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Scale 2921: "Pogian"

Scale 2921: Pogian, 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
Pogian
Dozenal
Opsian

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

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

Forte Number

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

7-31

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: 731

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.

0 (ancohemitonic)

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: 731

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, 2, 1, 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, 3, 6, 3, 3, 3>

Interval Spectrum

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

p3m3n6s3d3t3

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

Spectra Variation

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

1.714

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.

(4, 27, 84)

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 TriadsF{5,9,0}331.8
G♯{8,0,3}431.6
B{11,3,6}331.8
Minor Triadsfm{5,8,0}331.7
g♯m{8,11,3}331.7
Diminished Triads{0,3,6}231.9
d♯°{3,6,9}232
{5,8,11}232
f♯°{6,9,0}232
{9,0,3}231.9
Parsimonious Voice Leading Between Common Triads of Scale 2921. Created by Ian Ring ©2019 G# G# c°->G# B B c°->B d#° d#° f#° f#° d#°->f#° d#°->B fm fm f°->fm g#m g#m f°->g#m F F fm->F fm->G# F->f#° F->a° 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 2921 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 877
Scale 877: Moravian Pistalkova, Ian Ring Music TheoryMoravian Pistalkova
3rd mode:
Scale 1243
Scale 1243: Epylian, Ian Ring Music TheoryEpylian
4th mode:
Scale 2669
Scale 2669: Jeths' Mode, Ian Ring Music TheoryJeths' Mode
5th mode:
Scale 1691
Scale 1691: Kathian, Ian Ring Music TheoryKathian
6th mode:
Scale 2893
Scale 2893: Lylian, Ian Ring Music TheoryLylian
7th mode:
Scale 1747
Scale 1747: Mela Ramapriya, Ian Ring Music TheoryMela Ramapriya

Prime

The prime form of this scale is Scale 731

Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode

Complement

The heptatonic modal family [2921, 877, 1243, 2669, 1691, 2893, 1747] (Forte: 7-31) is the complement of the pentatonic modal family [587, 601, 713, 1609, 2341] (Forte: 5-31)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2921 is 731

Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2921 is chiral, and its enantiomorph is scale 731

Scale 731Scale 731: Alternating Heptamode, Ian Ring Music TheoryAlternating Heptamode

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> 2921       T0I <11,0> 731
T1 <1,1> 1747      T1I <11,1> 1462
T2 <1,2> 3494      T2I <11,2> 2924
T3 <1,3> 2893      T3I <11,3> 1753
T4 <1,4> 1691      T4I <11,4> 3506
T5 <1,5> 3382      T5I <11,5> 2917
T6 <1,6> 2669      T6I <11,6> 1739
T7 <1,7> 1243      T7I <11,7> 3478
T8 <1,8> 2486      T8I <11,8> 2861
T9 <1,9> 877      T9I <11,9> 1627
T10 <1,10> 1754      T10I <11,10> 3254
T11 <1,11> 3508      T11I <11,11> 2413
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 731      T0MI <7,0> 2921
T1M <5,1> 1462      T1MI <7,1> 1747
T2M <5,2> 2924      T2MI <7,2> 3494
T3M <5,3> 1753      T3MI <7,3> 2893
T4M <5,4> 3506      T4MI <7,4> 1691
T5M <5,5> 2917      T5MI <7,5> 3382
T6M <5,6> 1739      T6MI <7,6> 2669
T7M <5,7> 3478      T7MI <7,7> 1243
T8M <5,8> 2861      T8MI <7,8> 2486
T9M <5,9> 1627      T9MI <7,9> 877
T10M <5,10> 3254      T10MI <7,10> 1754
T11M <5,11> 2413      T11MI <7,11> 3508

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 2923Scale 2923: Baryllic, Ian Ring Music TheoryBaryllic
Scale 2925Scale 2925: Diminished, Ian Ring Music TheoryDiminished
Scale 2913Scale 2913: Senian, Ian Ring Music TheorySenian
Scale 2917Scale 2917: Nohkan Flute Scale, Ian Ring Music TheoryNohkan Flute Scale
Scale 2929Scale 2929: Aeolathian, Ian Ring Music TheoryAeolathian
Scale 2937Scale 2937: Phragyllic, Ian Ring Music TheoryPhragyllic
Scale 2889Scale 2889: Thoptimic, Ian Ring Music TheoryThoptimic
Scale 2905Scale 2905: Aeolian Flat 1, Ian Ring Music TheoryAeolian Flat 1
Scale 2857Scale 2857: Stythimic, Ian Ring Music TheoryStythimic
Scale 2985Scale 2985: Epacrian, Ian Ring Music TheoryEpacrian
Scale 3049Scale 3049: Phrydyllic, Ian Ring Music TheoryPhrydyllic
Scale 2665Scale 2665: Aeradimic, Ian Ring Music TheoryAeradimic
Scale 2793Scale 2793: Eporian, Ian Ring Music TheoryEporian
Scale 2409Scale 2409: Zacrimic, Ian Ring Music TheoryZacrimic
Scale 3433Scale 3433: Thonian, Ian Ring Music TheoryThonian
Scale 3945Scale 3945: Lydyllic, Ian Ring Music TheoryLydyllic
Scale 873Scale 873: Bagimic, Ian Ring Music TheoryBagimic
Scale 1897Scale 1897: Ionopian, Ian Ring Music TheoryIonopian

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.