_______ 10. The Great Work of Counting PDF Print E-mail
Written by D. Bond   
Tuesday, 25 May 2010 21:39

10. The Great Work of Counting

                       

            Everything that can be counted does not necessarily count;

            everything that counts can not necessarily be counted.

 

                        - Albert Einstein

 

Scale builders will want to know how the number of notes grows as our matrix becomes more complex. The max patch below calculates this [with a less than optimal correction for mantissa problems inherent within float32 mathematics].

 

            max patch: scale_builder

 

This chart summarizes the findings:

 

Matrix dimensions   Number of unique notes (1-2) (1/2 – 2/1)          

                                                one octave    median range (includes mirror doubles)           

            1                                              1         1                     

            2                                              1         1

            3                                              3         3                      2*1+1

            4                                              3         5                     

            5                                              7         10                   2*3+1

            6                                              7         14

            7                                              13       20                   2*2*3+1

            8                                              13       24

            9                                              19       30                   2*3*3+1

            10                                            19       34

            11                                            29       45                   2*2*7+1

            12                                            29       49

            13                                            41       62                   (2^3)*5+1

            14                                            41       69

            15                                            49       78                   (2^3)*3+1

            16                                            49       88      

            17                                            53?     89?**[1]

                       

 

 

Consider the ratios of the harmonic matrix. We consult Pythagoras again and employ the properties of right triangles to create a definition for relative pitch expressed as an angle:

 

            relative pitch =  atan(r) where r= the ratio.

 

Pitch Prism

 

 

Thus 1/1 gives an angle of .785398 radians = 45 degrees.

 

                                    radians                                  degrees

 

            1/2                  .463648                                 26.57

            2/3                  .588003                                 33.69

            1/ sqrt(2)        .615480                                 35.26

            3/4                  .643501                                 36.87

            1/1                  .785398                                 45.00

            4/3                  .927295                                 53.13

            sqrt(2)             .955317                                 54.74

            3/2                  .982794                                 56.31 

            2/1 =               1.10715                                 63.44

           

 

Not terribly useful at this point, perhaps, as it serves to further obscure the patterns and repetition in the matrix, but clearly demonstrates the celestial globe upon which our constellations are projected. Pitch can be seen as approaching a continuous spectrum as the matrix approaches its infinite dimensions. Another way to view this is that the rays emitting from the point [0,0] pass through each point in our matrix  and as the matrix grows bigger, the accumulation of new points making up the scale approach the continuous.

 

Two important concepts should be pointed out here. First, that this model parallels the way the ear perceives pitch. Discrimination is good in the middle of the range where there is greater size of a similar angle [1/1, 5/4 as compared to 4/1, 5/1] and poor at the extremes where the angle is small.

 

Also, while it may seem esoteric at this point [at least in terms of my own understanding], if we consider these matrix points as sound particles, as opposed to waves, and then by counting up the series of occurrences of any particular interval, we find it to be directly proportional to its perceived consonance.

           

            You Decide

 

            Particle or Waves? Pick and choose!

            But always account for both points of view.

           

            By this method, you never lose,

            viewing no view as completely true.

 

 

But I digress.

 

next - 11. As Above and Below...

[1] ** Technical note: above this, due to a limitations of my brain, higher counting is problematic. I know this is correct though: M[32] = 183. Computer geniuses are asked to help out on this.

 

Last Updated on Thursday, 10 June 2010 14:38