I'm not sure if this is even thread worthy or if I should even ask but...:

Does anyone know their post-tonal theory? My foreign() teacher was lecturing yesterday about it and I just daydreamed the whole time. Like....what the hell is "mod 12?"

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I'm not sure it has anything to do with music theory, but I know that "mod 12" is the arithmetic used in "clock maths".

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hahaThat might actually be right because he drew a big "clock" on the black board, and wrote in all half steps next to each of the 13 tallies.

....can you explain then "clock maths?"

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Well, it may seem obvious but not many people can identify it. Instead of counting to 10 and then starting again, you count to 12 and start again. The mathematics behind it can be explained from the following excerpt from "Number, Groups and Codes" by J.F. Humphreys (one of my old lecturers) and M.Y. Prest:

"Congruence Classes

Definition Suppose that n is an integer greater than 1, and let a,b be integers. We say that a is congruent to b mod(ulo) n if a and b have the same remainder when divided by n. If this is so, we write


The definition may be more usefully formulated as follows:


*i.e. no remainder after division

Examples

-1 ≡ 4 mod 5 (because -1-4=-5, which is divisible by 5)
6 ≡ 18 mod 12 (because 6-18=-12, which is divisible by 12)
19 ≡ -5 mod 12 (because 19-(-5)=24, which is divisible by 24)

The notion of two integers being congruent modulo some fixed integer is actually one with which we are familiar from special cases in everyday life. For example, if we count days from now, then day k and day m will be the same day of the week if, when divided by 7, k and m have the same remainder. Similarly a clock works, in hours, modulo 12 (or 24).

Christmas in 1988 falls on a Sunday: therefore Christmas 1989 will fall on a Monday since there are 365 days in 1989 (not a leap year), and 365 is congruent to 1 modulo 7 [365 ≡ 1 mod 7 ([365-1]/7=52)]: therefore the day of the week on which Christmas falls moves one day forward.

For another example take measurement of angles, where it is often appropriate to work modulo 360 degrees."

So, in the context of music theory, one would define:

1=C
2=C#/Db
3=D
4=D#/Eb
5=E
6=F
7=F#/Gb
8=G
9=G#/Ab
10=A
11=A#/Bb
12=B

So, for example, if you were to move 6 half steps up from G# what note would be played?

To go about solving, we express what we know from the information provided. Well, we know that, by our own definition, that G# is equivalent to 9 and the interval is 6 half steps and so a=9+6=15. We also know that there are 12 intervals altogether and so n=12. So we have

15 ≡ b mod 12

from which it can be seen that b=3 in order for the equation to hold (because 15-3=12, which is divisible by 12). Now, if we refer back to our definition, we see that 3=D, which is 6 half steps above G#.

It's quite counter-intuitive at first but that's largely down to the fact that we are taught to count "mod 10", i.e.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15...

"mod 12" is therefore:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1 0, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 1 9, 1 10, 1 11, 2 0, 2 1...

Hope that helps!

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Oh, by the way, number theory was not one of my interests at university (actually, I found it rather boring) so if anybody spots any errors in the above please feel free to post a correction.

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Oh my...like wow.....so there really is a connection between math and theory!!!!????Ahhhhh!!!!!!!!!!!!!
Dude, that is totally it, haha.

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my god.. i tried.. i just couldn't read halve of it before my mind wanted to explode

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enlightening but i dont have the patience!!!

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Shit congruences remind me of discrete math ARGH!!!!!!!!

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No way! Discrete maths is much more interesting than congruences!

Clock maths is easy - we can all add 5 hours on to 10:43am can't we? It's just the same with music theory - we all know what note we get if we move up 3 steps from F#. The mathematics behind it just formalises the process, which often has the effect of making it appear more difficult than it actually is.

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I finished my mod 12 homework but a couple days ago. Awesome. John, I think you made it sound harder in that post than it really is.
Since visual aids are so great, all I really needed to do was to draw a clock face and write in all the enharmonic half steps. Then is was all there...right in front of my face

When I get the "notes" handy, I'll post them all here to explain it in the music sense more clearly. Though they only tell how it is, not why and what it's used for......

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Haha, yep! Mathematicians have a real knack of making things seem harder than they actually are in order to make themselves appear more intelligent! It's a form of elitism, I guess.

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