The magic of 7 in the circle of fifths.Why is it like this?

Currently I am teaching our son to play the scales and teaching him how many sharps and flats each scale has.

It was during one of these sessions that I realized that if you start with C (C major scale has no sharps/flats) and move 7 steps up it takes you to G and the G major scale has one sharp. If you go another 7 steps you land at D and D major has two sharps... This pattern works all the way up. And it even works in reverse, if you go 7 steps down from C it takes you to F and F major has one flat, another 7 steps takes you to Bb and Bb major has two flats. This pattern works all the way down too. Does anybody know why it works out this way???

I am very curious to learn more about this interesting side of music theory so any inputs are very welcome!

you answered your own question in the title of your post: magic.
plain and simple musical sorcery, someone was trying to keep it from you, but you found out and will now be accepted into elite musical circles, well done sir.

Could you just run it by me again?

Circle of "Fifths" ---
A perfect fifth has 7 half steps (or 8 if you count the starting note).

Not much of a mystery here. Think of the Perfect Fifth to keep this simple. Otherwise, we'd have to rename this to the "Circle of 7 Half Steps"

And whats so wrong with the "magic circle of 7 half steps?" it sounds so mysterious and clever at the same time.lol

Great! Thanks for explaining this JazzWee!

Rostosky... too funny! Sounds like something from Stonehenge... Perhaps that is what Stonhenge was all about; Music Teory!

Now that you mention it, it does sound like some sort of "morris dance" can you believe that actually still goes on here?

"stupid noise of the week" or "morris dance " starts at approx 29 secs in...

In this clip they are using the "magic circle of 6 quarter steps" so it may be slightly off topic, but here it is anyways..

First I thought you were talking about seventh chords, the magic of them. Maybe the magic of them is that they are the quickest way to add some extra "color" to your chords. Besides, they are a sort of a bridge between major chords, when shifting from one chord to the next.

Actually, the first time I read it I thought it was really MAGIC. Amaruk said move 7 steps.

Well -- I had to think about that. Moving seven SCALE steps will always bring you back to the same note. C to C.

And then it dawned on me -- CHROMATIC steps (half steps).

Sorry to kill the magic

Circle Progressions:

http://www.angelfire.com/fl4/moneychords/circleprogressions.html

The perfect fifth interval represents a vibration frequency ratio of 3:2. With a tempered piano (where the frequency ratio of every half step is exactly the same) the perfect fifth is impossible.

The frequency ratio between the upper and lower note of the octave interval is 2:1 (you actually double the frequency by going one octave up). There are 12 half steps in one octave. Therefore the frequency ratio of the half step must be the 12th root of 2 which is approximately 1.0594631 . The fifth interval on the piano is the 12th root of 2 in the seventh power. This makes 1.4983071 - very close to, but not exactly 3:2.

The "magic" actually comes from the fact that you have to raise the 12th root of 12 to the power of seven to get as close as possible to the perfect ratio 3:2.

Simple, isn't it?

"Magical Circle of Seven Half Steps" I like it! Let's start a petition.

The first note or fundamental in the harmonic series has a fixed wavelength. The second note in the series, or first harmonic, has a 2:1 relationship with it. This is the octave. The next step has a 3:1 relationship this is an octave and a half, to the dominant, and is equivalent to 3:2. Because these notes are so close in their relationship, even though they're an octave and a fifth apart in distance, the scale based on them is only out by one accidental.

Conversely, C to C#, despite being adjacent on the keyboard differs by seven accidentals.

A major diatonic scale is made of two similar tetrachords tied together by a whole steps (a tetrachord is a group of four notes separated by a set of fixed intervals).

So, if you start on C you have:
[C-ws-D-ws-E-hs-F]-ws-[G-ws-A-ws-B-hs-C] (ws = whole step, hs = half step)

As you can see, the second tetrachord (starting on G) is structurally similar to the one starting on C (same interval arrangement)

This has for consequence that you can use this tetrachord as it is to start a new scale (so starting on the fifth degree of the scale, 7 half steps away from your original tonic):

[G-ws-A-ws-B-hs-C]-ws-[D-ws-E-hs-F-ws-G]

But as you can see, if you add the second tetrachord, there is a problem with the ws/hs arrangement and you have to raise the seventh degree by a half step to correct it

[G-ws-A-ws-B-hs-C]-ws-[D-ws-E-ws-F#-hs-G]

Now, the second tetrachord of this new scale (starting on D), can be used as a starting point for the D Major Diatonic scale

[D-ws-E-ws-F#-hs-G]-ws-[A-ws-B-hs-C-ws-D]

As you can see, you will once again have to sharpen the seven degree to comply with the interval rules of the major diatonic scale. This goes on until you reach the c# Major Diatonic scale where you run out of sharps.

The flat sides of things is fairly similar and I am sure you will be able to build it on yourself

Sorry if the explanations aren't crystal clear, English isn't my native language (as you probably noticed )

I wish the kids I help out with theory for choir could reach these sort of conclusions

By the way, if you use perfectly tuned steps by the time you get to B# you're higher than C by a quarter tone!

Jean-Luc, tes explanations me paraissent claires comme cristale. Merçi pour avoir donné l'explanation dont je cherchais les mots.

[Jean-Luc, your explanations seem to me to be clear as crystal. Thank you for having given the explanation for which I was seeking the words.]

/s/ PianoElève quatre-vingt huit

Circle of fourths for flats...

Being etheral after death, ghosts can fly.

@ PianoStudent88: Ton français est excellent

[@ PianoStudent88: your French is excellent :)]

I actually assumed Maine was in France. Had to look it up. Apparently it -was- in France! (Former colony)

Now try taking a diminished chord and learn how all the tones can be leading [ up or down by half step]....then try all the enharmonics applied in the same dim. chord and do the math.I've never done the math here myself.

rada

Rostosky, Did you not know that you can't be a Morris Dancer if you've been circumcised? PM me if you don't know why.
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Fortunately I am intacticus, but as I never ever want to even see or hear, let alone be a morris dancer.... Pm is on its way...

PM answered.
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Pmed answer so hilarious, I will use it in as many situations as possible.

It does have a very appealing ring to it. Perhaps more music students will be driven to piano given the cool name. Draw towards it. Like magic if you will.

It is magic! There's lot's of magic in music notation, there's no end to it, it is wonderous. That is why music can heal the savage breast.

Just one thing, though, the magic of the "circle of seven half steps" is not particular to the piano, it casts its spell on all musical instruments.