source file: mills2.txt
Date: Fri, 27 Sep 1996 13:39:32 -0700

Subject: The tonicity of tempered intervals and t

From: PAULE <ACADIAN/ACADIAN/PAULE%Acadian@mcimail.com>

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))<f(1)/f(2)<(p(j)+p(k))/(q(j)+q(k))=b(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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