RE: ENSONIQ

> From: "Paul H. Erlich"
>
> >The VFX-SD is wavetable based. Gary's Ensoniq sampler can already do
> >anything the VFX-SD can do, including Transwaves if he's got an EPS-16
> Plus
> >or ASR-10.
>
> You can't change patches in real-time on the samplers.
>
Yes, that's true. In fact it's one of the major reasons I dumped my EPS-16
Plus for a Kurzweil K2500 instead of upgrading to an ASR-10. But this has
nothing to do with the VFX being a sound-from-scratch synthesizer (or not
being one, I should say).

John Loffink
jloffink@pdq.net

More on this later; I'm in a rush. But:

On Fri, 12 Jun 1998, Patrick Ozzard-Low wrote:
> Is there an historically accepted limit to the size of
> what will work as an effective 'performance comma' - (that is, from a
> traditional harmonic point of view)? Is that a question which is too
> subjective to worry about? I have always assumed it could not be more
> than about 21-24 cents.

In addition to the 81/80, in traditional 12TET the 128/125 of ca. 41
cents also vanishes. Moreover, their combination, the 648/625, of over
62 cents, even vanishes. This comma is exploited each time someone uses
a diminished 7th chord in 12TET.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

After a long absence from the list due to the pressing need to complete
a long-overdue thesis, I noticed the current discussion of
syntonic-comma equivalents in ETs. I thought the following might be of
use to anyone interested in the topic:

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".)

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] )

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.

Since I haven't seen any similar formulations elsewhere, I trust these
are originals; but if they've appeared on the list before, do tell me
who else obtained them.

Jonathan Walker

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/

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Jonathan Walker,

Your formulae for commas in equal temperaments are equivalent to Paul
Rapoport's in various published articles. The consistency issue which
Graham Breed is referring to involves the traditional definition of the
syntonic comma as the difference between an octave and a major sixth up,
and three perfect fifths up. Many equal temperaments have a better
approximation to the major sixth than the best approximation of a major
third plus the best approximation of a perfect fourth. Therefore two
distinct syntonic commas can be defined for such equal temperaments.
5-limit consistency means that the best approximations of the perfect
fifth/fourth, the major third, and the major sixth are all compatible.
Given 5-limit consistency, then, there can be only one syntonic comma.
ETs that are not 5-limit consistent include 11, 17, 20, and 21. 64tET is
interesting in that the triads defined as you or Paul Rapoport would
define them are not even second-best, they are third-best (i.e.,
altering either the major third or the perfect fifth yields a minor
third [or major sixth] that is so much better that the entire triad is
improved).

Paul Erlich

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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

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Sorry people, I don't know why my e-mail is suddenly doing that MIME
stuff!

Jonathan Walker wrote,

>Thanks, Paul, for your definition of the "consistency" which was at
>issue. A 5-limit consistent ET, unless I misunderstand you, would then
>be definable as any ET of which the proposition

>abs ( 4f(3/2) modn - f(5/4) ) = abs ( f(4/3) + f(5/4) - f(5/3) )

>is true, where for any real x, f(x) = [1/2 + log_2 x], and presumably
>where the value of the expressions on each side is greater than 0. Is
>this what you effectively mean?

I think that is probably correct, but a much simpler and clearer
proposition to determine 5-limit consistency is

f(3/1)+f(5/3)=f(5/1).

The general definition for L-limit consistency is

f(b/a)+f(c/b)=f(c/a)

for all 1<=aL need be considered).

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Paul H. Erlich wrote:

> ... The consistency issue which
> Graham Breed is referring to involves the traditional definition of
> the syntonic comma as the difference between an octave and a major
> sixth up, and three perfect fifths up. Many equal temperaments have a
> better approximation to the major sixth than the best approximation of
> a major third plus the best approximation of a perfect fourth.
> Therefore two distinct syntonic commas can be defined for such equal
> temperaments. 5-limit consistency means that the best approximations
> of the perfect fifth/fourth, the major third, and the major sixth are
> all compatible. Given 5-limit consistency, then, there can be only one
> syntonic comma. ETs that are not 5-limit consistent include 11, 17,
> 20, and 21.

Thanks, Paul, for your definition of the "consistency" which was at
issue. A 5-limit consistent ET, unless I misunderstand you, would then
be definable as any ET of which the proposition

abs ( 4f(3/2) modn - f(5/4) ) = abs ( f(4/3) + f(5/4) - f(5/3) )

is true, where for any real x, f(x) = [1/2 + log_2 x], and presumably
where the value of the expressions on each side is greater than 0. Is
this what you effectively mean?

Jonathan Walker

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