An extension of the side conversation from the Michael Jackson tribute thread. Circle of Fifths is the first question. What is it? Why do I need to know it? 
Thanks dada. Itās as bland as I remember it, but at least I know how I can apply it now. My piano teacher just told me āMemorize this. Itās importantā Thatās when I lost interest, and quit piano lessons. A decision I will always regret making. 
Therein lies the problem with most music teachers Iāve encountered. Thereās precious little passion for the actual teaching part of it, and a lot of them are good musicians but poor communicators of their knowledge.*
My first piano teacher (I was seven or eight at the time) was a grumpy bastard who was more interested in paying for his med school than instilling any joy in performance, practice, theory, or composition. He taught me, for example, major and minor scales, but not how or why they work. At no time did he point out that all major scales follow the ātonic-tone-tone-semitone-tone-tone-tone-semitoneā pattern and that all they were was transpositions of C-Major. This is so eminently simple yet it was never communicated to me - something which would have made the process of learning so straightforward. Instead, it was āHere, learn thisā without reasons why.
Plus it was all dreadfully dull classical music. Now, you may love classical music, but I donāt. In fact, being forced to play Chopin, Beethoven and Haydn did more to inculcate me with a loathing for the form than any inborn distate for it. There are plenty of challenging songs in other genres (ragtime, prog rock, jazz, etc.) that wouldāve been far more relevant to me. Itās not like playing a minuet or waltz is somehow the One True Path to technical skill and virtuosity.
The sad part is that that awful teacher (and one who was nearly as bad who succeeded him) turned me off all music for several years thereafter. I was in my teens before I gave a damn about music again. By that time Iād lost whatever chops I did have in those three years of lessons.
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- This is a generalization that applies to the ones Iāve been trained by. Your experiences may vary.
I think many teachers become jaded because very few students arenāt all that interested in playing their instrument seriously. The vast majority of students arenāt interested in music theory, nor do they know that there is such a thing. Why would an instructor teach any theory when the student doesnāt even practice their arpeggios and scales? Teachers need to be motivators. Perhaps if the teacher found out what kind of music moved the pupil, maybe the teacher could convince the student that those early lessons are a means to that end: even if they are playing the clarinet.
However, look at the most popular music that young people see today. Performers on television are usually song and dance people and arenāt playing instruments. The role models of the music industry today are quite different than the guitar, keys, and drum playing heroes of the 60s-90s.
I would love to be able to play Bach or Musgorvski(sp?) But my piano teacher said, āYouāll get to thatā¦but right now, you need to learn such and such and such and suchā and I didnāt want to hear it. I wanted to actually PLAY SOMETHING, but instead, I had to learn other boring stuff, without being able to ask āHow and why is it like that?ā If I dared to ask, Iād be told, āThatās not important. Just learn it.ā I donāt learn well like that. I need to know WHY and HOW.
Circle of Fifths
Consider your thumb to be number one; for this example weāll call that C (the scale of C major is a natural starting point because it has no sharps or flats). Count up five notes on your fingers starting with your thumb, or CDEFG. If you look on the circle of fifths, youāll find that one pie piece clockwise (the direction for sharps) from C is G. Why? A fifth above C major is G major. Thatās the key that has one sharp. The note that is sharp is actually the fourth tone, or the note right before the name of the key. In other words G major has one sharp and that is F. To find what has two sharps, start with G as your thumb and count up a fifth, or GABCD. Now you know D major has two sharps: the F sharp we already had, and the C sharp that weāre adding. You can continue this way going up keys and adding sharps, bearing in mind that when you get up to six sharps, which is F sharp major, that you have to include the sharps in the name of the key.
Once youāve figured it out in your head, you can remember the sharp keys by using the mnemonic device āGo Down And Eat Breakfast,ā because the keys with sharps are GDAEB. Another way of looking at what we just counted is that if you want to know what key you are in, just look at the last sharp you see and go up a half step.
Each major key has a relative minor (the darker side of the scales). The relative minors have the same key signature as the major, so the one that shares C major on the pie slice is A minor. The only difference between them is that A minor starts a little lower and ends a little lower. All you have to do to find the relative minor keys is go down a minor third. A minor third is only a whole step and a half step, three frets on a guitar. In C major you go down CBA. For another example: G major has one sharp (F sharp) and so does E minor. If you go down a third from G, which is G F# E, you have E minor. Now if you get up to A major, youāll find that the relative minor is actually F sharp minor, because F sharp is in the key signature (you know, A has three sharps F, C, G).
You can use a similar technique to sort out the flats of the circle of fifths going in the counter-clockwise direction.
The C-Major (diatonic) Scale in Standard Notation
Every guitar player should be able to play it; with this scale you cover about 50 % of all folk, country, Christmas and other non-Blues music styles. You can also play simple songs for your children.
An often overlooked important musical fact is that the middle C played on a guitar (3rd fret, A-string) is actually one octave lower (130.8 Hz) in sound that the middle C on the piano (261.1 Hz). In other words, to enable useful working with standard notation for a guitar, the guitar is played one octave higher on the treble scale than tuned. That means that standard notation for a guitar is different to standard notation for a piano. You can use sheet music for a piano, but you play it one octave higher.
