12/7 7/4 BEATS

Bob,
There shouldn't be any beats in a just 12/7 or 7/4. The 7/6 interval
may be too narrow for some ears...sounds rough to some. I find a lot of
energy in a
6-7-8 and (utonic) 1/8, 1/7, 1/6 triad.

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> Can someone explain to me the neutral third, how it functions, how it has been
> used historically

I confess that I'm not very well-versed on historical usage of neutral thirds,
but it's probably fair to say that blues has the most common use of neutral
thirds. I have some personal experience with them though.

Neutral thirds are usually (as far as I'm aware) taken to be 11:9 ratio
intervals. 11:9 turns out to be remarkably close to the square root of 3:2 -
remarkably for such a simple ratio that is.

I wrote a short composition in 88CET tuning intended to explore neutral
thirds. Four steps in 88CET (the nonoctave equal temperament whose primitive
step size is 88 cents) is pretty close to 11:9. I don't have copies of that
composition on a web page or anything like that (yet anyway!), but I could
perhaps send you a cassette of it, or something to that effect.

But based upon my own experiences with it, here are some relevant comments:
1. Without a doubt, it's most valuable usage is for its own character, but
like it or not, the most likely reaction to it is instead as an ambiguous
third: We're so used to classifying thirds as major or minor that most
people, upon hearing a neutral third, will try to call it one or the other.
2. That in itself is a powerful musical tool: In that context the effect can
can really tantalyze your audience, because they can hear the same
interval ten times, and each time they hear it, their impression changes:
"it's major... no, it's minor... no, wait, it IS major..."
3. If that is your musical goal, then a third slightly flatter than 11:9 is
probably a better bet than 11:9 itself. Although 11:9 is indeed quite
close to the middle position between the two, our ears seem to
gravitate more toward major than minor. So to make it as difficult
to decide as possible, you have to bias it toward minor a little bit.
4. Neutral thirds, all in all to my ears at least, create an impression of
ambiguity and indecisiveness. Chords composed entirely of neutral
thirds have a fascinating "lost in space" unclarity to them.

Neutral thirds are often credited to Zalzal, a midieval 'oud player named
Zalzal that introudced the lute (l'oud) to Southern Europe). Now, the
'oud (which means "wood" in Arabic) goes through phases every 200 years or
so between fretted and unfretted fingerboards.

Zalzal had frets and he placed a fret between major and minor, as well as
between their corresponding sixths (major and minor). Modern Egyptian
music still uses these intervals, as well as others implied by them.

I first noticed the neutral third in the General Motors Word's Fair car
horn: D F+ A F+

One special characteristic of the neutral third is that no matter what
kind of 5th there is in a system, the neutral third is always in the same
place logarithmically.

My experience with it really began with a piece I composed called "Neo"
which has a be-bop-like quartertone romp set off by the previously
mentioned car horn motive,

Perhaps familiarity with the interval eases it into its own niche wherein
it is neither major nor minor. It simply is.

Johnny Reinhard
Director
American Festival of Microtonal Music
318 East 70th Street, Suite 5FW
New York, New York 10021 USA
(212)517-3550/fax (212) 517-5495
reinhard@idt.net
http://www.echonyc.com/~jhhl/AFMM

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One special characteristic of the neutral third is that no matter what
kind of 5th there is in a system, the neutral third is always in the same
place logarithmically.

Can you explain what it means to be in the same place logarithmically? Also,
I noticed in Helmhotz that there are several intervals that may pass for a
neutral third including - the 39/32 at 342 cents, the 72/59 interval at 345
cents, and Zalzal's middle finger for 303 and 408 cents which is 27/22 at 355
cents. I assume 11/9 is around 347 cents. Can we make any particular
distinctions about these different ratios or is it basically a matter of
context?

> One special characteristic of the neutral third is that no matter what
> kind of 5th there is in a system, the neutral third is always in the same
> place logarithmically.
>
> Can you explain what it means to be in the same place logarithmically?

Simply, that while there are widely divergent types of major and minor
thirds in different systems, they move proportionately in opposite
directions per specific system. Their midpoint is the same nonetheless.

The temperament of the perfect 3/2 fifth is the least of any interval to
be tempered, except perhaps the identity intervals (of unison and octave).
Its logarithmic middle is realtively stable and therefore recognizeable in
many different tunings.

Johnny Reinhard
AFMM

p.s. I'm off for Amsterdam now with 4 other musicians, and then to
Wolfenbuttal, Germany with 8 musicians (and where we'll do Carrillo's
Cristobal a Colon). Of course, I'm excited to hear my new quartet
"Trespass," and to meet all my Dutch friends.

Thanks to Johnny Reinhard for the fascinating historical and etymological info
on the lute and the neutral third!

Johnny, could you please clarify this intriguing statement:

>One special characteristic of the neutral third is that no matter what
>kind of 5th there is in a system, the neutral third is always in the same
>place logarithmically.

Do you mean that neutral thirds have the same relationship to the tonic
regardless of how the 5th is tuned, or that the pitch of the 3rd rises and
falls in logarithmic proportion to the height of the 5th, or...? Is this a
function of Arabic theory, something that you have observed universally,
something stated by someone else in a referencable document, ...?

David J. Finnamore

On Mon, 28 Sep 1998 DFinnamore@aol.com wrote:
> Johnny, could you please clarify this intriguing statement:
>
> >One special characteristic of the neutral third is that no matter what
> >kind of 5th there is in a system, the neutral third is always in the same
> >place logarithmically.
>
> Do you mean that neutral thirds have the same relationship to the tonic
> regardless of how the 5th is tuned, or that the pitch of the 3rd rises and
> falls in logarithmic proportion to the height of the 5th, or...?

