Yahoo Tuning Groups Ultimate Backup tuning Schoenberg again

> [me, monz, TD 457.19]
>> I think it's significant that Schoenberg in particular found
>> so much to explore among the 'augmented triads', 'diminished 7th
>> chords', and '4th-chords', all of which pretty much require
>> 12-tET, since it is the smallest ET which can produce them all
>> within a closed system.

> [Paul Erlich, TD 458.10]
> Don't see that for '4th-chords' unless they include 12
> different notes.
> '4th-chords' of 3-5 notes are especially cool in 22-tET
> or any tuning with ~709-cent fifths, since the intevals
> produced approximate 7:4, 7:3, and 7:9.
> '5th-chords' of 4-5 notes are especially nice in meantones,
> where you get approximations to 5:3 and 5:2.

Two reasons why I added the bit about '4th-chords':

1) Schoenberg placed quite a bit of emphasis on the use
of chords constructed out of a series of '4ths' both in
his book and (at least for a few years) in his compositions.
Since we were already discussing 'augmented triads' and
'diminished-7th' chords, and Schoenberg definitely noted
and made use of their ambiguities in 12-tET, I thought
that for completeness I should include the '4th-chords' too.

2) You've pointed out several times (in fact, once again
in this very thread), as one of the examples of 12-tET being
'necessary' for certain chords, the 'major 6/9 chord'
(major triad with added '6th' and '9th').

Your point in examining this chord is that when a harmony is
constructed out of a series of '5ths' their musical context
frequently demands that an equivalence must exist between
the note which functions simultaneously as (3/2)^4 [= 81/64
= ~408 cents] and 5/4 [=~386 cents].

I simply assumed that what's true for a series of '5ths' is
equally true for a series of '4ths'. Care to refute that?

Schoenberg certainly did construct '4th-chords' that had
9 tones, in which case the ninth '4th' would be very close
to a 5/4 (in a higher octave) above the 'root'. It seems
to me that this would entail exactly the same kind of
equivalences as in your 'major 6/9 chord' example. Actually,
the two different functions in the case of the '4th' chord
are much closer in pitch (only a skhisma [= ~2 cents] apart).

And as far as your first point goes, one reason Schoenberg
gravitated towards using '4th-chords' is certainly that
he recognized that all 12 notes of the chromatic scale could
be used in a chord before the cycle is closed, which is
*not* true of tertian harmonic practice, where the cycle
is closed after 7 different notes (unless altered and unaltered
chord-members are simultaneously admitted, as they may sometimes
be in jazz).

During the first decade of this century Schoenberg was
definitely interested in assuming the 12-tone scale in itself
as the tonal basis, rather than the more usual view of
the 7-tone diatonic scale with 5 available alterations.

> [Jerry Eskelin, TD 459.16]
>
> Speaking of Schoenberg, are you familiar with William
> Thompson's book "Schoenberg's Error"? Essentially, he
> proposes that Schoenberg appears to have noted the "trend"
> toward atonality and, because he didn't look back in
> history far enough for a larger perspective, simply jumped
> in front of the parade. His point, of course, is that the
> parade has now largely dissipated.
> (Thank goodness!!!)

Oh my, what a can of worms you've opened now...

Yes, not only am I familiar with _Schoenberg's Error_, but
I was already so interested in Schoenberg's theories (and
music), and found so much wrong with Thomson's approach,
that I decided when I read it a few years ago that my next
book is going to be a commentary built entirely around
Schoenberg's _Harmonielehre_, with a lot of arguing against
several points made by Thomson.

Along these same lines, Daniel Wolf has pointed out that
Martin Vogel has already written a book about Schoenberg
from a very biased JI perspective, _Scho"nberg und die Folgen_.
My German is very limited, but from what I've read so far,
I can say that Vogel makes many of the same points I was
going to, but that he doesn't look broadly enough at the
historical and personal context of Schoenberg's music and
theories.

Certainly Thomson is correct that Schoenberg had a very
narrow view of musical history, and the history of music-theory
in particular, and that Schoenberg's arguments are often
very twisted ways of making his own ideas appear as tho
they're the logical evolution of past trends in music.
But Thomson bases many of his arguments on premeses that
are common among music-theorists but were *not* held
by Schoenberg himself.

