Scale Adjacency, Sharps and Flats | Maths and Music | N J Wildberger
Vložit
- čas přidán 27. 07. 2024
- The almost uniform property of the major scale that we discussed in our last video has an important consequence which is responsible for the current staff notation system of using sharps and flats to create key signatures for the 12 major keys.
Here we look at this from a mathematical point of view, using 12 tone chromatic scale notation. The circle of fifths or fourths, which is better described as the circle of 7 steps or 5 steps, naturally makes an appearance. And there is an interesting connection with the geometry of the clockface.
This video anticipates a wider ranging discussion about alternative ways of notating music.
This is indeed part of the Maths and Music Playlist: please check out the other videos in the series at • Maths and Music
Artwork created with the help of Midjourney, an amazing Bot that stands ready to put half of the planet's graphic designers out of business in the next few years. It's a new world coming, and us pure mathematicians are not immune, as we shall discuss in our Sociology and Pure Maths series!
You can check out my TikTok (music) channel: www.tiktok.com/@njwildberger
And a big Thank You to all my Patreon supporters at / njwildberger
Video Contents:
00:00 Math: three sets from {1 2 3 4 5}
3:42 Circle of Adjacent Scales
12 :36 Major Scale
18:32 Using Traditional Notation
21:04 Sharps and Flats
***********************
My research papers can be found at my Research Gate page, at www.researchgate.net/profile/...
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
• MathHistory: A course ...
• WildTrig: Intro to Rat...
• Math Foundations
• Wild Linear Algebra
• Famous Math Problems
• Box Arithmetic
• Elementary Mathematics...
• Year9Maths
• Ancient Mathematics
• Wild West Banking
• Sociology and Pure Mat...
• Sociology and Pure Phy...
• Old Babylonian Mathema...
• Probability and Statis...
• Boole's Logic and Circ...
• Universal Hyperbolic G...
• Differential Geometry
• Algebraic Topology
• MathSeminars
• Playing Go
• Diffusion Symmetry: A ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
• Algebraic Calculus One
• Classical to Quantum
• Algebraic Calculus Two
• Advice for mathematics...
• Solving Polynomial Equ...
• The Hexagrammum Mystic...
• Algebraic Calculus and...
• Dynamics on Graphs
• Playlist
• Triangle Geometry
• Explorations with q-se...
• Six: An elementary cou...
• Maxel Inverses and Ort...
************************
Absolutely wonderful, Norman! I started on piano over 60 years ago, know all the sharps and flats, but never once did I realize the clock face with the evens being "right" and the odds being "anti-right" so to speak. Please keep up this series, I continue to learn and be amazed at how complicated it is in relation to mathematics!
Hugely enjoying this, as you also seem to be!
Another thing that seems arbitrary from a mathematical perspective is the preference for the Ionian mode. Dorian mode is symmetric and perhaps highlights certain relationships better. For example, the first adjacent scale either raises to a major third (#3) or lowers to a minor sixth (b9 = b(-3)); so there is an inverse relationship with respect to 0 mod 12. Also, it can highlight the relationship between the modes better - Dorian moves to Mixolydian by raising the third (#3) or lowers to Aeolian by lowering the sixth (9b).
Slide 3 shows the relationship between the raising note and the starting note of the scale - the Dorian version demonstrates a relationship where the notes leading into an augmented triad {x, x + 4, x - 4} are a raising or lowering of the same note. For Ionian, if the starting note is x then the raising/lowering note is x + 2. For Dorian, the raising/lowering note is x. For example, if the starting note of the scale is 2 then the notes leading into 6 & 10 are 4# & b4, respectively. But for Dorian, with starting note 2, the notes leading into 6 & 10 are 2# & b2, respectively.
love your work, thank you
Very unique stuff you mske
sir you're a legend
I'm still confuzzled, I'm not sure if this has to do with this video, but how do I do 4#8? It was in a mathcounts competition. I don't know how to do sharps, flats, and naturals in math...
Respected Professor, I need to talk to you. Cantor's work on infinity is incorrect and I have proof. I heard you retired, how can I send in a mail about this?
My Name is Ishit Kulshreshtha. I was in the PhD program at CSE Indian Institute of Technology Bombay.
The jump from sound to numbers was so sudden -- didn't exist, actually. The numbers represent intervals, I take it. That seems like a cop out. The sounds are frequencies that have been numerified. What frequencies are harmonic and not harmonic would be interesting. Are all the notes on a scale equidistant? Are all sharps and flats equidistant from actual notes? You're fiddling with arbitrary measurements or assigned numbers here.