source file: m1453.txt
Date: Sun, 21 Jun 1998 8:41 +0100 (BST)

Subject: Re: Commas and Consistency

From: gbreed@cix.compulink.co.uk (Graham Breed)

Jonathan Walker wrote:

>If the best approximations of 3/2 and 5/4 are consistent in n-ET, for
>any integer n, then

>      4[1/2 + nlog_2 3/2] mod n == [1/2 + nlog_2 5/4]

>but if the two sides are unequal then a syntonic-comma equivalent is
>present in that n-ET. (The square brackets denote the "integer value"
>function, i.e. "[x]" is the greatest integer less than any given real
>number "x".)

The equation can be simplified by defining {x} to be the nearest integer
and omitting the underscores.  Also, clarified by adding parentheses:

            (4{nlog2(3/2)}) mod n = {nlog2(5/4)}

>If a syntonic-comma equivalent is present, then the size of the comma in
>n-ET steps is, obviously enough:

>      abs ( 4[1/2 + nlog_2 3/2] mod n - [1/2 + nlog_2 5/4] )

This is only true for consistent temperaments -- otherwise, there is no
single definition of the syntonic comma.  Also, I think the abs is
redundant.  It is similar to the equation for a syntonic comma in my
matrix form: (4 -1)h', where h' is the tempered octave invariant 5-limit
plane.  The process I outline on the "Equal Temperaments" page in my
website is essentially the same, although there's nothing like Jonathan's
equation there.

In octave specific terms, the syntonic comma can be defined like so:

             -4n + 4{nlog2(3)} - {nlog2(5)}

Which corresponds to (-4 4 -1)H' with H' defined:

                         (    n     )
                H' = 1/n ({nlog2(3)})
                         ({nlog2(5)})

>Anyone who has read this far will no doubt be able to adapt the above to
>cover compatiblity between powers of any number of ratios. For example:

>      -2[1/2 + nlog_2 3/2] mod n == [1/2 + nlog_2 7/4]

>for compatibility between the best approximations in n-ET to 16/9 and
>the septimal seventh.

My spreadsheet defines MOD(-2,12) as -2.  As long as -2 mod 12 would
normally be 10, the equation works.  I still prefer to define this
septimal comma as (6 -2 0 -1)H', and go through the usual procedure for
getting H'.  (-2 0 -1)h' also works, but I've never defined exactly what
h' is.  Really, multiplying by it should always give an answer between 0
and 1, but this takes us outside of straight matrix arithmetic.

The spell checker wants to call that a sceptical/skeptical comma.  What is
it trying to say, I wonder?

I do, incidentally, have a Mathcad worksheet that calculates comma sizes
from the number of steps to an octave.

                     Graham Breed
     gbreed@cix.co.uk            www.cix.co.uk/~gbreed/