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Scale 2859: "Phrycrian"

Scale 2859: Phrycrian, 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
Phrycrian
Dozenal
Sagian

Analysis

Cardinality

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

7 (heptatonic)

Pitch Class Set

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

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

Forte Number

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

7-26

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: 2715

Hemitonia

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

3 (trihemitonic)

Cohemitonia

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

1 (uncohemitonic)

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 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.

6

Prime Form

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

no
prime: 699

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, 3, 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.

<3, 4, 4, 5, 3, 2>

Interval Spectrum

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

p3m5n4s4d3t2

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

Spectra Variation

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

2.286

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.

2.549

Polygon Perimeter

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

5.967

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.

(13, 34, 98)

Tertian Harmonic Chords

Tertian chords are made from alternating members of the scale, ie built from "stacked thirds". Not all scales lend themselves well to tertian harmony.

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}231.75
F{5,9,0}331.5
G♯{8,0,3}331.5
Minor Triadsfm{5,8,0}421.25
g♯m{8,11,3}242
Augmented TriadsC♯+{1,5,9}242
Diminished Triads{5,8,11}231.75
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 2859. Created by Ian Ring ©2019 C# C# C#+ C#+ C#->C#+ fm fm C#->fm F F C#+->F f°->fm g#m g#m f°->g#m fm->F G# G# fm->G# F->a° 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
Radius2
Self-Centeredno
Central Verticesfm
Peripheral VerticesC♯+, g♯m

Modes

Modes are the rotational transformation of this scale. Scale 2859 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3477
Scale 3477: Kyptian, Ian Ring Music TheoryKyptian
3rd mode:
Scale 1893
Scale 1893: Ionylian, Ian Ring Music TheoryIonylian
4th mode:
Scale 1497
Scale 1497: Mela Jyotisvarupini, Ian Ring Music TheoryMela Jyotisvarupini
5th mode:
Scale 699
Scale 699: Aerothian, Ian Ring Music TheoryAerothianThis is the prime mode
6th mode:
Scale 2397
Scale 2397: Stagian, Ian Ring Music TheoryStagian
7th mode:
Scale 1623
Scale 1623: Lothian, Ian Ring Music TheoryLothian

Prime

The prime form of this scale is Scale 699

Scale 699Scale 699: Aerothian, Ian Ring Music TheoryAerothian

Complement

The heptatonic modal family [2859, 3477, 1893, 1497, 699, 2397, 1623] (Forte: 7-26) is the complement of the pentatonic modal family [309, 849, 1101, 1299, 2697] (Forte: 5-26)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2859 is 2715

Scale 2715Scale 2715: Kynian, Ian Ring Music TheoryKynian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2859 is chiral, and its enantiomorph is scale 2715

Scale 2715Scale 2715: Kynian, Ian Ring Music TheoryKynian

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> 2859       T0I <11,0> 2715
T1 <1,1> 1623      T1I <11,1> 1335
T2 <1,2> 3246      T2I <11,2> 2670
T3 <1,3> 2397      T3I <11,3> 1245
T4 <1,4> 699      T4I <11,4> 2490
T5 <1,5> 1398      T5I <11,5> 885
T6 <1,6> 2796      T6I <11,6> 1770
T7 <1,7> 1497      T7I <11,7> 3540
T8 <1,8> 2994      T8I <11,8> 2985
T9 <1,9> 1893      T9I <11,9> 1875
T10 <1,10> 3786      T10I <11,10> 3750
T11 <1,11> 3477      T11I <11,11> 3405
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 699      T0MI <7,0> 2985
T1M <5,1> 1398      T1MI <7,1> 1875
T2M <5,2> 2796      T2MI <7,2> 3750
T3M <5,3> 1497      T3MI <7,3> 3405
T4M <5,4> 2994      T4MI <7,4> 2715
T5M <5,5> 1893      T5MI <7,5> 1335
T6M <5,6> 3786      T6MI <7,6> 2670
T7M <5,7> 3477      T7MI <7,7> 1245
T8M <5,8> 2859       T8MI <7,8> 2490
T9M <5,9> 1623      T9MI <7,9> 885
T10M <5,10> 3246      T10MI <7,10> 1770
T11M <5,11> 2397      T11MI <7,11> 3540

The transformations that map this set to itself are: T0, T8M

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 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
Scale 2861Scale 2861: Katothian, Ian Ring Music TheoryKatothian
Scale 2863Scale 2863: Aerogyllic, Ian Ring Music TheoryAerogyllic
Scale 2851Scale 2851: Katoptimic, Ian Ring Music TheoryKatoptimic
Scale 2855Scale 2855: Epocrain, Ian Ring Music TheoryEpocrain
Scale 2867Scale 2867: Socrian, Ian Ring Music TheorySocrian
Scale 2875Scale 2875: Ganyllic, Ian Ring Music TheoryGanyllic
Scale 2827Scale 2827: Runian, Ian Ring Music TheoryRunian
Scale 2843Scale 2843: Sorian, Ian Ring Music TheorySorian
Scale 2891Scale 2891: Phrogian, Ian Ring Music TheoryPhrogian
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2987Scale 2987: Neapolitan Major and Minor Mixed, Ian Ring Music TheoryNeapolitan Major and Minor Mixed
Scale 2603Scale 2603: Gadimic, Ian Ring Music TheoryGadimic
Scale 2731Scale 2731: Neapolitan Major, Ian Ring Music TheoryNeapolitan Major
Scale 2347Scale 2347: Raga Viyogavarali, Ian Ring Music TheoryRaga Viyogavarali
Scale 3371Scale 3371: Aeolylian, Ian Ring Music TheoryAeolylian
Scale 3883Scale 3883: Kyryllic, Ian Ring Music TheoryKyryllic
Scale 811Scale 811: Radimic, Ian Ring Music TheoryRadimic
Scale 1835Scale 1835: Byptian, Ian Ring Music TheoryByptian

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.