Temperament (Monz, Paul, Marc) -- was "wolf pack"

Hello, there, Monz and Paul and Marc and everyone.

Mainly I want to encourage a friendly dialogue about "What Temperament
Means to Different People" while expressing my appreciation for all of
your contributions.

For myself, I find that temperament can be connected with two kinds of
ideas, the second of which, curiously, can sometimes also apply to
just tunings:

(1) The use of irrational interval sizes, or to put it
another way, the intuitive use of sizes which could
not be expressed by rational ratios, although they
could, of course, be approximated as closely as
desired through the use of such ratios; and

(2) The element of "deliberate intonational compromise,"
which might be achieved through rational as well as
irrational ratios.

With you, Monz, I agree that the irrational element is important: this
was the issue that Prosdocimus de Beldemandis raised in rightly or
wrongly reading Marchettus of Padua to advocate an _equal_ fivefold
division of a 9:8 whole-tone, and then asserting the impossibility of
such a division (using rational ratios, we might add).

Also, while I haven't yet read Isacoff, I am familiar with Mark
Lindley's writings on the role of Henricus Grammateus (1518) in using
Euclid's geometry to calculate an equal division of the 9:8 into two
semitones. Lindley notes that some writings of this era use the
fractional exponents of Nicolas Oresme (14th century), a notation
which Lindley finds not especially helpful, most likely, to the
musicians of the time.

At the same time, Paul, I agree with you that temperaments and the
like are often distinguished in good part precisely by dispersing
certain commas or distinctions found in many just systems.

I'd consider it a very important feature of meantone that it tempers
out the 81:80 (4 fifths up = 5:4), of 22-EDO that it tempers out the
64:63 (4 fifths up = ~9:7), and of 46-EDO that it tempers out the
896:891 (4 fifths up = ~14:11).

Here we're doing something more than simply emulating a classic JI
system based on the applicable ratios: we're setting up certain
equivalences.

Anyway, the "temperament by ratio" concept really has me fascinated
nowadays: your famous Aristoxenian decision in one piece for a minor
third around 279 cents, then realized as a 75:64 (~274.58 cents), is a
fine example.

As far as this "tempering by ratio" goes, the English organ tuning
described in a treatise of 1373 could be a fine historical
example. Diatonic notes are Pythagorean, with accidentals added by a
division of 18:17:16.

One result is the "bisecting" of the Pythagorean comma into two parts,
although I'm not sure that this was the main motivation for the
tuning, and the two "odd" fifths are still impure by about 15 cents
and 9 cents, as I recall.

Since most fifths are still pure, one could call this "JI" from a
Keenan-like viewpoint (in a context where a complete 2:3:4 trine
represents saturated stable concord).

Maybe I could sum up three possible approaches to musical dilemmas of
the kind often inviting temperament:

(1) Leave as many intervals as possible pure,
and let the others be as impure as they
turn out with these arrangement;

(2) Let all relevant intervals be slightly
and equally impure, as in a regular
temperament; or

(3) Let some intervals remain pure, and others
be compromised more gently than in (1), but
more noticeably than in (2).

If (3) is realized through rational intonation (RI), then we have a
kind of tuning which might incorporate aspects of both JI and
temperament.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> (1) The use of irrational interval sizes, or to put it
> another way, the intuitive use of sizes which could
> not be expressed by rational ratios, although they
> could, of course, be approximated as closely as
> desired through the use of such ratios; and

Irrational intervals are used when they are not for temperament (Jacky does this, for example) and not used when they are; in other words, this is neither necessary nor sufficient for being a temperament.

> (2) The element of "deliberate intonational compromise,"
> which might be achieved through rational as well as
> irrational ratios.

This is temperament.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> > (1) The use of irrational interval sizes, or to put it
> > another way, the intuitive use of sizes which could
> > not be expressed by rational ratios, although they
> > could, of course, be approximated as closely as
> > desired through the use of such ratios; and
>
> Irrational intervals are used when they are not for temperament >
(Jacky does this, for example) and not used when they are; in other
>words, this is neither necessary nor sufficient for being a
>temperament.

i have to agree with gene here.

>
> > (2) The element of "deliberate intonational compromise,"
> > which might be achieved through rational as well as
> > irrational ratios.
>
> This is temperament.

that's right -- did you know that kirnberger II actually flattens the
D-A and A-E fifths by 160:161 and 161:162?

Hello Margo, and thanks for the commentary on this thread.

> From: M. Schulter <MSCHULTER@VALUE.NET>
> To: <tuning@yahoogroups.com>
> Sent: Monday, March 11, 2002 8:59 PM
> Subject: [tuning] Temperament (Monz, Paul, Marc) -- was "wolf pack"
>
>
> Anyway, the "temperament by ratio" concept really has me fascinated
> nowadays: your famous Aristoxenian decision in one piece for a minor
> third around 279 cents, then realized as a 75:64 (~274.58 cents), is a
> fine example.

Margo, since you were addressing paul right before this, i thought
i'd let everyone else know that you're referring to me again
here, in connection with my JI retuning of my piece _3 Plus 4_
http://www.ixpres.com/interval/monzo/3plus4/3_plus_4_by_monz.mp3
(~ 5 megs)

the harmony in question occurs just after 0:21 and in all
further recurrences of this chord (1:15, 2:08, 3:02, 3:55).

i've already posted quite a bit on the 64:75:96 "minor triad"
i used here. here are a few examples:
/tuning/topicId_7293.html#7324
/tuning/topicId_15732.html#15740
/tuning/topicId_15976.html#16015
/tuning/topicId_15976.html#16030

> Maybe I could sum up three possible approaches to musical dilemmas of
> the kind often inviting temperament:
>
> (1) Leave as many intervals as possible pure,
> and let the others be as impure as they
> turn out with these arrangement;
>
> (2) Let all relevant intervals be slightly
> and equally impure, as in a regular
> temperament; or
>
> (3) Let some intervals remain pure, and others
> be compromised more gently than in (1), but
> more noticeably than in (2).
>
> If (3) is realized through rational intonation (RI), then we have a
> kind of tuning which might incorporate aspects of both JI and
> temperament.

this use of (3) is exactly what intended in _3 Plus 4_,
and in much of what i was doing around 1996-98.

-monz

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