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Scale 645: "DUYian"

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of reflective symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Keyboard Diagram





Other diagrams coming soon!

Common Names

Names are messy, inconsistent, polysemic, and non-bijective. If you see a name with lots of citations beside it, that's a good measure of credulity.

Name Sources

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale. Cardinalities can be expressed as an adjective, e.g. pentatonic, hexatonic, heptatonic, and so on.

 4 (tetratonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11.

 {0,2,7,9}

Leonard Notation

As practiced in the theoretical work of B P Leonard, this notation for describing a pitch class set replaces commas with subscripted numbers indicating the interval distance between adjacent tones. Convenient when you are doing certain kinds of analysis. The superscript in parentheses is the sonority's Brightness.

[02257293](18)

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

 none

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.

 [4.5]

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

 no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

 no

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

 0 (anhemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

 0 (ancohemitonic)

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.

 1

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.

 4

Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

 no
prime: 165

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

 4-23

Generator

Indicates if the scale can be constructed using a generator, and an origin.

 generator: 5
origin: 9

Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.

 no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

 [2, 5, 2, 3]

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.

 <0, 2, 1, 0, 3, 0>

Hanson's Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson. Hanson categorizes all intervals as being one of six classes, and gives each a letter: p m n s d t, ordered from most consonant (p) to most dissonant (t). When an interval appears more than once in a sonority, it is superscripted with a number, like p2.

 p3ns2

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.

<0, 0.667, 0.25, 0, 1, 0>

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.

<1> = {2,3,5}
<2> = {5,7}
<3> = {7,9,10}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

 2

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

 no

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.

 no

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.

1.616

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.346

Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.

 no

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.

 0.224144

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.

 [9]

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".

Proper

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.

(0, 4, 14)

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.

0.429

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.

0.222

Brightness

Based on the theories of B P Leonard, Brightness is a measurement of interval content calculated by taking the sum of each tone's distance from the root. Scales with more tones at higher pitches will have a greater Brightness than those with fewer, lower pitches. Typically used to compare pitch class sets with the same cardinality.

18

Xenome

A naming schema invented by Qid Love, and published in their book The Book Of Xenomes. A xenome is a succinct encoding of the pitch class set as three hexadecimal characters, with reversed bits such that they read left to right as ascending pitches.

 A14

Lewin-Quinn FC Components

Conceived by David Lewin in 1959, and generalized by Ian Quinn; FC is a function that transforms the set by multiplying by a given number of semitones, then gives the total distance of the sum of the vectors produced by each tone when mapped to a circular plane. The result is a discrete Fourier transform that quantifies harmonic qualities. A more thorough and sensible explanation is coming in the book. FC Components can be used to analyze other tuning systems besides 12-TET. FC0 is the cardinality of the scale.

 FC0 = 4
FC1 = 0.896575
FC2 = 1.732051
FC3 = 0
FC4 = 1
FC5 = 3.346065
FC6 = 0

Generator

This scale has a generator of 5, originating on 9.

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.

Modes

Modes are the rotational transformation of this scale. Scale 645 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 1185		Genus Primum Inverse
3rd mode:
Scale 165		Genus PrimumThis is the prime mode
4th mode:
Scale 1065		Karen 4 Tone Type 1

Prime

The prime form of this scale is Scale 165

Scale 165Genus Primum

Complement

The tetratonic modal family [645, 1185, 165, 1065] (Forte: 4-23) is the complement of the octatonic modal family [1455, 1515, 1725, 1965, 2775, 2805, 3435, 3765] (Forte: 8-23)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 645 is 1065

Scale 1065Karen 4 Tone Type 1

Interval Matrix

Each row is a generic interval, cells contain the specific size of each generic. Useful for identifying contradictions and ambiguities.

Contradictions (0)

Ambiguities(4)

Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
11101000010100645k = 1h = 1
23([10][10]0)00([10][10]0)645k = 2h = 3
33([10][10]0)00([10][10]0)645k = 3h = 3
43([10][10]0)00([10][10]0)645k = 4h = 3
53([10][10]0)00([10][10]0)645k = 5h = 3

Center of Gravity

If tones of the scale are imagined as identical physical objects spaced around a unit circle, the center of gravity is the point where the scale is balanced.

Position

with origin in the center

(-0.158494, -0.158494)
Distance from Center0.224144
Angle in degrees

measured clockwise starting from the root.

315
Angle in cents

100 cents = 1 semitone.

1050

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 645       T0I <11,0> 1065
T1 <1,1> 1290      T1I <11,1> 2130
T2 <1,2> 2580      T2I <11,2> 165
T3 <1,3> 1065      T3I <11,3> 330
T4 <1,4> 2130      T4I <11,4> 660
T5 <1,5> 165      T5I <11,5> 1320
T6 <1,6> 330      T6I <11,6> 2640
T7 <1,7> 660      T7I <11,7> 1185
T8 <1,8> 1320      T8I <11,8> 2370
T9 <1,9> 2640      T9I <11,9> 645
T10 <1,10> 1185      T10I <11,10> 1290
T11 <1,11> 2370      T11I <11,11> 2580
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3585      T0MI <7,0> 15
T1M <5,1> 3075      T1MI <7,1> 30
T2M <5,2> 2055      T2MI <7,2> 60
T3M <5,3> 15      T3MI <7,3> 120
T4M <5,4> 30      T4MI <7,4> 240
T5M <5,5> 60      T5MI <7,5> 480
T6M <5,6> 120      T6MI <7,6> 960
T7M <5,7> 240      T7MI <7,7> 1920
T8M <5,8> 480      T8MI <7,8> 3840
T9M <5,9> 960      T9MI <7,9> 3585
T10M <5,10> 1920      T10MI <7,10> 3075
T11M <5,11> 3840      T11MI <7,11> 2055

The transformations that map this set to itself are: T0, T9I

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 647DUZian
Scale 641Lahuzu 3 Tone Type 1
Scale 643DUXian
Scale 649Byptic
Scale 653Dorian Pentatonic
Scale 661Major Pentatonic
Scale 677Scottish Pentatonic
Scale 709Rāga Shri Kalyan
Scale 517ALUian
Scale 581Bolic
Scale 773ESUian
Scale 901BOFian
Scale 133Suspended Second Triad
Scale 389CIXian
Scale 1157ALKian
Scale 1669Rāga Matha Kokila
Scale 2693Lahuzu 5 Tone Type 3

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages were invented by living persons, and used here with permission where required: notably collections of names by William Zeitler, Justin Pecot, Rich Cochrane, and Robert Bedwell.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (DOI, Patent owner: Dokuz Eylül University, Used with Permission.

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with naming the Carnatic ragas. Thanks to Niels Verosky for collaborating on the Hierarchizability diagrams. Gratitudes to Qid Love for the Xenomes. Thanks to B P Leonard for the Brightness metrics. Thanks to u/howaboot for inventing the Center of Gravity metrics.