Cool. No problem. The answer to the first question is...
Since the tone Dave is the tonic, the chords ROOT name is Dave, and since Dave is the Seventh tone in a diatonic scale, it hence becomes a:
DAVE HALF DIMINSIHED chord.
Since only human names were applied to unknown tones, we have no way of knowing a relative known pitch, such as 440 hz applied to a given tone.
If we ARBITRARILY apply 440hz to the tone Dave, we now have a relative position in a Chromatic scale to start from. Since "I" applied certain intervals to a FEW of the given names, I can now mathamatically deduce their frequency and interval distance, that is, IF we apply their relative pitches. That is, IF, we use EQUAL TEMPERED intonation whereby the scale(chromatic)
s determined by the formula:
fn = f0 * (a)n
where
f0 = the frequency of one fixed note which must be defined. A common choice is setting the DAVE above middle C on a piano, (DAVE4) at f0 = 440 Hz.
n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away.
a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...
The wavelength of the sound for the notes is found from Wn = c/fn
where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature."
Now we have a way to establish the frequency of all 12 tones in ONE OCTAVE of an Equal Tempered Chromatic Scale. However, we only have applied a name to one tone...ie...DAVE. Now lets figure out the rest. So far, our GIVEN tones and intervals from 440hz are as follows:
Sally(m3rd), Suzan(b5), Ebeneezer(b7) and Dave(tonic)
For instance
Since all Chromatic pitches or tones are a HALF STEP(one fret), or an interval of a minor second, and our first pitch is 440hz, it is relatively easy to calculate the pitches of ALL 12 of the Chromatic tones, which is the same as applying a pitch to each fret on the fretboard. Since 440hz is a given, and 440hz is above middle C on a piano, and assuming this "C" is the fourth
octave, then this C=C4, then using Equal Tempered tuning, the rest of the pitches can be defined thus.
Examples using DAVE = 440 Hz:
I have already applied the name "SALLY" to an interval of a Min3(b3) above Dave.
Since a Min3 is 3 halfsteps(3 frets) above a given note, and our given is 440hz, then it follows that our Min3 is=C4(above the middle C on a piano, and "our" name for this tone is "SALLY", it
follows that Sally5 =C5, as well as middle C (C4/Sally4)on a piano.
Hence
Sally5 = the Sally an octave above middle Sally. This is 3 half steps above Dave4 and so the frequency is
f3 = 440 * (1.059463..)3 = 523.3 Hz
If your calculator does not have the ability to raise to powers, then use the fact that
(1.059463..)3 = (1.059463..)*(1.059463..)*(1.059463..)
That is, you multiply it by itself 3 times.
Middle Sally is 9 half steps below Dave4 and the frequency is:
f -9 = 440 * (1.05463..)-9 = 261.6 Hz
If you don't have powers on your calculator, remember that the negative sign on the power means you divide instead of multiply. For this example, you divide by (1.05463..) 9 times.
Using the same logic, our "Susan", being a b5, is 6 half steps(six frets) above DAVE4, and DAVE4=440hz
Now, let me apply "our" names to the calculated frequencies. However, since we only have 5 names, out of a possible 12 chromatic tones, then we must come up with 7 more names. I will arbitrarily apply some. Maybe you can figure out where they are on the fretboard. Here is a clue.
DAVE4 is on the skinnyest string at the 5th fret. . Also, for the sake of this chart, I will abbreviate the Names as follows.
DAVE=Dv (=A note)
SALLY=S (=C)
SUSAN=Sn1/Sn2 (=Eb/D#) Note that Sally uses S, and now we have another name beginning with "S", soooooooo...
since, standard notation requires the use of b's and #'s to denote
ENHARMONIC tones, the use of SUSAN requires TWO SUSANS!!! fuck! And since these enharmonic tones are based on the tone below and above, now we are REALLY confusing the shit out of nameing these tones...ie. If the tone Sn1 is actually a sharped note of the tone below it, now it becomes something else. Fuck. See below.
Ebeneezer=E (=G note)
For the sake coherence, I would like to pick the rest of the names using the standard "note" names as the beginning letter of the rest. Unfortunately, this becomes difficult as we have already used SIMILAR names. Sooooooooooooo...
George=G= fuck, G note is already defined by Ebeneezer and is named (E) which is what I was going to use for the note "E", which I was going to define by....Ethan, BUT NOW I CAN"T
FUCK! FUCK! FUCK!
Ok, nevermind, lets do this:
SALLY=C note
FRANK1/FRANK2=C#/Db note......SHIT! That means C#(SALLY#) is the same as Db(BILLb) which is the same as FRANK! fuck.......
BILL=D note
SUSAN=D#/Eb note......SHIT! That means D#(Bill#) is the same as Eb(CAROLEb) which is the same as SUSAN! fuck.......
CAROLE=E note
MIKE=F note
LISA=F#/Gb ..SHIT! That means F#(MIKE#) is the same as Gb(EBENEEZERb) which is the same as LISA! fuck.......
