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Scale 2009: "Stacryllic"

Scale 2009: Stacryllic, 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
Stacryllic
Dozenal
EGGian

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

 {0,3,4,6,7,8,9,10}

Forte Number

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

 8-12

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: 893

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.

 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.

 8

Prime Form

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

 no
prime: 763

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.

 [3, 1, 2, 1, 1, 1, 1, 2]

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, 6, 5, 4, 3>

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, 0.5, 0.333, 0, 0.5>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

 p4m5n6s5d5t3

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> = {3,4,6}
<4> = {4,5,7,8}
<5> = {6,8,9}
<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.

 2.5

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.

(46, 63, 142)

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.

0.59

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.

0.276

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}342
D♯{3,7,10}342
G♯{8,0,3}342
Minor Triadscm{0,3,7}441.83
d♯m{3,6,10}342.17
am{9,0,4}342.17
Augmented TriadsC+{0,4,8}342
Diminished Triadsc°{0,3,6}242.17
d♯°{3,6,9}242.33
e°{4,7,10}242.33
f♯°{6,9,0}242.33
a°{9,0,3}242.33
Parsimonious Voice Leading Between Common Triads of Scale 2009. Created by Ian Ring ©2019 c° c° cm cm c°->cm d#m d#m c°->d#m C C cm->C D# D# cm->D# G# G# cm->G# C+ C+ C->C+ e° e° C->e° C+->G# am am C+->am d#° d#° d#°->d#m f#° f#° d#°->f#° d#m->D# D#->e° f#°->am a° a° G#->a° a°->am

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
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale.Scale 2009 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 763		Scale 763: Doryllic, Ian Ring Music TheoryDoryllicThis is the prime mode
3rd mode:
Scale 2429		Scale 2429: Kadyllic, Ian Ring Music TheoryKadyllic
4th mode:
Scale 1631		Scale 1631: Rynyllic, Ian Ring Music TheoryRynyllic
5th mode:
Scale 2863		Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
6th mode:
Scale 3479		Scale 3479: Rothyllic, Ian Ring Music TheoryRothyllic
7th mode:
Scale 3787		Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
8th mode:
Scale 3941		Scale 3941: Stathyllic, Ian Ring Music TheoryStathyllic

Prime

The prime form of this scale is Scale 763

Scale 763Scale 763: Doryllic, Ian Ring Music TheoryDoryllic

Complement

The octatonic modal family [2009, 763, 2429, 1631, 2863, 3479, 3787, 3941] (Forte: 8-12) is the complement of the tetratonic modal family [77, 833, 1043, 2569] (Forte: 4-12)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2009 is 893

Scale 893Scale 893: Pycryllic, Ian Ring Music TheoryPycryllic

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
111001101111102009k = 1h = 1
211001101111102009k = 2h = 1
311001101111102009k = 3h = 1
4300([1][1]0[1])11([1][1]0[1])2009k = 4h = 3
5300([1][1]0[1])11([1][1]0[1])2009k = 5h = 3

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2009 is chiral, and its enantiomorph is scale 893

Scale 893Scale 893: Pycryllic, Ian Ring Music TheoryPycryllic

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> 2009       T0I <11,0> 893
T1 <1,1> 4018      T1I <11,1> 1786
T2 <1,2> 3941      T2I <11,2> 3572
T3 <1,3> 3787      T3I <11,3> 3049
T4 <1,4> 3479      T4I <11,4> 2003
T5 <1,5> 2863      T5I <11,5> 4006
T6 <1,6> 1631      T6I <11,6> 3917
T7 <1,7> 3262      T7I <11,7> 3739
T8 <1,8> 2429      T8I <11,8> 3383
T9 <1,9> 763      T9I <11,9> 2671
T10 <1,10> 1526      T10I <11,10> 1247
T11 <1,11> 3052      T11I <11,11> 2494
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2909      T0MI <7,0> 1883
T1M <5,1> 1723      T1MI <7,1> 3766
T2M <5,2> 3446      T2MI <7,2> 3437
T3M <5,3> 2797      T3MI <7,3> 2779
T4M <5,4> 1499      T4MI <7,4> 1463
T5M <5,5> 2998      T5MI <7,5> 2926
T6M <5,6> 1901      T6MI <7,6> 1757
T7M <5,7> 3802      T7MI <7,7> 3514
T8M <5,8> 3509      T8MI <7,8> 2933
T9M <5,9> 2923      T9MI <7,9> 1771
T10M <5,10> 1751      T10MI <7,10> 3542
T11M <5,11> 3502      T11MI <7,11> 2989

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 2011Scale 2011: Raphygic, Ian Ring Music TheoryRaphygic
Scale 2013Scale 2013: Mocrygic, Ian Ring Music TheoryMocrygic
Scale 2001Scale 2001: Gydian, Ian Ring Music TheoryGydian
Scale 2005Scale 2005: Gygyllic, Ian Ring Music TheoryGygyllic
Scale 1993Scale 1993: Katoptian, Ian Ring Music TheoryKatoptian
Scale 2025Scale 2025: Mixolydyllic, Ian Ring Music TheoryMixolydyllic
Scale 2041Scale 2041: Aeolacrygic, Ian Ring Music TheoryAeolacrygic
Scale 1945Scale 1945: Zarian, Ian Ring Music TheoryZarian
Scale 1977Scale 1977: Dagyllic, Ian Ring Music TheoryDagyllic
Scale 1881Scale 1881: Korian, Ian Ring Music TheoryKorian
Scale 1753Scale 1753: Hungarian Major, Ian Ring Music TheoryHungarian Major
Scale 1497Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini
Scale 985Scale 985: Mela Sucaritra, Ian Ring Music TheoryMela Sucaritra
Scale 3033Scale 3033: Doptyllic, Ian Ring Music TheoryDoptyllic
Scale 4057Scale 4057: Phrygic, Ian Ring Music TheoryPhrygic

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.

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