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Scale 4011: "Styrygic"

Scale 4011: Styrygic, 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
Styrygic
Dozenal
Zifian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

9 (enneatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,3,5,7,8,9,10,11}

Forte Number

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

9-6

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.

[4]

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.

no

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

6 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

5 (multicohemitonic)

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 does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

8

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 1407

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.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 2, 2, 2, 1, 1, 1, 1, 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.

<6, 8, 6, 7, 6, 3>

Interval Spectrum

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

p6m7n6s8d6t3

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

Spectra Variation

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

2

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.

yes

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.

2.799

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

6.106

Myhill Property

A scale has Myhill Property if the Interval Spectra has 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.

[8]

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.

(54, 115, 200)

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}242.31
D♯{3,7,10}242.54
F{5,9,0}342.15
G♯{8,0,3}442
Minor Triadscm{0,3,7}242.31
fm{5,8,0}442
g♯m{8,11,3}342.15
a♯m{10,1,5}242.54
Augmented TriadsC♯+{1,5,9}342.31
D♯+{3,7,11}342.31
Diminished Triads{5,8,11}242.31
{7,10,1}242.62
{9,0,3}242.31
Parsimonious Voice Leading Between Common Triads of Scale 4011. Created by Ian Ring ©2019 cm cm D#+ D#+ cm->D#+ G# G# cm->G# C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F a#m a#m C#+->a#m D# D# D#->D#+ D#->g° g#m g#m D#+->g#m f°->fm f°->g#m fm->F fm->G# F->a° g°->a#m g#m->G# G#->a°

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 4011 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 4053
Scale 4053: Kyrygic, Ian Ring Music TheoryKyrygic
3rd mode:
Scale 2037
Scale 2037: Sythygic, Ian Ring Music TheorySythygic
4th mode:
Scale 1533
Scale 1533: Katycrygic, Ian Ring Music TheoryKatycrygic
5th mode:
Scale 1407
Scale 1407: Tharygic, Ian Ring Music TheoryTharygicThis is the prime mode
6th mode:
Scale 2751
Scale 2751: Sylygic, Ian Ring Music TheorySylygic
7th mode:
Scale 3423
Scale 3423: Lothygic, Ian Ring Music TheoryLothygic
8th mode:
Scale 3759
Scale 3759: Darygic, Ian Ring Music TheoryDarygic
9th mode:
Scale 3927
Scale 3927: Monygic, Ian Ring Music TheoryMonygic

Prime

The prime form of this scale is Scale 1407

Scale 1407Scale 1407: Tharygic, Ian Ring Music TheoryTharygic

Complement

The enneatonic modal family [4011, 4053, 2037, 1533, 1407, 2751, 3423, 3759, 3927] (Forte: 9-6) is the complement of the tritonic modal family [21, 1029, 1281] (Forte: 3-6)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4011 is 2751

Scale 2751Scale 2751: Sylygic, Ian Ring Music TheorySylygic

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

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 4011       T0I <11,0> 2751
T1 <1,1> 3927      T1I <11,1> 1407
T2 <1,2> 3759      T2I <11,2> 2814
T3 <1,3> 3423      T3I <11,3> 1533
T4 <1,4> 2751      T4I <11,4> 3066
T5 <1,5> 1407      T5I <11,5> 2037
T6 <1,6> 2814      T6I <11,6> 4074
T7 <1,7> 1533      T7I <11,7> 4053
T8 <1,8> 3066      T8I <11,8> 4011
T9 <1,9> 2037      T9I <11,9> 3927
T10 <1,10> 4074      T10I <11,10> 3759
T11 <1,11> 4053      T11I <11,11> 3423
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2751      T0MI <7,0> 4011
T1M <5,1> 1407      T1MI <7,1> 3927
T2M <5,2> 2814      T2MI <7,2> 3759
T3M <5,3> 1533      T3MI <7,3> 3423
T4M <5,4> 3066      T4MI <7,4> 2751
T5M <5,5> 2037      T5MI <7,5> 1407
T6M <5,6> 4074      T6MI <7,6> 2814
T7M <5,7> 4053      T7MI <7,7> 1533
T8M <5,8> 4011       T8MI <7,8> 3066
T9M <5,9> 3927      T9MI <7,9> 2037
T10M <5,10> 3759      T10MI <7,10> 4074
T11M <5,11> 3423      T11MI <7,11> 4053

The transformations that map this set to itself are: T0, T8I, T8M, 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 4009Scale 4009: Phranyllic, Ian Ring Music TheoryPhranyllic
Scale 4013Scale 4013: Raga Pilu, Ian Ring Music TheoryRaga Pilu
Scale 4015Scale 4015: Phradyllian, Ian Ring Music TheoryPhradyllian
Scale 4003Scale 4003: Sadyllic, Ian Ring Music TheorySadyllic
Scale 4007Scale 4007: Doptygic, Ian Ring Music TheoryDoptygic
Scale 4019Scale 4019: Lonygic, Ian Ring Music TheoryLonygic
Scale 4027Scale 4027: Ragyllian, Ian Ring Music TheoryRagyllian
Scale 3979Scale 3979: Dynyllic, Ian Ring Music TheoryDynyllic
Scale 3995Scale 3995: Ionygic, Ian Ring Music TheoryIonygic
Scale 4043Scale 4043: Phrocrygic, Ian Ring Music TheoryPhrocrygic
Scale 4075Scale 4075: Katyllian, Ian Ring Music TheoryKatyllian
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 3947Scale 3947: Ryptygic, Ian Ring Music TheoryRyptygic
Scale 3755Scale 3755: Phryryllic, Ian Ring Music TheoryPhryryllic
Scale 3499Scale 3499: Hamel, Ian Ring Music TheoryHamel
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 1963Scale 1963: Epocryllic, Ian Ring Music TheoryEpocryllic

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. 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.