a Pythagorean question--Margo?

I was wondering if there were established names for any of the
extended Pythagorean subcommatic intervals other than the schisma; the
33554432/33480783 for example?

And historically has anyone ever spelled anything as strangely as--or
stranger than--a 1-b4-5-bb8?

thanks,

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> I was wondering if there were established names for any of the
> extended Pythagorean subcommatic intervals other than the schisma;
the
> 33554432/33480783 for example?

According to http://www.xs4all.nl/~huygensf/doc/intervals.html, this
is called Beta 2 (Eduardo Sabat-Garibaldi's term) or "septimal
schisma" (Margo Schulter's term). I don't know if that counts
as "established" yet, but there you go.

Yes, septimal schisma, of course... I like it.

----- Original Message -----
From: "Paul Erlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, October 11, 2001 4:04 PM
Subject: [tuning] Re: a Pythagorean question--Margo?

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > I was wondering if there were established names for any of the
> > extended Pythagorean subcommatic intervals other than the schisma;
> the
> > 33554432/33480783 for example?
>
> According to http://www.xs4all.nl/~huygensf/doc/intervals.html, this
> is called Beta 2 (Eduardo Sabat-Garibaldi's term) or "septimal
> schisma" (Margo Schulter's term). I don't know if that counts
> as "established" yet, but there you go.
>
>
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Hello, there, Dan Stearns and Paul and everyone.

First, Paul, please let me agree that the recognized terms in Scala
for the Pythagorean schisma involving ratios of 2-3-7-9 are either
Eduardo Sa/bat's "Beta-2," or "septimal schisma."

Just to clarify: this ratio of 33554432:33480783 (~3.80 cents) is the
amount, for example, by which 16 pure 3:2 fifths up less nine octaves
(~431.38 cents) fall short of a pure 9:7 major third (~435.08 cents).

While I have often described this interval as the "septimal schisma,"
Graham Breed pointed out that such a term could also be taken as
referring to difference between a regular Pythagorean diminished fifth
at 1024:729 (~588.27 cents) and a pure 7:5 (~582.51 cents), a ratio of
5120:5103 or ~5.76 cents.

For this reason, I have come to use the term "3-7 schisma" for the
ratio mainly under discussion here, and "5-7" schisma for the ratio to
which Graham called my attention.

Sa/bat refers to these two schismas as "Beta-2" and "Beta-5," the
first name evidently referring to another definition for what I term
the "3-7" schisma: the difference between 14 fifths up and a pure
8:7. In Sa/bat's definition, the relevant factors here are 2 and 7,
and for what I term the "5-7" schisma they are 5 and 7.

There's also what I now term the "3-5" schisma, the difference for
example between eight fifths down or fourths up (8192:6561, ~384.36
cents) and a pure 5:4 (~386.31 cents), a ratio of 32805:32768 or ~1.95
cents. Sa/bat calls this schisma "Alpha," and tempers his 53-note
Dinarra by 1/9 of this schisma for pure 6:5 minor thirds.

Here's a table of these names and ratios:

-------------------------------------------------------------------
Name by factors Sa/bat Ratio Cents
-------------------------------------------------------------------
3-5 schisma Alpha 32805:32768 ~1.95
...................................................................
3-7 schisma Beta-2 33554432:33480783 ~3.80
...................................................................
5-7 schisma Beta-5 5120:5103 ~5.76
-------------------------------------------------------------------

As Sa/bat notes, the difference between Beta-5 and Beta-2 is equal to
Alpha (with a bit of rounding error in this table).

Dan, I'm not sure about the notation for the spellings you give at the
end of your article, "1-b4-5-bb8" -- are these notes, intervals, or
degrees of some scale? -- but I'd be really interested to understand
it, and maybe reply to that question.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hi Margo,

By "1-b4-5-bb8" I just meant a 4:5:6:7 schismatic harmonic seven chord
(so if 1 were C, that would be a c-fb-g-cbb Pythagorean spelling), and
generally I was wondering how far out there these sorts of things
might've got... so in other words, historically speaking, I'm
interested to know what odd schismatic spellings have occurred in
theory or practice--and who better to ask!

thanks,

--Dan Stearns

----- Original Message -----
From: "mschulter" <MSCHULTER@VALUE.NET>
To: <tuning@yahoogroups.com>
Sent: Thursday, October 11, 2001 7:25 PM
Subject: [tuning] Re: a Pythagorean question--Margo?

