The tonicity of tempered intervals and t

The following is excerpted from an old version of a paper that will appear
in Xenharmonikon:

***********************************PART
1***************************************
[13]

Parncutt's theory of harmony is essentially an extension of the theory of
complex pitch perception, i.e., the waywe synthesize the harmonic partials
of a musical tone into a single sensation, whise pitch is that of the
fundamental component -- even if that component is physically absent from
the stimulus[14]:

According to Terhadt, the root of a chord is a virtual pitch, i. e., a
complex tone
sensation. This observation alone is not very useful, as there are
virtual pitches at _all_
the notes of the chord. The root is different in that the spectral
pitches in its harmonic
pitch pattern arise from more than one complex tone (note). In other
words, the root is
the implied fundamental of a group of pure tone components belonging to
different
complex tones.

For two notes in a harmonic relationship, whose frequency ratio in lowest
terms is given by p/q, with

p>=q (0)

the "fundamental frequency" is given by

f0=f(p)/p=f(q)/q (1)

Now according to any of the theories of complex pitch perception
(Goldstein[15], Terhardt[16], Wightmann[17], the relationship becomes
difficult to perceive for f0 very low (or, what will amount to the same
thing, for p and q very large); let us tentatively require

f0>=F (2)

F being the lower bound for any f0. We now find

F<=f(p)/p and F<=f(q)/q (3)

or

p<=f(p)/F and q<= f(q)/F (4)

If we are comparing intervals by fixing f(p) and varying f(q)[18], the
fractions will satisfy

p,q<=N (5)

where

N=f(p)/F (6)

Now we will assume that N is large compared with the p and q of fractions we
are actually trying to represent in our tuning. This is reasonable (I am
able to tune just intervals such as 17:13 by ear, though it could hardly be
considered a consonant interval); the question is to what extent these more
complex ratios disturb the representation of simpler ones. In fact, we will
let N approach infinity (by letting F go to zero), causing the particular
frequency at which f(p) is fixed to cease to have any bearing on the result,
so that we can judge intervals solely by their size and not by the register
in which they are played.
Given any N, the set of fractions satisfying (5) and (0), arranged in
ascending order, is called the Farey series of order N. For example, the
Farey series of order is

1/1,6/5,5/4,4/3,3/2,5/3,2/1,5/2,3/1,4/1,5/1,6/1 (
7)

This series has the property that any two consecutive fractions p(i)/q(i)
and p(j)/q(j) (j=i+1) satisfy[19]

p(j)q(i)-p(i)q(j)=1 (8)

If the next fraction after p(j)/q(j) is denoted by p(k)/q(k) (k=j+1) we find

1=p(j)q(i)-p(i)q(j)=p(k)q(j)-p(j)q(k)

p(j)q(i)+p(j)q(k)=p(k)q(j)+p(i)q(j)

Say the interval one is trying to represent is p(j)/q(j); then p(j) is small
compared with N. Then

p(k)~p(i)~N (9)

because:
(a) membership in the Farey series requires that they be <= N; and
(b) All three fractions being similar in magnitude implies that (9) also
implies

q(k)~q(i)~Nq(j)/p(j) (9c)

therefore if p(j) were <= N-p(j), the fraction (p(i)+p(j))/(q(i)+q(j)),
which lies between p(i)/q(i) and p(j)/q(j), would also belong to the Farey
series of order N, contradicting the assumption that p(i)/q(i) and p(j)/q(j)
are consecutive. By (8) we know that

cp(j)=p(k)+p(i), cq(j)=q(i)+q(k) (10)

for some c, so

p(k)=cp(j)-p(i),q(i)=cq(j)-q(k) (11)

whence

q(k)=cq(j)-q(i) (12)

>From (11),

p(k)q(i) =
(c^2)p(j)q(j)-cp(i)q(j)-cp(j)q(k)+p(i)q(k) (13)

and using (12),

p(k)q(i)-p(i)q(k)=(c^2)p(j)q(j)-cp(i)q(j)-cp(j)(cq(j)-q(i))

p(k)q(i)-p(i)q(k)=-cp(i)q(j)+cp(j)q(i)

p(k)q(i)-p(i)q(k)=c (14)

by virtue of (7). Now (9) and (10) tell us that

c~2N/p(j)

so

p(k)q(i)-p(i)q(k)~2N/p(j) (15)

While N is still finite, there is a range of intervals f(1)/f(2) which could
be interpreted as p(j)/q(j). A natural set of bounds for this range is[20]

a(j)=(p(i)+p(j))/(q(i)+q(j)) (16)

which unambiguously ascribes to any f(1)/f(2) one and only one fraction from
th e Farey series, namely p(j)/q(j). Now the width of this range on a
logarithmic scale, such as a scale of cents, is given by

W(j)=log(((p(i)+p(j))/(q(i)+q(j)))/((p(j)+p(k))/(q(j)+q(k))))=log(a(j)/
b(j)) (17)

Since in our case p(j),q(j) are relatively small,

W(j)=log(p(k)q(i)/p(i)q(k))

and since the range is small,

W(j)=p(k)q(i)/p(i)q(k)-1

or

W(j)=(p(k)q(i)-p(i)q(k))/p(i)q(k)

Substituting (15), (9), and (9c),

W(j)=(2N/p(j))/((N^2)q(j)/p(j))

or

W(j)=2/Nq(j) (18)

Now, if we want to take octave equivalence into account, we would like
all inversions and octave expansions of an interval to be equal in tonicity
to the individual interval. Denote the lowest-terms reduction of
(2^n)p(j)/q(j) by p(j,n)/q(j,n). L(j), the "limit" of p(j)/q(j), is the
greater of the largest odd factor of p(j) and the largest odd factor of
q(j). Obviously, L(j)=L(j,n). Since q(j,n) is often equal to, but never
greater than L(j),

W(L(j))=W(j,n)=2/NL(j)<=W(j) (19)

is a lower bound on the "width" of all ratios of the L(j) limit. Harry
Partch, using the word "identity" with the same meaning as our "limit,"
states essentially the same thing in his "Observation One": what he calls
the "field of attraction"[21] is proportional to our W(L(j)) (or equal with
N~2.9).

To be continued next
week...................................................................


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