symmetric chords and ETs

Joe Monzo wrote [about dim7 chords and aug triads]:

> Some have argued that composers would not have used these
> two particular chords in the particular ways they did (and do),
> especially around 100 years ago (say from around Wagner to
> Schoenberg and Debussy), if 12-tET had not become the accepted
> norm and thus influenced these composers's thinking.
>
> 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.

There's another take on Schoenberg's and Debussy's use of symmetrical
chords, quite opposite from Joe's and Daniel Wolf's: Yasser's. Lots of you
will probably remember the chapters in question, but some might never have
come across these ideas so I'll give a quick overview (with the caveat
that I don't 'buy' the ideas, though I think they're extremely interesting
and the products of a very fertile mind) --I just reread Yasser a few
weeks ago and he's great.

In Yasser's book *A Theory of Evolving Tonality*, (from the 30's I think)
he suggests that musical systems tend towards the more complex over time,
developing according to a Fibonacci-like sequence that has given rise to
pentatonic, heptatonic and dodecaphonic systems in the West, and that the
next logical step should be 19-tones per octave (arguments for this point
of view take up most of the book; I haven't done the idea justice).

Towards the end of the book Yasser explains that certain composers are
sensitive to this historical development and its implications; they
recognise, dimly, the potential of Yasser's "supra-diatonic," and their
inner ear gropes towards the sonorities of the 19-tone system. When
Schoenberg uses all twelve chromatic tones, says Yasser, it is because he
*wants* (subconsciously) to hear the 12-out-of-19 collection (which would
replace our 7-out-of-12 as 'basic' subset) but the closest he can
approximate it, in 12tet, is to use all 12 chromatic notes. Likewise when
Debussy uses a wholetone scale, it is because he intuits the 'basic hexad'
of 19tet (I think it's built by taking every other note of the
12-within-19 scale, analogous to diatonic 7th chords in 12tet) but the
closest he can approximate it is by the wholetone scale. Perhaps most
interesting is that Yasser explains Scriabin's 'mystic chord' (which can
be seen as a wholetone collection with one note shifted by a semitone) as
Scriabin's *own* approximation of the basic 19tet hexad, technically not
as accurate as Debussy's but capturing the essential assymetry of the
19-tone version.

So Yasser, then, believed that no one would use these symmetrical, sterile
chords for their own sake. Rather *in spite* of their symmetry
("sterility") they are chosen because they approximate particular
*asymmetrical* chords from the next musical system, just around the corner
(which alas we're still waiting for, as far as the mainstream is
concerned).

best wishes --jon wild

Hi Jon, and thanks for bringing up Yasser. His book was a great inspiration
for me, but unfortunately Yasser's logic rests essentially on numerology (in
that he searches for number patters without any underlying reason as to
_why_ they should occur), and a very _selective_ use of acoustical ratios
(when they support his theory, not when they don't).

An appoach inspired by, and similar to, Yasser's was taken by David
Kraehenbuehl and Christopher Schmidt in "On the Development of Musical
Systems", _Journal of Music Theory_ vol. 6 no. 1 (1962) pp. 32-65. Unlike
Yasser they put everything in just intonation, but like Yasser they use
arbitrary manipulations of numbers based on patterns without giving any
reasons for them.

You may enjoy reading my paper _Tuning, Tonality, and Twenty-Two-Tone
Tempermant_ (Xenharmonikon 17 or
http://www-math.cudenver.edu/~jstarret/22ALL.pdf) which combines a
Yasser-like proposal of a hierarchy of simple to complex tonal systems with
a Partch/Vogel view of what the "next level of complexity" would be for the
basic harmonic unit after the familiar 5-limit triad (namely, the 7-limit
tetrad). Although I've written it very concisely, I hope you'll agree it
provides ample justification for all its assumptions, rather than simply
following perceived number patters. (Please ignore page 20; it is in error).

Anyway, as far as chords like the augmented triad and diminished seventh, I
think the relevant point is simply that, by any of the measures of
dissonance in the psychoacoustic literature, a JI tuning would not minimize
dissonance, even though it does so for most other chords.

Paul E wrote:

> thanks for bringing up Yasser. His book was a great inspiration for
> me, but unfortunately Yasser's logic rests essentially on numerology
> (in that he searches for number patters without any underlying reason
> as to _why_ they should occur),

hi Paul, Yasser was a great inspiration for me too - I found, and read,
him and Mandelbaum, quite by accident, the first time I wandered into the
music library at the college I was at at the time, before I had any clue
what music theory was. I'd say, though, that his patterns aren't as empty
and numerological as you seem to say - for example the Fibonacci-like
sequence 5-7-12-19-31 etc occurs because each stage conflates the two
earlier stages: the chromatic came about by combining the diatonic with a
configuration of the earlier pentatonic (the "black keys", slotted in
between the usual diatonic). (Just to be safe: while I think it's an
appealing theory in some sense, it's definitely a pretty kooky one and not
one I'd defend as being true -- but Yasser *does* present a rationale for
the patterns, one that is more than mere numerology.)

> An appoach inspired by, and similar to, Yasser's was taken by David
> Kraehenbuehl and Christopher Schmidt in "On the Development of Musical
> Systems", _Journal of Music Theory_ vol. 6 no. 1 (1962) pp. 32-65.
> Unlike Yasser they put everything in just intonation, but like Yasser
> they use arbitrary manipulations of numbers based on patterns without
> giving any reasons for them.

