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Scale 3783: "Phrygyllic"

Scale 3783: Phrygyllic, 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
Phrygyllic

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

{0,1,2,6,7,9,10,11}

Forte Number

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

8-4

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: 3183

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.

7

Prime Form

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

no
prime: 447

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, 1, 4, 1, 2, 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, 5, 5, 5, 5, 2>

Interval Spectrum

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

p5m5n5s5d6t2

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> = {1,2,4}
<2> = {2,3,5}
<3> = {3,4,6,7}
<4> = {4,5,7,8}
<5> = {5,6,8,9}
<6> = {7,9,10}
<7> = {8,10,11}

Spectra Variation

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

3

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.

2.366

Polygon Perimeter

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

5.838

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.

(75, 56, 136)

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 TriadsD{2,6,9}231.78
F♯{6,10,1}331.56
G{7,11,2}252.33
Minor Triadsf♯m{6,9,1}341.89
gm{7,10,2}341.78
bm{11,2,6}242
Augmented TriadsD+{2,6,10}431.44
Diminished Triadsf♯°{6,9,0}152.67
{7,10,1}231.89
Parsimonious Voice Leading Between Common Triads of Scale 3783. Created by Ian Ring ©2019 D D D+ D+ D->D+ f#m f#m D->f#m F# F# D+->F# gm gm D+->gm bm bm D+->bm f#° f#° f#°->f#m f#m->F# F#->g° g°->gm Parsimonious Voice Leading Between Common Triads of Scale 3783. Created by Ian Ring ©2019 G gm->G G->bm

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 VerticesD, D+, F♯, g°
Peripheral Verticesf♯°, G

Modes

Modes are the rotational transformation of this scale. Scale 3783 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3939
Scale 3939: Dogyllic, Ian Ring Music TheoryDogyllic
3rd mode:
Scale 4017
Scale 4017: Dolyllic, Ian Ring Music TheoryDolyllic
4th mode:
Scale 507
Scale 507: Moryllic, Ian Ring Music TheoryMoryllic
5th mode:
Scale 2301
Scale 2301: Bydyllic, Ian Ring Music TheoryBydyllic
6th mode:
Scale 1599
Scale 1599: Pocryllic, Ian Ring Music TheoryPocryllic
7th mode:
Scale 2847
Scale 2847: Phracryllic, Ian Ring Music TheoryPhracryllic
8th mode:
Scale 3471
Scale 3471: Gyryllic, Ian Ring Music TheoryGyryllic

Prime

The prime form of this scale is Scale 447

Scale 447Scale 447: Thyphyllic, Ian Ring Music TheoryThyphyllic

Complement

The octatonic modal family [3783, 3939, 4017, 507, 2301, 1599, 2847, 3471] (Forte: 8-4) is the complement of the tetratonic modal family [39, 897, 2067, 3081] (Forte: 4-4)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3783 is 3183

Scale 3183Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3783 is chiral, and its enantiomorph is scale 3183

Scale 3183Scale 3183: Mixonyllic, Ian Ring Music TheoryMixonyllic

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

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 3781Scale 3781: Gyphian, Ian Ring Music TheoryGyphian
Scale 3779Scale 3779, Ian Ring Music Theory
Scale 3787Scale 3787: Kagyllic, Ian Ring Music TheoryKagyllic
Scale 3791Scale 3791: Stodygic, Ian Ring Music TheoryStodygic
Scale 3799Scale 3799: Aeralygic, Ian Ring Music TheoryAeralygic
Scale 3815Scale 3815: Galygic, Ian Ring Music TheoryGalygic
Scale 3719Scale 3719: Xofian, Ian Ring Music TheoryXofian
Scale 3751Scale 3751: Aerathyllic, Ian Ring Music TheoryAerathyllic
Scale 3655Scale 3655: Mathian, Ian Ring Music TheoryMathian
Scale 3911Scale 3911: Katyryllic, Ian Ring Music TheoryKatyryllic
Scale 4039Scale 4039: Nonatonic Chromatic 7, Ian Ring Music TheoryNonatonic Chromatic 7
Scale 3271Scale 3271: Mela Raghupriya, Ian Ring Music TheoryMela Raghupriya
Scale 3527Scale 3527: Ronyllic, Ian Ring Music TheoryRonyllic
Scale 2759Scale 2759: Mela Pavani, Ian Ring Music TheoryMela Pavani
Scale 1735Scale 1735: Mela Navanitam, Ian Ring Music TheoryMela Navanitam

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.