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Scale 3051: "Stalygic"

Scale 3051: Stalygic, 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
Stalygic
Dozenal
Tatian

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

{0,1,3,5,6,7,8,9,11}

Forte Number

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

9-8

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: 2811

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.

4 (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: 1503

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, 1, 1, 1, 1, 2, 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, 7, 6, 7, 6, 4>

Interval Spectrum

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

p6m7n6s7d6t4

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

1.556

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.

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.

(8, 94, 180)

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}242.47
F{5,9,0}442.2
G♯{8,0,3}442.07
B{11,3,6}342.47
Minor Triadscm{0,3,7}342.33
fm{5,8,0}442.07
f♯m{6,9,1}342.47
g♯m{8,11,3}342.33
Augmented TriadsC♯+{1,5,9}342.4
D♯+{3,7,11}342.4
Diminished Triads{0,3,6}242.67
d♯°{3,6,9}242.53
{5,8,11}242.47
f♯°{6,9,0}242.53
{9,0,3}242.33
Parsimonious Voice Leading Between Common Triads of Scale 3051. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B D#+ D#+ cm->D#+ G# G# cm->G# C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F f#m f#m C#+->f#m d#° d#° d#°->f#m d#°->B g#m g#m D#+->g#m D#+->B f°->fm f°->g#m fm->F fm->G# f#° f#° F->f#° F->a° f#°->f#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 3051 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 3573
Scale 3573: Kaptygic, Ian Ring Music TheoryKaptygic
3rd mode:
Scale 1917
Scale 1917: Sacrygic, Ian Ring Music TheorySacrygic
4th mode:
Scale 1503
Scale 1503: Padygic, Ian Ring Music TheoryPadygicThis is the prime mode
5th mode:
Scale 2799
Scale 2799: Epilygic, Ian Ring Music TheoryEpilygic
6th mode:
Scale 3447
Scale 3447: Kynygic, Ian Ring Music TheoryKynygic
7th mode:
Scale 3771
Scale 3771: Stophygic, Ian Ring Music TheoryStophygic
8th mode:
Scale 3933
Scale 3933: Ionidygic, Ian Ring Music TheoryIonidygic
9th mode:
Scale 2007
Scale 2007: Stonygic, Ian Ring Music TheoryStonygic

Prime

The prime form of this scale is Scale 1503

Scale 1503Scale 1503: Padygic, Ian Ring Music TheoryPadygic

Complement

The enneatonic modal family [3051, 3573, 1917, 1503, 2799, 3447, 3771, 3933, 2007] (Forte: 9-8) is the complement of the tritonic modal family [69, 321, 1041] (Forte: 3-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3051 is 2811

Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3051 is chiral, and its enantiomorph is scale 2811

Scale 2811Scale 2811: Barygic, Ian Ring Music TheoryBarygic

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> 3051       T0I <11,0> 2811
T1 <1,1> 2007      T1I <11,1> 1527
T2 <1,2> 4014      T2I <11,2> 3054
T3 <1,3> 3933      T3I <11,3> 2013
T4 <1,4> 3771      T4I <11,4> 4026
T5 <1,5> 3447      T5I <11,5> 3957
T6 <1,6> 2799      T6I <11,6> 3819
T7 <1,7> 1503      T7I <11,7> 3543
T8 <1,8> 3006      T8I <11,8> 2991
T9 <1,9> 1917      T9I <11,9> 1887
T10 <1,10> 3834      T10I <11,10> 3774
T11 <1,11> 3573      T11I <11,11> 3453
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2811      T0MI <7,0> 3051
T1M <5,1> 1527      T1MI <7,1> 2007
T2M <5,2> 3054      T2MI <7,2> 4014
T3M <5,3> 2013      T3MI <7,3> 3933
T4M <5,4> 4026      T4MI <7,4> 3771
T5M <5,5> 3957      T5MI <7,5> 3447
T6M <5,6> 3819      T6MI <7,6> 2799
T7M <5,7> 3543      T7MI <7,7> 1503
T8M <5,8> 2991      T8MI <7,8> 3006
T9M <5,9> 1887      T9MI <7,9> 1917
T10M <5,10> 3774      T10MI <7,10> 3834
T11M <5,11> 3453      T11MI <7,11> 3573

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 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic
Scale 3053Scale 3053: Zycrygic, Ian Ring Music TheoryZycrygic
Scale 3055Scale 3055: Messiaen Mode 7, Ian Ring Music TheoryMessiaen Mode 7
Scale 3043Scale 3043: Ionayllic, Ian Ring Music TheoryIonayllic
Scale 3047Scale 3047: Panygic, Ian Ring Music TheoryPanygic
Scale 3059Scale 3059: Madygic, Ian Ring Music TheoryMadygic
Scale 3067Scale 3067: Goptyllian, Ian Ring Music TheoryGoptyllian
Scale 3019Scale 3019: Subian, Ian Ring Music TheorySubian
Scale 3035Scale 3035: Gocrygic, Ian Ring Music TheoryGocrygic
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2795Scale 2795: Van der Horst Octatonic, Ian Ring Music TheoryVan der Horst Octatonic
Scale 2539Scale 2539: Half-Diminished Bebop, Ian Ring Music TheoryHalf-Diminished Bebop
Scale 3563Scale 3563: Ionoptygic, Ian Ring Music TheoryIonoptygic
Scale 4075Scale 4075: Katyllian, Ian Ring Music TheoryKatyllian
Scale 1003Scale 1003: Ionyryllic, Ian Ring Music TheoryIonyryllic
Scale 2027Scale 2027: Boptygic, Ian Ring Music TheoryBoptygic

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.