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Hi AMers,
Time for your regular dose of music theory, we are never too old to learn. This video finally helped me understand the practical reason we have 12 notes in Western music. As well, I finally understand “equal temperament” or the tuning compromise that allows us to play music in any key such that they all sound the same. I was amazed to find out how large a compromise that is for some of our most common tones. Hope you find it interesting.
John -
That is a very informative video John I am very glad you found it. I have often wondered what the frequency relationships were I only knew A at 440 or its octave at 220. A very well presented piece. I dont know if its really of any practical use but very definitely of great interest and maybe if I ever get enough understanding and skill there prove very useful.
Many thanks for posting it has cast some light!
JohnStrat -
Although most of the information in this video is correct, it actually doesn’t explain why there are twelve notes in the western chromatic scale! It’s not at all because “it is the most sensible and convenient way to divide up the octave”! Other cultures in the world have a totally different yet equally sensible approach and make music with scales that have little to do with our chromatic scale.
In order to understand why we settled for 12 notes, we need to go back to the works of Pyhagoras (approx. 500 B.C.). He discovered the fundamental relationship between pitches an octave apart (corresponding to a frequency doubling), and between pitches a 5th apart (corresponding to a multiplication par 3/2).
Starting with a pitch of arbitrary frequency f, we have three pitches: f, 3/2f, 2f
Let’s call them C0, G and C1 for the sake of simplicity…Pythagoras then wondered what would happen if you repeat the process, i.e. you multiply the frequency of G by 3/2 again, and again, and again…
You end up with this:
C = f
G = 3/2f
D = (3/2)^2f
A = (3/2)^3f
E = (3/2)^4f
B = (3/2)^5f
F# = (3/2)^6f
C# = (3/2)^7f
G# = (3/2)^8f
D# = (3/2)^9f
A# = (3/2)^10f
E# = (3/2)^11f
B# = (3/2)^12fYes, absolutely: this is the circle of fifths! However in the circle of fifths there is no B#…. That final B# is very close to (an octave of) C1… but not exactly equal to it.
In fact the mathematical equation (3/2)^n = (2)^m doesn’t have any solution for integer values n and m. So no matter how many fifths you stack up, you’ll never get a multiple of an octave. The closest you can get is by taking n = 12 and m = 7 (you’ll have to trust Pythagoras and me on this!).
Put differently, 12 fifths correspond to almost 7 octaves.
So Pythagoras decided that from now on B# would be equal to C1.All those fifths actually define a scale; just divide each frequency by the appropriate power of 2 to map it back between f and 2f (I.e. between 1 and 2).
You get the Pythagorean scale, one that was used in the early Middle-Ages:
C C# D D# E E# F# G G# A A# B (B# = C)So there you have it: twelve notes! That’s where they come from.
Unfortunately, in making B# equal to C, Pythagoras shortened the last fifth; the ratio between the frequencies of the two notes is 1.0136 and this is called the Pythagorian comma, also called “the wolf”. It’s small, but enough to make this wolf sound terribly off!
…during centuries, musicians have been looking for the best way to get rid of (or deal with) this comma and the way they have done this is through various temperaments (the video is correct about this).
It is only at the turn of the 20th century that the equal temperament has gained widespread acceptance, but even today some musicians still use various “well temperaments” also called “mean-tone temperaments”.And other cultures in the world as well. In fact, I may post something about this as it has a direct connection with… the blues!
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So I guess the E# from Pythagoras scale was renamed to F?
It’s amazing how everything in this universe can be explained through Mathematics!
🎸JoLa
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Almost, JoLa ! In fact, E# was not just renamed to F, it was replaced by F.
If you generate a 5th below the initial frequency f (i.e. you divide f by 3/2), you get 2f/3. Map that pitch back between the boundaries of the octave, and you get 4f/3, which is a 4th; let’s call this new pitch “F”.
This “F” is very close to E#, but differs from it by a comma.
However, because generating F is much easier than generating E#, Pythagoras decided to use F instead. -
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Many years ago when I was in university, I had to take some elective courses, so I chose Music Theory, because there were some fine looking young women in that course. Plus, I did actually learn a few things, like this: Back in the times of JS Bach, musicians used to agonize and debate over this very subject. Bach was a big believer in the use of equal temperament scales, and as a professional piano tuner, he was well aware of the compromises necessary to achieve that goal. Toward that end, he produced a series of works called “The Well Tempered Clavier”. It contains songs in all 24 keys (major and minor), and the idea is you could actually play any song in any key, and it would still sound good. Exactly 300 years later, we are still pondering over equal temperament tuning.
https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier
Sunjamr Steve
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Such an interesting subject! I have always been fascinated by the expressive power of dissonance in familiar and unfamiliar music and also the particular issue of microtonality. Also so interesting for musicians is the juxtaposition of artistic and theoretical/technical considerations which must be addressed. Thanks for raising this topic, John, and for the very interesting contributions above.
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