Those interested in my work on Schoenberg can find some of
the non-technical stuff here:
http://www.ixpres.com/interval/monzo/schoenberg/Vienna1905.htm

(BTW, it's 'Thomson' without the 'p'.)

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

> [Paul Erlich, TD 460.14]
> Joe, you're remembering incorrectly.

Oops... my bad.

> I say any meantone is fine for a 'major 6/9' chord, while
> the augmented triad and diminished seventh chord definitely
> cry out for something close to 12-tET. What these chords have
> in common, though, is that their ideal tuning is something
> different from JI.

Yes, in fact it's more specific than that: what those chords
all have in common is that the syntonic comma [81/80 = ~22 cents]
must vanish, which is accomplished in both 12-tET and meantone.

At any rate, a 'major 6/9' works fine in 12-tET too, whereas it
doesn't in simple 5-limit JI, which is your point.

My reference there to 'simple' JI begs a new question:
'extended JI' or 'expanded JI' is, I believe, fairly well
accepted for describing JI tunings which exploit higher
prime/odd factors (if I'm not mistaken, it was made current
mainly by Ben Johnston).

And just plain old 'JI' is almost universally used to describe
5-limit JI systems of from 7 to about 12 (and perhaps as many
as 19?) different pitches/intervals, all of which are quite well
defined in their relationship to 1/1.

What to call 5-limit systems which exploit the more remote
lateral lattice connections, such as the use of a 64:75:96
'minor triad' as in my last posting? If the Johnston meaning
of higher-limits is characterized by 'extended' and not by
'expanded', then could this type of larger lateral low-limit
system (aha... more alliteration!) be called 'expanded'?

Any other ideas?

-monz

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

I wrote,

>> I say any meantone is fine for a 'major 6/9' chord, while
>> the augmented triad and diminished seventh chord definitely
>> cry out for something close to 12-tET. What these chords have
>> in common, though, is that their ideal tuning is something
>> different from JI.

Joe Monzo wrote,

>Yes, in fact it's more specific than that: what those chords
>all have in common is that the syntonic comma [81/80 = ~22 cents]
>must vanish, which is accomplished in both 12-tET and meantone.

Meantone doesn't do any better than JI for augmented triads or diminished
seventh chords. You're still left with a relatively dissonant diminished
fourth in the former chord, and augmented second in the latter chord. 12-tET
smooths these out nicely.

>What to call 5-limit systems which exploit the more remote
>lateral lattice connections, such as the use of a 64:75:96
>'minor triad' as in my last posting?

I haven't examined your example closely, but my suspicion is that a whole
slew of irrational renditions of this triad would have satisfied you just as
well, and that you're clinging to low-prime ratios purely for idealogical
reasons.

>The big surprise to me was that there was such an audible
>difference between 75/64 and 7/6 [= ~8 cents difference],
>more so than between 19/16 and 75/64, which are ~23 cents apart!

I think this is explained by the fact that 7/6 can be tuned quite accurately
by ear (to within a fraction of a cent) with most timbres; moving up from
there, the next such interval is 6/5, so between 7/6 and 6/5, you pretty
much have an undifferentiated expanse of mildly dissonant intervals. Look at
the harmonic entropy graph in that region.

>Is 75/64 close enough to some other ratio with larger (but
>not 'too' large) prime/odd-factors, that it is emulating it?

How about smaller odd factors (which is what I think is relevant)? 34/29?
But no, I think you were looking for a qualitative solution, to get
something with a certain "shade of darkness" that Robert Johnson used. I'd
say you've gone beyond the realm where any JI theory can help you and you
just have to trust your ears in picking the right cents value. Forget about
ratios.

You may remember my theory of the minor triad -- it derives its "tonalness"
from the perfect fifth alone, and the minor third is really an "added note".
Although 6:5 may be the minimum-roughness choice and 19:16 the
maximum-tonalness choice, anything can work (though the character is greatly
affected). I've been convinced of that ever since I heard Blackwood's etude
in "17-tone equal tuning", where the minor triad is constructed as
282�+424�. I love that piece!

By the way, I was just reading Mathieu's _Harmonic Experience_, and though
he think we can hear some pretty darn complex ratios in the 5-limit lattice,
he happens to consider 75:64 a "wrongness" (in the context of Mozart).