JI errors in 72ET, for Paul

Paul,

I've been reading with interest the tables and whatnot of the
differences between the various subsets of 72 and equivalent JI
intervals, and I have a basic (probably elemenatary) question:

I can see, easily, where the differences in an interval being only a
max of 3 cents off wouldn't be objectionable (or even noticeable?),
but would this be compounded as you added intervals to a harmony, or
would you just pick the next nearest tone in the 72ET galaxy.

In other words, if you had a seventh or ninth chord (traditional
terminology) and all of the intervals are off by as much as 3 cents,
does this make the chord less stable, make it wobble, or in general
be a lot less pleasant than a strict JI chord?

Does the question even make sense?

Cheers,
Jon

--- In tuning@y..., JSZANTO@A... wrote:
> Paul,
>
> I've been reading with interest the tables and whatnot of the
> differences between the various subsets of 72 and equivalent JI
> intervals, and I have a basic (probably elemenatary) question:
>
> I can see, easily, where the differences in an interval being only a
> max of 3 cents off wouldn't be objectionable (or even noticeable?),
> but would this be compounded as you added intervals to a harmony,

Not if the harmony is constructed along Partchian lines. For example, a 5-limit triad has three
5-limit consonances. A 7-limit tetrad has six 7-limit consonances. A 9-limit pentad has ten 9-limit
consonances. An 11-limit hexad has fifteen 11-limit consonances. Nothing gets compounded --
every single interval in every one of these chords is within 4 cents of JI in 72-tET.

If the harmony is constructed differently, then yes, errors can be compounded in the _dissonant_
intervals. For example, if 5:4 is 2.5 cents flat, then 25:16 will be 5 cents flat. But that doesn't
matter, since 25:16 is dissonant anyway, and can't be tuned directly by ear. So whether it's just
or 5 cents flat or 5 cents sharp, 25:16 will have basically the same effect. In a chord composed
of stacking two 5:4 intervals, it's the 5:4 intervals that give the chord its consonance, not the outer
25:16.

> or
> would you just pick the next nearest tone in the 72ET galaxy.

Nope -- then one of the consonances may have to suffer a greatly

>
> In other words, if you had a seventh or ninth chord (traditional
> terminology)

Traditional terminology is completely ambiguous . . . do you mean a simple otonal 7- or 9-limit
chord?

> and all of the intervals are off by as much as 3 cents,
> does this make the chord less stable, make it wobble, or in general
> be a lot less pleasant than a strict JI chord?

This is independent of what I wrote above, so treat it separately:

In my experience, much larger errors can be tolerated in a complete 7- or 9-limit otonal chord
than in an isolated 7- or 9-limit interval. If, say, an isolated 9:7 is, say, 10 cents out-of-tune, there
is considerable uncertainty as to what is actually being heard, as compared with a just 9:7, in a
favorable register and timbre. In this context, a 10-cent out-of-tune 9:7 is not a 9:7 in any way . .
. it's simply a dissonant interval. However, if the chord is 4:5:6:7:9, the brain will quickly
understand that the 7:9 interval represents the 7th and 9th harmonics of an implied fundamental
-- even if the 9:7 is 10 cents out-of-tune.

Paul,

Thanks for the reply!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Nothing gets compounded -- every single interval in every one of
> these chords is within 4 cents of JI in 72-tET.

Yes, I figured that, I just wondered if multiple discrepancies of a
few cents had more, negligable, or less effect -- on the listener --
than a simple interval.

> If the harmony is constructed differently, then yes, errors can be
> compounded in the _dissonant_ intervals.

And I realize that I was mostly concerned with the consonant ones.
For some reason, in the kind of music I would imagine wanting to
make, I'd like the consonances be just that, and then have a lot of
different kinds (or degrees) of dissonance.

> Traditional terminology is completely ambiguous . . . do you mean a
> simple otonal 7- or 9-limit chord?

Yes, sorry for not being more clear.

> In my experience, much larger errors can be tolerated in a complete
> 7- or 9-limit otonal chord than in an isolated 7- or 9-limit
> interval.

That's interesting. Once I get all the pieces of my setup happening,
that will be one of the first things I play around with. See if it
is "in my experience" as well.

Thanks in arears,
Jon

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, June 09, 2001 10:39 AM
> Subject: [tuning] Re: JI errors in 72ET, for Paul
>

>
> In my experience, much larger errors can be tolerated in a
> complete 7- or 9-limit otonal chord than in an isolated
> 7- or 9-limit interval. If, say, an isolated 9:7 is, say,
> 10 cents out-of-tune, there is considerable uncertainty
> as to what is actually being heard, as compared with a
> just 9:7, in a favorable register and timbre. In this
> context, a 10-cent out-of-tune 9:7 is not a 9:7 in any way . . .
> it's simply a dissonant interval. However, if the chord
> is 4:5:6:7:9, the brain will quickly understand that the
> 7:9 interval represents the 7th and 9th harmonics of an
> implied fundamental -- even if the 9:7 is 10 cents out-of-tune.

This is my experience too.

One might say that the more members there are in the gang,
the greater the gang's "fundamentals" come into focus.
Just a bit of popularization for those having trouble understanding...

-monz
http://www.monz.org
"All roads lead to n^0"

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