> Hello, there, Dan Stearns and Paul and everyone.
>
> First, Paul, please let me agree that the recognized terms in Scala
> for the Pythagorean schisma involving ratios of 2-3-7-9 are either
> Eduardo Sa/bat's "Beta-2," or "septimal schisma."
>
> Just to clarify: this ratio of 33554432:33480783 (~3.80 cents) is
the
> amount, for example, by which 16 pure 3:2 fifths up less nine
octaves
> (~431.38 cents) fall short of a pure 9:7 major third (~435.08
cents).
>
> While I have often described this interval as the "septimal
schisma,"
> Graham Breed pointed out that such a term could also be taken as
> referring to difference between a regular Pythagorean diminished
fifth
> at 1024:729 (~588.27 cents) and a pure 7:5 (~582.51 cents), a ratio
of
> 5120:5103 or ~5.76 cents.
>
> For this reason, I have come to use the term "3-7 schisma" for the
> ratio mainly under discussion here, and "5-7" schisma for the ratio
to
> which Graham called my attention.
>
> Sa/bat refers to these two schismas as "Beta-2" and "Beta-5," the
> first name evidently referring to another definition for what I term
> the "3-7" schisma: the difference between 14 fifths up and a pure
> 8:7. In Sa/bat's definition, the relevant factors here are 2 and 7,
> and for what I term the "5-7" schisma they are 5 and 7.
>
> There's also what I now term the "3-5" schisma, the difference for
> example between eight fifths down or fourths up (8192:6561, ~384.36
> cents) and a pure 5:4 (~386.31 cents), a ratio of 32805:32768 or
~1.95
> cents. Sa/bat calls this schisma "Alpha," and tempers his 53-note
> Dinarra by 1/9 of this schisma for pure 6:5 minor thirds.
>
> Here's a table of these names and ratios:
>
> -------------------------------------------------------------------
> Name by factors Sa/bat Ratio Cents
> -------------------------------------------------------------------
> 3-5 schisma Alpha 32805:32768 ~1.95
> ...................................................................
> 3-7 schisma Beta-2 33554432:33480783 ~3.80
> ...................................................................
> 5-7 schisma Beta-5 5120:5103 ~5.76
> -------------------------------------------------------------------
>
> As Sa/bat notes, the difference between Beta-5 and Beta-2 is equal
to
> Alpha (with a bit of rounding error in this table).
>
> Dan, I'm not sure about the notation for the spellings you give at
the
> end of your article, "1-b4-5-bb8" -- are these notes, intervals, or
> degrees of some scale? -- but I'd be really interested to understand
> it, and maybe reply to that question.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net
>
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
> FREE COLLEGE MONEY
> CLICK HERE to search
> 600,000 scholarships!
> http://us.click.yahoo.com/Pv4pGD/4m7CAA/ySSFAA/RrLolB/TM
> --------------------------------------------------------------------
-~->
>
> You do not need web access to participate. You may subscribe
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>
>

Hello, there, Dan, and thanks for explaining that notation -- for
example, a Pythagorean C-Fb-G-Cbb as a near-4:5:6:7 sonority using the
3-7, 3-5, and 5-7 schismas all at the same time.

Historically, at least in Pythagorean tuning, I'm aware only of the
3-5 schisma playing a significant role, and that largely "unwritten,"
with major thirds such as E-G#, A-C#, and D-F# often being realized on
early 15th-century keyboards as E-Ab, A-Db, and D-Gb in the Gb-B
tuning for a 12-note instrument.

In a 31-note meantone cycle such as that of Nicola Vicentino (1555) or
Fabio Colonna (1618), we get some steps and sonorities with all kinds
of remote spellings, were we to use a conventional notation like that
of your Pythagorean example. However, both in that era and for at
least some of us in the early 21st century, there's a tendency to use
some simpler spelling, maybe based on the keyboard layout familiar to
a notator.

