Diaschisma article

Someone put up a Wikipedia article on the diaschisma a few months
back. It it I find: "Medieval theorists Boethius and Tinctoris
described the diaschisma as one-half of the semitonium, or smaller
semitone, of Pythagorean tuning." This doesn't make a lot of sense to
me, but if you take the diaschisma to be one-half of a diesis we are
back into the "did Boethius assume schismatic tempering" thread, since
(128/125)/(2048/2045)^2 = 32805/32768.

Any help on this would be appreciated. The Wikipedia article is here:

http://en.wikipedia.org/wiki/Diaschisma

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Someone put up a Wikipedia article on the diaschisma a
> few months back. It it I find: "Medieval theorists
> Boethius and Tinctoris described the diaschisma as
> one-half of the semitonium, or smaller semitone, of
> Pythagorean tuning."

That's wrong -- it wasn't *Boethius* describing the
diaschisma as 1/2 semitonium, it was Boethius quoting
*Philolaus*.

> This doesn't make a lot of sense to me, but if you
> take the diaschisma to be one-half of a diesis we are
> back into the "did Boethius assume schismatic tempering"
> thread, since (128/125)/(2048/2045)^2 = 32805/32768.
> Any help on this would be appreciated. The Wikipedia
> article is here:
>
> http://en.wikipedia.org/wiki/Diaschisma

I guess you missed this?

http://tonalsoft.com/enc/p/philolaus.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software

monz, I read your excellent article on Philolaus, bravo! But I do not
understand how Boethius commits an error when he says that the whole tone is
larger than 8, smaller than 9 commas. If it is a question of the Pythagorean
comma, that certainly is the case.

It is also notable that you have chosen 106-tET to explain Philolaus's
tetrachordal division. I myself seperately found out that all the theories
on Maqam Music from 13th century up to 1920s can be explained in terms of
106 equal divisions of the octave with less than 1 cents error for each
pitch.

Cordially,
Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 06 Aral�k 2005 Sal� 1:34
Subject: [tuning] Re: Diaschisma article

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> > Someone put up a Wikipedia article on the diaschisma a
> > few months back. It it I find: "Medieval theorists
> > Boethius and Tinctoris described the diaschisma as
> > one-half of the semitonium, or smaller semitone, of
> > Pythagorean tuning."
>
>
> That's wrong -- it wasn't *Boethius* describing the
> diaschisma as 1/2 semitonium, it was Boethius quoting
> *Philolaus*.
>
>
> > This doesn't make a lot of sense to me, but if you
> > take the diaschisma to be one-half of a diesis we are
> > back into the "did Boethius assume schismatic tempering"
> > thread, since (128/125)/(2048/2045)^2 = 32805/32768.
> > Any help on this would be appreciated. The Wikipedia
> > article is here:
> >
> > http://en.wikipedia.org/wiki/Diaschisma
>
>
>
> I guess you missed this?
>
> http://tonalsoft.com/enc/p/philolaus.aspx
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>
>
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--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> I guess you missed this?
>
> http://tonalsoft.com/enc/p/philolaus.aspx

That suggests to me that 2048/2025 is supposed to be half of 256/243,
which would mean tempering out (256/243)/(2048/2025)^2 = 16875/16384,
the negri comma. This still doesn't make a whole lot of sense to me,
as 2048/2025 is between 1/4 and 1/5 of 256/243. On the other hand, we have
(256/243)/(128/125)^2 = 15625/15552 = (250/243)^2/(256/243). Do maybe
we are really talking about a diesis.

Hi Oz,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> monz, I read your excellent article on Philolaus, bravo!

Thanks. I expanded it into a fairly lengthy paper and
submitted it to John Chalmers over a year ago for
publication in _Xenharmonikon 18_ ... and am still waiting
for its appearance.

> But I do not understand how Boethius commits an error
> when he says that the whole tone is larger than 8,
> smaller than 9 commas. If it is a question of the
> Pythagorean comma, that certainly is the case.

No, you've got that wrong ... perhaps i used some
misleading wording on my webpage ... don't have time
to check now, and will have to do that when i return
home next week.

The error Boethius commits is that he uses an arithmetic
string-length (i believe it's 7153, not sure) and mulitplies
it instead of treating it as a ratio raised to exponents,
which is the proper way to handle the calculation.

I posted something on this here several years ago,
and gave the complete analysis of the correct calculation.
By the end of it, the terms of the ratio have something
like 50 digits. And the error is tiny, so presumably
Boethius was aware of the error and knew that it wouldn't
matter all that much if he intentionally did it wrong.

> It is also notable that you have chosen 106-tET to
> explain Philolaus's tetrachordal division. I myself
> seperately found out that all the theories on Maqam
> Music from 13th century up to 1920s can be explained
> in terms of 106 equal divisions of the octave with
> less than 1 cents error for each pitch.

Now that *is* interesting!