EBENEEZER=G note
LINDA=G#/Ab.....SHIT! That means G#(EBENEEZER#) is the same as Ab(DAVEb) which is the same as....good grief....LINDA!!....fuck me
DAVE=A
RONDA=A#/Bb......crap, here we go again. This means A#(DAVE#) is the same as Bb(HOLLYb) which is the.same as......fuck it.
HOLLY=B
Now lets build SEVEN OCTAVES of these 12 notes.
Note
Frequency (Hz)
SALLY0/C0
16.35
FRANK0/C#0/Db0
17.32
BILL0/D0
18.35
SUSAN0/D#0/Eb0
19.45
CAROLE0/E0
20.60
MIKE0/F0
21.83
LISA0/F#0/Gb0
23.12
EBENEEZER0/G0
24.50
LINDA0/G#0/Ab0
25.96
DAVE0/A0
27.50
RONDA0/A#0/Bb0
29.14
HOLLY0/B0
30.87
SALLY1/C1
32.70
FRANK0/C#1/Db1
34.65
BILL1/D1
36.71
SUSAN1/D#1/Eb1
38.89
CAROLE1/E1
41.20
MIKE1/F1
43.65
LISA1/F#1/Gb1
46.25
EBENEEZER0/G1
49.00
LINDA1/G#1/Ab1
51.91
DAVE1/A1
55.00
RONDA1/A#1/Bb1
58.27
HOLLY1/B1
61.74
SALLY2/C2
65.41
FRANK2/C#2/Db2F
69.30
BILL2/D2
73.42
SUSAN2/D#2/Eb2
77.78
CAROLE2/E2
82.41
MIKE2/F2
87.31
LISA2/F#2/Gb2
92.50
EBENEEZER2/G2
98.00
LINDA2/G#2/Ab2
103.83
RONDA2/A#2/Bb2
116.54
HOLLY1/B2
123.47
SALLY3/C3
130.81
FRANK3/C#3/Db3
138.59
BILL3/D3
146.83
SUSAN3/D#3/Eb3
155.56
CAROLE3/E3
164.81
MIKE3/F3
174.61
LISA3/F#3/Gb3
185.00
EBENEEZER3/G3
196.00
LINDA3/G#3/Ab3
207.65
DAVE3/A3
220.00
RONDA3/A#3/Bb3
233.08
HOLLY3/B3
246.94
SALLY4/C4
261.63
FRANK4/C#4/Db4
277.18
BILL4/D4
293.66
SUSAN4/D#4/Eb4
311.13
CAROLE4/E4
329.63
MIKE4/F4
349.23
LISA4/F#4/Gb4
369.99
EBENEEZER4/G4
392.00
LINDA4/G#4/Ab4
415.30
DAVE4/A4
440.00
RONDA4/A#/Bb4
466.16
HOLLY4/B4
493.88
SALLY5/C5
523.25
FRANK5/C#5/Db5
554.37
BILL5/D5
587.33
SUSAN5/D#5/Eb5
622.25
CAROLE5/E5
659.26
MIKE5/F5
698.46
LISA5/F#5/Gb5
739.99
EBENEEZER5/G5
783.99
LINDA5/G#5/Ab5
830.61
DAVE/A5
880.00
RONDA5/A#5/Bb5
932.33
HOLLY5/B5
987.77
SALLY6/C6
1046.50
FRANK6/C#6/Db6
1108.73
BILL6/D6
1174.66
SUSAN6/D#6/Eb6
1244.51
CAROLE6/E6
1318.51
MIKE6/F6
1396.91
LISA6/F#6/Gb6
1479.98
EBENEEZER6/G6
1567.98
LINDA6/G#6/Ab6
1661.22
DAVE6/A6
1760.00
RONDA6/A#6/Bb6
1864.66
HOLLY5/B6
1975.53
SALLY7/C7
2093.00
FRANK7/C#7/Db7
2217.46
BILL7/D7
2349.32
SUSAN7/D#7/Eb7
2489.02
CAROLE7/E7
2637.02
MIKE7/F7
2793.83
LISA7/F#7/Gb7
2959.96
EBENEEZER7/G7
3135.96
LINDA7/G#7/Ab7
3322.44
DAVE7/A7
3520.00
RONDA7/A#/Bb7
3729.31
HOLLY7/B7
3951.07
SALLY8/C8
4186.01
FRANK8/C#8/Db8
4434.92
BILL8/D8
4698.64
SUSAN8/D#8/Eb8
4978.03
Now we have a 7 octaves of chromatic scales.
So, to answer the second question:
"but the real question is when played as a second inversion, what key is it in and what scale tone 7th chord does it become?"
IS:
Given the formula for a HALF DIMISHED chord is: 1 b3 b5 b7 based on its OWN Diatonic scale. Since the (1) is "A"(DAVE), build a Diatonic scale in the key of DAVE.
