In classical music harmony analysis, we see the chord inversions notated like I6/4, I6, V4/3.
What are those numbers? How do they define the inversion? What's the theory behind using those specific numbers?
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Just to expand on Pat's answer, there is a figured bass symbols for all type of inversion including root position.
The picture above shows the complete figured bass symbol and how it will be denoted in analysis. As you can see root position triads and 7th chords have their own complete figured bass symbols, but reduce drastically because how common they are.
The numbers are determined by the interval between the bass note(the lowest note) and the other chord members. So if there was a B in the bass and the chord reduced to a G major in the key of C, the Roman Numeral would be V figured bass would be 6/3 which would reduce to 6 . Only the exact voicing of the bass matters.
The reason why the distinctions are made in analysis is because the bass defines the harmonic progression and different inversions will function differently. For example, a second inversion triad is rather dissonant and has to be approached and resolved correctly in order to sound correct. The figured bass along with the Roman Numeral analysis tells almost everything you need to know about the harmonic progression of a piece.
They are figured bass numbers, as used under bass lines in (primarily) the Baroque era to indicate harmonic content to be improvised by continuo players such as the harpsichordist. The numbers refer to the diatonic intervals above the bass. Look at your 6/4 example above. The two distinct notes above the bass note are a sixth and a 4th above, hence 6/4. This is true for all of them, although some intervals are left out. For instance, 6/5 (first inversion seventh chord) is technically 6/5/3, but we generally omit the 3rd. 6 (first inversion triad) is technically 6/3, etc.
The roman numerals indicate which chord we are using in the scale. If the Roman numerals are written with capital letters then they indicate a Major Chord. If they are written in lower case letters then they indicate a minor chord. A Major chord has a major third and a perfect fifth. A minor chord has a minor third and a Perfect fifth.
I(i) - Tonic
II(ii) - Super Tonic
III(iii) - Mediant
IV(iv) - Sub Dominant
V(v) - Dominant
VI(vi) - Sub Mediant
VII(vii) - Leading Tone
You also sometimes get + and °. The plus sign means we have a augmented chord which means a major third and an augmented fifth. The little degree sign means we have diminished chord whihc implies a minor third and an diminished fifth.
See those two as an accordian going in and out with the augmented chord we have both the third and the fifth going away from the root just like an expanding accordian. With the diminished we have bot the third and the fifth coming closer to the root.
Then lastly we have the inversions. When we talk about inversions we want to know what notes in the chord is at the bottom. The other notes can be arranged differently for specific reasons but inversion tells us what note is on the bottom.
If you have your basic triad you have three notes. Lets say we are in C Major's tonic chord. The notes in our chord would be C-E-G.
If it is in root position the C is at the bottom. If we count from C. We have C(1)d(2)E(3)f(4)G(5) hence the notation of 5/3 for the triad in root position.
If we now go to first inversion we now have the E at the bottom. So we have E(1)f(2)G(3)a(4)b(5)C(6) hence the notate the triad in first inversion as 6/3
The triad in second inversion now has the G at the bottom. So we have the notes G(1)a(2)b(3)C(4)d(5)E(6) hence the notation 6/4
There is also notes with there sevent add. Usually done on the Dominant or Super Tonic chords. So our Triads just get another noted added to them.
So for the seveth chord on the dominant step of C Major we have the notes G-B-D-F
If we are in root position the G is at the bottom So we have the notes G(1)a(2)B(3)c(4)D(5)e(6)F(7) hence the notation of 7/5/3 or more commonly G7
If we are in first inversion of the dominant seventh chord of C Major we now have the notes B(1)c(2)D(3)e(4)F(5)G(6) hence the notations 6/5/3 or more commonly just 6/5
If we are in second inversion of the dominant seventh chord of C Major we have D(1)e(2)F(3)G(4)a(5)B(6) hence the notation 6/4/3 but because our triad in second inversion is called 6/4 we use the notation 4/3 for the dominant seventh chord in second inversion.
Lastly the seventh chord in third inversion here we would have the notes F(1)G(2)a(3)B(4)c(5)D(6) hence the notation of 6/4/2 which we generally call 4/2.