The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2427: "Katoryllic"

Scale 2427: Katoryllic, Ian Ring Music Theory

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

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

 no

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 TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC♯{1,5,8}252.5
E{4,8,11}331.7
G♯{8,0,3}331.7
B{11,3,6}252.5
Minor Triadsc♯m{1,4,8}242.1
fm{5,8,0}341.9
g♯m{8,11,3}341.9
Augmented TriadsC+{0,4,8}431.5
Diminished Triadsc°{0,3,6}242.3
f°{5,8,11}242.1
Parsimonious Voice Leading Between Common Triads of Scale 2427. Created by Ian Ring ©2019 c° c° G# G# c°->G# B B c°->B C+ C+ c#m c#m C+->c#m E E C+->E fm fm C+->fm C+->G# C# C# c#m->C# C#->fm f° f° E->f° g#m g#m E->g#m f°->fm g#m->G# g#m->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.

Diameter5
Radius3
Self-Centeredno
Central VerticesC+, E, G♯
Peripheral VerticesC♯, 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		Scale 3261: Dodyllic, Ian Ring Music TheoryDodyllic
3rd mode:
Scale 1839		Scale 1839: Zogyllic, Ian Ring Music TheoryZogyllic
4th mode:
Scale 2967		Scale 2967: Madyllic, Ian Ring Music TheoryMadyllic
5th mode:
Scale 3531		Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
6th mode:
Scale 3813		Scale 3813: Aeologyllic, Ian Ring Music TheoryAeologyllic
7th mode:
Scale 1977		Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic
8th mode:
Scale 759		Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllicThis is the prime mode

Prime

The prime form of this scale is Scale 759

Scale 759Scale 759: Katalyllic, Ian Ring Music TheoryKatalyllic

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 3027Scale 3027: Rythyllic, Ian Ring Music TheoryRythyllic

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
111101111010012427k = 1h = 1
23([11]0[11])00([11]0[11])2427k = 2h = 3
33([11]0[11])00([11]0[11])2427k = 3h = 3
43([11]0[11])00([11]0[11])2427k = 4h = 3
53([11]0[11])00([11]0[11])2427k = 5h = 3

Enantiomorph

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

Scale 3027Scale 3027: Rythyllic, Ian Ring Music TheoryRythyllic

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 2425Scale 2425: Rorian, Ian Ring Music TheoryRorian
Scale 2429Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
Scale 2431Scale 2431: Gythygic, Ian Ring Music TheoryGythygic
Scale 2419Scale 2419: Raga Lalita, Ian Ring Music TheoryRaga Lalita
Scale 2423Scale 2423: Thorcryllic, Ian Ring Music TheoryThorcryllic
Scale 2411Scale 2411: Aeolorian, Ian Ring Music TheoryAeolorian
Scale 2395Scale 2395: Zoptian, Ian Ring Music TheoryZoptian
Scale 2363Scale 2363: Kataptian, Ian Ring Music TheoryKataptian
Scale 2491Scale 2491: Layllic, Ian Ring Music TheoryLayllic
Scale 2555Scale 2555: Bythygic, Ian Ring Music TheoryBythygic
Scale 2171Scale 2171: NEGian, Ian Ring Music TheoryNEGian
Scale 2299Scale 2299: Phraptyllic, Ian Ring Music TheoryPhraptyllic
Scale 2683Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
Scale 2939Scale 2939: Diminishing Nonamode 2nd Rotation, Ian Ring Music TheoryDiminishing Nonamode 2nd Rotation
Scale 3451Scale 3451: Garygic, Ian Ring Music TheoryGarygic
Scale 379Scale 379: Aeragian, Ian Ring Music TheoryAeragian
Scale 1403Scale 1403: Espla's Scale, Ian Ring Music TheoryEspla'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. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

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 the Carnatic ragas.