From: fields@zip.eecs.umich.edu (Matthew Fields)
Subject: GEMS 2 [repost]
Organization: University of Michigan EECS Dept., Ann Arbor, MI
Date: Tue, 17 Nov 1992 21:14:33 GMT

GEMS 2
==== =

Matthew H. Fields

I mentioned a willingness to post some general and specific observations
regarding music composition, and so far, I've received an enthusiastic
response.  Therefore, this is the second such posting.

In my first GEM article (named after the phrase 'gems of wisdom' that
was passed around a great deal in the discussion that preceeded the
first such posting), I discussed dramatic shape and climax-building,
and passed on several famous hints for building better climaxes to
dramatic musical works.

Today's presentation is a bit more philosophical, and takes a more
round-about route towards being helpful to composers.

The topic for today is:

PARALLEL FIFTHS AND OCTAVES --- WHY I BOTHER ABOUT THEM

I have chosen to present this topic here in my sequence because most
of my later proposed articles will be written assuming you have some
idea of my biases regarding countrapuntal issues.  This article will
not contain any hints or suggestions regarding composition, but will
instead talk about some meta-issues of perception.

Disclaimers: I am presenting the material here mainly as my opinion.
If you try to make use of my suggestions and they don't help you write
fabulous music, I don't accept any liability.  Likewise, it is
strictly to your credit and none of mine if you do write fabulous
music before or after reading these posts.  Plenty of the ideas I will
be discussing in this series have been mentioned before, and some
theorists may even wish to lay copyright claim or patent claim to some
of them.  However, I claim that the core ideas have been known to
composers and used by them long before anybody published any writings
on them, and these ideas are therefore basically in the public domain.
In fact, some of these ideas have even been bandied about on
rec.music.compose in recent weeks, often quite well.

On the other hand, I actually sat down and wrote the text of this
posting, and it took me a bit of time and thought, so if anybody were
to exploit this text as a commodity without consulting me, I might get
very mad (standard disclaimer).  Furthermore, to the best of my
knowledge, at no time have I herein explicitly quoted anybody's
special article: this prose is all mine.

All that having been said, I am interested in getting some feedback
on how interesting or useful you find this article.

ABSTRACT 
In this article, I will describe a perceptual basis for being careful
concerning the use of parallel octaves and fifths.  I don't expect to
convince anybody to take on such a concern, and I most especially will
not hand out any rules, generative or proscriptive, on this matter.
On the other hand, it is my intent to argue that this concern is not
obsolete but current, and not a matter of abstract rule-making, nor a
matter of mystical invocation of physics, but rather a matter of
hearing and musical expression.


INTRODUCTION 
Parallel octaves and fifths: we hear of a 'proscription' against them
in our music theory classes.  Then we find out that Bach's organs had
8-foot, 4-foot, and 3-foot stops, so that every melody he played could
be sounded out in parallel octaves and fifths.  Even worse, we
discover that parallel octaves are ubiquitous in ensemble music and
piano music.  And then, as we delve into musical history, we discover
early forms of organum in which singers always sang in parallel
fifths.

Why, then, is a big deal made about these things in theory classes?
and why these intervals, only, and not thirds, sixths, and sevenths?
What is the role of dogma and propaganda in this matter?


As I so often do -- perhaps it's a Jewish habit? -- I'll begin my
answer with a story.  No, not "we were slaves in the land of
Mitzrayim", but rather: once, I was teaching the rudiments of aural
skills to a total beginner, and he was working on the game of "name
that interval", meaning that given the sound of two pitches played
either sequentially or simultaneously, he was to name the interval
between them.  He complained at one point that he was having a bit of
trouble hearing octaves and fifths when the notes were played
simultaneously, and he said it sounded like the upper pitch was
somehow 'hiding' behind the lower pitch.  I probed him a bit on this
observation: had he noticed this phenomenon outside of his work with
the aural-skills software?  Yes, he had started noticing it in all the
music he heard.  Did it apply to other intervals?  Yes, especially
strongly to the unison, and quite weakly to the major third.

