Re: [tuning] Digest Number 3545

There is something to what you felt implied in my post.
Most JI scales involve more than one generator , But we cannot say they have too.
There is something in the way we work with the material and organize it though. In that at a certain point there is a tendency to add generating intervals to fill in the gaps.
As an activity, JI implies ways of approaching one problems and developments.
Other systems imply others.
I think more is gained by keeping them developing along the lines they do and where they meet with others, so be it. But that doesn't mean we should call them the same thing.
If they truly were, we would not discussing the differences here.

>
>Message: 3 > Date: Tue, 07 Jun 2005 15:19:13 +0200
> From: klaus schmirler <KSchmir@online.de>
>Subject: Over and Under [was: Definitions of JI]
>
>
>lI would have made a strict division between generator chains and >interval division. They are often mixed (as in common practice 5 limit >with its chains of many fifths but only up to two thirds), but only an >interval constellation arrived at by division would be pure JI (divide >the octave: 3/2 and 4/3; divide each of these: 5/4 and 6/5 and 7/6 and >8/7; divide the 5/4: 9/8 and 10/9). I think that no interval is >repeated. Now compare it to a chain of fifths. There are only two >different intervals; the first division is identical, the second >division can be mapped to different interval sizes; but the 10/9 is >mapped to another 9/8, so the "imitation of justness" fails at this >point. For the 5/4 and 7/6, you'd acknowledge the attempt and call it >just as well, but add badly tuned or well enough or something of that >sort.
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>I thought this was uniqueness, but it isn't.
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--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> There is something to what you felt implied in my post.
> Most JI scales involve more than one generator , But we cannot say
they
> have too.

If they don't have more than one generator, then to get much of a
scale the best way would seem to be to for the generator should give
an approximate equal temperament. Otherwise you just end up with
something like all octaves, or stacks of minor thirds, etc, or maybe
something goofy like the division of a slightly sharp doubled octave
into nine parts by intervals of 7/6.

> There is something in the way we work with the material and
organize it
> though. In that at a certain point there is a tendency to add
generating
> intervals to fill in the gaps.

That doesn't seem like a very good method of gap-filling. Why wouldn't
the tendency be to fill the gaps via the generators already in use?

> I think more is gained by keeping them developing along the lines they
> do and where they meet with others, so be it. But that doesn't mean we
> should call them the same thing.
> If they truly were, we would not discussing the differences here.

Certainly a rank 3 subgroup of the reals under multiplication is in
some sense the same sort of thing as another such subgroup. However,
if one subgroup consists solely of rational numbers, that *is* the
difference. It isn't a difference in the structure of the group, but a
difference in that rational numbers are usually (but with exceptions,
like Lucy) considered normative. If you define a regular temperament
by a mapping *from* a group of rational numbers, then when the mapping
is trivial, it's clearly different than when it isn't. It seems to me
you could define this "structural" JI not in terms of the image of the
mapping, which is tuning, but purely by the mapping. If the mapping is
an isomorphism, then we have "JI"; if not, even if the numbers in
question are all rational numbers, we don't.

Let's say we start with 5-limit JI. We could have

2-->2, 3-->3, 5-->5; this would be "just intonation".

2-->2, 3-->3, 5-->24/5; this transforms major to minor. Asking if this
counts as "just intonation" isn't even a question for most people, but
from this point of view it has to be asked.

2-->2, 3-->32768/10935, 5-->2^15 3^(-7) 5^(-1); despite the fact that
the mapping is from the 5-limit and to the 5-limit, like the
major/minor transformation above, this clearly is not JI and in fact
is a disguised form of 12-et.

2-->2, 3-->5^(1/4), 5-->5; this is 1/4 comma meantone, not JI.

2-->2, 3-->2^(49/31), 5-->2^(72/31); in spite of the fact that this
maps to a rank one group, most likely it would be used more or less
like the 1/4 comma meantone mapping. Again, not JI.

2-->2, 3-->2^(1623/1024), 5-->2^(114127/49152); this is the 5-limit
interpreted in terms of pitch bends. Structurally, we pay no attention
to this mapping, and do not consider it to be in effect. Instead, we
consider the result of pitch bending to be JI.

Note that the above definition does not depend on what the actual
measured pitch is, but what the mapping intends for it to be. It gets
around the problem of defining "JI" purely in terms of the tuning, and
therefore has no problem rejecting many counterexamples as not being
JI, such as the pseudo 12-et above. However, it now asks the new and
slightly goofy question of whether playing a major key piece notated
in 5-limit JI but transposing to a minor key still counts as JI, since
you are playing the wrong third.