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Scale 2427: "Katoryllic"

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: Katoryllic

Dozenal: ONSian

Analysis

Cardinality Cardinality is the count of how many pitches are in the scale.	8 (octatonic)
Pitch Class Set The tones in this scale, expressed as numbers from 0 to 11	{0,1,3,4,5,6,8,11}
Forte Number A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.	8-14
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: 3027
Hemitonia A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.	5 (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.	2
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.	8
Prime Form Describes if this scale is in prime form, using the Starr/Rahn algorithm.	no prime: 759
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, 2, 1, 1, 1, 2, 3, 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.	<5, 5, 5, 5, 6, 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.	<0.333, 0.333, 0.25, 0.333, 0.667, 0>
Interval Spectrum The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.	p⁶m⁵n⁵s⁵d⁵t²
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> = {1,2,3} <2> = {2,3,4,5} <3> = {3,4,5,6} <4> = {5,7} <5> = {6,7,8,9} <6> = {7,8,9,10} <7> = {9,10,11}
Spectra Variation Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.	2.25
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.	2.616
Polygon Perimeter Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.	6.002
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.	0.188246
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.	(34, 56, 136)
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.	0.662
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.	0.306

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♯	{1,5,8}	2	5	2.5
	E	{4,8,11}	3	3	1.7
	G♯	{8,0,3}	3	3	1.7
	B	{11,3,6}	2	5	2.5
Minor Triads	c♯m	{1,4,8}	2	4	2.1
	fm	{5,8,0}	3	4	1.9
	g♯m	{8,11,3}	3	4	1.9
Augmented Triads	C+	{0,4,8}	4	3	1.5
Diminished Triads	c°	{0,3,6}	2	4	2.3
Diminished Triads	f°	{5,8,11}	2	4	2.1

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	5
Radius	3
Self-Centered	no
Central Vertices	C+, E, G♯
Peripheral Vertices	C♯, B

Modes

Modes are the rotational transformation of this scale. Scale 2427 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode: Scale 3261	Dodyllic
3rd mode: Scale 1839	Zogyllic
4th mode: Scale 2967	Madyllic
5th mode: Scale 3531	Neveseri
6th mode: Scale 3813	Aeologyllic
7th mode: Scale 1977	Dagyllic
8th mode: Scale 759	Katalyllic	This is the prime mode

Prime

The prime form of this scale is Scale 759

Scale 759

Katalyllic

Complement

The octatonic modal family [2427, 3261, 1839, 2967, 3531, 3813, 1977, 759] (Forte: 8-14) is the complement of the tetratonic modal family [141, 417, 1059, 2577] (Forte: 4-14)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2427 is 3027

Scale 3027

Rythyllic

Interval Matrix

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

Contradictions (34)

Ambiguities(56)

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	`110111101001`
2	3	`([11]0[11])00([11]0[11])`
3	3	`([11]0[11])00([11]0[11])`
4	3	`([11]0[11])00([11]0[11])`
5	3	`([11]0[11])00([11]0[11])`

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2427 is chiral, and its enantiomorph is scale 3027

Scale 3027

Rythyllic

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	(0.1875, 0.016747)
Distance from Center	0.188246
Angle in degrees measured clockwise starting from the root.	95.104
Angle in cents 100 cents = 1 semitone.	317.013333

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>	2427	T₀I	<11,0>	3027
T₁	<1,1>	759	T₁I	<11,1>	1959
T₂	<1,2>	1518	T₂I	<11,2>	3918
T₃	<1,3>	3036	T₃I	<11,3>	3741
T₄	<1,4>	1977	T₄I	<11,4>	3387
T₅	<1,5>	3954	T₅I	<11,5>	2679
T₆	<1,6>	3813	T₆I	<11,6>	1263
T₇	<1,7>	3531	T₇I	<11,7>	2526
T₈	<1,8>	2967	T₈I	<11,8>	957
T₉	<1,9>	1839	T₉I	<11,9>	1914
T₁₀	<1,10>	3678	T₁₀I	<11,10>	3828
T₁₁	<1,11>	3261	T₁₁I	<11,11>	3561
Abbrev	Operation	Result	Abbrev	Operation	Result
T₀M	<5,0>	507	T₀MI	<7,0>	3057
T₁M	<5,1>	1014	T₁MI	<7,1>	2019
T₂M	<5,2>	2028	T₂MI	<7,2>	4038
T₃M	<5,3>	4056	T₃MI	<7,3>	3981
T₄M	<5,4>	4017	T₄MI	<7,4>	3867
T₅M	<5,5>	3939	T₅MI	<7,5>	3639
T₆M	<5,6>	3783	T₆MI	<7,6>	3183
T₇M	<5,7>	3471	T₇MI	<7,7>	2271
T₈M	<5,8>	2847	T₈MI	<7,8>	447
T₉M	<5,9>	1599	T₉MI	<7,9>	894
T₁₀M	<5,10>	3198	T₁₀MI	<7,10>	1788
T₁₁M	<5,11>	2301	T₁₁MI	<7,11>	3576

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 2425				Rorian
Scale 2429				Kadyllic
Scale 2431				Gythygic
Scale 2419				Raga Lalita
Scale 2423				Thorcryllic
Scale 2411				Aeolorian
Scale 2395				Zoptian
Scale 2363				Kataptian
Scale 2491				Layllic
Scale 2555				Bythygic
Scale 2171				NEGian
Scale 2299				Phraptyllic
Scale 2683				Thodyllic
Scale 2939				Diminishing Nonamode 2nd Rotation
Scale 3451				Garygic
Scale 379				Aeragian
Scale 1403				Espla's Scale

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.