I was, of course, surprised to hear a beginner mentioning such a
phenomenon.  He had never heard of any rule which made a big deal
about parallel octaves and fifths, and was quite surprised by it
when it came up in his theoretical studies---after all, parallel
octaves are ubiquitous in piano music.  But he was a dilligent student,
and promptly proposed an abstract theory in which parallel octaves
and fifths were somehow purely timbral events of physics, while other
parallel intervals were events of multiple melodies.

Many authors continue to describe the harmonic series and say, without
further explanation, that it is the cause of the concern with parallel
fifths and octaves.  I think that such a description of the physical
world is not sufficient to describe how certain composers have treated
these materials, but coupling that description with some purely
/SUBJECTIVE/ observations (like the ones my student complained of) may
actually bring us closer to an understanding of the matter.  Even that
will not be enough to explain the concern with parallelisms, though,
since parallelism is a matter of melodic motion, not of how we
perceive individual intervals.

DEFINITIONS
Before I go any further, let's make sure we're all talking about the
same things.

When I say that two parts are in /unison/, I mean that they are
sounding the same pitch at the same time, i.e. in the same octave.
For the acoustically-minded out there, this means that within
tolerances that our ears define, they are sounding the same
fundamental frequency (where applicable).

When I say that one note is /an octave higher/ than the other, I mean
that it sounds the eighth ascending diatonic step from the other, or
is at an ascending distance of seven diatonic steps, or twelve
half-steps (in 12-tone equal temperment).  For the acoustically-
minded, this means tolerably close to a frequency ratio of 2:1, so
A-880 is an octave above A-440, and A-1760 is an octave higher than
that.  Naturally, if I say that a note is an octave /lower/ than a
second note, this means just that the second one is an octave higher
than the first.  Carrying out the arithmetic, we find that the first
note is seven diatonic steps below the second note, or twelve
half-steps below the second note, or tolerably-close to a frequency
ratio of 1:2 with the second.

When I say that one note is a perfect fifth higher than another, I
mean that there is an ascending distance of 7 half-steps between them.
I don't give this definition in diatonic steps, because while the
fifth diatonic step in the C-major scale over C is G, at a distance of
7 half-steps, the fifth diatonic step over B is F, at a distance of
only 6 half-steps.  So, I'm saying that I care about the distance
being 7 half-steps, regardless of where it sits in the scale.  For the
acoustically-minded, the frequency ratio this time is 3:2.  In 12-tone
equal temperment, this ratio (which can be precisely expressed in
decimal form as 1.5) is approximated by the seven-twelveths power of 2
(~~1.498307077, or a little more than 1% flat).

Finally, by /compound interval/, I mean an interval augmented by the
addition of one or more octave to its distance.  In the case of a
perfect fifth, the first few compoundments of it are the perfect twelveth
and the perfect 19th, or distances of 12+7=19 and 24+7=31 half-steps,
or frequency ratios of 3:1 and 6:1 (within tolerances).

The tolerances I mention above have been the topic of quite a lot of
debate over the years, so I'm not going to pin them down, partly
because doing so would not add any vital information to this article.
Mathematicians out there are asked to please refrain from the
temptation to say 'Let epsilon be any positive real number'.  If
anybody is tempted to do that, would they please agree that our
tolerances are less than 2% of the lower frequency for the sake of
this article?  Ok.  I'm not going to talk about quantitative acoustics
much more in this article, because I think it's time to talk about
psychological phenomena.


SO WHAT'S THE BIG DEAL?
All right, we're getting to that.  But first, let's talk about melody.
I THOUGHT THIS WAS ABOUT PARALLEL FIFTHS.
Yes, but we're coming to that, and we have to back up and visit melody
and polyphony on the way.

