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I'm looking at the formulae used to create the Major and Minor scales, with 'W' meaning whole step and 'H' meaning half step.

Major scale:       W-W-H-W-W-W-H
Minor scale:   W-H-W-W-H-W-W

Is there a fundamental relationship between the formulae, and if so what is the reason for it?

As displayed above, they seem to effectively be the same formula, but offset by 2.

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Yes, the are related exactly as you say - offset by two. In fact, every major key has a relative minor and each minor key has its corresponding relative major. For example, if you play only the white keys on a piano starting from C and going up to the next C, that is the C major scale. If you want to play in the relative minor to C major, start from A and play the white keys up to the next A and that is A minor. Relative pairs stick together, so the relative minor of C major is A minor, and the relative major of A minor is C major.

There's even a pattern of relative major/minors. The relative minor of any major key is three half steps below, and the relative major of any minor key is three half steps above. So the relative minor of G major is E minor, and the relative major of D minor is F major.

  • Very interesting - so the 'offset by two' is actually an offset of 3 semitones, due to the value of the 'W' and 'H' which it is offset by adding up to 3 semitones? – rick Mar 16 '15 at 14:40
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    Umm... yeah I guess so. I was just going along with your concept of "two" meaning "two of the steps that are either whole or half" but you could also say offset by three semitones. – Todd Wilcox Mar 16 '15 at 18:31
  • I would just like to mention that you need to teach the relative keys better than just telling your students to count three semi tones down / up. How are they going to know E major's relative minor is not Db? – Neil Meyer Mar 17 '15 at 14:52
  • I'm not a teacher and I have no students, so problem solved. :-) Although now that I think about it... I think a student thinking the relative minor of E major is Db minor offers a great teaching opportunity: Have the student in question spell out Db minor and then ask them if there is an enharmonically equivalent spelling with a simpler key signature. This is assuming this is a music theory student and not a performance student. In some performance situations, if the student can see the scale on the instrument, it doesn't matter how it's spelled. – Todd Wilcox Mar 17 '15 at 14:56
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Yes, they are derived from each other. In particular, they are modes of each other. If you don't know about modes, here is a good starting point from the site. Basically, you're getting a different scale by using the same set of notes and treating a different note as the root. If we're in C major, our set of notes is the naturals (C D E F G A B) and your major scale starts on C while your minor scale starts on A. Then there are 5 more. They all have specific names apart from the major and minor names, but are generally discussed as being major or minor types of scales anyway. This is determined by their third. The only exception to this is Locrian, which is a diminished mode.

C Ionian     W-W-H-W-W-W-H  Major
D Dorian     W-H-W-W-W-H-W  Minor (♮6)
E Phrygian   H-W-W-W-H-W-W  Minor (♭2)
F Lydian     W-W-W-H-W-W-H  Major (♯4)
G Mixolydian W-W-H-W-W-H-W  Major (♭7)
A Aeolian    W-H-W-W-H-W-W  Minor
B Locrian    H-W-W-H-W-W-W  Diminished

Also, as a side note, you may noticed that the Dorian scale is symmetrical in its formula. There's a lot more there, but it's interesting by itself.

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You are right, in that the major scale uses notes from its relative minor scale - PROVIDED we're talking about the Aeolian mode or natural minor.

Be aware that a relative minor set of notes will vary with the other two minor scales. In the MELODIC minor, (classical), the 6th and 7th notes are raised by a semitone, usually on the way up, and it reverts to the natural minor notes descending. The HARMONIC minor only raises the 7th (leading) note. Thus, if you follow the WWHWWWH circular pattern in your question, it won't fit for any except the natural minor(relative)/Aeolian mode.

You asked the reason - it's because in the major, the notes gravitate towards the root, say, of C, whereas using the same notes, with a 'minor' feel, they gravitate towards A. This actually works better from our human point of view when there is a G# rather than the G natural found in the parent relative major.

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