A more rational

Let's use our logical mathematical notation for the 12 tones of the chromatic scale to discuss various important scales that are found in a variety of cultures and idioms, including the major, minor, blues, pentatonic and Indian or uniform scales. But we have to let go of the unfortunate "whole tone" and "half-tone" or "semitone" terminology. But this has important advantages in clarifying the underlying mathematical patterns behind a lot of music.

This is part of the Mathematics and Music Playlist at this channel. I would like to thank all my Patreon supporters for their continued help and encouragement --- your support is really appreciated ! :)

Here is a link to that Tik Tok improv video where I use the uniform 2 step scale:

https://vt.tiktok.com/ZSRvoX3Ha/

CORRECTION: The minor scale that I discuss here is the harmonic, not the melodic, minor scale.

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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 scales

0:04

which are fundamental melodic frameworks for music

0:08

we have our chromatic scale of 12 tones going from one note to the next note one octave above

0:16

in this case from c to c but that's a lot of notes and if we just use all of those

0:23

notes it's easy for the listener to feel a lack of sort of center for example if i make up a tune

0:40

you know you you have a hard time perhaps it's not unpleasant but it's a little bit um

0:48

too free so most cultures end up restricting the the notes that are used for for first songs and

0:56

typically certain notes are chosen to create a scale and so we'll talk

1:00

about those today but crucially we're going to introduce a notation to describe these scales

1:05

which is more mathematically and pleasantly organized than the usual story so a good

1:11

starting point is with a major scale okay going from here to here looks like this one

1:18

very easy just the white notes now how would we describe this

1:23

well we don't really want to describe the actual absolute positions because we recognize that

1:28

this same scale could easily be translated for example we could move it up by one step

1:35

or by another step right so there's these 12 equivalent scales all having the same sort of

1:42

relative structures so we want to describe them all at the same time so what we're going to do

1:46

is we're going to describe the relations between successive notes we're going to just record the

1:52

intervals to go from one note to another so starting from here to here that's a two step

1:58

traditional nomenclature is calling that a whole tone okay and calling this

2:05

a half tone which is unfortunate because it's really this step from here to here which is

2:12

the fundamental unit here right we have 12 and this is one two three et cetera so we're going

2:17

to stick with that we're going to say that this is a one step and from here to here is a two step

2:22

so our major scale consists of a two step a two step a one step a two step a two step a two step

2:30

and a one step so in short two two one two two two one that's a major scale

2:41

the next most important one is the minor scale and there's a couple of different

2:45

minors i'm going to introduce talk about this one here the melodic minor okay so

2:56

there's a two step there's a one step a two step a two step a one step a three step and a one step

3:07

okay so it's two one two two one three one the major and minor scales had

3:20

seven notes in them there's other ones which have fewer for example a blues scale

3:30

has one two three four five six notes before we get back to the the tonic so what are the

3:37

intervals here well from here to here that's a minor third which sort of characteristic of blues

3:44

and that's a three step okay so we have a three step and then a two step and then a

3:50

one step followed by another one step so we have two consecutive one steps there

3:54

and then another three step and a two step so all together three two one one three two

4:13

okay blue is very important very nice okay but there are also shorter scales for example there's

4:20

pentatonic scales found throughout africa and asia china japan other places so

4:30

the major pentatonic scale consists just of those five notes

4:38

or if you want to keep carrying on to the to the octave

4:45

so how would we describe that well again just in terms of intervals two two step two step

4:53

three step two step three step so there's three two steps involved and two three steps involved

5:07

that's a pentatonic or major pentatonic scale there's also a minor pentatonic scale

5:18

so that's a three step two step two step three step two step

5:29

so again we have two three steps and three two steps

5:37

now these pentatonic scales can also be thought of in a different way on the piano okay conveniently

5:44

the piano has these black notes also which i'm not using currently but if we just shift

5:48

to the black notes very temporarily then we can realize the the major pentatonic scale

5:58

by playing the black notes but starting with uh this one here say

6:02

the the leftmost of the group of three

6:07

or maybe up here is that the sound the same as it's the same right so that's another way

6:14

of playing a pentatonic scale and in fact the minor pentatonic scale can also be found in the

6:22

black notes okay but you just have to sort of start at a different spot you start here okay

6:36

by the way the black notes are quite convenient

6:40

for sort of improvising especially if you don't have a lot of experience playing the piano

6:45

someone just asks you to sit down on the piano and play something and you've never

6:49

done this before my advice to you is ignore the white notes and just play on the black notes

7:06

it's not so hard to sound pleasant on the black notes

7:10

because the black notes don't have any one tones amongst them so the white notes have these two

7:20

these two notes here which are adjacent so there's a one step between them and also these

7:23

two notes which have a one step in between them and if you play them both at the same time then

7:28

that tends to have a bit of a jarring discordant effect okay sometimes it's effective in in special

7:36

things but even if you play um these two notes like this this one here and here if you play this

7:42

one an octave down play these two notes it also has a slightly discordant kind of sound although

7:50

actually it's quite important in modern jazz and major seven but anyway so anyway when you play

7:56

with the black notes you sort of automatically are removed from having to worry about that so

8:01

that's just a nice thing so anyway so it's interesting that the two pentatonic scales can

8:05

be found purely in the black notes and one more uh one more important scale that i'll mention is um

8:13

perhaps arguably the simplest one actually from a mathematical point of view and that's when we say

8:20

okay we go from here to here which is uh a range of 12. and let's just go up two at a time okay so

8:26

two at a time just take two steps two step two step two step two step two step two step so that's

8:35

sometimes called a whole tone scale it's called an indian scale because it's used in indian music

8:46

my tick tock improvisation called even six number one is all based on this whole tone uh scale

8:58

which is uh we might say it's a sort of a uniform two-tone scale okay so that's uh

9:06

at least an introduction and i especially want to to promote this idea that we can express these

9:12

scales mathematically in a pleasant way using this interval arithmetic which is quite natural

9:18

based on the underlying chromatic scale and this also lends itself as we'll see to understanding

9:26

chords in a very nice way but also allows us to understand modes of scales okay so it turns

9:32

out that if you take a scale like this here the major scale then there's various variants

9:38

of it called modes and we can get at these modes quite pleasantly with this arithmetical

9:45

nomenclature that we've introduced i'm Norman Wildberger thanks for listening