Re: For Manuel Op de Coul -- septimal comma and 25-cent symbol

Hello, there, Manuel Op de Coul and everyone.

Thank you for the idea of a 48-tET/144-tET notation with the symbols
^/v showing a note raised or lowered by 1/8-tone or 25 cents, indeed a
close approximation of the neo-Gothic "septimal comma" interval or
also the Pythagorean comma shown by the same symbols.

Of course, since the neo-Gothic "^/v" may cover a range of around
15-27 cents, no one 48-tET/144-tET notation symbol will _always_
correspond, but I would agree that the most typical neo-Gothic uses
often do correspond very nicely with a 25-cent step:

septimal comma proper: ~27.26 cents 64:63
Pythagorean comma proper: ~23.46 cents 531441:524288
e-based septimal comma: ~21.68 cents

As an aside, I might suggest that in a neo-Gothic setting, the 36-tET
interval of 1/6-tone or 33.33... cents might belong to a different
category because it represents not only the "septimal comma," but also
the 459:448 diesis (~41.99 cents) between a Pythagorean minor third at
32:27 (~294.13 cents) and a 17:14 third (~336.13 cents). In 36-tET,
these intervals are approximated as 300 cents and 333.33... cents.

In practice, I tend to use a generic "*" or "diesis" sign for this
interval and various larger ones in the range of around 35-70 cents, a
usage somewhat connected to Vicentino's original diesis sign (actually
a dot above a note) more specifically suggesting an interval of
1/5-tone in a circulating 31-note tuning such as 1/4-comma meantone or
31-tET.

Especially in a gentler timbre, 36-tET can be used as both a
neo-Gothic and "Xeno-Renaissance" tuning, with the 1/6-tone interval
serving for the latter type of style as a kind of small diesis. It is
actually not too much smaller than the smaller fifthtone of 1/4-comma
meantone, the difference between the chromatic semitone (~76.05 cents)
and the 128:125 diesis (~41.06 cents), or about 34.99 cents.

Thus while there is no neat and invariable mapping between neo-Gothic
"septimal comma" and any 48-tET/144-tET interval, I would warmly agree
that the 1/8-tone of 25 cents would be a nice approximation in some of
the most characteristic tunings.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello, there, Joe Pehrson, and it's a pleasure to be discussing theory
in an ideal place: the Alternate Tuning List.

First of all, apart from identifying myself as a self-proclaimed
"tuning.kook" (it's time, in my view, to associate the word "kook"
with fun and zaniness and creative wackiness rather than with
incivility and other unpleasantries), I wouldn't attempt to say who
has the most or least "unconventional" view on this issue.

Looking specifically at European music, I would concur in part and
dissent in part from a statement that this music "is based on
small integer ratios."

If you were to say that stable concords in medieval-Romantic music are
generally based on some ratios or reasonably close approximations, I
would agree: the 2:3:4 trine and 4:5:6 triad indeed fit this paradigm.

However, for example, Pythagorean tuning in its historical Gothic form
mostly has _low_ integer ratios like 2:1, 3:2, 4:3, and 9:8, plus some
octave extensions or complements (e.g. a 16:9 minor seventh, which I
consider "just" but some other people regard as psychoacoustically
complex) -- but also lots of _large_ integer ratios.

Further, the _large_ integer ratios of Pythagorean tuning include some
very important "partial concords" (e.g. major and minor thirds at
81:64 or ~407.82 cents, and 32:27 or ~294.13 cents) quite vital to the
cadential and coloristic qualities of 13th-14th century music. Just
try removing all of the thirds and sixths from Perotin or Machaut, for
example -- let's say when playing them on a Pythagorean keyboard, to
avoid issues of shifts in either direction that could occur with
non-fixed-pitch ensembles.

Also, I'd say that melody is important, and melodic semitones often
tend to have complex ratios.

Thus I'd say we have a mixture of simple and complex ratios, with
various degrees of concord or discord -- 13th-century theorists may
offer four, five, or six categories from the purest concords to the
most acute discords.

Curiously, "Pythagorean" and "JI" are often taken almost as opposites,
since "JI" often implies "5-limit or higher," likely reflecting the
complexities of trying to mesh two or more incommensurate prime
factors like 3 and 5 (the Renaissance dilemma).

For me, "JI" mainly implies that _some_ intervals have pure and
reasonably simple ratios, like 3:2 and 4:3 in Pythagorean, and also
9:7, 7:6, and 7:4 in Pythagorean-derived systems based on factors of 3
and 7.

What I often favor might be called "kooky JI," because it doesn't
necessarily follow a "low ratios _whenever possible_" approach, but
seeks a partnership of small and large ratios.

Maybe the dilemma for JI with two or more primes other than 2 is about
as follows:

(1) Seek low ratios everywhere, and run into complications
with melodic unevenness or comma shifts where you
might not want them;

(2) Temper in some way -- or maybe "adaptively tune" so as
to split melodic differences while maintaining pure
and simple vertical concords;

(3) Mix low and high JI (integer-based) ratios, seeking
evenness or uneveness, vertical "purity" or "complexity,"
and lots of contrasts -- as desired.

Please let me add that forms of "strict" 2/3/7 JI with a consistent
(in the usual nontechnical sense) use of sonorities such as
12:14:18:21 or 14:18:21:24 can be very charming. However, I like
54:64:81:96 or 64:81:96:108 also, and a tuning with two Pythagorean
manuals a 64:63 or 7:6 apart makes it easy to shift between these
flavors.

CAUTION: Another basic cultural distinction may affect the reading of
this last paragraph. For me, as for medieval European theorists,
citing a ratio simply says that these are the integers or string
lengths defining the interval. However, some people take these ratios
to imply relations involving the harmonic series -- something I would
assume for 3:2 or 9:8, but not necessarily for 81:64 or 256:243, etc.

