source file: m1454.txt
Date: Mon, 22 Jun 1998 03:25:42 +0000

Subject: Re: Commas and Consistency

From: kollos@cavehill.dnet.co.uk (Jonathan Walker)

Graham Breed wrote:
> 
> 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.

Nothing of substance, but it's best to use notation according to
majority conventions: "{x}" is generally taken to mean "the fractional
value of 'x'", i.e. x - [x]. As for the function [1/2 + x], yes it is
the same as the rounding function, but I prefer not to multiply notation
or functions beyond necessity, especially when writing in ASCII, where
confusion can easily arise. I think you'll also find that the underscore
notation for log bases is normal in ASCII.

>  Also, clarified by adding parentheses:

I used to employ parentheses only (as I would in Mathematica) until
someone complained that expressions could be read more quickly in
messages if spaces were used judiciously.

I wrote:

> >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] )

Graham wrote:

> This is only true for consistent temperaments -- otherwise, there is no
> single definition of the syntonic comma.

I don't know what you mean here. The consistency I was talking about was
that between the best approximations for 5/4 and 81/64 in an equal
temperament, i.e. where there is no syntonic comma equivalent. Where
such consistency obtains, the above expression will, of course, be equal
to 0. From your last comment, it would seem that we are talking at cross
purposes. I also prefer to leave the historic "81/80" definition of the
syntonic comma undisturbed; this is why I have been using the phrase
"syntonic comma equivalent" in the context of equal temperaments. Unlike
the terms "major third" or "perfect fifth", which have functional
definitions independent of particular tuning systems, the term "syntonic
comma" was defined only in tuning terms from the start; but we've had
this discussion on the list before -- I don't want to provoke a rerun. 

>  Also, I think the abs is redundant.

Not in the context of my sentence: I gave it as the expression for the
*size* of the syntonic comma equivalent.

> My spreadsheet defines MOD(-2,12) as -2.  As long as -2 mod 12 would
> normally be 10, the equation works.

Sorry, but your spreadsheet sucks.

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

while your spell checker evidently has hidden depths.

I must take another look at your website.

Best,

Jonathan Walker