PCTD

2019cof15
Pitch class Tenney distance (PCTD) and Tenney distance for small intervals from D.

A simple proof that for many commonly-met intervals, Tenney distance is approximated by 2xPCTD, with an error equal in magnitude to the interval’s pitch height

We prove the result first for intervals in a pitch system with period 2 and generator 3.

In this system the interval vector (m,n) (mϵZ, nϵZ) generates an interval with pitch height

H=log2(2m.3n)=m+n.log23

By definition the Tenney distance of (m,n) is

D = log2(2|m|.3|n|) =|m|+|n|log23

Define the pitch-class Tenney distance (PCTD) of (m,n) as

D’=log2(3|n|) =|n|log23

Then  2D’-D=|n|log23 -|m|

We can think of 2D’-D as the error incurred by taking 2D’ as an estimate of D. Four cases cover all possibilities:

error 2D’-Dn>0n0
m>0-m + n.log23-m – n.log23 = -H
m0m + n.log23 = Hm – n.log23
  • The smallest (in H) interval with m>0 and n>0 is (1,1)
  • The smallest interval with m<0 and n<0 is (-1,-1)
  • For both of these, |H|= 1+ log23 = 2.58… octaves
  • For any interval smaller than this, m and n are of opposite sign, and the absolute error |2D’-D|=|H|

Thus for ‘small’ intervals so defined, the error in estimating Tenney distance by this method is equal, up to sign, to the pitch height of the interval. This equality is valid for intervals from a given centre pitch, to other pitches within a range of more than five octaves. For intervals of less than an octave, the error will be numerically less than 1.

Since D’ is independent of m, it is reasonable to regard D’ not only as a distance between two pitches of the system, but more simply as a distance between their pitch classes.

A similar proof would apply to other pairs of small, coprime period and generator.