Visualizations of Tonality
CCRMA Affiliates, 10 May 2001
Craig Stuart Sapp (craig@ccrma.stanford.edu)

Key determination algorithm

Pitch Space

Basic Pitch Space		Earliest know use of this configuration is by Leonhard Euler, 18th century (*)
Essential Properties		Minor 3rds at +45° angle Major 3rds at -45° angle All reasonable possible enharmonic spellings present (e.g.: C#/Db/B##)
Compact Triads		Major and Minor triads in most compact spatial configuration.
Interval Space with root on C		Treating C as the root, organize other pitches with their relative significance to a chord with a root on C. (35 total pitches).
Relative Interval Space		Generalized intervals organized with respect to a hypothetical root. (35 possible intervals, octave transposed) Horizontal axis scaled by factor alpha. alpha = 0.578 gives equal distance for M3 and P5.

Sample Graphical Determination of Root

Sample Music		C minor triad in second inversion. Root is on weak metrical position
C test root		Smallest sum of interval vectors -- therefore assumed to be most likely root.
Eb test root
G test root
B test root		Largest sum of inverval vectors -- worst possible root of 4 given hypothetical roots.

Rhythmic Weightings

Duration scaling
- Long notes scaled away from root
  - if note was far from root originally, then the pitch is moved very far from root.
  - If note was close to root, then still not too far away.
- Short notes scaled towards root
  - Non-harmonic notes are usually resolved in the chord, so bringing these notes closer to the root is reasonable.
Metric scaling
- Analygous to duration scaling
- Strong metrically attacked notes scaled away from root
- Weak metrically attacked notes scaled towards root

4/4 Meter Metric Levels

Samples root scores for above musical example

C       10.87
E-      16.90
G       17.45
A-      18.55
A       18.97
F       23.13
E       24.80
C-      26.59
F-      27.70
F#      28.04
B       29.44
D-      30.34
D       30.50
C#      33.13
B-      36.14
A--     36.23
D--     37.27
D#      37.49
G#      37.77
B--     39.64

A#      42.44
G-      43.99
E#      46.32
B#      46.49
G--     50.19
E--     52.74
C--     62.69
F##     1866.
C##     1871.
G##     1871.
X       2124.
F--     2125.
X       2134.
X       2144.
D##     4207.
X       5771.
X       6026.
B##     6030.
E##     6034.
A##     7844.

Basic Algorithm Calculations

 
@vectorx = ( 0, 7, 7, 1000, 4, 4, 4, 4, 4, 1000, 8, 1, 1, 8, 8, 5, 5,
                5, 5, 5, 1000, 9, 2, 2, 2, 9, 1000, 6, 6, 6, 6, 6, 1000, 3,
                3, 3, 3, 10, 7, 7 );
 
@vectory = ( 0, -2, -4, 1000, 4, 2, 0, -2, -4, 1000, 3, 1, 1, -3, -5, 5,
                3, 1, -1, -3, 1000, 4, 2, 0, -2, -4, 1000, 4, 2, 0, -2, -4,
                1000, 3, 1, -1, -3, -5, 4, 2 );
$delta = -4;
$lambda = -3;
$alpha = 0.65;

For every hypothetical root, measure the distance to all input notes. Choose the hypothetical root with the lowest distance score as the most likely root.

sub analysis {
   my $count = @pitch;
   my @analysis;
   for ($i=0; $i<40; $i++) {
      $asum = 0.0; $bsum = 0.0; $dsum = 0.0; $lsum = 0.0;
      for ($j=0; $j<$count; $j++) {
         $p = ($pitch[$j] - $i + 40) % 40;
         $ipn = sqrt($alpha * $alpha * $vectorx[$p] * $vectorx[$p] +
                      $vectory[$p] * $vectory[$p]);
         $asum += $ipn * ($delta + log($duration[$j]));
         $bsum += $ipn * ($lambda + log($level[$j]));
      }
      $analysis[$i] = ($asum + $bsum)/$count;
   }
   return @analysis;
}