# Re: CANTUS FIRMUS

clicketi pshh wrote:
>
> Hi, it's the first time that I post to this newsgroup,so excuse me
> in advance if my question was already ansewered a 1000 times :)
>
> What exactly is an acoustic root?

Intervals of pitch exist at various frequency ratios.

The freq ratios for the pure intervals are tied up with the partial
numbers of the harmonic overtone series.

Here are the tones in the overtone series of a fundamental tone A, along
with the partial numbers.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A A E A C# E G A B C# D# E F/F# G G# A

>From this we can see that:
Octaves have a freq ratio of 2:1.
Pure P12ths have a freq ratio of 3:1.
Pure P5ths have a freq ratio of 3:2.
Pure Maj 10ths have a freq ratio of 5:2.
Pure Maj 3rds have a freq ratio of 5:4.
Pure Min 3rds have a freq ratio of 6:5.
Pure P4ths have a freq ratio of 4:3.

The acoustical root of any interval is the note that is at "1" relative
to the interval's freq ratio, i.e. the root of the ratio is the root of
the interval.

So, the acoustical root of A-E, at 3:2 (a P5th), is A an octave below
the A in the interval. I.e. "1" = A.
The ac rt of E-A, at 4:3 (a P4th), is A 2 octaves below the high A in
the interval.
The ac rt of C#-E, at 6:5 (a min 3rd), is a low A.
Etc., etc.

This is fairly simple for basic intervals but can get quite a bit more
complicated for chords.

A chord voicing in which the spacing of the component tones has a strong
proportional mapping to the overtone series of a single fundamental tone
will impart a strong sense that said fundamental tone is the acoustical
root of the entire chord.
A chord voicing in which the spacing of the component tones does not
have a strong proportional mapping to the overtone series of any single
fundamental tone, or in which there appear to be several different
fundamental tones, will not possess a strong sense that any particular
tone is the acoustical root of the entire chord.
I.e. Some chords do not have a strong sense of root at all.
Chords that are spaced very closely to the spacing in the overtone
series have the strongest sense of acoustical root.

> And by the way what exactly is a root? - I mean a non-acoustic root....

The constructional root of a chord is the tone from which the other
component tones of the chord are calculated according to a predetermined
intervallic formula. Sometimes the constructional root and the
acoustical root are not the same tones.

Eg.
A min7b5 chord is defined by the following intervallic formula:
1 b3 b5 b7

So, C#m7b5 would be C# E G B.

But the acoustical root of this chord voicing is A.
I.e. C# E G B = partial numbers 5 6 7 9 in the overtone series of a low A.

> And like the Fermat corollary, given a cantus firmus, how do
> you determine the acoustic or the non-acoustic root that goes below?
>
> TIA!

I'm not sure that this topic relates to species counterpoint in the way
that you seem to be guessing that it might.

An acoustical root is a phenomenon that exists outside of any musical context.
So are constructional roots.
But functional roots are contextually dependant.

--
Joey Goldstein
http://www.joeygoldstein.com
joegold AT sympatico DOT ca
.

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