Special Way

Paul Erlich wrote...

>The 7-limit harmony in barbershop is not merely "stuck on" to the
>diatonic scale; it consists of dominant seventh chords which are >diatonic
in function and origin, and the 27-cent adjustments required >to achieve JI
are not large enough to disturb the essential >diatonicity of the music.

You're nitpicking what I meant by "Stuck on"?

>As for the pitch sets being different for melody and harmony, don't >tell
me that the sevenths of these dominant seventh chords never occur >in the
melody!

They do, but not often. If the lead is singing the 7, then usually the
tenor has the melody. There are many songs where the melody contains no
pitches outside the 3-limit. What is important is that the melody is not
really part of the voice leading in the traditional sense, except in the
case of sustained notes. Chords are built _on_ it. What follows in this
post may help to clear the issue up.

>>}I think that most people recognize that barbershop harmonies sound
>>better than piano harmonies. If you can hear the difference, doesn't
>>that make it a "perceptually distinct pitch set"? Maybe I'm just
>>splitting hairs...
>>
>I think that many people will hear the difference but not know whether
>any particular pitch is being adjusted upwards or downwards -- they >will
simply detect a difference analagous to a change in timbral >quality on the
chordal level. An adjustment large enough to evoke a >non-diatonic pitch (I
mean diatonic in the sense of several closely >related heptatonic keys),
however well-motivated harmonically, will >have a different effect,
probably not one that most people would find >pleasant.

I usually use "pitch set" to mean "all the pitches you need to play the
thing". Mr. Erlich's "pitch set", as used above, is something very
different. If I am not mistaken, he uses the term "pitch set" above to
touch on a model that describes a certain aspect of music: Where harmony
and melody interact in a Special Way.

What is this Special Way? Krumhansl, quoted by Mr. Erlich, seems to be
touching on it: "If chord construction is determined in some principled way
by scale structure, then this further serves to maintain the tonal
framework for encoding pitch information."

Most western classical music happens to work in this Special Way, and the
thing it uses to do it is the diatonic scale. Mr. Erlich's goal, if I'm
not mistaken, is to find a new scale suitable for use in this Special Way.

To recap so far, I am using "Special Way" to mean a way of explaining
beauty in music from how melody and harmony are related, a construct
heretobefore unknown to music theory, at least at the level Mr. Erlich
takes it. In his paper, it is called "Generalizing diatonicity". I
suggest this is an unfortunate term, as it implies the diatonic scale,
which is exactly what he wants to replace, kind of how "octave" isn't a
good way to say "2/1", since it implies 12-tone...

>I think Schonberg may have been right that diatonicism had been played
>out, but in abandoning the concepts of tonality and scale he threw the
>baby out with the bathwater.

..Here, Mr. Erlich uses the words "tonality and scale" to mean Special
Way. His post then translates: "I think Schonberg may have been right that
the diatonic scale has been exhausted as a tool for making music in the
Special Way, but he threw out the Special Way, which wasn't what was used up."

Now follows a review of his paper, which hopes, it seems, to show just how
the Special Way isn't used up, and just how you can tune an instrument
that'll make the Special Way seem as new as 1998.

---------------------------------------------------------------

The author starts by observing that the Western Music is 5-limit, and,
following the "music evolves up the harmonic series" thing, sets his sights
on the 7-limit.

He then defines a set of scales as candidates for representing the 7-limit.
These scales are assumed to be root of 2 equal-step tunings, with only one
pitch per 7-limit approximation.

The one pitch per approximation thing is necessary for the Special Way to
work, as far as I can tell. I can't quite tell you how, but I have a gut
feeling it is, and since the Special Way is what we're after, I won't argue
it. Besides, it keeps the size of the pitch set manageable.

The root of 2 part is understandable, considering that we need strong low
identies for our 7's to work. This seems contradictory to the rule that
the higher identities are more sensitive to mistuning, since there are more
low-numbered fractions near them. Indeed it is, and perhaps it is a
counterbalance to this principle. The paper offers only that "octave
equivalence seems pervasive" and that it is "universally perceived, even by
some animals".

The matter of mistuning is far from clear, even touching back on our old
bone about unknowingly passing low-numbered ratios when measuring cent
detuning of an interval. Paul's paper addresses this by making the
standard deviation in log-frequency detuning inversely proportional to the
limit of the interval. Not ideal, but better than any other method I've
been able to think of, and good enough for root of 2 equal tunings from 12
to 31.

The equal-step part is the most dubious. The Special Way has always been
based on temperament of some kind, and it seems that only Bog or God can
decide if all those schkissmas need to disappear for it to work, insomuch
as Mr. Erlich has never shown it. He has mentioned, on this digest, some
modulatory effects that require two D's to be the same note, etc, but this
is a problem of trying to retune music already written. That Special Way
music cannot be written for JI, complete with its own list of modulatory
effects impossible in a temperament, has not been demonstrated.

So the list of scales comes down to 22, 26, 27, and 31 tone equal temperament.

Then are listed criteria for determining a scale's usefulness for the
Special Way. Since the Special Way is a relationship between a scale's
melodic and harmonic usefulness, the criteria are separated in these two
groups...

1. Melodic. In Paul's paper we have maximal evenness and tetrachordality.
In Gibson-land, we have the melodic limen. Jules Siegel proposed that the
intervals should get smaller as you go up the scale. None of these
convince me in the least even that a scale can "work" melodically. Maybe
the most useful thing I've heard for describing the melodic properties of a
scale is "symmetry". I will accept, however, that the criteria used by Mr.
Erlich are, "enough to ensure an intelligible melodic framework".

2. Harmonic. Now we're talking. This is where the real relations between
melodic and harmonic are drawn...
(a)"There exists a pattern of intervals" ... "which produces a complete,
consonant chord on most scale degrees"
(b)"The majority of consonant chords have a root that lies" the best
approximation of a 3/2 "away from another consonant chord"
(c)"A chord progression of no more than three consonant chords is
required to cover the entire scale"

..in what would be letter "d", Mr. Erlich makes use of a term I do not
understand: "characteristic dissonance". He defines it to be any dissonant
interval that shares the same number of scale steps as a consonant
interval. Shadings of dissonance aside, what kind of scale steps we
talkin'? The example of the diminished 5th is given, but why it should be
considered a type of 5th, or why the P5 should not be considered a 7th is
not made clear.

(e)The rarest intervals in the scale should be located next to notes in
the tonic chord. Like leading tones.

What would be "f" makes use again of the term "characteristic dissonance".

The number of scales fitting these criteria is shown to be few. Among them
are the 5-limit diatonic and the 3-limit pentatonic, two scales proven over
hundreds of years to be great for the Special Way. While this shouldn't
come as much of a surprise, considering the author admits to basing his
criteria on the success of these scales, any time such a unique set of
properties can be found, they're probably worth looking into.

Finally, it is shown that the decatonic scale in 22TET fits the criteria well.

So has the paper provided a good definition of the Special Way? Yes. Has
it succeeded in convincing me that the Special Way is still good, and that
the 22TET decatonic scale is a fresh vehicle for it? Without a doubt.
What it hasn't provided is a name for the Special Way. Maybe "tonality" is
best, but I didn't use it because it has so many other meanings.