Re: Saturated sonorities

Paul H. Erlich wrote in Tuning Digest 227

> Margo's used the word "saturated" several times in her post, and it
> seemed not incompatible with my definition of "saturated" (see
> Graham Breed's http://www.cix.co.uk/~gbreed/ass.htm for more
> info). For reference, here are all the saturated chords at each
> harmonic limit through 13 (octave equivalence is assumed):

Hello, there, and we are evidently in agreement on the matter of
"saturated" sonorities.

The list of saturated sonorities leads me to offer a comment on one
matter from a medieval (or medievalist) viewpoint: the definition of a
saturated 3-limit sonority.

The list which you quote proposes:

> 3-limit:
> ratios cents
> 3:2 (=1/3:1/2) 0 702

If one considers only nonoctaval intervals, then this is quite
correct, and of course in much traditional music it is customary to
treat the octave as a kind of "compound unison" rather than a distinct
interval in its own right.

From a medieval viewpoint, however, there are _three_ simple
_symphoniae_ or stable concords: the octave (2:1), fifth (3:2), and
fourth (4:3). According to Johannes de Grocheio (c. 1300), the _trina
harmoniae perfectio_ or "threefold perfection of harmony" is
manifested when _all three_ of these concords are sounded at once:
that is in a sonority such as D3-A3-D4. This "trine" (borrowed from
Grocheio's _trina_) has an outer 2:1 octave, a lower 3:2 fifth, and an
upper 4:3 fourth.

As Coussemaker's Anonymous I (c. 1300) and Jacobus of Liege (c. 1325)
remark, the fourth can also be placed below the fifth, e.g. D3-G3-D4,
but this arrangement is not as harmonious as the first.

If one follows this medieval approach and recognizes the 3-limit trine
as the "saturated harmony" of the era, then our table might be revised
to read as follows:

3-limit:
ratios cents
2:3:4 0 702 1200
1/2:1/3:1/4 0 498 1200

Of course, such recognition of the octave as a "real" interval
specifically in a 3-limit (or specifically medieval European 3-limit
setting) might raise the question of why this interval might be
treated as more of a duplication in 5-limit or higher systems. There
are at least two possible answers, other than the argument from
"historical correctness" in following medieval theorists themselves.

First, the complete 3-limit trine has the interesting property of
consisting of three successive partials, with the lowest partial an
even power or two (i.e. an even octave) of the fundamental --
2:3:4. It shares this property with the 5-limit major triad. In
contrast, a 5-limit triad plus octave (4:5:6:8) does not share this
"adjacency" property.

Secondly, the octave (2:1) is only one intuitive "consonance category"
removed from the most richly stable trinic (3-limit) concords, the
fifth and fourth. We might possibly say that the octave is "2-limit,"
the fifth and fourth "3-limit." In contrast, an octave added to a
5-limit triad is _two_ such categories removed from the richest stable
intervals (the fifth of the triad being one category removed, like the
octave of a trine).

More experientially speaking, I do tend to hear that moving from a
simple 3:2 fifth to a complete 2:3:4 trine (including octave and upper
fourth) "adds more" than moving from a complete 4:5:6 triad to 4:5:6:8
(with the octave).

The first point might be more theoretically neat than the second,
since the 6:4 fifth of 4:5:6:7 is also two categories removed from the
richest stable intervals of this 7-limit sonority, the 7:4 minor
seventh and the 7:5 diminished fifth. Yet it's unlikely that anyone
would argue that this fifth is not a "real" interval.

In a medieval setting, I would go with the complete trine as the
definition of 3-limit saturation; for a more general kind of theory,
the issue may be more controversial. One possible argument in favor of
considering the trinic octave as a "real" interval is that we then
have a 3-limit otonality (2:3:4) and utonality (1/2:1/3:1/4). This
would permit a certain gratifying symmetry with 5-limit major and
minor triads.

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo Schulter wrote,

>> 3-limit:
>> ratios cents
>> 3:2 (=1/3:1/2) 0 702

>If one considers only nonoctaval intervals, then this is quite
>correct, and of course in much traditional music it is customary to
>treat the octave as a kind of "compound unison" rather than a distinct
>interval in its own right.

I did say "octave equivalence is assumed" so all trinic variants fall under
a single category, the 3-limit diad.

Clearly Renaissance music was "saturated" with saturated 5-limit sonorities;
Vogel has interpreted the first few bars of Wagner's _Tristan_ in terms of
the two saturated 7-limit sonorities. As 12-tET is incapable of
distinguishing the two saturated 9-limit pentads from one another, and also
merges the two saturated 9-limit tetrads into a single chord, does a
historical application of this theoretical construct necessarily stop at the
7-limit?

Nachricht geschrieben von INTERNET:tuning@onelist.com

>Clearly Renaissance music was "saturated" with saturated 5-limit
sonorities;
Vogel has interpreted the first few bars of Wagner's _Tristan_ in terms of
the two saturated 7-limit sonorities. As 12-tET is incapable of
distinguishing the two saturated 9-limit pentads from one another, and also
merges the two saturated 9-limit tetrads into a single chord, does a
historical application of this theoretical construct necessarily stop at
the
7-limit?<

The history of tonal music in 12tet certainly does! While there are some
registral or spacing techniques avaiable which might assist the listener in
disambiguating the sonorities mentioned, the last attempts to extend the
representation of higher identities by 12tet collapsed under their own
inconsistancies. Witness the static (non-transposable) pseudo-harmonic
constructs of Skryabin and Obuchov, Schoenberg's letter to Yasser, and the
attempts by Schoenberg and Hindemith to describe 12tet intervals in terms
of harmonic series. Even in recent years, in such works as James Tenney's
_Chorales_ for orchestra, one finds music in 12tet where the representation
of segments of the harmonic series collapses under the limits of the
approximation.

