Re: New Topic: Music Idiosyncracy

ESchluessel · Tue, 25 Apr 2000 06:30:23 -0700

Well, I haven't used this knowledge in about
three years, but here goes:

In the
Pre-Baroque/Baroque/Classical era, not all keys are equal. Some math to
follow.

A scale is divided into twelve chromatic intervals
from do to do' (octave higher). There are several ways
to determine how to space these intervals; the term
for the way the pitches in a scale are defined is
'temperament.' There are three basic systems to be discussed:
Even, odd, and just.

Odd-tempering is the oldest
system; the term is a catchall, but generally refers to
Pythagorean tuning. To build an odd-tempered scale: Start
with a base frequency for C. Define C' as 2C, define
C'' as 4C, etc. following C^n = C2^n. Now things get
more difficult. Pythagorean tuning is based around the
interval C-G, the perfect fifth, defined such that G =
3C/2 (in other words, the frequency of G is 1.5 times
that of C). By the same token, F below C can be
defined 'F (F an octave lower) = 2C/3. Therefore, by the
octave definition established earlier, G' = 2G = 3C, ''F
= 1/3 C, etc. You have now defined the "tonic,"
"dominant," and "subdominant." To define the other intervals,
rely on the definition of the perfect fifth.
Thus:
C = 1C
G = 3C/2
D' = a fifth above G =
(3C/2) = 9C/4
A' = 27C/8
E'' = 81C/16

And
so on until you've defined at least one occurence of
the "dominant series," C-G-D-A-E-B-F#. Divide their
frequencies using the octave definition to bring them back
into the range of the scale (thus D is 9C/8, A is
27C/4, etc.). Now, establish the "subdominant series" by
multiplying the tonic by the subdominant definition 'F =
2C/3. You will obtain the series C-F-Bb-Eb-Ab-Db-Gb.
Raise those tones to insert them into the octave. Those
of you following along will have already noticed a
problem -- F# is supposed to be enharmonic to Gb, yet
violates the definition of 'Gb = 1F#/2. You will
furthermore note that the discrepancy increases the more
iterations from your base C you are. However, all notes are
defined relative to C (actually, all notes are defined
relative to A, but that's a separate issue). [more]