Yahoo Tuning Groups Ultimate Backup tuning For Joe Pehrson -- 24-of-72-tET (1/12-tone) subset

Hello, there, Joe, and thank you for your gracious words, which remind
me of a very important responsibility.

That responsibility is to make distinction between historical medieval
intonation systems, in Continental Western Europe meaning Pythagorean
tunings of up to 17 notes, and also some clues about flexible
intonation systems given by Marchettus of Padua and a few others --
and all kinds of 21st-century spinoffs and adaptations.

For example, to put the point concisely, I would _not_ promote 13-tET
as an "Historically Informed Performance" (HIP) practice, although
using an intonation for ensemble performance approximating
Pythagorean, but with some cadential major thirds and sixths near 9:7
and 12:7, might be one _possible_ HIP interpretation based on
Marchettus plus a lot of "intuition."

For some people to get the "historically correct" intonation right at
least some of the time is very important: not only is it beautiful,
but it gives us standard to follow, emulate with variations, or even
honor by the antithesis of an "unconventional" interpretation.

Also, an awareness both of historical practice and of recent
innovations can open up new creative possibilities for those of
interested in writing new music drawing inspiration from old styles.

For example, I regard the use of a sonority like 14:17:21 (about
0-336-702 cents) as a _new_ variation on standard 14th-century
cadences, something I happened to run into in a tuning chosen without
realizing that this would be one of the available "features."

Designing temperaments with fifths wider than pure to get this kind of
sonority (often from "diminished fourths" and "augmented seconds"), or
even finding a close equivalent in a subset of a large enough
Pythagorean tuning, are fun, and at least for me musically very
pleasing -- but that doesn't make it an authentic approach of the
Gothic era itself.

Also, while Marchettus gives some examples of direct chromaticism
which show lots of possibilities, I wouldn't want to give the
impression that this was typical 14th-century practice, comparable to
the very widespread element of chromaticism in the 16th century; it
appears to be a kind of "local" practice among a circle of composers.

In the summer of 2000, this moved me to ask the question "What if?" --
how might we use tunings with diesis-like steps (e.g. with fifths
around 704 cents) to develop a "neo-medieval chromaticism" further?
Since then, I've come upon some surprising possibilities, with many
more out there to be discovered and developed.

Looking at different historical practices can also suggest modern
tuning systems that curiously seem to combine elements more or less
suggesting different practices. Since I know that you like 72-tET, a
24-out-of-72-tET subset occurs to me here.

While 24-out-of-36-tET with two 12-tET keyboard at 1/6-tone apart is
great for neo-Gothic styles with near-just 12:14:18:21 and 14:17:21
sonorities, let's look at a different kind of subset with features
suggesting both "neo-medieval" and "neo-Renaissance" elements.

Suppose we take two 12-tET keyboards at 1/12-tone apart, getting
something like this:

117 217 617 817 1017
C#'3 Eb'3 F#'3 G#'3 Bb'3
C'3 D'3 E'3 F'3 G'3 A'3 B'3 C'4
17 217 417 517 717 917 1117 1217
---------------------------------------------------------
100 200 600 800 1000
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 200 400 500 700 900 1100 1200

Here I've used an apostrophe ('), resembling Vicentino's "comma" sign,
to show the smallest interval in 72-tET at 16-2/3 cents. Of course,
people are free to substitute the symbols they consider most
appropriate, and my choice here is intended mainly to prevent a
conflict with other symbols in a neo-Gothic style of notation:

Let's consider first some neo-Gothic possibilities maybe not so often
discussed in dialogues on 72-tET, but well worthy of recognition.

If we play a major third like D3-F#'3, we get an interval of 416.67
cents, very close to 11:14 (~417.51 cents); likewise F#'3-A3 gives us
a minor third of 283.33 cents, close to 28:33 (~284.45 cents). Playing
a major sixth like D3-B'3, we get about 916.67 cents, close to 10:17
(~918.64 cents) or to the familiar tempered interval of 13/17 octave in
17-tET (~917.65 cents).

Something like

B'3 C4
A3 G3
F#'3 G3
D3 C3

has an "exaggerated Pythagorean" kind of quality with compact 83-cent
semitones, and might be especially affable in a gentle timbre where
the thirds and sixths have a "relatively blending" effect.

