The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 4057: "Phrygic"

Scale 4057: Phrygic, 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
Phrygic
Dozenal
Zuhian

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

{0,3,4,6,7,8,9,10,11}

Forte Number

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

9-3

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: 895

Hemitonia

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

7 (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: 895

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.

[3, 1, 2, 1, 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.

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

Interval Spectrum

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

p6m7n7s6d7t3

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}
<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> = {9,10,11}

Spectra Variation

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

2.222

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.

2.683

Polygon Perimeter

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

6.038

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.

(64, 107, 194)

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}342.24
D♯{3,7,10}342.35
E{4,8,11}342.35
G♯{8,0,3}442.24
B{11,3,6}342.35
Minor Triadscm{0,3,7}442.12
d♯m{3,6,10}342.53
em{4,7,11}442.24
g♯m{8,11,3}342.24
am{9,0,4}342.53
Augmented TriadsC+{0,4,8}442.24
D♯+{3,7,11}542
Diminished Triads{0,3,6}242.59
d♯°{3,6,9}242.76
{4,7,10}252.71
f♯°{6,9,0}242.76
{9,0,3}252.71
Parsimonious Voice Leading Between Common Triads of Scale 4057. Created by Ian Ring ©2019 cm cm c°->cm B B c°->B C C cm->C D#+ D#+ cm->D#+ G# G# cm->G# C+ C+ C->C+ em em C->em E E C+->E C+->G# am am C+->am d#° d#° d#m d#m d#°->d#m f#° f#° d#°->f#° D# D# d#m->D# d#m->B D#->D#+ D#->e° D#+->em g#m g#m D#+->g#m D#+->B e°->em em->E E->g#m f#°->am g#m->G# 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.

Diameter5
Radius4
Self-Centeredno
Central Verticesc°, cm, C, C+, d♯°, d♯m, D♯, D♯+, em, E, f♯°, g♯m, G♯, am, B
Peripheral Verticese°, a°

Modes

Modes are the rotational transformation of this scale. Scale 4057 can be rotated to make 8 other scales. The 1st mode is itself.

2nd mode:
Scale 1019
Scale 1019: Aeranygic, Ian Ring Music TheoryAeranygic
3rd mode:
Scale 2557
Scale 2557: Dothygic, Ian Ring Music TheoryDothygic
4th mode:
Scale 1663
Scale 1663: Lydygic, Ian Ring Music TheoryLydygic
5th mode:
Scale 2879
Scale 2879: Stadygic, Ian Ring Music TheoryStadygic
6th mode:
Scale 3487
Scale 3487: Byptygic, Ian Ring Music TheoryByptygic
7th mode:
Scale 3791
Scale 3791: Stodygic, Ian Ring Music TheoryStodygic
8th mode:
Scale 3943
Scale 3943: Zynygic, Ian Ring Music TheoryZynygic
9th mode:
Scale 4019
Scale 4019: Lonygic, Ian Ring Music TheoryLonygic

Prime

The prime form of this scale is Scale 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

Complement

The enneatonic modal family [4057, 1019, 2557, 1663, 2879, 3487, 3791, 3943, 4019] (Forte: 9-3) is the complement of the tritonic modal family [19, 769, 2057] (Forte: 3-3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 4057 is 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 4057 is chiral, and its enantiomorph is scale 895

Scale 895Scale 895: Aeolathygic, Ian Ring Music TheoryAeolathygic

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> 4057       T0I <11,0> 895
T1 <1,1> 4019      T1I <11,1> 1790
T2 <1,2> 3943      T2I <11,2> 3580
T3 <1,3> 3791      T3I <11,3> 3065
T4 <1,4> 3487      T4I <11,4> 2035
T5 <1,5> 2879      T5I <11,5> 4070
T6 <1,6> 1663      T6I <11,6> 4045
T7 <1,7> 3326      T7I <11,7> 3995
T8 <1,8> 2557      T8I <11,8> 3895
T9 <1,9> 1019      T9I <11,9> 3695
T10 <1,10> 2038      T10I <11,10> 3295
T11 <1,11> 4076      T11I <11,11> 2495
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3037      T0MI <7,0> 1915
T1M <5,1> 1979      T1MI <7,1> 3830
T2M <5,2> 3958      T2MI <7,2> 3565
T3M <5,3> 3821      T3MI <7,3> 3035
T4M <5,4> 3547      T4MI <7,4> 1975
T5M <5,5> 2999      T5MI <7,5> 3950
T6M <5,6> 1903      T6MI <7,6> 3805
T7M <5,7> 3806      T7MI <7,7> 3515
T8M <5,8> 3517      T8MI <7,8> 2935
T9M <5,9> 2939      T9MI <7,9> 1775
T10M <5,10> 1783      T10MI <7,10> 3550
T11M <5,11> 3566      T11MI <7,11> 3005

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 4059Scale 4059: Zolyllian, Ian Ring Music TheoryZolyllian
Scale 4061Scale 4061: Staptyllian, Ian Ring Music TheoryStaptyllian
Scale 4049Scale 4049: Stycryllic, Ian Ring Music TheoryStycryllic
Scale 4053Scale 4053: Kyrygic, Ian Ring Music TheoryKyrygic
Scale 4041Scale 4041: Zaryllic, Ian Ring Music TheoryZaryllic
Scale 4073Scale 4073: Sathygic, Ian Ring Music TheorySathygic
Scale 4089Scale 4089: Decatonic Chromatic Descending, Ian Ring Music TheoryDecatonic Chromatic Descending
Scale 3993Scale 3993: Ioniptyllic, Ian Ring Music TheoryIoniptyllic
Scale 4025Scale 4025: Kalygic, Ian Ring Music TheoryKalygic
Scale 3929Scale 3929: Aeolothyllic, Ian Ring Music TheoryAeolothyllic
Scale 3801Scale 3801: Maptyllic, Ian Ring Music TheoryMaptyllic
Scale 3545Scale 3545: Thyptyllic, Ian Ring Music TheoryThyptyllic
Scale 3033Scale 3033: Doptyllic, Ian Ring Music TheoryDoptyllic
Scale 2009Scale 2009: Stacryllic, Ian Ring Music TheoryStacryllic

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.