I've been trying to understand the theory behind dominant sevenths and have been reading a lot of articles. There's just one thing I couldn't understand-- the basic relationship of the minor 7th to the 5th.
In music theory, a dominant seventh chord, or major minor seventh chord,[a] is a seventh chord, usually built on the fifth degree of the major scale, and composed of a root, major third, perfect fifth, and minor seventh. Thus it is a major triad together with a minor seventh, denoted by the letter name of the chord root and a superscript "7". An example is the dominant seventh chord built on G, written as G7, having pitches G–B–D–F:
What do they mean by it being built on the fifth degree of the major scale? I played around on the keyboard trying to figure out a relationship of Bb to G and I couldn't make one out.
Take key of C and the C7. The dominant 7th is root-major 3rd-perfect 5th-minor 7th. That's easy, but what's the relationship of Bb to G? G major triad is just 5-7-9 of the C and Gmaj7 adds an 11. Moreover, since we're talking C, it's guaranteed to be all white keys, yet the minor 7th is black.
It seems like the minor 7th is just injected in there arbitrarily and doesn't seem connected to anything else in the key, which is alright, but I know there's supposed to be a relationship.
I maybe figured it out by playing around but I won't answer my own question because I'm not sure and don't want to spread errors, but here's what I got.
If you think in the key of C only you can't figure out where the minor 7th comes from and therefore what builds C7, but you can figure out what G7 will be and where that minor comes from. The knowledge of the key of C gives you knowledge of G7. It appears to be just 3 degree-stepovers (i.e. 1 to 3 or 2 to 4 is a stepover) of the notes in the key of C, starting at the 5th. I'm trying to be careful not to call them whole steps because that has a different meaning, and it indeed makes the dominant chord sound when played. Since the 5-7-9 of a given key is guaranteed to be a major chord in the 5th position (perfect fifth triad that many know from the I-ii-iii-IV-V-vii-dim model), that gives us the triad base we need. Now all we need is to bring in the 7th, specifically a minor 7th, which if instead of switching keys to the key of the fifth, we confine ourselves to the key that our fifth is the fifth of, the 7th is a natural minor (I thought this had to do with 'relative minor' but now I don't think so). BTW I apologize if I sound both naive and then over the top. I'm learning this all on my own by reading and thinking about it.
So to get the C7 dominant, we have to think backwards and find what C is the perfect 5th of, and that would be F. Now stay in the key of F and take 3 "steps-overs" or hops, from it's 5th ('-') => C-E-G-Bb. There's our Bb, linked to both a 5th and to a tonic. It's linked to the tonic F and C is the fifth of F, so there's a union there, the union of root, fifth and minor 7th note I was looking for. That's the best I've been able to figure it out. Maybe I'm making it overly complex but at least I found this connection.
I'm certain this has everything to do with the circle of fifths but I'm not good at learning or remembering things top down if I haven't build up to it from simpler primitives and working through logic, but basically to find the X7 chord, walk back p steps to key of Y where X is Y(5). Make an ordered set of all the note degrees n of Y (do re mi) and then play the odd degree n of Y 4 times (i.e. to the 7th; 4*2-1=7) starting at Y(5), or in symbols:
for z=0 to z=3: note n = y(tonic) as 0 + 5 + 2*z // returns 5-7-9-11 of F or 1-3-5-7min of C ==> we got our translation
So again, simplified, walking back until you find what C is the fifth of -> F. Then taking the 5-7-9-11 of F (staying in key), is the same as the 1-3-5-7 of perfect 5th (skewed scale of C) or the 1-3-5-min7 of pure C (breaking keys or crossing keys if you look at it from that way, as I did in the beginning)
BTW yes I'm just figuring this out in real time and yes the math notation makes it easier for me to understand. Maybe someone else shares my frame and it helps them. Of course if I'm wrong, correct me and stop the error propagation.