BTW, it might be useful to find an alternative term to _saturation_, which
is in current use in another area of music theory. _Fullness_ or _harmonic
density_, perhaps?

Daniel Wolf
Frankfurt am Main

Daniel Wolf wrote,

>BTW, it might be useful to find an alternative term to _saturation_, which
>is in current use in another area of music theory. _Fullness_ or _harmonic
>density_, perhaps?

Yuck. Saturation expresses the concept much more correctly, with a meaning
similar to that in organic chemistry ("saturated fats"), for example. When
Graham Breed speaks of "anomalous saturated suspensions", he is referring to
yet another chemical analogy. What is the other music-theoretic meaning of
saturation?

Nachricht geschrieben von INTERNET:tuning@onelist.com
>What is the other music-theoretic meaning of
saturation?<

Someone from the Princeton camp can probably provide a more official
definition, but I'll give it a try. A piece of music is saturated to the
degree that all of the parameters project the same underlying structure,
for example, an ordered set of intervals with all possible combinatorial
partitionings.

I may be completely wrong about this, but it is my impression that the term
arose as a prefered alternative to 'complexity' during the complexity
discussions earlier in this decade. Works by Ferneyhough, for example, were
notationally complex, but represented a low level of saturation, while
works by Babbitt were highly saturated if not as complex notationally.

This is not irrelevant to the tuning list! Some of Kraig Grady's pieces,
especially those based on CPSes, exhibit a high degree of saturation, and
are definitely more audibly saturated than works by Babbitt, if not
determined in as many parameters. Some of Partch's works (Dark Brother,
Even Wild Horses) are much more saturated with the diamond then others
(Castor & Pollux -- which he called 'atonal' -- or Petals). Pieces by
Ferneyhough, on the other hand, consume sets of pitches, usually
microtonal, without regard to any underlying structure. Through the use of
strong gestures, he is sometimes able to evoke a kind of coherence similar
to that found in some free improvisation, but the lack of saturation is
ultimately tiring.

> From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>
> Pieces by
> Ferneyhough, on the other hand, consume sets of pitches, usually
> microtonal, without regard to any

Dan - do you mean the 30-tone thing in the flute piece? Or otherinflective
microtonal use in the ensemble pieces? Both?

> strong gestures, he is sometimes able to evoke a kind of coherence similar
> to that found in some free improvisation, but the lack of saturation is
> ultimately tiring.

Ultimately tiring to the performer, some might say.

Rick

I have had a hard time understanding this but your definition is getting me
closer. I think. Would Walter O'Connells use of all interval sets fall into the
catagory of maximun saturation. in 12 et these are:
C E F# G
C Db EB G
C D F Gb
C Db E F#

All 12 intervals can be generated bu these 4 note sets

Daniel Wolf wrote:

>
>
> Someone from the Princeton camp can probably provide a more official
> definition, but I'll give it a try. A piece of music is saturated to the
> degree that all of the parameters project the same underlying structure,
> for example, an ordered set of intervals with all possible combinatorial
> partitionings.

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Nachricht geschrieben von INTERNET:tuning@onelist.com
>
> From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>
> Pieces by
> Ferneyhough, on the other hand, consume sets of pitches, usually
> microtonal, without regard to any

Dan - do you mean the 30-tone thing in the flute piece? Or otherinflective
microtonal use in the ensemble pieces? Both?

> strong gestures, he is sometimes able to evoke a kind of coherence
similar
> to that found in some free improvisation, but the lack of saturation is
> ultimately tiring.

Ultimately tiring to the performer, some might say.

Rick
<

In one of the solo flute pieces, Ferneyhough uses pitches selected from
31tet, 24tet and 12tet. Most of his other pieces imply 24tet from the
notation, but my understanding is that he is not after precise
24th-root-of-two intervals, but rather microtonal inflections of 12. At
Tenney's Darmstadt lecture, Ferneyhough was rather pointed in saying that
he heard 12tet intervals as broadly defined, i.e. inflectable, without
reference to their rational neighbors.

This is quite different from the approach of either Babbitt or Balzano, who
both are able to hear their chosen temperaments (12 and 20, respectively)
by focusing on group symmetries. Anyone with doubts about this ought to
meet either of these guys -- they both have fantastic ears and get around
in their respective tunings with incredible ease. (Balzano chose 20tet
because it was a division of the octave with the same number of factors as
12tet).

As to Ferneyhough's scores and performer fatigue, I can only say that there
are quite a few players who find his pieces tremendously rewarding to play,
in part because the impossibility of exactly executing the notation forces
the player to become creative within the disciplined environment of the
notation. This has always struck me as a bit dishonest, but it doesn't seem
to be an issue to either F. or his players, and the world's certainly big
enough to acommodate disagreement on this point.

Daniel Wolf