Math and Music Pt5/5

Math and music is the topic of this interactive lecture by Waldorf math educator and author of the Making Math Meaningful curriculum books. This lecture complements Jamie York's high school math curriculum found in his SOURCE BOOK TEACHING HIGH SCHOOL MATH, which contains a unit on MATH AND MUSIC. It is available at www.jamieyorkpress.com.

[@insightfool]

Really thorough coverage of "basic" music theory that I have NEVER seen anywhere else. Thank you very much for going over these details without any hand waving.

[@uomodibassamorale]

Very nice lesson! By the way, a quick way to explain why there are 12 notes in an octave is the following: start by recalling that doubling the frequency means going up an octave, and that multiplying the freq. by three means going up a twelfth (one octave plus a fifth). So if the base note has frequency F, going up octaves you get 2F, 4F, 8F.... (F times a power of two). On the other hand going up twelfts you get 3F, 9F, 27F etc (F times a power of three). Now we would like that, after a certain number of "going up twelfths", we obtain the same note as F: in other words we want that 3^n * F = 2^m * F, for some numbers n and m. This is mathematically impossible (assuming we want a finite number of notes in an octave...) but a good approximation can be found. If you make the calculations, the problem boils down to find an approximation to the base2 logarithm of three. It turns ou that 19/12 is a good approximation, and this mirrors the fact that 19 semitones and 12 semitones are a twelfth and an octave, respectively. 65/41 is even better, but maybe 41 notes in an octave is a bit of an overkill... cheers

[@cosimobaldi03]

nice

[@bemdav]

Just finished it, very interesting. I've been playing guitar for 5 years but had no clue how the harmonics worked. Thank you!

[@gauravmusics]

Thank you so much for such a great lesson....god bless you

[@IliyaPopovski]

It is very simple. You probably know, that every note can take a # - sharp. B#=C and E#=F, e.a. they don`t include half step (or a black key on the piano if you wish) just because the distance between these notes is half step, not a whole step as C-D, G-A etc. Dividing the octave on 12 equal steps shows what musical theory has determined as ideal for musical notes, and every step here equals a half step (o.0) when talking about notes.

[@1rsalc]

Great explanation! Here's a question... octave means eight in Latin. Who in history came up with the idea that the 2:1 pitch was located 8 above the 1:1 pitch?

[@andreasihlen6244]

It's because of the major scale and the fact that it contains 7 unique notes or pitches, and the 8th is simply a repeat of the route or tonic.

[@timamalsinaidi5713]

Awesome videos !! very helpful , thanks much appreciated .

[@IIexeII]

Thank you for this lesson! I really appreciate it

[@IliyaPopovski]

In other words C-D is a whole step because there is a half step in between - C-C#-D. but when going to E there practically is no E# and that`s why musicians continue with F :)

[@chounoki]

We choose 12 steps because 12 has rich divisors: 2, 3, 4, 6, so there are more chances to have the notes harmonic (in order to have chords). Although we could have more divisors if we make the steps larger like 48, it would be hard for 10 fingers to play.

[@SZG65]

I have an old tuning fork that says C - 256 on it. You start with C at 264. Why is there a difference?

[@YourMumIsBurning]

this video explains it, drawing C off of B and C# as per this video, it's approx 524.15 = 262.075 Due to the process they use in making tuning forks, it was easier to replicate 256 Hz rather than 262 Hz. The Human ear in most cases, can't pick up the 6 Hz difference. Plus, when tuning stringed instruments such as a guitar, the motto is to use your theory knowledge to perfect the tuning by analyzing the dissonance between two strings pitched the same.

[@TEZZOR8]

Can anyone explain why between B-C and E-F there is no middle note? I get all the math and frequency stuff, but when dividing an octave into 12 equal steps, why did they not include half steps between these two intervals?

[@1rsalc]

I'll attempt to answer it, but I am not a pro at this! Long ago, they determined that a root note (1:1) and what we call octave (2:1) were pleasing to the ear. Then they discovered that a 3/2 note (now called fifth) and a fourth (I forget the ratio). Were pleasing. You can't change the position of root, 4th 5th and octave (ex: C F G C'). When they decided to divide the scale into 12 notes, they left the F where it is. It so happened that 4 notes fit between C and F, so they called them C#, D, D# E. As you can see, E sits right next to F. Since the top C' is fixed, they kept adding notes between F and C' It so happens that they had F#, G, G#, A, A#, B but there was no extra note to add between B and C. C(fixed) C# D D# E (these 4 were added) F(fixed) F#(added) G(fixed), G# A A# B(added) C'(fixed) That's my understanding of it. Plz take it with a grain of salt!

[@danielgb7891]

Thanks! This answered many questions I had

[@reggyh5186]

it seems to me that a violin can play in just intonation and change keys without a problem since the musician can put their finger anywhere on the bridge, is this right? and is it because of keyed instruments that they invented equal tempermant?

[@joppevanwetswinkel4636]

correct, the only thing that's unchangeable on a violin is the ratio between strings

[@gordonwilliamshaw]

I find it surprising that the 3rd has an error of 13.3%, I think I want to read more about this. Does anyone know a good book on this? it wasn't clear from the presentation.

[@ForPropertyInvestors]

So if you really wanted to could a guitarist for example have a guitar made with the early Pythagorean’s way tunes to C major. And 12 of the same guitars each tuned to a different key. Not practical but say they wanted the perfect sounds for recording a song in a key.

[@vanyamba]

Why 1/12th. Audible range of freqs is 20 Hz - 22 kHz and these are 10 octaves.

[@andreasihlen6244]

It's because of equal temperament. They fill in the gaps between the harmonic pitches and make key changes possible.

[@brennengrimes]

+TEZZOR08 When they decided to have 12 steps, 2 notes out of the 7 wouldn't have a black key (half step) with it otherwise it will be 14 steps. And the reason they can't do 6 notes with with each having a black key since it would be Impossible to know what note is which on a piano. We use the grouping of 2 and 3 black keys to figure out the notes. For example a C is left of the 2 black key grouping. And a B and a C is a Half step apart as well as an E and an F. We can say a B is a C flat and an F is an E sharp. These technical stuff can be difficult for beginners. I advise you to learn piano before learning all of this stuff.

[@brennengrimes]

+TEZZOR8 read this