What I'd emphasize is that "JI" is not a monolithic bloc. For example,
consider a point of lattice making.

One thing I notice is that if making a lattice with a 3/7-JI kind of
tuning, I tend to take the 3:2 as the horizontal dimension, and the
7:6 as the vertical dimension, so that we get quadrangles or "quads"
of 12:14:18:21 (or 14:18:21:24, depending how you read). Here I'll a
use a neo-Gothic "v" to show a note lowered by a septimal comma of
64:63 from Pythagorean, or ~27.26 cents (close to the 25-cent "v" or
1/8-tone of Manuel Op de Coul's 144-tET notation):

Gbv -- Dbv -- Abv -- Ebv ...
/ / / /
/ / / /
Eb -- Bb -- F -- C ...

Interestingly, I'd guess, the majority of JI folks tend to look for
lattice patterns based on 4:5:6:7, with 7:4 rather than 7:6 as the
main "7-based" interval. Such are the varieties of tastes and
outlooks.

A curious aside, as long as we're talking about the nonmonolithic
nature of "JI" -- with temperaments, also, I find it very cool to have
"two versions of the same note" about 15-30 cents apart.

I'm not saying that every 24-note temperament I find interesting has
to do this: Vicentino's, whether taken as 1/4-comma meantone or some
conceptual model of 31-tET, doesn't, and it's a great tuning! (The
smallest step is ~34.99 cents, the "small fifthtone," in 1/4-comma, or
of course 1/31 octave or ~38.70 cents, I guess, in 31-tET).

Sometimes evenness can be very important: I'd say that Manneristic
"fifthtone" music is indeed meant to have reasonably equal fifthtones
or dieses, and that means something close to 1/4-comma or 31-tET. In
moving from a minor third to a "proximate minor third" to a major
third (in JI terms, 6:5 - 11:9 - 5:4), those steps should be even.

A difference of 6 cents, like in 1/4-comma, is no problem; but
2/7-comma, for example, has a different feel and "mood" (to speak in a
Darregian manner). It's a great 24-note system, but has a different
temperament (to borrow a pun from Mark Lindley) than the style of
16th-century enharmonicism.

Of course, I favor high-integer "rational intonation" or RI as a kooky
and pleasantly subversive approach to questioning some of these set
categories and factions. Jacky Ligon is my treasured comrade in this,
and I would be honored to be called a "numerologist" for my pains
<grin>.

Sometimes I suspect that some of the "JI debates" involve a lot of
implicit points of agreement between the "sides" where some us might
have different views.

For example, people in a "JI vs. temperament" debate may already
pretty much agree that "low integer ratios are where it's at," and are
debating whether to use untempered ratios or maybe something like
72-tET. There's a lot of common ground on general philosophy, and
maybe on stylistic assumptions also.

What I'd say is that there's a big continuum out there, and that JI/RI
is one very interesting way to approach it. It doesn't have to be
"right" or "wrong," only musically engaging in its many quite
nonmonolithic forms as _one_ of the available choices.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_26020.html#26424

> Hello, there, Joe Pehrson, and it's a pleasure to be discussing
theory in an ideal place: the Alternate Tuning List.
>

Thank you very much, Margo, and this does *indeed* seem to be the
place for such discussion... unless, of course, it gets excessively
mathematical, in which case it goes to the *math tuning* list, *or*
it gets exceedingly compositional, in which case it goes to Jon
Szanto's new CreateMicroMusic list... :)

I guess these days we live in an era of specialization. It wasn't
particularly *my* idea but I am, as they say, "getting used to it."

> First of all, apart from identifying myself as a self-proclaimed
> "tuning.kook" (it's time, in my view, to associate the word "kook"
> with fun and zaniness and creative wackiness rather than with
> incivility and other unpleasantries), I wouldn't attempt to say who
> has the most or least "unconventional" view on this issue.
>

I also appreciate this viewpoint. I believe, however, as can be seen
by Graham Breed's citations on the _Metatuning_ list (which is still
functional, but not particularly active) there is a defined
*Internet* usage for the term "kook." There seem to be plenty of
them out there and, regrettably, their aims are not so benign.

> If you were to say that stable concords in medieval-Romantic music
are generally based on some ratios or reasonably close
approximations, I would agree: the 2:3:4 trine and 4:5:6 triad indeed
fit this paradigm.
>

I guess this kind of generalization is all we can go for... but it
still is a powerful observation.... Maybe obvious, but still
powerful in the overview...

From your discussion it seems as though one of the more
*controversial* points is whether Pythagorean tuning "should" be
considered just intonation.

Logic would state that it should be, since the generating principle
of it stems from pure intervals.

However, it differs, of course, with definitions such as Ben
Johnston's which are based specifically on 5-limit just intonation.

Most of this is, naturally, a matter of *semantics* so I guess it
really matters little one way or the other, as long as we ALL know
what we're talking about.

In the case of Ben Johnston, however, some have suggested that his
notation might be simpler if he had started with the more *logical*
Pythagorean basis...

I'm sure to Johnston, though, that basis *wouldn't* be more
intuitive... so there are the semantics again...

> What I'd say is that there's a big continuum out there, and that
JI/RI is one very interesting way to approach it. It doesn't have to
be "right" or "wrong," only musically engaging in its many quite
> nonmonolithic forms as _one_ of the available choices.
>

Well, this is truly the *root* of the discussion and it seems, as
usual, that the more one knows about any subject, the harder it is to
be dogmatic in one particular view...

Thank you for sharing these valuable insights...

_________ _______ __________
Joseph Pehrson