A long time ago, somebody first started coming up with the notion of
'a nice melody' or 'a nice melodic shape' that some of us still use
today (it's the first thing you now study when you learn species
counterpoint).  The basics of this concept were things like: it had
one and only one climax point, which was typically its highest note,
or sometimes its lowest note; it started on, ended on, and generally
circled around a main note which was supposed to express a sense of
repose;  it moved mainly by step, occasionally by third, and rarely
by fourth or fifth --- any time a string of notes was constructed that
leaped a lot up and down, this was perceived not as a single melody but
rather as a sort of time-sharing between two or more melodies, each of
which moved stepwise (/compound melody/).

Long before people were experimenting with what we now call harmony,
they had gotten pretty good at building interesting and exciting
things that were single melodic lines.  After a while, folks tried two
crucial experiments that forever changed the way people made music: 1)
Two folks got together and sang the same melody at the same time; 2)
Two folks got together and sang different melodies at the same time.
Of course, this last sentence is a gross oversimplification of
history, and is not a documented event anywhere in the world.  But
let's consider the consequences of the two experiments anyhow.  In the
first case, perhaps the people had the same voice range the first time
they tried this, in which case they sang in unison, and the sound
reverberated larger than either of them.  Or perhaps, the first time
they tried this, they had such different voice ranges that they sang
in octaves (Perhaps an evolutionary theorist could explain our ability
to recognize melodic content after transposition in terms of our
needing to recognize the same intonation pattern from adults and
children?).  Now, the first people to try singing two different
melodies together had a much more complicated result.  Certain
combinations of tones came to be called pleasing-sounding, and others,
anxious-sounding; from these basic notions, a variety of complex
systems of consonance and dissonance were developed---which were
different in different eras---and plans were developed for ways in
which various consonances and dissonances could be strung together to
express something vaguely analogous to a sentence-structure.  Meanwhile,
folks were listening to, and enjoying, two melodic shapes at once.
At one point, the two shapes crossed through the same note, perhaps.

The listeners became confused, because just after the crossing, it was
hard to tell whether the voices had bounced off each other like this

i  ---\v/--- i
       *
ii ---/^\--- ii

or crossed through each other like this:

i  ---\ /--- ii
       X
ii ---/ \--- i

Some folks complained that trying to keep the melodies clear in their
heads detracted from their appreciation of the individual melodies as
well as their appreciation of the consonances and dissonances that
arose between them.  So some musicians tried to find pairs of melodies
that eliminated the second possibility altogether, so after a while,
everyone would get used to hearing things the first way anyhow.

Sooner or later, it was bound to happen: the two melodies passed through
two notes in a row exactly the same:
             -----  i
i  ---\__  *<
        _>*  \____  ii
      _/
ii /\/


People had gotten used to keeping the two melodies clear in their
heads for one shared note, but two in a row was just too hard for many
people.  It sounded like one of the melodies had momentarily gone
silent while the other had momentarily gotten stronger or louder.  At
about the same time, ideas of perspective, shadows, and oclusion were
being developed in the visual arts, and people had analogous ideas
brewing regarding making foreground and background shapes all equally
visible and readily enjoyable.  So, some musicians decided that in
their compositions, one was the largest number of consecutive notes in
a row on which two melodic lines would sound in unison, the better to
allow the listeners to follow the shapes of each of the lines up and down.

But the situation in music was more complex.  Some folks, like my
talented student, felt a sense of conjunction and aural oclusion at
not just the unison, but the octave as well, and its compoundments.
These folks decided that when two players were supposed to be playing
different musics, they'd never have two consecutive octaves with each
other, again so the melodies wouldn't seem to hide one behind the
other for too long for their enjoyment of each melodic shape by itself
as well as the overall composite.  Some folks had the same experience
with the fifth and its compoundments, and foreswore parallel fifths
from their multiple-melody expression (counterpoint).  Perhaps some
folks even experienced the same perception with parallel fourths,
thirds, and sixths; if so, those folks probably got disgusted with the
whole thing and went into something like mathematics or geography,
where great new things were being uncovered every day.

