Thank you, that was a detailed answer that gave me a lot of new things and I have a new question and some old questions-
You are saying there are 8 notes in an octave, but we only selected 7 notes after selecting 7th semi tone 7 times. I think you said the 8 note is 2x the original. Is that correct understanding?
So I got the point about 2^(1/12), what I didn't get is the staring point for A major and the jump structure. I thought all jumps are selecting 7th semi tone. So what is meant by "jump down one 7-semitone step and up 5"?
You partially answered my second question I think- saying that for chords it's okay to go from n <-> 2n scale to 2n <-> 4n scale. My question is for non chord music is this a common occurrence?
The music tradition sees the notes rather as intervals between two notes.
C to the same C is "unison" (You may need to instruments to play two identical C notes at the same time.)
C to C#/Db (this note has two names) is "small second"
C to D is "large second". And so on.
C to the next higher C is "perfect octave".
So if we take the C major scale, it has 7 different notes. But if we also include the next higher up C, you can pair the base C with 8 different notes, when we include pairing with itself, and pairing with the higher C. When you have two instruments playing, these are the pairings you can make when you play two notes at the same time.
> C to the same C is "unison" (You may need to instruments to play two identical C notes at the same time.)
> C to C#/Db (this note has two names) is "small second"
While C# and Db are the same note (in equal temperament [1]), the intervals C/C# and C/Db have different names: C# is called 'augmented unison' [2]. For the name, you start from the basic interval (e.g. C/C) and apply the accidentals (# or b) [3].
This is an example of why the traditional approach to music theory can be cryptic for a beginner. After the Western music moved to well and equally tempered scales (starting from the early 1700's), the context in which there is a difference between C# and Db has disappeared. But we still use terminology and notation from 500-800 years ago.
I actually didn't realize that the notes of the major scale and it's modes were contiguous on the circle of fifths until seeing guitardashboard.com last night. That's what the down one, up five is saying.
A fifth is seven semi-tones above the root. It has a 3:2 frequency ratio to the initial note. It so happens that if you jump up by seven semi-tones five times, you've got most of the major scale, albeit strewn over 3 doublings of the initial frequency. Usually, the major scale is thought of as occurring within a single doubling so in the key of C you'd have to divide the frequencies D and A by 2; E and B by 4 to get back into the original n<->2n range. Any time you double or halve the frequency you get a note with the same mood/purpose/name but one octave higher or lower. The frequency ratios end up something like this:
C = 1
G = 3/2
D = (3/2)(3/2) (/2 to get back to the initial range)
A = (3/2)(3/2)(3/2) /2
E = (3/2)(3/2)(3/2)(3/2) /2/2
B = (3/2)^5 /2/2
There's one note missing, though--the one with a 4:3 frequency ratio: F. To get that one, we invert the 3/2 relationship and take the frequency that's 2/3 of our initial frequency. That's in the halved range, n/2<->n, though so we've gotta double to get back into our starting range. This is the one time we go down seven semi-tones instead of the five times that we go up.
I'm not sure how useful it is to think in these terms but it does show that you can derive the major scale by using simple ratios which, psycho-acoustically speaking, are generally considered more pleasant than complex ratios when played at the same time. The relationship between B and C, (3/2)^5 == 243/32 is already pretty tense. E.g., you could alternate between two notes with that ratio to make it sound like Jaws is lurking somewhere nearby:
You are saying there are 8 notes in an octave, but we only selected 7 notes after selecting 7th semi tone 7 times. I think you said the 8 note is 2x the original. Is that correct understanding?
So I got the point about 2^(1/12), what I didn't get is the staring point for A major and the jump structure. I thought all jumps are selecting 7th semi tone. So what is meant by "jump down one 7-semitone step and up 5"?
You partially answered my second question I think- saying that for chords it's okay to go from n <-> 2n scale to 2n <-> 4n scale. My question is for non chord music is this a common occurrence?
Thank you again!