Which means, based on the formula for a Diatonic scale in intervals

W= Whole tone/1/2=Half Tone)
W W 1/2 W W W 1/2
In the Key of A gives the following notes:
1 2 3 4 5 6 7
A B C# D E F# G#
OR
DAVE, HOLLY, FRANK, BILL, CAROLE, LISA, and LINDA
Now, flat the 3rd, 5th and 7th, and you get the following tones, which make up a HALF DIMINISHED or Min7b5 chord.
DAVE, SALLY, SUSAN and EBENEEZER.
Now, given that scale tone Chords are built from the formulas as follows:
1=MAJ7 1 3 5 7 Intervals(MAJ3rd,Min3rd, MAJ3rd)
2=MAJ7 1 3 5 7 Intervals(MAJ3rd,Min3rd, MAJ3rd)
3=Min7 1 b3 5 b7 Intervals(Min3rd, MAJ3rd, Min3rd)
4=MAJ7 1 3 5 7 Intervals(MAJ3rd,Min3rd, MAJ3rd)
5=Dom7 1 3 5 b7 Intervals(MAJ3rd,Min3rd, Min3rd)
6=Min7 1 b3 5 b7 Intervals(Min3rd, MAJ3rd, Min3rd)
7=HalfDim. 1 b3 b5 b7 Intervals(Min3rd, Min3rd,MAJ3rd)
then it follows that a HALF DIMISHED is built on the 7th tone of a Diatonic Scale. BTW, these are called CHORD QUALITIES.
Therefore, what Key does an "A"(DAVE) Half Dimished Chord belong to? Working backwards.
1/2 W W W 1/2 W W
7. A(DAVE)
6. G#/Ab(LINDA)
5. F#/Gb(LISA)
4. D#/Eb(SUSAN)
3. D(BILL)
2. C(SALLY)
1. A#/Bb(RONDA)
VOILA! An "A"or DAVE Half Diminshed chord is the 7th scale tone chord in the key of A#/Bb, or the KEY of RONDA!
Now, invert the chord
b3 b5 b7 1 which gives you the same tones in another scale tone 7th chord in ANOTHER KEY. But what chord and what Key?
Since the interval distance from a 1 to a b3 is a min 3rd, and the distance from a b3 to a b5 is a MAJ 3rd, and the distance from a b5 to a b7 is a MAJ 3rd, it would seem that when inverted, the intervals might match one of the other scale tone chords...maybe?
Apply a little CHORD ANALYSIS. Use the same tones against another Scale tone 7th Chord QUALITY.
Since we can eliminate a Half Diminished since its already been defined,
lets try a Major. IF, the same tones, A, C, Eb, G(ie-DAVE, SALLY, SUSAN and EBENEEZER.) were applied to a MAJ chord, they would be
1?
3=A(DAVE)
5=C
7=Eb oooops....the interval distance from a 5th to a 7th is a maj third. From C to Eb is a minor third....it doesn't work.. besides, where does the G note fit in?
Next, try a Min. chord
1?
b3=A
5=C ....hmmm, oooops....the interval distance from a b3rd to a 5th is a maj third. From A to C is a minor third....it doesn't work..besides, where does the G note fit in?
b7=Eb
Well, lets try a dominant
1?
3=A
5=C
b7=Eb
Since a Dominant chord has the following intervals...MAJ3rd, Min3rd, MAJ3rd, and from A to C is a Min3rd, and from C to Eb is a MAJ3rd, we are half way there. However, where does that FUCKING "G" note fit? Well, lets see. If 1 to 3 is a MAJ3rd, then from what note to "A" is a MAJ3rd? LOOK at the diatonic scale formula.
W W 1/2 W W W 1/2
Well, lets see. If "A"(DAVE) is the 3rd tone, and its two WHOLE STEPS from 1, working backwards, the second note in the scale would be a WHOLE STEP backwards, or ...VOILA! the "G" tone, or EBENEEZER! Now, working backwards again, one WHOLE STEP...VOILA! The 1 becomes...yup, you got it....."F". But where does the "G" fit in????
Well, count up from 1 through TWO Octaves and this is what you have
1 2 3 4 5 6 7 1 9 10 11 12 13 1 or
F G A A#/Bb C D E F G A A#/Bb C D....E and F
If you count up to the 9th note, what do you have? Yup...."G"
Now lets look at that Dominant formula again.
1
3
5
b7
Now add the 9th and you have a Dominant 9th chord
1=F(Mike)
3=A(Dave)
5=C(Sally)
b7=Eb(SUSAN)
9=G(EBENEEZER)
And there you have it. An "A" Half Diminished Chord is the same as a second inversion of an F dom9th Chord....and since Dominant Chords are the Fifth Scale tone Chord in a Diatonic Scale, and F is the 5th of Bb, it is in the Key of....VOILA!! A#/Bb OR the Key of RONDA.
Have a nice day.

fitZ