Expert: Marbeth - 6/7/2007

Question
Hi, I've just found out that it was common for sacred Mediaeval music to be sung in parallel 5ths, and that it was common for 'chords' to have no third, but I also know that all this was later forbidden under the rules of voiceleading used by Bach and co. What happened in between to make this change possible?
Thanks very much for your help, I can't find out about this anywhere!
Sophie

Answer
Let me preface these remarks by saying that just because something [ex.:  polygamy] seems to be a "DNA imperative" [the male of a species wanting to contribute to and dominate the species propagation] doesn't mean that it is acceptable behavior in a given society and/or a given era, despite its currently being practiced elsewhere [Senegal] or at another time [Moses, Abraham, David, and other Old Testament patriarchs].

I know you are wondering how polygamy and the harmonic series - - and parallel fifths - - are related topics, so let us proceed.

It wasn't until the Renaissance (about the time of Josquin) that our "modern" dominant-tonic
(V > I ) tonal relationship was moving fairly well toward being established as common practice.  This transition was completed in the early 1600s.  Along the way, other tonal organizing relationships flourished - - modal music of the English virginalist school, for example.  

Medieval chant was doubtless meant to be sung in unison.  As we know, all singers are not created equal.  Most of the monks sang the same note (unison).  The boys sang the same note but higher (octave).   But - - and this is my speculation - - SOME of the monks couldn't carry a tune in a bucket with a lid on it, so they fumbled around until their throats produced a pitch that was at least related to the one everyone else was singing.  Related how?  At the fifth.

I am sure you've been singing with a crowd - - let's say "Happy Birthday" warbled at your local pub - - and are aware that some people are not "in tune" with the rest of you.  How?  At the fifth.

When the singing continues beyond one note, this creates parallel fifths.

There is a recording of a 1965 civil rights march in Selma (AL) in which we hear "We Shall Overcome" sung on the correct notes AND on a fifth above them.  

Why the fifth?  Enter the harmonic series.  

The harmonic series is a "ladder" of higher pitches (called "harmonics") derived from the main pitch  ("fundamental").   Note: The higher pitches on the ladder are also called "partials" and even "overtones," though the latter is an incorrect use for this instance.

As the ladder ascends, the partials farther away from the fundamental are not related to the fundamental as mathematically-elegantly as the earlier ones.  They are also more closely spaced, as measured by half-steps and, compared with our modern equally-tempered scale, are somewhat out of tune.

The harmonic series is all about wave lengths and cycles per second.  Let me make a coarse summary by saying that the fundamental and all its octaves are numerical multiples of each other.  The other partials in the series do not have such a tidy mathematical relationship with the fundamental, though some are still simple ratios; 3:1, for example.  Therefore, the farther up the ladder the pitch is, the less consonant it is with the fundamental that spawned it.  More on that in a moment.

If you look at the harmonic series, you see some very interesting things.  

The second note in the series - - called the "second harmonic" - -  is an octave above the fundamental (also called the "first harmonic"), and the third note - - the "third harmonic " - - is a *fifth* above the previous pitch.  ( V > I !)

From strictly a physics point of view, octave and fifth displacements seem almost ordained in the human larynx, as well as in the music of the spheres!

example:  
middle C (fundamental)
C on the next-to-top space on the treble staff (second harmonic)
G just above the top line on the treble staff (third harmonic)  - - G is V when C is I.

Now, then, suppose we go on to the fourth note in the series ("fourth harmonic").  It lies at an interval of a 4th above the third harmonic.  And guess where that lands us?  Right!  On a note that's has an octave relationship with the fundamental!

To expand the previous example, this new pitch is another C - - the one that is a 4th above G.   The intervals of a 4th and 5th are inversions of each other.  So, if you turn a 4th upside down, you get a 5th.  Everything is still I and V!

So, the first four notes of the harmonic series on C are Cs, with one G:  middle C, C an octave above, the G above that one, and the C above (commonly called "high C," a note considered the Holy Grail for sopranos).

complete example:  
middle C (fundamental)
C on the next-to-top space on the treble staff (second harmonic)
G just above the top line on the treble staff (third harmonic)
high C above (fourth harmonic)  

I know you are curious, so let me put you out of your misery:  the interval of the Major 3rd appears next in the harmonic series (fifth harmonic).  It is interesting to note that the fundamental and harmonics two through five form a triad.  Or is a triad formed because of the placement of these?

You are wondering how polygamy and Bach have anything to do with chant and first-year music theory, yes?  

To answer your next question, I don't think Bach indicated anything about using or not using parallel intervals.  I think he avoided them instinctively because by this time parallel fifths were deemed unattractive, even though the math in the harmonic series shows that fifths are consonances and a "natural" part of sound production.  

As so often happens, documentation follows practice much later.  Using the great master's examples, we examined what he did.  And what he did not do:  parallel fifths.

And so it was written in theory books.

While tin-eared monks sang in parallel fifths, theory harmonization exercises should not reflect "historical performance practice," even though the math of the matter is supportive!
mb

Polygamy sidebar:  I know the anthropologists reading this essay will take issue with my careless   use of polygamy here, rather than polygyny.