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> Someone put up a Wikipedia article on the diaschisma a few months
> back. It it I find: "Medieval theorists Boethius and Tinctoris
> described the diaschisma as one-half of the semitonium, or smaller
> semitone, of Pythagorean tuning." This doesn't make a lot of sense to
> me, but if you take the diaschisma to be one-half of a diesis we are
> back into the "did Boethius assume schismatic tempering" thread, since
> (128/125)/(2048/2045)^2 = 32805/32768.

Be careful, Gene. "Diaschisma" has been used to refer to a lot of
different things. The chances that it referred to 2048/2045 in this
context are slim at best.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I guess you missed this?
> >
> > http://tonalsoft.com/enc/p/philolaus.aspx
>
> That suggests to me that 2048/2025 is supposed to be half of 256/243,
> which would mean tempering out (256/243)/(2048/2025)^2 = 16875/16384,
> the negri comma.

You're still interpreting "Diaschisma" to mean 2048/2025, which is
almost certainly inappropriate in this context. It was only a couple
centuries ago that "Diaschisma" was first used to mean 2048/2025 --
before that, the word had completely different meanings.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> You're still interpreting "Diaschisma" to mean 2048/2025, which is
> almost certainly inappropriate in this context.

This is what the Wikipedia article asserted. It seems to be wrong, but
I'd like to know what is right before fixing it.

It was only a couple
> centuries ago that "Diaschisma" was first used to mean 2048/2025 --
> before that, the word had completely different meanings.

Which were?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > You're still interpreting "Diaschisma" to mean 2048/2025, which is
> > almost certainly inappropriate in this context.
>
> This is what the Wikipedia article asserted. It seems to be wrong, but
> I'd like to know what is right before fixing it.
>
> It was only a couple
> > centuries ago that "Diaschisma" was first used to mean 2048/2025 --
> > before that, the word had completely different meanings.
>
> Which were?

According to the Monzopedia, Helmholtz used "diaschisma" it in 1875 to
mean 2048/2025, but Rameau in 1722 called 2048/2025 "diminished comma"
instead, and the ancient and medieval usage (where prime 5 was of very
limited relevance) of "diaschisma" specifically referred to an interval
of 45-47 cents, not <20 cents.

Hi Gene (and Paul),

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > You're still interpreting "Diaschisma" to mean
> > 2048/2025, which is almost certainly inappropriate
> > in this context.

I pointed you to my Philolaus page and explained that
there were older meanings of "diaschisma" which were
different.

> This is what the Wikipedia article asserted. It seems to
> be wrong, but I'd like to know what is right before fixing
> it.
>
> > It was only a couple centuries ago that "Diaschisma"
> > was first used to mean 2048/2025 -- before that, the
> > word had completely different meanings.
>
> Which were?

When i posted the URL for my Philolaus page, i was
assuming that you'd *read* it. :)

About 1/3 of the way down the page i tabulate the values
of all the small intervals exactly as described by Philolaus
and quoted by Boethius, and i provide the actual Boethius
quotes at the bottom of the page.

>> The diesis, he [Philolaus] says, is the interval
>> by which a sesquitertian ratio [4:3] is larger than
>> two tones [(9:8)^2].
>>
>> <snip>
>>
>> The diaschisma is half a diesis -- that is, half a
>> minor semitone.

The "sesquitertian ratio" = 4/3 = 2,3-monzo [2 -1>.
"Two tones" = = (9/8)^2 = 2,3-monzo [-3 2> * 2 = [-6 4>.

.... [+2 -1> sesquitertian = 4/3
.. - [-6 +4> two tones = (9/8)^2
.. ---------
.... [+8 -5> diesis (i.e., small semitone)

Thus, diesis = 2,3-monzo [8 -5> = ~90.22499567 cents, and

diaschisma = 2,3-monzo [4 -5/2> = ~45.11249784 cents

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Paul and Gene,

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> According to the Monzopedia,

Guys, i do appreciate that you honor me by attaching
my name to the Tonalsoft Encyclopedia ... but i'd prefer
it if you use the official name. Thanks.

> Helmholtz used "diaschisma" it in 1875 to mean 2048/2025,
> but Rameau in 1722 called 2048/2025 "diminished comma"
> instead, and the ancient and medieval usage (where
> prime 5 was of very limited relevance) of "diaschisma"
> specifically referred to an interval of 45-47 cents,
> not <20 cents.

Thanks for mentioning that i have a "diaschisma" page
in the Tonalsoft Encyclopedia, Paul -- i should have
referenced it along with my Philolaus page.

http://tonalsoft.com/enc/d/diaschisma.aspx

Indeed, it's only since about 1900 or so that
music-theorists have used the names of intervals
smaller than a semitone with any kind of regularity,
with regard to their actual pitch size. Before that,
apparently "anything goes".

For other examples, whose values varied even more
widely, consult the pages on "schisma" and "comma".

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 07 Aral�k 2005 �ar�amba 4:26
Subject: [tuning] Re: Diaschisma article

> Hi Paul and Gene,
>
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > According to the Monzopedia,
>
>
> Guys, i do appreciate that you honor me by attaching
> my name to the Tonalsoft Encyclopedia ... but i'd prefer
> it if you use the official name. Thanks.
>
>

I'm the culprit for this trend.

SNIP!

Cordially,
Oz.