The Circle of Fifths & Itās Use for Blues Guitar
Also called the cycle of fifth, itās a very common graphic that illustrates the key signatures. The outer circle is moving clockwise in the dominant direction that means each note is followed by its fifth note. The inner circle is moving counterclockwise in the subdominant direction. If you start with C the next note in dominant direction is G, in subdominant direction F. You can go through the whole circle and end after 12 steps (the 12 notes) again at the starting note. Note the enharmonic notes, which have the same pitch but different names like G - Abb. The important thing is the sound of the fifth chord of each scale - played as 7th it will always resolve back to the root (tonic) chord.
For Blues guitar based on the I-IV-V progression this circle makes it simple to find the right chords. If you play a Blues in E you need the chords E(I), A(IV) and B(V). Locate the E note and youāll find A in subdominant and B in dominant direction - thatās all. If you really want to know how a Blues in G# may sound, take C#(IV) and D#(V) - easy!
Itās getting even better - the pentatonic scale can also be described from the circle of fifths (= penta!). Letās take the major pentatonic scale in C. Notes are C-D-E-G-A-C. Now start from the root note C and go up using the circle of fifths: C-G-D-A-E. Order them and you get the right scale!
Farmer Brown Eats Apples Daily Good C# I think is the circle of fourths I believe.
Thanks, DeFrag. Nice and easy to remember. Makes it easy to count keys, as well. dada, If I had learned on a Moog and not a rented Yamaha digital piano, I would have stuck with it. 
While I agree with what you say, bear in mind that my experiences with that awful instructor were 1979-1982 (i.e. before MTV, Autotune, YouTube and thus preceding the period when āimage over craftā took hold of music). Plus he really wasnāt old enough to be jaded. He couldnāt have been more than 23 or 24. I guess the real problem is that anyone can put out their shingle and call themselves a music teacher; there isnāt really a quality control aspect.
I dunno. The piano teachers I had were universally good. In addition to scale and arpeggio learning, we also got to improvise in class. This tended to freak out the musicians coming from a stricter (playing only what was written) background, but I had no problem with it. Of course, this was a requirement in Music School (BA in Music Comp/Theory from the University of New Orleans). While a bit of theory was taught in piano class, I got most of my theory from Composition classes. Unfortunately, due to Hurricane Katrina, there isnāt a Composition/Theory degree @ UNO anymoreā¦ 
Then again, my composition teacher absolutely hated synthesizersā¦
I am sure that others here have covered the topic of the circle of 5th well. Its really just a way to map of all the keys expressed as a cycle of 5th (or 4ths) depending on which direction you are travelling on the circle.
Regarding music theory. Many donāt like it and want to play in a more intutive way. Frankly, I am not into the whole idea of dumbing down everthing. Bascialy, music theory is a tool. Its a way to get somewhere. It can help a composer/musican to travel in a different drection that others have.
It really depends as well on the type of music. Not so useful in hip hop but if you are playing jazz or playing or composing classical, it very useful. Itās really not t hat hard to learn and you donāt have to learn it from grumpy piano teachers. There is a lot you can find on the internet.
My music theory:
If someone else likes the noise I produce, it might be music!
I have fumbled about with synths in the past and i finally got an LP in May
and was determined to learn some scales and stuff, so for the last two
months all i have done in my spare time is practice scales and learn
formulas from major to minor, pentatonic, gypsy and so on, until they
are programmed into my brain, the blues scales are some of the funnest
scales i have came across.
Here is a formula which i came across which is derived from the circle of
fifths which i found quite useful.
Father Charles Goes Down And Ends Battle
C major is all the white keys, no flats or sharps
G major has F sharp only
D major has F and C sharp
A major has F,C and D sharp
E major has F,C,D and A sharp
B major has F,C,D,A and B sharp
Hereās an Interactive Circle of Fifths:
Iām embarrassed by my lack of theoretical knowledge. Ignorance is nothing to be proud of. I read music very poorly. In another world Iād be considered functionally illiterate. Iām sure there is someone out there who canāt read or write who has dictated a fabulous novel but a miniscul vocabulary is rarely seen as an advantage.
Music is strange: itās rife with people who are proud of their lack of knowledge. āI donāt know any chords but I am a geniusāā¦
The bottom line is that some of these people do very well BUT itās in SPITE of their shortcomings, not because of them. I still take lessons when I can! Itās an amazing adventure!
You want to be a complete intellectual anarchist? Refuse to learn anything about your Moog. Donāt learn the difference between a VCF and a VCA. Just turn knobs until you like the sound. Better yet use presets. Oh Wait. Lists of people do that.
Since that works off-line, I see it as being very handy. Thanks.
My 0.02:
Donāt forget the flat scales!
Battle Ends And Down Goes Charles Father (the flats used in the scales).
So the scales are: C, F, Bb, Eb, Ab, Db, Gb, Cb (B), Fb (E).
C Major: no flats
F Major: Bb
Bb Major: Bb, Eb
etc.
Like the sharp scales, you add a flat each time you go DOWN a perfect fifth (which is an octave lower than a perfect forth ABOVE the tonic/root (itās a frequency thingā¦)).
Itās easy to remember which scale youāre in, because the second to last flat used (excluding C Major and F Major) is the name/tonic of the scale.
Confused? 
You can actually save a copy of that page with IE into a .MHT file for offline use.
I learned the naughty versions in my high school jazz band class:
Fred Can Get Drunk At Every Bar (order of sharps)
Can Girls Do Anything Else But Flirt Constantly (circle of fifths)
I do like that interactive Circle of Fifths. Very nice find!