Yes, I came across this article once but was disappointed in it. My memory
tends to associate it with a Carlton Gamer article of similar vintage, in
the same journal, which I enjoyed much more. It looked at deep scales in
different ETs, but I can't remember whether there were hierarchical ideas
like Yasser's, and Kraehenbuehl/Schmidt's.

> You may enjoy reading my paper _Tuning, Tonality, and Twenty-Two-Tone
> Tempermant_ (Xenharmonikon 17 or
> http://www-math.cudenver.edu/~jstarret/22ALL.pdf) which combines a
> Yasser-like proposal of a hierarchy of simple to complex tonal systems
> with a Partch/Vogel view of what the "next level of complexity" would
> be for the basic harmonic unit after the familiar 5-limit triad

Thanks - I read it a couple of years ago, and liked it - have there been
any recent changes? The other proposal of a simple-to-complex hierarchy of
systems that comes to mind is Carey and Clampitt's paper on Well-formed
Scales, in Music Theory Spectrum, maybe '89? or '87? Their sequence goes
...7, 12, 17... and I can't remember what comes next, but that too
reminded me of something Yasseresque. You know, even lots of John
Clough-type scale theory, with the ideas of "sub-diatonic" and
"hyper-diatonic", seems to owe a greater debt to Yasser ("infra-diatonic",
"ultra-diatonic") than anyone ever acknowledges. Or how about Clough's
"second-order maximally-even" business - these principles are basically
what Yasser follows in choosing his 6-within-12 hexad from the
12-within-19 scale of 19tet.

One last thing tangentially about Yasser - someone asked me if I knew what
was the earliest use of 'Locrian' to name the last mode of the major
scale. I'm sure it's older, but the earliest I could find at the time was
in Yasser! Does anyone have an earlier source?

best wishes --jon

p.s. by the way Paul, I'm out of town now, but I live around Somerville so
when you're playing again in Cambridge/Somerville drop me a line.

I wrote,

>> You may enjoy reading my paper _Tuning, Tonality, and Twenty-Two-Tone
>> Tempermant_ (Xenharmonikon 17 or
>> http://www-math.cudenver.edu/~jstarret/22ALL.pdf)

Jon Wild wrote,

>Thanks - I read it a couple of years ago, and liked it - have there been
>any recent changes?

Nothing major, besides the omission of page 20. I can mail you the latest
version if you like.

>Or how about Clough's
>"second-order maximally-even" business - these principles are basically
>what Yasser follows in choosing his 6-within-12 hexad from the
>12-within-19 scale of 19tet.

I had a very interesting exchange with Clough on those ideas, where I
proposed an alternate definition of second-order maximal evenness which more
efficiently produces the 'correct' scales in his Indian music paper, though
ultimately I think the Indian 7-out-of-22 (unequal) scales are better
understood acoustically.

>p.s. by the way Paul, I'm out of town now, but I live around Somerville so
>when you're playing again in Cambridge/Somerville drop me a line.

Sure -- you're also welcome to stop by and play with my instruments anytime.

Jon Wild wrote,

>Yasser *does* present a rationale for
>the patterns, one that is more than mere numerology.

Although it's not numberology in the conventional sense, it's tantamount to
the same thing in a few places, for example, where he tries to determine the
number of notes in the basic chord of the next system. He ends up choosing a
hexad because 6 equals 2 (the number of notes in the basic infra-diatonic
chord) times 3 (the number of notes in the basic diatonic chord). Kooky!

Jon Wild wrote,

>The other proposal of a simple-to-complex hierarchy of
>systems that comes to mind

Don't forget Balzano. Like Yasser, he makes an extremely elegant
presentation for his case, which ultimately boils down to number patterns
that, though the author makes them _seem_ musically relevant, it's more an
act of intellectual sleight-of-hand.

Daniel Wolf wrote,

>I really beg to differ on this one. Balzano really trained himself to hear

>12tet and 20tet by systematically learning all of the simple divisions
(i.e.
>12 by 2,3,4,6; 20 by 2,4,5,10) and then the cyclic patterns (i.e. by
>relatively prime divisions like 5 or 7 in 12). This is basically the same
>mode of hearing as that demanded by Princeton-style 12-tone music, so your

>evaluation of whether it is "musically relevant" or not may be essentially
an
>aesthetic one. I find that with a small bit of practice, one can learn to
>recognize basic tri- and hexachords, as well as complete aggregates, _by
ear_
>in 12tet pieces by Hauer, Webern, Schoenberg, Babbitt. I can't imagine why

>it would be otherwise in 20, which shares the same number of divisors as
12.

That's all well and good, but doesn't address the point. This mode of
hearing would work for any ET, but Balzano singles out 12, 20, 30, 42, . . .
. Balzano claimed that _tonal_, _diatonic_, _triadic_ behavior was governed
by the fact that the size of the major third (4) times the size of the minor
third (3) is equal to the size of the whole set (12). His presentation of
this case is extremely elegant but unfortunately abstract enough to conceal
its true nature -- a sly fallacy. If Costeley or Vicentino had won out, our
tonal, diatonic, triadic music would be in 19 or 31, not 12, and the Balzano
theory falls flat on its face.