For example, how about something in Vicentino like a sonority with a
"proximate minor third," which he finds rather concordant and for
which he gives an approximate ratio of 5-1/2:4-1/2, or 11:9.

Using Vicentino's actual notation, which I generally follow, we have
something like C3-Eb*3-G3, with the "*" (a dot above a note in
Vicentino) showing a note raised by an enharmonic diesis of around
1/31 octave or 1/5-tone. In my 1/4-comma meantone version of a 24-note
subset of his tuning, this is a diesis of 128:125 (~41.06 cents).

Using a conventional spelling, this would be C3-Fbb3-G3.

Moving to modern music, however, I would say that some really
"far-out" Pythagorean spellings are now in use -- if we choose to
follow conventional notation, rather than a more "system-oriented"
style of naming the notes and intervals.

Consider, for example, the Pythagorean tricomma tuning, with two
12-note manuals at the distance of three Pythagorean commas or a
"tricomma," about 70.38 cents. This is a beautiful tuning, and in
practice I would simply use Vicentino's diesis symbol, or its ASCII
equivalent (*), to show the notes on the upper manual raised by a
tricomma.

The tricomma is equal to 36 fifths up or fourths down, which means
that conventional spellings would get rather complicated. In fact, I
hadn't considered this until your post called my attention to the
possibilities for such complications.

Here I'll focus on the two main kinds of sonorities, other than the
usual Pythagorean ones available within either keyboard, which this
tuning was designed to obtain.

First we have a near-14:17:21, with the supraminor third equal to a
regular major third less a tricomma, e.g. C*4-E4-G*4, an interval of
32 fourths up or fifths down.

That gives me G#####3-E4-D#####4 as a conventional spelling.

Either way you spell it, it's around 0-337.44-701.96 cents.

There's also a not-too-close approximation of 12:14:18:21 -- in my
usual notation, a sonority such as C3-D*3-G3-A*3. Here the near-7:6
C3-D*3 is 38 fifths up, and the near-7:4 C3-A*3 is 39 fifths up.

In conventional notation, this is C3-A#####2-G3-E#####3.

Here I'm trying to stick to things of clear practical use. The
tricomma tuning is a method for getting approximations of both
12:14:21:24 and 14:17:21 in a single 24-note tuning set together with
Pythagorean JI for the basic intervals (6:8:9:12) plus the usual more
complex Pythagorean ratios found within a 12-note tuning.

I might really call it a kind of "temperament by ratio" of the
approximate ratios of 2-3-7-9 and 2-7-17, with a tradeoff leaning
toward the latter ratios.

At the same time, it does have the conceptual kind of aesthetic
satisfaction of using only "native" Pythagorean ratios.

Again, at least for myself, I would call these things fairly routine,
and in fact only the stimulus of your post makes me consider what the
conventional spellings would look like.

What I do tend to notice in categorizing the tricomma comma is that
it's a "metachromatic" tuning in the general sense: a near-7:4 is
equal to a regular major sixth plus the distance between the two
keyboards, with a location of D3-B*3, for example, and so forth.

The near 17:14 and 21:17 intervals are also "metachromatic," with
the supraminor third equal to major third less the distance between
keyboards, and the submajor third to the minor third plus this
distance -- e.g. C*3-E3 and E3-G*3.

Thus I call it a "double metachromatic" tuning. (A "single
metachromatic" tuning of the kind I'm familiar with would derive the
2-3-7-9-type intervals in the same way, but the 17:14 and 21:17 as
respectively minor third plus diesis or major third less diesis.)

It's curious that these things look and feel much simpler on a
keyboard than in a conventional notation, and Vicentino's 31-note
meantone notation of 1555 may also reflect this kind of notational
simplification to fit a given keyboard arrangement.

Of course, how singers or players of flexible-pitch instruments would
prefer to notate it is another question, and an interesting one.

Most appreciatively,

Margo Schulter
mschulter@value.net

you have got the wrong email address quit emailing me please!

people are emailing me with questions, stop this