The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2223: "Arcadian Minor"

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.

 {0,1,2,3,5,7,11}

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.

[011121325274111](29)

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: 3747

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.

 3 (tricohemitonic)

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.

 4

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: 351

Forte Number

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

 7-9

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, 1, 2, 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, 5, 3, 4, 3, 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.

 p3m4n3s5d4t2

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, 0.75, 0.25, 0.333, 0.25, 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> = {2,3,4,5,6}
<3> = {3,4,5,6,7,8}
<4> = {4,5,6,7,8,9}
<5> = {6,7,8,9,10}
<6> = {8,10,11}

Spectra Variation

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

 3.429

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 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.

 0.342024

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.

(46, 40, 104)

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.

0.386

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.

0.175

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.

29

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.

 F51

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 = 2.39417
FC2 = 1.732051
FC3 = 1
FC4 = 1
FC5 = 1.505971
FC6 = 3

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 TriadsG{7,11,2}221
Minor Triadscm{0,3,7}131.5
Augmented TriadsD♯+{3,7,11}221
Diminished Triadsb°{11,2,5}131.5

The following pitch classes are not present in any of the common triads: {1}

Parsimonious Voice Leading Between Common Triads of Scale 2223. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ Parsimonious Voice Leading Between Common Triads of Scale 2223. Created by Ian Ring ©2019 G D#+->G b° b° G->b°

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.

Diameter3
Radius2
Self-Centeredno
Central VerticesD♯+, G
Peripheral Verticescm, b°

Modes

Modes are the rotational transformation of this scale. Scale 2223 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3159		Alt 763
3rd mode:
Scale 3627		Kalian
4th mode:
Scale 3861		Ionian 654
5th mode:
Scale 1989		Dominant +34
6th mode:
Scale 1521		Lydian 7632
7th mode:
Scale 351		Infra-Alt 567This is the prime mode

Prime

The prime form of this scale is Scale 351

Scale 351Infra-Alt 567

Complement

The heptatonic modal family [2223, 3159, 3627, 3861, 1989, 1521, 351] (Forte: 7-9) is the complement of the pentatonic modal family [87, 1473, 1797, 2091, 3093] (Forte: 5-9)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2223 is 3747

Scale 3747Superlydian Augmented 245

Interval Matrix

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

Contradictions (46)

Ambiguities(40)

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
111111010100012223k = 1h = 1
211111010100012223k = 2h = 1
311111010100012223k = 3h = 1
411111010100012223k = 4h = 1
52(1)(1)(1)(1)0(1)0(1)000(1)2223k = 5h = 2

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2223 is chiral, and its enantiomorph is scale 3747

Scale 3747Superlydian Augmented 245

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

(0.266575, -0.214286)
Distance from Center0.342024
Angle in degrees

measured clockwise starting from the root.

51.206
Angle in cents

100 cents = 1 semitone.

170.686667

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> 2223       T0I <11,0> 3747
T1 <1,1> 351      T1I <11,1> 3399
T2 <1,2> 702      T2I <11,2> 2703
T3 <1,3> 1404      T3I <11,3> 1311
T4 <1,4> 2808      T4I <11,4> 2622
T5 <1,5> 1521      T5I <11,5> 1149
T6 <1,6> 3042      T6I <11,6> 2298
T7 <1,7> 1989      T7I <11,7> 501
T8 <1,8> 3978      T8I <11,8> 1002
T9 <1,9> 3861      T9I <11,9> 2004
T10 <1,10> 3627      T10I <11,10> 4008
T11 <1,11> 3159      T11I <11,11> 3921
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3243      T0MI <7,0> 2727
T1M <5,1> 2391      T1MI <7,1> 1359
T2M <5,2> 687      T2MI <7,2> 2718
T3M <5,3> 1374      T3MI <7,3> 1341
T4M <5,4> 2748      T4MI <7,4> 2682
T5M <5,5> 1401      T5MI <7,5> 1269
T6M <5,6> 2802      T6MI <7,6> 2538
T7M <5,7> 1509      T7MI <7,7> 981
T8M <5,8> 3018      T8MI <7,8> 1962
T9M <5,9> 1941      T9MI <7,9> 3924
T10M <5,10> 3882      T10MI <7,10> 3753
T11M <5,11> 3669      T11MI <7,11> 3411

The transformations that map this set to itself are: T0

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 2221Rāga Madhuranjani
Scale 2219Phrydimic
Scale 2215Ranimic
Scale 2231Arcadian
Scale 2239Dacryllic
Scale 2191Thydimic
Scale 2207Byzantine Minor
Scale 2255Alt 7346
Scale 2287Lodyllic
Scale 2095MUMian
Scale 2159Quixotic Minor
Scale 2351Alt 7345
Scale 2479Harmonic Neapolitan Minor
Scale 2735Gynyllic
Scale 3247Aeolonyllic
Scale 175BEWian
Scale 1199Alt 3456

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.