3-D matrix determinant

I tried the formula for 3-D matrix
determinant given by Gary Morrison,
and was puzzled by a result I got.

The exponents I used are to make
3 commas vanish for a "dominant 7th"
chord - the three types illustrated
on Mark Nowitzky's webpage. I illustrate
them on this lattice - notes not used in
this chord are marked by ( ).

( )---------( )--------15:8
/ \ /|\ /|\
/ \ / | \ / | \
/ \ / | \ / | \
/ \ / ( )--------21:16 \
/ \ /.-' \'-.\ /.-'/ '-.\
4:3---------( )------\--3:2--/------9:8
\ / \ \ /|\ / / \
\ / \ \ | / / \
\ / \ / \|/ \ / \
\ / \ / ( ) \ / \
\ / \ / \ / \
( )---------( )---------( )--------27:20

The commas in question are:
the 81/80 between 27/20 and 4/3
the 64/63 between 4/3 and 7/4
the 36/35 between 27/20 and 7/4

Disregarding powers of 2,
81/80 = 3^ 4 * 5^-1(* 7^ 0)
64/63 = 3^-2 *(5^ 0 *)7^-1
36/35 = 3^ 2 * 5^-1 * 7^-1

In matrix form, this is:

| 4 -1 0 |
|-2 0 -1 |
| 2 -1 -1 |

But when I used the formula:
(4*0*-1)+(-1*-1*2)+(0*-2*-1)
-(4*-1*-1)-(-1*-2*-1)-(0*0*2)

the result my spreadsheet gave me
was 0. What does this mean?

I tried it for some other commas,
and got 12, which at least was
something I could expect.

- Monzo
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Now that is a lattice I can use!!

Joseph L Monzo wrote:

> From: Joseph L Monzo <monz@juno.com>
>
> I tried the formula for 3-D matrix
> determinant given by Gary Morrison,
> and was puzzled by a result I got.
>
> The exponents I used are to make
> 3 commas vanish for a "dominant 7th"
> chord - the three types illustrated
> on Mark Nowitzky's webpage. I illustrate
> them on this lattice - notes not used in
> this chord are marked by ( ).
>
> ( )---------( )--------15:8
> / \ /|\ /|\
> / \ / | \ / | \
> / \ / | \ / | \
> / \ / ( )--------21:16 \
> / \ /.-' \'-.\ /.-'/ '-.\
> 4:3---------( )------\--3:2--/------9:8
> \ / \ \ /|\ / / \
> \ / \ \ | / / \
> \ / \ / \|/ \ / \
> \ / \ / ( ) \ / \
> \ / \ / \ / \
> ( )---------( )---------( )--------27:20
>
> The commas in question are:
> the 81/80 between 27/20 and 4/3
> the 64/63 between 4/3 and 7/4
> the 36/35 between 27/20 and 7/4
>
> Disregarding powers of 2,
> 81/80 = 3^ 4 * 5^-1(* 7^ 0)
> 64/63 = 3^-2 *(5^ 0 *)7^-1
> 36/35 = 3^ 2 * 5^-1 * 7^-1
>
> In matrix form, this is:
>
> | 4 -1 0 |
> |-2 0 -1 |
> | 2 -1 -1 |
>
> But when I used the formula:
> (4*0*-1)+(-1*-1*2)+(0*-2*-1)
> -(4*-1*-1)-(-1*-2*-1)-(0*0*2)
>
> the result my spreadsheet gave me
> was 0. What does this mean?
>
> I tried it for some other commas,
> and got 12, which at least was
> something I could expect.
>
> - Monzo
> ___________________________________________________________________
> You don't need to buy Internet access to use free Internet e-mail.
> Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
> or call Juno at (800) 654-JUNO [654-5866]
>
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