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Scale 2531: "Danian"

Scale 2531: Danian, 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
Danian
Dozenal
Pilian

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

{0,1,5,6,7,8,11}

Forte Number

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

7-7

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: 2291

Hemitonia

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

5 (multihemitonic)

Cohemitonia

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

3 (tricohemitonic)

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.

2

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: 463

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, 4, 1, 1, 1, 3, 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.

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

Interval Spectrum

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

p5m3n2s3d5t3

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

Spectra Variation

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

2.571

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.

2.183

Polygon Perimeter

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

5.734

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.

(22, 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 TriadsC♯{1,5,8}121
Minor Triadsfm{5,8,0}210.67
Diminished Triads{5,8,11}121

The following pitch classes are not present in any of the common triads: {6,7}

Parsimonious Voice Leading Between Common Triads of Scale 2531. Created by Ian Ring ©2019 C# C# fm fm C#->fm f°->fm

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.

Diameter2
Radius1
Self-Centeredno
Central Verticesfm
Peripheral VerticesC♯, f°

Modes

Modes are the rotational transformation of this scale. Scale 2531 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3313
Scale 3313: Aeolacrian, Ian Ring Music TheoryAeolacrian
3rd mode:
Scale 463
Scale 463: Zythian, Ian Ring Music TheoryZythianThis is the prime mode
4th mode:
Scale 2279
Scale 2279: Dyrian, Ian Ring Music TheoryDyrian
5th mode:
Scale 3187
Scale 3187: Koptian, Ian Ring Music TheoryKoptian
6th mode:
Scale 3641
Scale 3641: Thocrian, Ian Ring Music TheoryThocrian
7th mode:
Scale 967
Scale 967: Mela Salaga, Ian Ring Music TheoryMela Salaga

Prime

The prime form of this scale is Scale 463

Scale 463Scale 463: Zythian, Ian Ring Music TheoryZythian

Complement

The heptatonic modal family [2531, 3313, 463, 2279, 3187, 3641, 967] (Forte: 7-7) is the complement of the pentatonic modal family [199, 451, 2147, 2273, 3121] (Forte: 5-7)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2531 is 2291

Scale 2291Scale 2291: Zydian, Ian Ring Music TheoryZydian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2531 is chiral, and its enantiomorph is scale 2291

Scale 2291Scale 2291: Zydian, Ian Ring Music TheoryZydian

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> 2531       T0I <11,0> 2291
T1 <1,1> 967      T1I <11,1> 487
T2 <1,2> 1934      T2I <11,2> 974
T3 <1,3> 3868      T3I <11,3> 1948
T4 <1,4> 3641      T4I <11,4> 3896
T5 <1,5> 3187      T5I <11,5> 3697
T6 <1,6> 2279      T6I <11,6> 3299
T7 <1,7> 463      T7I <11,7> 2503
T8 <1,8> 926      T8I <11,8> 911
T9 <1,9> 1852      T9I <11,9> 1822
T10 <1,10> 3704      T10I <11,10> 3644
T11 <1,11> 3313      T11I <11,11> 3193
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2291      T0MI <7,0> 2531
T1M <5,1> 487      T1MI <7,1> 967
T2M <5,2> 974      T2MI <7,2> 1934
T3M <5,3> 1948      T3MI <7,3> 3868
T4M <5,4> 3896      T4MI <7,4> 3641
T5M <5,5> 3697      T5MI <7,5> 3187
T6M <5,6> 3299      T6MI <7,6> 2279
T7M <5,7> 2503      T7MI <7,7> 463
T8M <5,8> 911      T8MI <7,8> 926
T9M <5,9> 1822      T9MI <7,9> 1852
T10M <5,10> 3644      T10MI <7,10> 3704
T11M <5,11> 3193      T11MI <7,11> 3313

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 2529Scale 2529: Pikian, Ian Ring Music TheoryPikian
Scale 2533Scale 2533: Podian, Ian Ring Music TheoryPodian
Scale 2535Scale 2535: Messiaen Mode 4, Ian Ring Music TheoryMessiaen Mode 4
Scale 2539Scale 2539: Half-Diminished Bebop, Ian Ring Music TheoryHalf-Diminished Bebop
Scale 2547Scale 2547: Raga Ramkali, Ian Ring Music TheoryRaga Ramkali
Scale 2499Scale 2499: Pirian, Ian Ring Music TheoryPirian
Scale 2515Scale 2515: Chromatic Hypolydian, Ian Ring Music TheoryChromatic Hypolydian
Scale 2467Scale 2467: Raga Padi, Ian Ring Music TheoryRaga Padi
Scale 2403Scale 2403: Lycrimic, Ian Ring Music TheoryLycrimic
Scale 2275Scale 2275: Messiaen Mode 5, Ian Ring Music TheoryMessiaen Mode 5
Scale 2787Scale 2787: Zyrian, Ian Ring Music TheoryZyrian
Scale 3043Scale 3043: Ionayllic, Ian Ring Music TheoryIonayllic
Scale 3555Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
Scale 483Scale 483: Kygimic, Ian Ring Music TheoryKygimic
Scale 1507Scale 1507: Zynian, Ian Ring Music TheoryZynian

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.