Kees tuning pages

My 1978 paper; A Theory of Equal-Tempered Scales

The Bohlen scale

List of 5 limit intervals

List of 7 limit intervals

List of 7 limit interval in the tritave


Searching small intervals

My interpretation of good periodicity blocks is not only that they do not contain smaller intervals than the defining unison intervals, but also that they do not contain intervals only expressible in larger integers then those unison intervals. A measure of this ‘expressibility’ of a 5-limit interval (m,n) can be defined as:

|m|*log(3) + |n|*log(5)            when m and n have the same sign and

max( |m|*log(3), |n|*log(5) )    otherwise.

(Look here for a reaction to Paul Erlich's observations about this measure)

We only have to consider cases where m is positive, because (m,n) and (-m,-n) are equivalent. So to search for small intervals of a certain incremental level of complexity l, we have to take into account all (m,n) where

m*log(3) + n*log(5) < l

and all (m,-n) where

m*log(3) < l and n*log(5) < l

This can be illustrated in the following picture:

The results of a search along these lines, growing the blue and green lines outward as it were, can be found here.

The same for 7-limit in the ‘1:3 tritave’ is here.

To extend this in the third dimension for 7-limit in the octave is slightly more complicated. The picture becomes this:

The search has to take place in the one ‘blue’ plane in the all positive (+++) ‘octant’ (3-D quadrant) and in the two ‘outward’ ‘green’ planes in each of the three (++-) (+-+) (-++) octants. The problem here is that it is much more difficult to traverse these planes algorithmically keeping the correct relative logarithmic relation. And to be honest I didn’t do that in getting these results. I increased the search equally along all axes, putting the intervals manually in the right order in the first part, where the integer calculation was still feasible. To do it really right is still an exercise to be undertaken.

Another approach to find best-fitted Equal Temperaments is calculating by brute force the relative errors of the primes for each possible octave division. Here it also seems logical to apply weights to the errors relative to the above-mentioned logarithms:

f3 + log(3)/log(5)*f5

              1   4.70e-01
              2   2.97e-01
              3   2.46e-01
              7   1.97e-01
             12   9.54e-02
             53   4.26e-02
            118   2.70e-02
            612   1.40e-02
           1783   1.20e-02
           2513   1.14e-02
           4296   2.38e-03
          73709   1.70e-03
          78005   7.23e-04
         229719   7.03e-04
        1251764   2.93e-04
        6796263   5.46e-05

One of the reasons that a number of well-known possibilities does not occur in this list is the following: there is no account of the fact that, when the two errors, before they are taken to be absolute values, have equal signs, they compensate each other partially for the representation of the major sixth 3:5. If we let this play along in the combined error calculation as follows, we get:

  f3 + log(3)/log(5)*f5 + log(3)/log(5)*f(5/3)

              1   5.03e-01
              2   4.39e-01
              3   2.85e-01
              7   2.25e-01
             12   1.43e-01
             19   1.39e-01
             34   1.24e-01
             53   5.87e-02
            118   3.74e-02
            612   2.10e-02
           1171   1.98e-02
           1783   1.37e-02
           2513   1.20e-02
           4296   3.72e-03
          73709   2.67e-03
          78005   1.08e-03
         229719   8.40e-04
        1251764   2.94e-04
        6796263   8.49e-05

Equivalent tables for the 7-limit are:

              1   4.82e-01
              2   3.68e-01
              3   3.43e-01
              5   2.78e-01
             10   2.16e-01
             12   2.00e-01
             22   1.96e-01
             31   1.35e-01
             53   1.26e-01
            130   1.14e-01
            171   5.50e-02
            441   4.48e-02
           3125   2.10e-02
          14789   1.93e-02
          18355   1.49e-02
          81689   1.43e-02
          84814   1.15e-02
         103169   3.77e-03
        5700509   1.58e-03

              1   5.96e-01
              2   5.51e-01
              3   4.90e-01
              4   4.21e-01
             10   3.65e-01
             12   3.10e-01
             19   2.97e-01
             31   1.84e-01
             99   1.38e-01
            171   5.94e-02
           3125   3.61e-02
          11664   3.45e-02
          14789   3.35e-02
          18355   1.76e-02
          84814   1.36e-02
         103169   4.90e-03
        5700509   2.75e-03

And for the 7-limit in (1:3)

              1   5.02e-01
              2   3.85e-01
              4   1.57e-01
             13   4.96e-02
            258   4.01e-02
            271   9.76e-03
           1342   9.35e-03
           8979   3.39e-03
          18229   2.99e-03
          27208   4.52e-04
        1123707   3.02e-04
        5800012   1.62e-04

              1   5.62e-01
              2   5.01e-01
              3   4.77e-01
              4   2.43e-01
             13   5.19e-02
            258   4.27e-02
            271   9.78e-03
           8979   3.40e-03
          18229   2.99e-03
          27208   5.49e-04
        1123707   4.52e-04
        5772804   3.42e-04
        5800012   2.40e-04