Diatonicity in a nutshell                             rev. 13 Feb. 2003.
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Define...

  pitch - a musical sound characterized by a log-frequency value.

  scale - an ordered list of pitches.

  interval - a distance between two pitches, given as their difference.

  generic interval - a distance between two scale members, given as the
     number of scale members subtended, inclusive of the outer members.

  interval class - a list of intervals produced by moving a given
     generic interval sequentially over all positions in a scale.

  melody - an ordered list of signed generic intervals and a starting
     position in a scale, which generates a series of pitches when
     applied iteratively in the scale.

Then define a scale's diatonicity with four quantities...

  (1) Pitch Tracking
  Generalized octave equivalence; Miller limit

     The scale is periodic with respect to a strong consonance, and
     addressing the pitches in a period stretches but does not exhaust
     human working memory.

        k, where k is the number of pitches in a period of the scale and
        4 < k < 11.

  (2) Mode Autonomy
  Rothenberg Efficiency (higher values are better)

     A relatively long melody must be heard before the listener can
     unambiguously determine its starting position.

        Rothenberg, David. "A Model for Pattern Perception with Musical
        Applications." Parts I & II.
        Mathematical Systems Theory vol. 11, 1978.

           Note that for equal temperaments, Rothenberg considers a
           single element sufficient, and assigns minimal efficiency
           1/k.  These scales should have maximal mode autonomy however,
           and for our purposes efficiency 1.

  (3) Modal Transposition
  Stability (higher values are better)

     The relative sizes of the intervals produced by a melody may change
     when the melody's starting position is changed.  A stable scale is
     one for which these changes are slight or nonexistent.

        The portion, in log-frequency space, of the scale's period that
        is not covered by the spans of its interval classes is the raw
        stability.  We divide this by a factor based on the portion that
        is redundantly covered.

           s/(1+i), where s is the "Lumma stability" and i the
           "impropriety factor" as given by Scala's "show data".

  (4) Diatonic harmony
  Diatonic Property (higher values are better)

     There is at least one generic interval at which a melody may be
     harmonized without the voices becoming timbre-fused.  Such a
     harmony is parallel with respect to generic intervals but not with
     respect to actual intervals.

        (k+1-c)(d^2)/(k^3) for a given interval class, where d is the
        number of times it produces a consonant triad with the interval
        of equivalence and c is the length of the longest consecutive
        run of any single interval through it.


                                                           ~ Carl Lumma
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