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Scale 2627: "Qerian"

Scale 2627: Qerian, 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

Dozenal
Qerian

Analysis

Cardinality

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

5 (pentatonic)

Pitch Class Set

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

{0,1,6,9,11}

Forte Number

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

5-Z36

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: 2123

Hemitonia

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

2 (dihemitonic)

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.

4

Prime Form

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

no
prime: 151

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, 5, 3, 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.

<2, 2, 2, 1, 2, 1>

Interval Spectrum

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

p2mn2s2d2t

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

Spectra Variation

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

4

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.

1.683

Polygon Perimeter

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

5.381

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, 7, 38)

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
Minor Triadsf♯m{6,9,1}110.5
Diminished Triadsf♯°{6,9,0}110.5

The following pitch classes are not present in any of the common triads: {11}

Parsimonious Voice Leading Between Common Triads of Scale 2627. Created by Ian Ring ©2019 f#° f#° f#m f#m f#°->f#m

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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2627 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3361
Scale 3361: Vatian, Ian Ring Music TheoryVatian
3rd mode:
Scale 233
Scale 233: Bigian, Ian Ring Music TheoryBigian
4th mode:
Scale 541
Scale 541: Demian, Ian Ring Music TheoryDemian
5th mode:
Scale 1159
Scale 1159: Hasian, Ian Ring Music TheoryHasian

Prime

The prime form of this scale is Scale 151

Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian

Complement

The pentatonic modal family [2627, 3361, 233, 541, 1159] (Forte: 5-Z36) is the complement of the heptatonic modal family [367, 1777, 1931, 2231, 3013, 3163, 3629] (Forte: 7-Z36)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2627 is 2123

Scale 2123Scale 2123: Nacian, Ian Ring Music TheoryNacian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2627 is chiral, and its enantiomorph is scale 2123

Scale 2123Scale 2123: Nacian, Ian Ring Music TheoryNacian

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> 2627       T0I <11,0> 2123
T1 <1,1> 1159      T1I <11,1> 151
T2 <1,2> 2318      T2I <11,2> 302
T3 <1,3> 541      T3I <11,3> 604
T4 <1,4> 1082      T4I <11,4> 1208
T5 <1,5> 2164      T5I <11,5> 2416
T6 <1,6> 233      T6I <11,6> 737
T7 <1,7> 466      T7I <11,7> 1474
T8 <1,8> 932      T8I <11,8> 2948
T9 <1,9> 1864      T9I <11,9> 1801
T10 <1,10> 3728      T10I <11,10> 3602
T11 <1,11> 3361      T11I <11,11> 3109
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 737      T0MI <7,0> 233
T1M <5,1> 1474      T1MI <7,1> 466
T2M <5,2> 2948      T2MI <7,2> 932
T3M <5,3> 1801      T3MI <7,3> 1864
T4M <5,4> 3602      T4MI <7,4> 3728
T5M <5,5> 3109      T5MI <7,5> 3361
T6M <5,6> 2123      T6MI <7,6> 2627
T7M <5,7> 151      T7MI <7,7> 1159
T8M <5,8> 302      T8MI <7,8> 2318
T9M <5,9> 604      T9MI <7,9> 541
T10M <5,10> 1208      T10MI <7,10> 1082
T11M <5,11> 2416      T11MI <7,11> 2164

The transformations that map this set to itself are: T0, T6MI

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 2625Scale 2625, Ian Ring Music Theory
Scale 2629Scale 2629: Raga Shubravarni, Ian Ring Music TheoryRaga Shubravarni
Scale 2631Scale 2631: Macrimic, Ian Ring Music TheoryMacrimic
Scale 2635Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 2659Scale 2659: Katynimic, Ian Ring Music TheoryKatynimic
Scale 2563Scale 2563: Pofian, Ian Ring Music TheoryPofian
Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic
Scale 2691Scale 2691: Rahian, Ian Ring Music TheoryRahian
Scale 2755Scale 2755: Rivian, Ian Ring Music TheoryRivian
Scale 2883Scale 2883: Savian, Ian Ring Music TheorySavian
Scale 2115Scale 2115: Muyian, Ian Ring Music TheoryMuyian
Scale 2371Scale 2371: Omoian, Ian Ring Music TheoryOmoian
Scale 3139Scale 3139: Towian, Ian Ring Music TheoryTowian
Scale 3651Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
Scale 579Scale 579: Giyian, Ian Ring Music TheoryGiyian
Scale 1603Scale 1603: Juxian, Ian Ring Music TheoryJuxian

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.