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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