating this note with this note then we can also call that 0

4:34

and start our naming again if we didn't want to do that if we wanted to maintain

4:39

the distinction between this note and this oh well that's simple enough we just call this one

4:42

zero and this one twelve this one then would be twenty-four this would be thirty-six what about

4:48

this one here well that would have to be minus twelve so of course this convention depends on

4:56

the starting note that the zero note okay and in principle the zero starting note could be anywhere

5:04

and that's one of the beautiful aspects of this from a mathematical point of view that

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that is an intrinsic translation symmetry here there's actually some group theory

5:12

okay um which is actually quite interesting that's uh supporting this this musical structure and one

5:19

of the aspects of that is that it's translation and variance so for example if we wanted to start

5:24

with f okay as the zero note we could do that we could say okay that's node zero that's note one

5:31

two three four five six seven eight nine ten eleven and then twelve or perhaps zero

5:36

okay we so this is a very flexible um uh method and it leads to some interesting um

5:44

insights note and notational simplifications i think sometimes are for chords which is which

5:50

are sort of the building blocks of harmony in music okay so let's go back to starting at c

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this note c which now we're going to call this note zero okay now usually we start with middle

6:03

c this this particular c is sort of the central c but i'm just going to do it down here because it's

6:08

a little bit easier in terms of the camera okay so this is going to be zero and what about the other

6:15

main notes well we've talked about the importance of the the fourth and the fifth in traditional

6:20

notation but now we see that the fourth has this alternate name one two three four five that's

6:26

actually note five in this sequential ordering of the notes and the fifth is actually note

6:35

7 in this sequential ordering so we have this note is the note 5 this note is

6:42

note 7. another important one is the third well that's going to be note four the minor third is

6:51

three and the six is two above the seven is nine so let's just get used to that just to

6:58

get our heads around this okay three four five seven nine

7:08

those are you know very important notes uh in the scale of c

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although actually the three is not actually in the scale it's in the minor scale

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great okay so now we have notes we could get familiar with those notes

7:28

and now we can talk about intervals okay so the intervals relationship between two notes

7:34

so let's say uh between this note and and this note well it's obvious what that is that's a an

7:40

interval of five one two three four five we have to take five steps one two three four five to

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get from the start to the to the end okay so we could say that's an interval of step five

7:55

that makes sense but that that relative aspect can be applied to any two notes in principle here in

8:03

this range so let's say we want to go from e to to this a okay well that's also a five one two

8:11

three four five it's a five chord or five step interval the perfect fifth we said was a seventh

8:20

okay so this is also a step seven interval this is a step seven interval this is a step

8:26

seven interval and if you didn't know that of course if you play the piano a lot you

8:32

you you learn this is a perfect fifth you learn that you know if you go a perfect fifth up from

8:36

the e you get b okay sort of automatic eventually but if you didn't know and you just remember that

8:41

the name of this note is 4 and the name of this note is 11 then you could calculate the

8:47

interval just with a bit of mathematics because 11 minus 4 is 7. so you see the the mathematical

8:58

structure the way we set this thing up makes it easier to describe relationships between

9:05

notes also mathematically so we free ourselves from the c e sharp a a flat blah blah blah okay

9:15

so okay that's great so now um so then we have these these important intervals which really

9:20

correspond to the notes like that's a major third okay that's a step four chord perfect

9:26

fourth is a step five chord and a perfect fifth is a step seven chord sixth is major six is a step

9:36

nine chord okay there are others but those are the main ones i guess also the minor third is

9:41

very important that's the step three so step three step four step five step seven step nine

9:52

okay great so what do we do after we've understood intervals well we put them together to form

9:59

chords and the smallest chords are triads involving three notes say those three notes

10:05

usually called c e and g and that's a major triad and it sounds presumably the same as say this one

10:14

that's another major triad but starting on this this note here here's another major triad

10:21

major triad okay so we can analyze the major triad because it really consists of two intervals

10:27

so from here to here is from zero to four that's a major third major third from zero

10:36

to four and then from four to seven that's a difference of three okay so that's a minor third

10:43

okay so the major chord consists of a major third followed by a minor third

10:52

or we would say a step four interval followed by a step

10:55

three interval so we could say that this chord is described by the numbers four and three the

11:01

steps that we have to take to get from one note to the next it's a step four three chord

11:10

okay what's the step four three chord starting on there okay one two three

11:15

four that's the step four and then we have to go step three one two three that's up to there so

11:22

that is the major chord on this note here and what is this note here this one

11:29

well this is one that's actually exactly halfway between the original note and the octave 12.