While the thirds accordian proportionately to the fifth (however large is
one third, the other is condensed), the neutral third is always dead
center, dividing the fifth into 2 identical parts. Just as the tritone
bisects the octave, the neutral third bisects the perfect fifth.

The neutral thirds of all the meantones, 12-TET, 31-TET, quartertones,
Egyptian music, and others are roughly all in the same place because the
fifth is so etched in its shape. The difference of a few cents in either
direction from 351 cents (at perfect fifth of 702 cents) is negligible for
musical purposes.

> Is this a
> function of Arabic theory, something that you have observed universally,
> something stated by someone else in a referencable document, ...?
>
> David J. Finnamore

Consider it anecdotal in that I've come to recognize the place of the
neutral third as a distinct relationship in the order of musical sounds,
common to more than its share of tunings.

Johnny Reinhard
AFMM





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Thanks for the explanation, Johnny R.

>the neutral third is always dead
>center, dividing the fifth into 2 identical parts. Just as the tritone
>bisects the octave, the neutral third bisects the perfect fifth.

That's a very interesting observation. It seems reasonable to identify some
neutral thirds as fifth-based tritones, so to speak. But is "always" the best
term here? Would you contend that they're not really "thirds" in the same
sense that 5/4 is, that they're really based on the fifth? That would seem to
fly in the face of rational analyses based on higher primes, unless you mean
to say that rational intervals that fall approximately midway are something
other than neutral thirds.

>The neutral thirds of all the meantones, 12-TET,...

There are, of course, no neutral thirds in historical 12-tone meantone
tunings, least of all 12-tET. Whatever could you mean?

>[snip] The difference of a few cents in either
>direction from 351 cents (at perfect fifth of 702 cents) is negligible for
>musical purposes.

Hmm. So an otherwise 11-limit sonority would be just as consonant with a
16/13 as with an 11/9? They're about 12 cents apart. That seems equivalent
to saying that there's no musical difference between a 12-tET major 7th and a
just 15/8.

I like the concept of dividing the fifth equally like the octave has long
been. It could be extended to any x/2^y, or maybe even to any rational
interval. Maybe those should be called "the perfect fifth tritone," "the
major third tritone," etc. as distinguished from "the octave tritone." But
should the definition of the neutral third be limited to "the perfect fifth
tritone"? To me, its potential musical usage seems to be broader than that.

David J. Finnamore
Just tune it!

On Tue, 29 Sep 1998 DFinnamore@aol.com responded to my comment below that:
> >the neutral third is always dead
> >center, dividing the fifth into 2 identical parts. Just as the tritone
> >bisects the octave, the neutral third bisects the perfect fifth.
>
> That's a very interesting observation. It seems reasonable to identify some
> neutral thirds as fifth-based tritones, so to speak. But is "always" the best
> term here?

There is no easy way to name the bisection of the perfect other than the
neutral third. Its usage is certainly "thirdishable" compositionally.
When I speak of the neutral third of of 12 ET, I am basically plotting its
location -- as the bisection of the perfect fifth. As a melodic interval
it easily suffers up to 5 cents tempering. As such it even includes just
plottings, to a point. Perhaps harmonically someone might object. Not I,
apparently.

> Would you contend that they're not really "thirds" in the same
> sense that 5/4 is, that they're really based on the fifth? That would seem to
> fly in the face of rational analyses based on higher primes, unless you mean
> to say that rational intervals that fall approximately midway are something
> other than neutral thirds.

They seem to function for me as "thirds" regardless of their derivation.
Perhaps there is a special quality available to this particular interval
that is located midway within nature's dominant interval.

> Hmm. So an otherwise 11-limit sonority would be just as consonant with a
> 16/13 as with an 11/9? They're about 12 cents apart. That seems equivalent
> to saying that there's no musical difference between a 12-tET major 7th and a
> just 15/8.

I'm not quite sure what you're getting at here. 11/9 (347.4 cents) and
16/13 (359.5 cents) are certainly distinctive intervals and each has a
special relationship within its own number constellation (e.g. 11-limit
and 13-limit). Melodically, they would seem acceptable neutral thirds.
I'm note sure there is much of a difference harmonically, at least in
terms of function.

The "neutral" quality is sort of like limbo, a DMZ zone, Switzerland (as
we once believed it to be).

Johnny Reinhard
AFMM

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Johnny:

The fifth-bisected neutral third may well be the most common kind; it would
obviously be the kind used whenever the third equally bisects a
very impure fifth. I simply recoil at calling it the only kind. But
it does seem like an interesting coincidence that both of the simplest
13-limit rational neutrals fall within a few cents of dead center of
3/2.

I wrote
>> 16/13 ... 11/9? They're about 12 cents apart. That seems equivalent
>> to saying that there's no musical difference between a 12-tET major 7th and
a
>> just 15/8.

You responded
>I'm not quite sure what you're getting at here. 11/9 (347.4 cents) and
>16/13 (359.5 cents) are certainly distinctive intervals and each has a
>special relationship within its own number constellation (e.g. 11-limit
>and 13-limit). Melodically, they would seem acceptable neutral thirds.

I'd agree that in most melodic situations 11/9 and 16/13 are virtually
the same thing. Further, it's probably sufficient simply to refer to
the whole melodic zone in the middle as "neutral," and call it a
bisected fifth.

>I'm note sure there is much of a difference harmonically, at least in
>terms of function.

That's easy; in octave-reduced terms, one functions as the 13th subharmonic of
1/1 and the other as the 11th harmonic of 16/9.
(Like you didn't already know that :-) Theoretically distinct but practically
the same? It would be possible to construct situations
in which the "wrong" one would produce unwanted beating - 12 cents
is plenty for that; that's all I was getting at. Perhaps in practice those
situations have been very rare.

David J. Finnamore
Just tune it!