skhismas, schisms & the importance of this forum

With access to Ivor Darreg's writings now,
one article I've found was a tribute to A.J.
Ellis, the translator and copious appender
of Helmholtz's "On the Sensations of Tone".

Darreg pointed out that Ellis, who coined the
name "skhisma" for the interval 3^8 * 5^1
[= 32805/32768 = approx. 1.9537 cents],
spelled it purposely with a "k" so that it
would not allude to the word "schism", which
had religious and hostile connotations.
Obviously, Ellis's spelling has not survived.
(Any comments on what _that_ connotes?)

I've only now read the thread begun by Margo
Schulter a few weeks back on the Tuning Forum
about the "Septimal schisma as xenharmonic
bridge?". I'm struck by the fact that Margo,
as well as the others who contributed to the
discussion, "discovered" this: that 3-limit
ratios, calculated out far enough, come close
enough to not only 5-limit ratios (which has
been known possibly for centuries and recognized
since Ellis), but even close enough to 7- and
higher-prime ratios, that they can possibly
allude to or substitute for each other.

My "discovery" of all sorts of commas, schismas,
and other small "anomalies" (Scott Wilkinson's
term) throughout the prime-number relationships
is what led me to be so bold as to intuit that,
for example, as far back as 1318 Marchetto of
Padua may have been trying to notate not only
5- but also 7-limit ratios that he could discern
by ear.

I see by the discussion of the septimal schisma
that this idea may not be as foreign to other
microtonally-informed theorists as I had
previously thought. The idea is not just
incomprehsible to regular theorists steeped
in the 12-equal scale, but also seems absolutely
ludicrous to those who understand ratios but
have accepted historical theorizing (and bad
translations of same) without questioning it.
These theorists/ historians would say that it's
utter nonsense to think that European listeners
of the 1300s ever heard 7-limit ratios -- this
is barely around the time that these theorists
would begin to admit that the 5-limit could
have been in use.

I'm only a little sorry to bust that bubble
that surrounds European tuning history -- it
seems pretty likely to me that Marchetto _was_
describing 5- _and_ 7-limit usage in the 1300s,
and I'm convinced that 5-limit ratios have been
in more-or-less constant use in practice ever since
the time of Boethius (the 500s!) and possibly
even earlier than that, in Greece before the
Roman conquest (c. 200 AD).

This is but one reason why this Forum is so
important. For so many years many microtonalists
have worked hard in complete or almost-complete
isolation, and so many important concepts have
been "discovered" over and over again. That
disparate researchers keep stumbling upon these
same ideas tells me that there is much validity
in them, but an open discussion seems to me so
much more fruitful, especially as it gives the
opportunity for those with different views to
call these assertions, or even speculations, into
question, and offers those who have access to
information that may be denied the single researcher
the opportunity to publicly post it.

(Stay "tuned" to my website for the appearance
of the Marchetto article -- hopefully this week)

- Joe Monzo
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

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On Thu, 12 Nov 1998 Manuel.Op.de.Coul@ezh.nl wrote:
> I compared it with my scale archive, but found nothing close to it.
> That's not really surprising since well-temperaments with pure fifths
> and a few flattened ones [snip]

I _also_ note that the original message (which I can't seem to lay hands
on right now) described most of the fifths not as just, but
undertempered even for ET, about 1 cent as I recall. That would spread
the tempering around enough to prevent harmonic waste.

--pH http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

NOTE: dehyphenate node to remove spamblock. <*>

I compared it with my scale archive, but found nothing close to it.
That's not really surprising since well-temperaments with pure fifths
and a few flattened ones never have all the flattened ones in a consecutive
row, but rather a few pure, then one flat, a few pure, etc.

Manuel Op de Coul coul@ezh.nl

On Thu, 12 Nov 1998 Manuel.Op.de.Coul@ezh.nl wrote:
> I compared it with my scale archive, but found nothing close to it.
> That's not really surprising since well-temperaments with pure fifths
> and a few flattened ones never have all the flattened ones in a consecutive
> row, but rather a few pure, then one flat, a few pure, etc.

Er--Vallotti? Plus, many popular ones--Kirnberger, Werckmeister III,
Kellner, Barnes--have _most_ of the flat fifths clumped together with
only one outlier.

--pH http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

NOTE: dehyphenate node to remove spamblock. <*>

>> That's not really surprising since well-temperaments with pure fifths
>> and a few flattened ones never have all the flattened ones in a consecutive

>Er--Vallotti? Plus, many popular ones--Kirnberger, Werckmeister III,
>Kellner, Barnes--have _most_ of the flat fifths clumped together with
>only one outlier.

Ok, that's right. But I was talking about temperaments with only a few
tempered fifths. Let's say three or less. Werckmeister III has four, the other
ones you mention five or six.
At least I'm not aware of any with three or less which have consecutive tempered
fifths. This would lead to harmonic waste, doesn't it?

Manuel Op de Coul coul@ezh.nl

On Thu, 12 Nov 1998, Manuel Op de Coul wrote:
> Ok, that's right. But I was talking about temperaments with only a few
> tempered fifths. Let's say three or less.

Oh, okay. That's different.

> At least I'm not aware of any with three or less which have consecutive tempered
> fifths. This would lead to harmonic waste, doesn't it?

Yes, it does--eight consecutive just fifths would result in a major
third narrow of just by a skhisma, which meets the definition of
harmonic waste as any thirds narrower or fifths wider than just.

--pH http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

NOTE: dehyphenate node to remove spamblock. <*>

On Thu, 12 Nov 1998, Paul Hahn wrote:
> On Thu, 12 Nov 1998, Manuel Op de Coul wrote:
>> At least I'm not aware of any with three or less which have consecutive tempered
>> fifths. This would lead to harmonic waste, doesn't it?
>
> Yes, it does--[snip]

Note, however, that harmonic waste does not make a temperament unusable,
except perhaps among German speakers. 8-)> The French "temperament
ordinaire" is very harmonically wasteful, for example. I think some of
the temperaments John Sankey customized for specific Scarlatti sonatas
had some harmonic waste in them too.

--pH http://library.wustl.edu/~manynote
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

NOTE: dehyphenate node to remove spamblock. <*>

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> I _also_ note that the original message (which I can't seem to lay hands
> on right now) described most of the fifths not as just, but
> undertempered even for ET, about 1 cent as I recall. That would spread
> the tempering around enough to prevent harmonic waste.

Correct. For the comparison I took that into account by taking 9 fifths
of 701 cents and 3 of 697 cents, on Eb, Bb and F. No harmonic waste and
no similar temperaments found.

Manuel Op de Coul coul@ezh.nl

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