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Scale 2231: "Macrian"

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 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).

Common Names

Zeitler: Macrian

Dozenal: NORian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	7 (heptatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,2,4,5,7,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	7-Z36
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
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>
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>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p⁴m³n⁴s⁴d⁴t²
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 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.	0.285714
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

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 Type	Triad^*	Pitch Classes	Degree	Eccentricity	Closeness Centrality
Major Triads	C	{0,4,7}	2	3	1.4
Major Triads	G	{7,11,2}	2	3	1.4
Minor Triads	em	{4,7,11}	2	2	1.2
Diminished Triads	c♯°	{1,4,7}	1	4	2
Diminished Triads	b°	{11,2,5}	1	4	2

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.

Diameter	4
Radius	2
Self-Centered	no
Central Vertices	em
Peripheral Vertices	c♯°, 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	Rogian
3rd mode: Scale 3629	Boptian
4th mode: Scale 1931	Stogian
5th mode: Scale 3013	Thynian
6th mode: Scale 1777	Saptian
7th mode: Scale 367	Aerodian	This is the prime mode

Prime

The prime form of this scale is Scale 367

Scale 367

Aerodian

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 3491

Tharian

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.

k	Hierarchizability	Breakdown Pattern
1	1	`111011010001`
2	1	`111011010001`
3	1	`111011010001`
4	1	`111011010001`
5	2	`(1)(1)(1)0(1)(1)0(1)000(1)`

Enantiomorph

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

Scale 3491

Tharian

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 Center	0.285714
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 2229				Raga Nalinakanti
Scale 2227				Raga Gaula
Scale 2235				Bathian
Scale 2239				Dacryllic
Scale 2215				Ranimic
Scale 2223				Konian
Scale 2199				Dyptimic
Scale 2263				Lycrian
Scale 2295				Kogyllic
Scale 2103				MURian
Scale 2167				NEDian
Scale 2359				Gadian
Scale 2487				Phroptyllic
Scale 2743				Staptyllic
Scale 3255				Daryllic
Scale 183				BEBian
Scale 1207				Aeoloptian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and 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 are ©2005 William Zeitler (http://allthescales.org) used with permission.

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. Thanks to u/howaboot for inventing the Center of Gravity metrics.