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Scale 2231: "Arcadian"

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.

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.

 7 (heptatonic)

Pitch Class Set

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

 {0,1,2,4,5,7,11}

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.

[011122415274111](30)

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.

 none

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.

 yes
enantiomorph: 3491

Hemitonia

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

 4 (multihemitonic)

Cohemitonia

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

 2 (dicohemitonic)

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.

 3

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.

 7

Prime Form

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

 no
prime: 367

Forte Class

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

 7-Z36

Generator

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

 none

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.

 [1, 1, 2, 1, 2, 4, 1]

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.

 <4, 4, 4, 3, 4, 2>

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.

 p4m3n4s4d4t2

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.

<0.5, 0.5, 0.5, 0, 0.5, 0.5>

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> = {1,2,4}
<2> = {2,3,5,6}
<3> = {3,4,5,6,7}
<4> = {5,6,7,8,9}
<5> = {6,7,9,10}
<6> = {8,10,11}

Spectra Variation

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

 3.143

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.

2.299

Polygon Perimeter

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

5.803

Myhill's Property

A scale has Myhill's 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.

 0.285714

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.

 none

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".

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.

(38, 34, 100)

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.

0.486

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.

0.206

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.

30

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.

 ED1

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 = 7
FC1 = 2
FC2 = 2
FC3 = 1
FC4 = 2
FC5 = 2
FC6 = 1

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

Generator

This scale has no generator.

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC{0,4,7}231.4
G{7,11,2}231.4
Minor Triadsem{4,7,11}221.2
Diminished Triadsc♯°{1,4,7}142
b°{11,2,5}142
Parsimonious Voice Leading Between Common Triads of Scale 2231. Created by Ian Ring ©2019 C C c#° c#° C->c#° em em C->em Parsimonious Voice Leading Between Common Triads of Scale 2231. Created by Ian Ring ©2019 G em->G b° b° G->b°

view full size

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter4
Radius2
Self-Centeredno
Central Verticesem
Peripheral Verticesc♯°, b°

Modes

Modes are the rotational transformation of this scale. Scale 2231 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3163		Alt 76
3rd mode:
Scale 3629		Melodic 65
4th mode:
Scale 1931		Dorian +24
5th mode:
Scale 3013		Ionian +34
6th mode:
Scale 1777		Lydian 732
7th mode:
Scale 367		Infra-Alt 67This is the prime mode

Prime

The prime form of this scale is Scale 367

Scale 367Infra-Alt 67

Complement

The heptatonic modal family [2231, 3163, 3629, 1931, 3013, 1777, 367] (Forte: 7-Z36) is the complement of the pentatonic modal family [151, 737, 1801, 2123, 3109] (Forte: 5-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2231 is 3491

Scale 3491Superlydian Augmented 24

Interval Matrix

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

Contradictions (38)

Ambiguities(34)

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
111110110100012231k = 1h = 1
211110110100012231k = 2h = 1
311110110100012231k = 3h = 1
411110110100012231k = 4h = 1
52(1)(1)(1)0(1)(1)0(1)000(1)2231k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2231 is chiral, and its enantiomorph is scale 3491

Scale 3491Superlydian Augmented 24

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

(0.247436, -0.142857)
Distance from Center0.285714
Angle in degrees

measured clockwise starting from the root.

60
Angle in cents

100 cents = 1 semitone.

200

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> 2231       T0I <11,0> 3491
T1 <1,1> 367      T1I <11,1> 2887
T2 <1,2> 734      T2I <11,2> 1679
T3 <1,3> 1468      T3I <11,3> 3358
T4 <1,4> 2936      T4I <11,4> 2621
T5 <1,5> 1777      T5I <11,5> 1147
T6 <1,6> 3554      T6I <11,6> 2294
T7 <1,7> 3013      T7I <11,7> 493
T8 <1,8> 1931      T8I <11,8> 986
T9 <1,9> 3862      T9I <11,9> 1972
T10 <1,10> 3629      T10I <11,10> 3944
T11 <1,11> 3163      T11I <11,11> 3793
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3491      T0MI <7,0> 2231
T1M <5,1> 2887      T1MI <7,1> 367
T2M <5,2> 1679      T2MI <7,2> 734
T3M <5,3> 3358      T3MI <7,3> 1468
T4M <5,4> 2621      T4MI <7,4> 2936
T5M <5,5> 1147      T5MI <7,5> 1777
T6M <5,6> 2294      T6MI <7,6> 3554
T7M <5,7> 493      T7MI <7,7> 3013
T8M <5,8> 986      T8MI <7,8> 1931
T9M <5,9> 1972      T9MI <7,9> 3862
T10M <5,10> 3944      T10MI <7,10> 3629
T11M <5,11> 3793      T11MI <7,11> 3163

The transformations that map this set to itself are: T0, T0MI

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 2229Rāga Nalinakanti
Scale 2227Rāga Gaula
Scale 2235Exotic Minor
Scale 2239Dacryllic
Scale 2215Ranimic
Scale 2223Arcadian Minor
Scale 2199Dyptimic
Scale 2263Phlegmatic Minor
Scale 2295Kogyllic
Scale 2103MURian
Scale 2167Alt 7356
Scale 2359Unorthodox Minor
Scale 2487Phroptyllic
Scale 2743Adi Basant
Scale 3255Daryllic
Scale 183BEBian
Scale 1207Alt 356

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.