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Loc: New England, USA
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!
Edited by Amaruk (01/21/1307:11 PM)
My piano channel on YouTube: Link
Loc: Lost in cyberspace.in the UK.
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?
Rise like lions after slumber,in unvanquishable number. Shake your chains to earth like dew which in sleep has fallen on you. Ye are many,they are few. Shelley
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.
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.
Loc: Ireland (ex England)
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):
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
Now, the second tetrachord of this new scale (starting on D), can be used as a starting point for the D Major Diatonic scale
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 )
Edited by Jean-Luc (01/22/1304:21 PM)
- Please, forgive my bad English
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.