Cardinality
Cardinality is the count of how many pitches are in the scale.
Pitch Class Set
The tones in this scale, expressed as numbers from 0 to 11
Forte Number
A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.
Rotational Symmetry
Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.
Reflection Axes
If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.
Palindromicity
A palindromic scale has the same pattern of intervals both ascending and descending.
Chirality
A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.
enantiomorph: 1715
Hemitonia
A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.
Cohemitonia
A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.
Imperfections
An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.
Modes
Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.
Prime Form
Describes if this scale is in prime form, using the Starr/Rahn algorithm.
prime: 859
Generator
Indicates if the scale can be constructed using a generator, and an origin.
Deep Scale
A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.
Interval Structure
Defines the scale as the sequence of intervals between one tone and the next.
Interval Vector
Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.
Proportional Saturation Vector
First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.
Interval Spectrum
The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.
Distribution Spectra
Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.
<2> = {3,4}
<3> = {4,5,6}
<4> = {6,7,8}
<5> = {8,9}
<6> = {9,10,11}
Spectra Variation
Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.
Maximally Even
A scale is maximally even if the tones are optimally spaced apart from each other.
Maximal Area Set
A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.
Interior Area
Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.
Polygon Perimeter
Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.
Myhill Property
A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.
Centre of Gravity Distance
When tones of a scale are imagined as physical objects of equal weight arranged around a unit circle, this is the distance from the center of the circle to the center of gravity for all the tones. A perfectly balanced scale has a CoG distance of zero.
Ridge Tones
Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.
Propriety
Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".
Heteromorphic Profile
Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.
Coherence Quotient
The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.
Sameness Quotient
The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.