Something like

G#'4 A4
E4 D4
C#'4 D4

could be charming in lots of timbres, since the minor third often
tends to keep a relatively concordant quality across a wide spectrum
of intonations. This isn't necessarily to say that minor thirds around
283 cents (about 11 cents narrower than Pythagorean) were likely heard
in the 14th-century, although the variability of ensemble intonation
doesn't make it seem too improbable, only that we have a nice a 72-tET
resource for related styles.

For Renaissance or later tertian styles, we have those 5-limit
sonorities of 0-383-700 cents, e.g. C'3-E3-G'3. Here the main
complication is fingering for octave voicings on two conventional
keyboards, the same problem one runs into with Vicentino's adaptive JI
system in real-time performance in playing just fifths combining notes
from the two manuals of his 36-note instrument.

Also, for a 7-limit style, we get approximations of 4:5:6:7 at
0-383-700-983 cents (optimizing the 4:5), or 0-400-700-983 (optimizing
the 5:7). With a full 72-tET, one could do 0-383-700-967, consistently
optimizing all the intervals.

This may be an example of how a _subset_ of an n-tET, or of an open
tuning for that matter, might be "inconsistent" although a large set
would permit a more consistent optimization for a given sonority or
set of ratios.

I guess that a "Classic-like" resolution of 0-400-700-983 cents to a
near-just 0-383-700 cents might look like this:

B'4 C'5
F4 E4
D'3 G'3
G'3 C'3

Here the upper pair of voices contract from a near-5:7 at 583.33 cents
to a near-4:5 at 383.33 cents, using 100-cent semitones as efficient
as in 12-tET.

It's not consistent in this subset, but I suspect that some people
might consider it preferable to 12-tET, and if so, we have an example
of how "inconsistency" doesn't exclude practical musical advantages
for even a relatively small tuning set like this one.

Joe, thank you for all your contributions here, and also for sharing
with us your experiences as a musician and a resident of New York in
these tragic weeks.

Your activities as a global ambassador of xenharmonic diversity may
represent and promote our yearnings for world peace.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗jpehrson@rcn.com

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

/tuning/topicId_29485.html#29485

Hello Margo!

Thank you so very much for your post which, I believe, could be the
first article in history that projects Medieval and Renaissance
styles onto a subset of 72-tET!

When I first read your post I must confess that I had to view some of
my previous comments with a bit of humor. I know you are frequently
writing about "relativity" in esthetic issues and backgrounds and
this certainly was a good example of it:

In talking about John deLaubenfels interesting experiments
in "altering" traditional Classical and Romantic repertoire, I had
said that I praised such variations because I was rather saturated
with those periods. On the other hand, I was saying that I liked my
Medieval and Renaissance music "straight" as it were, and
historically accurate.

The humorous part of this, of course, is the fact that someone like
*yourself* who is so knowledgeable about those periods is *also*
ready for creative variations in *that* repertoire, in the same way
that *I* am for the more "standard" periods...

The overriding point being, naturally, that one must respect and
recognize the incredible variety of individual responses, depending
upon personal background and experience.... something that I know you
have emphasized time and again in your writings....

thanks again!

Joe

__________ _______ _______
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_29485.html#29550
>
> > --- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
> >
> > > Suppose we take two 12-tET keyboards at 1/12-tone apart,
> >
> > That's one of the first tunings I mentioned on this list, since
> it's an easy way to get 12 wafso-just major triads and 12 wafso-
just
> minor triads, each completing a full circle of fifths. I did not
> consider
> > the neo-Gothic possibilities, so thanks for bringing them to my
> > attention.
> >
>
> Paul... is there any clear relationship between Margo Schulter's
set
> of two keyboards in 72-tET and Vicentino's arrangement??

Sure . . . they're rather analogous, in a sense.
>
> Vicentino used a "meantone," correct, with a second set of just
> intervals that were a comma higher, yes?

Uh . . . Vicentino's second tuning of 1555 was 19-tone meantone with
a second set of 17 meantone pitches which were 1/4-comma higher. The
result is lots of just intervals _between_ the two sets.
>
> Could you please run a short comparison of the two by me again??

I don't think this would be "again", as I don't recall doing such a
comparison before, but here goes . . .

