How to make a diatonic chord replacement of a G7 chord with a different seventh chord that has B as root, in the key of C

During my reading of Complete Idiot's Guide to Musical Composition, I was doing an exercise in diatonic chord substitution in the key of C (major) (aka "the C major scale").


----------
FYI: diatonic, in this context, means that only notes from the scale are used. In the C major scale the notes are: C D E F G A B

An example of a diatonic C chord in the key of C major is the "C major chord": C E G.
An example of a chromatic C chord in the key of C major would be a Cm (minor) chord: C Eb G, because Eb (flat) is not in the C major scale.
----------


In the particular exercise I was working on, I was going to replace a G7 chord (notes G B D F) with a different, diatonic seventh chord in the same key.


----------
FYI: A seventh chord is a triad (3-note chord) plus a note forming an interval of a seventh above the chord's root (the note the chord is built upon, e.g. G in the G7 chord). An "interval of a seventh" means that there are 7 diatonic notes, counting from the root note to the seventh note, e.g.:
G A B C D E F

The "interval of a seventh" in the G7 chord--diatonic notes G to F--is a major seventh because it is an interval of 11 semitones (half steps):
G G# A A# B C C# D D# E F
----------


I chose to try to substitute the G triad portion of a G7 chord (notes G B D + F) with a B triad from the key of C major: the Bdim (diminished) chord (notes B D F). The next step was to find out which note I would need to place on top of the triad, so that the result would be a diatonic seventh chord.

Is the correct replacement a Bdim7? Bdim7 actually means: Bdim + dim7 and is composed of the following notes: B D F Ab.


----------
FYI: A diminished seventh chord is a diminished triad and an interval of a diminished seventh (9 semitones) over the root. It can also be seen as a a stack of 3 minor third intervals.

Using the words "diminished seventh" by themselves can lead to confusion for someone not familiar with common music theory contexts, since it might not be clear whether you mean the chord or the interval--which are 2 completely different things.

Chords can also be represented using integer notation. Values represent semitones and the number "0" is the root. So, if we look at the Bdim7 chord, highlighting the notes of the chord:
B C C# D D# E F F# G Ab

Bdim7 has the following integer notation: {0, 3, 6, 9}
----------


Bdim7 can't be the correct diatonic chord replacement for G7, since Ab isn't in the C major scale.

Here are all the possible seventh intervals (except for the harmonic seventh interval) with B as root:
B-Ab: diminished seventh interval
B-A: minor seventh interval
B-A#: major seventh interval
B-A##: augmented seventh interval


----------
FYI: The harmonic seventh interval can't be played on equal temperament instruments like the piano because its size is somewhere between the diminished seventh interval and the minor seventh interval, so in our case, somewhere between the Ab and the A keys.

A## (aka double sharp) is really the B note on the piano, but we use ## (also denoted as ) to show that it's the seventh staff position from-and-including the B staff position (the A note staff position) that's being augmented. If you were to write B, then it would be the 8th staff position, so to make it clear that we're talking about an augmented seventh interval, it's written with two sharps.
----------


The minor seventh interval, B-A, is the only diatonic seventh framework we can work with for a diatonic chord replacement of G7, using the root of the Bdim triad (B), so now we need to find the chord name for the following notes: B D F A

The half-diminished seventh chord is composed of a diminished triad and a minor seventh interval. It occurs naturally on the seventh staff position of any major scale. So we've found one possible chord name for our replacement chord: B half-diminished seventh chord

It's written like this: B^ø7

It's also known as a minor seventh flat five (m7b5), so it can also be written like this: Bm7b5

because a "B minor chord" (Bm):
B D F#

with a flatted fifth (b5):
F# -> F

is the same as a Bdim chord:
B D F