11:35

this note here this f sharp it's usually called is note 6 between the perfect fourth and between

11:42

the perfect fifth which is the number note 5 and note 7. so note 6 is exactly halfway in between

11:49

so these kind of mathematical um sensitivities are going to be more important for us now we're

11:55

sort of maybe alert to to this kind of thing okay what's next after a major chord well

12:02

probably the the minor the minor triad okay the minor tragedy is the variation where you replace

12:08

the major third here with a minor third so instead of going four three we're going three four

12:15

that's step three one two three and from here to here is a step four one two three four

12:23

and in terms of the notes that's zero that's three that's seven so the differences are three

12:30

and four zero to three to seven is a three four chord or triad that's a minor

12:40

a minor triad so suppose we wanted to start it on here okay so that little it gets extended up

12:48

beyond the original scale but that's fine so if we do a minor triad there we get this

12:54

okay so in terms of numbers this would be 8 9 10 11 and then 4 more 12 13 14

13:02

15. that's sort of interesting here right so if we just kept on linearly going this would be 15.

13:08

but this is the same as this note which is a three one two three so there is a a re repetitive aspect

13:18

to this every 12 it repeats so this note here we could give it the same name as this one if

13:25

we want to sort of just look at the the tone of it and disregard its position but if we want to put

13:32

the position in as well then this is maybe 3 and this is then 15. and there's a difference of 12

13:38

between them okay that's an octave and so going back to this minor triad that i built up from

13:44

this note this note number eight we went eight eleven fifteen step three followed by step four

13:54

now there's two other triads which i'll just finish with um that are that are commonly used uh

14:00

one is the augmented triad we take a major triad and augment that one there okay so what does that

14:10

consist of well it consists of this interval which is a what kind of interval once it's a four step

14:16

classically known as a major third well we can say it's a four step and from here to

14:20

here that's another four step so this triad is a four four chord or step four four chord that's

14:30

called an augmented chord it's a little bit uh an unusual chord we could build it on lots of things

14:42

okay and then there's a correspondingly a diminished

14:46

accord which is this one just a diminished triad i should say okay because there's only three notes

14:52

here okay and this is node 0 note 3 and note 6. remember that's 6 there 0 3 6 and obviously

15:03

that's step 3 3 a minor 3rd followed by another minor third and of course we can

15:09

build that anywhere so we could build it say on on this a there there there or on this e

15:19

maybe we should call this note four alright so we're going up three from four to seven to ten

15:28

so the the the numerals that we're assigning to these notes helps us in you know quickly saying

15:36

something about the relative positions of them and happily this entire story

15:41

translates this is the crucial you know part of the symmetry here it translates when we just move

15:48

up and base our scale not on c but on some other note so for example if we started on f okay then

15:57

that would be 0 1 2 3 4 5 6 7 8 9 10 11 12. okay so if we wanted to build a diminished chord on

16:03

that in terms of referring to this as zero okay it's still going to be a three three step chord so

16:10

it's going to go up three there's three and another three there's another three so there's the

16:16

diminished chord built on zero in the scale that's starting or referenced to this particular note

16:26

so the the absolute names of the notes then become less important and the the

16:33

relative names of the notes become more important especially with reference to

16:38

a fixed sort of tonic or or starting point which is the the zero note okay but of course

16:45

the standard default uh although there are many scales usually all beginners start with the c

16:50

scale maybe not not in terms of learning piano but but you know it's visually the easiest and

16:56

so that's perhaps the default zero although i'm starting down here but that would probably be

17:01

the default zero if you were looking at the entire piano i'm Norman Wildberger, thanks for listening.