In 1/4-comma meantone temperament, the major third is just, while the
minor third and the perfect fifth are both 1/4-comma flat. Thus, by
adding a second meantone chain 1/4-comma higher than the first, one
has access to perfectly just major triads (root and third from lower
chain, fifth from higher chain) and minor triads (root from lower
chain, third and fifth from higher chain) at every position in which
you have a tempered triad in both of the meantone chains (13 major
and 13 minor triads, at least).

In 12-tET, the perfect fifth is nearly just, which the major third is
sharp by about 15 cents, and the minor third is flat by about 15
cents. By superimposing two 12-tET circles about 15 cents apart, one
can get near-just major triads (root and fifth from higher chain,
third from lower chain) and near-just minor triads (root and fifth
from lower chain, third from higher chain) at every position in which
you had a tempered triad in both of the 12-tET circles (12 major and
12 minor triads).

The former system is decidedly superior for pre-Beethoven triadic
diatonic music, because the maximum "artificial pitch shift" caused
by switching between the two chains is 1/4-comma, or 5+ cents. In the
latter system it is of course 15 cents. But triadic musics that
require the vanishing of, say, the diaschisma or Pythaogean comma,
such as much of Schubert's music, would of course work much better in
the latter system, since these intervals become a full diesis of
about 40 cents in the former system.

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_29485.html#29604

> In 1/4-comma meantone temperament, the major third is just, while
the minor third and the perfect fifth are both 1/4-comma flat. Thus,
by adding a second meantone chain 1/4-comma higher than the first,
one has access to perfectly just major triads (root and third from
lower chain, fifth from higher chain) and minor triads (root from
lower chain, third and fifth from higher chain) at every position in
which you have a tempered triad in both of the meantone chains (13
major and 13 minor triads, at least).
>
> In 12-tET, the perfect fifth is nearly just, which the major third
is sharp by about 15 cents, and the minor third is flat by about 15
> cents. By superimposing two 12-tET circles about 15 cents apart,
one can get near-just major triads (root and fifth from higher chain,
> third from lower chain) and near-just minor triads (root and fifth
> from lower chain, third from higher chain) at every position in
which you had a tempered triad in both of the 12-tET circles (12
major and 12 minor triads).
>
> The former system is decidedly superior for pre-Beethoven triadic
> diatonic music, because the maximum "artificial pitch shift" caused
> by switching between the two chains is 1/4-comma, or 5+ cents. In
the latter system it is of course 15 cents. But triadic musics that
> require the vanishing of, say, the diaschisma or Pythaogean comma,
> such as much of Schubert's music, would of course work much better
in the latter system, since these intervals become a full diesis of
> about 40 cents in the former system.

Well this was a neat explanation... and I understand all of it except
for the very end... Are you saying that the second system with the
two keyboards 15 cents different works better to swallow the comma
because the 15 cents travels a larger distance than only 5+ cents in
the first system?? Could you briefly outline a progression to show
how this works, if you have time??

__________ ________ ______
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_29485.html#29639

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Well this was a neat explanation... and I understand all of it
> except for the very end... Are you saying that the second system
with the two keyboards 15 cents different works better to swallow the
comma because the 15 cents travels a larger distance than only 5+
cents in the first system??
>
> Not quite . . . the second system works better to swallow the
> diaschisma, or Pythagorean comma, or diesis, since all of these UVs
> vanish in the second system, while they become ~40¢ in meantone
and
> the first system.
>
> > Could you briefly outline a progression to show
> > how this works, if you have time??
>
> G--G#=Ab--Ab----G
> E----E----Eb----E
> C----B----C-----C
>
> In the first system, there would have to be either a shift of
40¢
> from the G# to the Ab (or something similar), or (if all common
tones are observed) an overall drift of 40¢ downward between the
first chord and the last chord.
>

Hi Paul...

So with all the common tones maintained, this is, essentially,
the "standard" meantone implementation, where one "wolf" fifth
absorbs the entire comma... (??)

> In the second system, one would have shifts of 15¢ at each chord
> change but no shifts of 40¢ and no overall drift.
>

And, I guess you are saying in the *second* system the common tones
are *not* maintained, yes, but they are reiterated 15 cents
differently each time... thereby dispersing the comma throughout...

Am I getting that??

Thanks

________ _______ _________
Joseph Pehrson

For Joe Pehrson -- 24-of-72-tET (1/12-tone) subset

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>