Meanwhile, the consequences of experiment number 1 above were still
brewing.  Having worked out several melodies to sound simultaneously,
people sometimes had more resources than melodies.  They quickly found
that two violins playing the same melody could balance one bass or
cello playing another melody better than one of each (due to the
differences in inherent size and loudness of the instruments).
Furthermore, individual melodies could be played by pairs of players
playing in octaves, often without changing much about the effect of
the music except its perceived loudness and strength.  Harpsichord
builders and organ builders made automatic doubling at the upper
octave a feature of their instruments, essentially a simple way of
getting a stronger sound with the same number of perceived melodies.
Orchestrators eventually decided on a rule for groups of players,
which still seems to work pretty well: octave doubling above the
highest melody, and below the lowest melody, but no octave-doubling of
inner melodies, as such doubling was perceived as still confusing to
the ears---except when it was provided by highly-controlled, automated
means, like organ stops, harpsichord stops, or 12-string lutes and
guitars.  Organs even came to have extra pipes to produce parallel
12ths (compoundments of fifths) for an even brighter, stronger tone.

So, for a great deal of western polyphonic (multi-melody) music,
parallel octaves and fifths were considered as falling into two
categories: features of a single melody--often highly-desireable
reinforcements of a melody that contributed to its tone color and
perceived loudness; and momentary interactions between two
melodies--usually considered undesirable, because they interfered with
/some/ listeners' ability to enjoy both melodies to the fullest.

Some people continue to hear in these terms, and find ways to treat
these 'sensitive' parallelisms as either constant features of their
music or things that rarely or never occur in their music.

Composers of the classical era worked out some highly elaborate ways
of constructing contrapuntal music so that it avoided parallel octaves
and fifths---yet didn't sound (to them) highly artificial.  The study
of the methods and tricks used by these composers (which involved the
resolution of a lot of other preferences and conventions as well as
the avoidance of or isolation and control of these special
parallelisms) eventually blossomed into our modern discipline of
classical counterpoint and harmonic theory.  This field and course of
study is now so loaded with interesting tidbits of musical thought
that the concept of parallel octaves and fifths is often dismissed
with the shorthand comment "they're forbidden"---occasionally with a
brief mention of the harmonic series, or of the vague idea that they
interfere with independent motion.  But, of course, the truth of the
matter is a bit more subtle.

LISTENING ASSIGNMENT
Once again, the assignments are purely optional.

Give serious consideration to playing around with parallel fifths and
octaves.  Do your ears tell you anything about them?  Do you have an
attitude about them?  How do you perceive music that avoids them?
(try the first or second fugue from Book One of the Well-Tempered
Clavier of JS Bach) music that uses them constantly? (try the
sarabande from Pour le piano by Claude DeBussy) music that uses them
indifferently? (supply your own example) music that uses them
constantly for long stretches, then not at all, but never
indifferently (try the Tenth fugue in e minor from book one of the
Well Tempered Clavier of JS Bach) ?  See if you can find sources and
recordings documenting the effect of different tuning systems on the
sound of the music. Do your discoveries suggest anything for your own
compositional preferences?

WRITTEN ASSIGNMENT
No written assignment this time.  Go compose.


CONCLUSION
I hope this article was interesting.  In writing it, I've tried to
condense an enormous amount of information and ideas into a small
space.  While the resulting article is still rather long, some of the
topics treated--especially the musical history--are quite eliptical,
abbreviated, and abstract.  However, I hope that for those readers who
find the article too hurried in its descriptions, the subject matter
may at least be intriguing, and those readers may wish to look into it
further, starting perhaps with the New Grove Dictionary of Music and
Musicians s.v. /counterpoint/.

For at least a while I will be keeping a copy of this article here
in my disk directory.  As long as the volume of "reprint" requests
is reasonably manageable, I will offer to send copies out by e-mail.

I can't really tell you when the next article in this series will be
ready for posting, since I haven't written it yet.  The next article
will be aimed at the student enrolled in the typical undergraduate
theory course, who has been asked to demonstrate proficiency at 18th-
century harmonic counterpoint.  It will consist of a very short list
of things to try as shortcuts, so that the reader might finish their
theory homework earlier and have more time available for composing.




4 September 1992  Matthew H. Fields, D.M.A.