Simple ratio

Hi.

I've read a lot here and there about these "simple ratios" but can
anybody explain me what is a "simple ratio"? I mean, intuitively I
understand that 3/2 is simpler than 5647/583 but it's possible to
quantify the simplicity of a ratio with an equation?

Maybe if the ratio is a/b the "simplicity coefficient" would be a*b?
It's stupid?

P.S. Always in my post don't pay attention to my english...

Hi!!!

>I've read a lot here and there about these "simple ratios" but can
>anybody explain me what is a "simple ratio"? I mean, intuitively I
>understand that 3/2 is simpler than 5647/583 but it's possible to
>quantify the simplicity of a ratio with an equation?

>Maybe if the ratio is a/b the "simplicity coefficient" would be a*b?
>It's stupid?

No, it's not stupid! I've recently made the amazing discovery that many very
different assumptions lead to exactly that conclusion -- the complexity goes
as a*b. Look over my posts from the last few months in the archives.

If there are no other notes and just the dyad is being played, then I would
say, if a*b>105, then what you're dealing with is most definitely _not_ a
simple ratio, even for the best ears.

An ordinary ear would consider any ratio with a*b<67 a simple ratio, if the
tones are, or have harmonics in, the 3000 Hz frequency range. In a more
typical range, a*b<~45 might be a better criterion.

Essentially, if a/b is not a simple ratio, then you don't perceive it as
essentially any more "in tune" than small mistunings of a/b.

-Paul

>Look over my posts from the last few months in the archives.

I've tried but I've found nothing. Is'nt a "search for author"
function?

>I would say, if a*b>105, then what you're dealing with is most
>definitely _not_ a simple ratio, even for the best ears.

> Essentially, if a/b is not a simple ratio, then you don't perceive
>it as essentially any more "in tune" than small mistunings of a/b.

15/8 gives 120 and his inversion 16/15 gives 240. Anyway I don't
think that these ratios are bad. When I hear this interval (a
semitone) in a chord very often I'm happy! Especially in major
seventh chord.

Anyway you have not written that complex ratios are bad but for this
reason I don't think is useful to distinguish between simple or
complex.
But maybe there's some reason that I don't know.

Lorenzo

>>Look over my posts from the last few months in the archives.

>I've tried but I've found nothing. Is'nt a "search for author"
>function?

The discussion was about "harmonic entropy". We've since set up a new
mailing list which I just sent you an invitation to join.

>>I would say, if a*b>105, then what you're dealing with is most
>>definitely _not_ a simple ratio, even for the best ears.

>> Essentially, if a/b is not a simple ratio, then you don't perceive
>>it as essentially any more "in tune" than small mistunings of a/b.

>15/8 gives 120 and his inversion 16/15 gives 240. Anyway I don't
>think that these ratios are bad. When I hear this interval (a
>semitone) in a chord very often I'm happy! Especially in major
>seventh chord.

You left out the part where I said

"If there are no other notes and just the dyad is being played"

The major seventh chord clearly does not satisfy that condition.

Besides, the other five intervals in the major seventh chord _are_ simple
ratios, and you only need to be able to tune three of them to tune the whole
chord.

>Anyway you have not written that complex ratios are bad but for this
>reason I don't think is useful to distinguish between simple or
>complex.
>But maybe there's some reason that I don't know.

There is a reason. For simple ratios, the discordance grows with the
complexity (n*d). For complex ratios, that is no longer true.

n*d means numerator times denominator. I should have stuck with a*b.

"Besides, the other five intervals in the major seventh chord _are_ simple
ratios, and you only need to be able to tune three of them to tune the whole
chord."

That's why it's Crunchy!

Annoyingly short and useless posts by:
Keenan P. :)

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13814

>
> No, it's not stupid! I've recently made the amazing discovery that
many very different assumptions lead to exactly that conclusion --
the
complexity goes as a*b. Look over my posts from the last few months
in
the archives.
>
> If there are no other notes and just the dyad is being played, then
I would say, if a*b>105, then what you're dealing with is most
definitely not_ a simple ratio, even for the best ears.
>

This is really interesting... thanks!

_________ ___ __ _ _
Joseph Pehrson

Paul,

Just trying to clarify what you are saying.

I'd have thought one would omit all factors of two in denominator and
denumerator. So 15/16 = 15/8 = 15/4 = 15/2 = 15, all same ease of
recognition or not. So if the a*b measure worked, 15/16 should have measure
15 rather than 240.

I'm not agreeing or disagreeing with your observation, just trying to figure
out what it is.

Robert

--- In tuning@egroups.com, "Robert Walker"
<robert_walker@r...> wrote:
> Paul,
>
> Just trying to clarify what you are saying.
>
> I'd have thought one would omit all factors of two in denominator
and
> denumerator. So 15/16 = 15/8 = 15/4 = 15/2 = 15, all same ease of
> recognition or not. So if the a*b measure worked, 15/16 should have
measure
> 15 rather than 240.
>
> I'm not agreeing or disagreeing with your observation, just trying
to figure
> out what it is.

Robert,

One is most definitely _not_ to omit factors of
2 in calculating a*b. The model in question
does not assume octave-equivalence. This
seems to agree well with experience -- for
example, 7:3 is more concordant than 7:6
which is more concordant than 12:7.

If one does assume octave-equivalence, in
order to evaluate chords independently of
inversion or octave-repeating tuning
systems, one gets the result that the _odd-
limit_ of a dyad measures its complexity. The
odd limit is the largest odd number in either
the numerator or denominator (after
removing all factors of 2). Then "simple
ratios" in the ideal case are those with odd
limit no greater than 11, and usually much
less. This measure of dissonance agrees
exactly with what Partch puts forward in his
"One-Footed Bride".

I came to both of the conclusions above
through the harmonic entropy model (and
lots of listening). If you'd like me to review
how I came to those conclusions, you should
probably go to the
harmonic_entropy@egroups.com list.

Just to show you why I think your idea
doesn't work, compare 5:3 and 15:8. If you
remove all powers of two and then multiply
numerator times denominator, you get 15
for both. But 5:3 is clearly much more
consonant than 15:8.