The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2891: "Phrogian"

Scale 2891: Phrogian, 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
Phrogian

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

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

Forte Number

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

7-25

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: 2651

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.

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.

6

Prime Form

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

no
prime: 733

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 Formula

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

[1, 2, 3, 2, 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, 5, 3, 4, 2>

Interval Spectrum

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

p4m3n5s4d3t2

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

(19, 28, 92)

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 TriadsG♯{8,0,3}331.63
B{11,3,6}331.63
Minor Triadsf♯m{6,9,1}231.88
g♯m{8,11,3}231.75
Diminished Triads{0,3,6}231.75
d♯°{3,6,9}231.75
f♯°{6,9,0}231.88
{9,0,3}231.75
Parsimonious Voice Leading Between Common Triads of Scale 2891. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B d#° d#° f#m f#m d#°->f#m d#°->B f#° f#° f#°->f#m f#°->a° g#m g#m g#m->G# g#m->B 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.

Diameter3
Radius3
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2891 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3493
Scale 3493: Rathian, Ian Ring Music TheoryRathian
3rd mode:
Scale 1897
Scale 1897: Ionopian, Ian Ring Music TheoryIonopian
4th mode:
Scale 749
Scale 749: Aeologian, Ian Ring Music TheoryAeologian
5th mode:
Scale 1211
Scale 1211: Zadian, Ian Ring Music TheoryZadian
6th mode:
Scale 2653
Scale 2653: Sygian, Ian Ring Music TheorySygian
7th mode:
Scale 1687
Scale 1687: Phralian, Ian Ring Music TheoryPhralian

Prime

The prime form of this scale is Scale 733

Scale 733Scale 733: Donian, Ian Ring Music TheoryDonian

Complement

The heptatonic modal family [2891, 3493, 1897, 749, 1211, 2653, 1687] (Forte: 7-25) is the complement of the pentatonic modal family [301, 721, 1099, 1673, 2597] (Forte: 5-25)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2891 is 2651

Scale 2651Scale 2651: Panian, Ian Ring Music TheoryPanian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2891 is chiral, and its enantiomorph is scale 2651

Scale 2651Scale 2651: Panian, Ian Ring Music TheoryPanian

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> 2891       T0I <11,0> 2651
T1 <1,1> 1687      T1I <11,1> 1207
T2 <1,2> 3374      T2I <11,2> 2414
T3 <1,3> 2653      T3I <11,3> 733
T4 <1,4> 1211      T4I <11,4> 1466
T5 <1,5> 2422      T5I <11,5> 2932
T6 <1,6> 749      T6I <11,6> 1769
T7 <1,7> 1498      T7I <11,7> 3538
T8 <1,8> 2996      T8I <11,8> 2981
T9 <1,9> 1897      T9I <11,9> 1867
T10 <1,10> 3794      T10I <11,10> 3734
T11 <1,11> 3493      T11I <11,11> 3373
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 761      T0MI <7,0> 1001
T1M <5,1> 1522      T1MI <7,1> 2002
T2M <5,2> 3044      T2MI <7,2> 4004
T3M <5,3> 1993      T3MI <7,3> 3913
T4M <5,4> 3986      T4MI <7,4> 3731
T5M <5,5> 3877      T5MI <7,5> 3367
T6M <5,6> 3659      T6MI <7,6> 2639
T7M <5,7> 3223      T7MI <7,7> 1183
T8M <5,8> 2351      T8MI <7,8> 2366
T9M <5,9> 607      T9MI <7,9> 637
T10M <5,10> 1214      T10MI <7,10> 1274
T11M <5,11> 2428      T11MI <7,11> 2548

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 2889Scale 2889: Thoptimic, Ian Ring Music TheoryThoptimic
Scale 2893Scale 2893: Lylian, Ian Ring Music TheoryLylian
Scale 2895Scale 2895: Aeoryllic, Ian Ring Music TheoryAeoryllic
Scale 2883Scale 2883, Ian Ring Music Theory
Scale 2887Scale 2887: Gaptian, Ian Ring Music TheoryGaptian
Scale 2899Scale 2899: Kagian, Ian Ring Music TheoryKagian
Scale 2907Scale 2907: Magen Abot 2, Ian Ring Music TheoryMagen Abot 2
Scale 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2827Scale 2827, Ian Ring Music Theory
Scale 2859Scale 2859: Phrycrian, Ian Ring Music TheoryPhrycrian
Scale 2955Scale 2955: Thorian, Ian Ring Music TheoryThorian
Scale 3019Scale 3019, Ian Ring Music Theory
Scale 2635Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic
Scale 2763Scale 2763: Mela Suvarnangi, Ian Ring Music TheoryMela Suvarnangi
Scale 2379Scale 2379: Raga Gurjari Todi, Ian Ring Music TheoryRaga Gurjari Todi
Scale 3403Scale 3403: Bylian, Ian Ring Music TheoryBylian
Scale 3915Scale 3915, Ian Ring Music Theory
Scale 843Scale 843: Molimic, Ian Ring Music TheoryMolimic
Scale 1867Scale 1867: Solian, Ian Ring Music TheorySolian

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.