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Scale 543: "Denian"

Scale 543: Denian, 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
Denian

Analysis

Cardinality

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

6 (hexatonic)

Pitch Class Set

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

{0,1,2,3,4,9}

Forte Number

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

6-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: 3849

Hemitonia

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

4 (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.

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.

5

Prime Form

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

no
prime: 159

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, 1, 1, 1, 5, 3]

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.

<4, 3, 3, 2, 2, 1>

Interval Spectrum

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

p2m2n3s3d4t

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

Spectra Variation

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

4.333

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.

1.75

Polygon Perimeter

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

5.417

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.

(41, 9, 55)

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 TriadsA{9,1,4}121
Minor Triadsam{9,0,4}210.67
Diminished Triads{9,0,3}121

The following pitch classes are not present in any of the common triads: {2}

Parsimonious Voice Leading Between Common Triads of Scale 543. Created by Ian Ring ©2019 am am a°->am A A am->A

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 Verticesam
Peripheral Verticesa°, A

Modes

Modes are the rotational transformation of this scale. Scale 543 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2319
Scale 2319: Oduian, Ian Ring Music TheoryOduian
3rd mode:
Scale 3207
Scale 3207: Ucoian, Ian Ring Music TheoryUcoian
4th mode:
Scale 3651
Scale 3651: Wuqian, Ian Ring Music TheoryWuqian
5th mode:
Scale 3873
Scale 3873: Yoyian, Ian Ring Music TheoryYoyian
6th mode:
Scale 249
Scale 249: Boqian, Ian Ring Music TheoryBoqian

Prime

The prime form of this scale is Scale 159

Scale 159Scale 159: Bamian, Ian Ring Music TheoryBamian

Complement

The hexatonic modal family [543, 2319, 3207, 3651, 3873, 249] (Forte: 6-Z36) is the complement of the hexatonic modal family [111, 1923, 2103, 3009, 3099, 3597] (Forte: 6-Z3)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 543 is 3849

Scale 3849Scale 3849: Yikian, Ian Ring Music TheoryYikian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 543 is chiral, and its enantiomorph is scale 3849

Scale 3849Scale 3849: Yikian, Ian Ring Music TheoryYikian

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> 543       T0I <11,0> 3849
T1 <1,1> 1086      T1I <11,1> 3603
T2 <1,2> 2172      T2I <11,2> 3111
T3 <1,3> 249      T3I <11,3> 2127
T4 <1,4> 498      T4I <11,4> 159
T5 <1,5> 996      T5I <11,5> 318
T6 <1,6> 1992      T6I <11,6> 636
T7 <1,7> 3984      T7I <11,7> 1272
T8 <1,8> 3873      T8I <11,8> 2544
T9 <1,9> 3651      T9I <11,9> 993
T10 <1,10> 3207      T10I <11,10> 1986
T11 <1,11> 2319      T11I <11,11> 3972
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1833      T0MI <7,0> 669
T1M <5,1> 3666      T1MI <7,1> 1338
T2M <5,2> 3237      T2MI <7,2> 2676
T3M <5,3> 2379      T3MI <7,3> 1257
T4M <5,4> 663      T4MI <7,4> 2514
T5M <5,5> 1326      T5MI <7,5> 933
T6M <5,6> 2652      T6MI <7,6> 1866
T7M <5,7> 1209      T7MI <7,7> 3732
T8M <5,8> 2418      T8MI <7,8> 3369
T9M <5,9> 741      T9MI <7,9> 2643
T10M <5,10> 1482      T10MI <7,10> 1191
T11M <5,11> 2964      T11MI <7,11> 2382

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 541Scale 541: Demian, Ian Ring Music TheoryDemian
Scale 539Scale 539: Delian, Ian Ring Music TheoryDelian
Scale 535Scale 535: Dejian, Ian Ring Music TheoryDejian
Scale 527Scale 527: Dedian, Ian Ring Music TheoryDedian
Scale 559Scale 559: Lylimic, Ian Ring Music TheoryLylimic
Scale 575Scale 575: Ionydian, Ian Ring Music TheoryIonydian
Scale 607Scale 607: Kadian, Ian Ring Music TheoryKadian
Scale 671Scale 671: Stycrian, Ian Ring Music TheoryStycrian
Scale 799Scale 799: Lolian, Ian Ring Music TheoryLolian
Scale 31Scale 31: Pentatonic Chromatic, Ian Ring Music TheoryPentatonic Chromatic
Scale 287Scale 287: Gynimic, Ian Ring Music TheoryGynimic
Scale 1055Scale 1055: Gihian, Ian Ring Music TheoryGihian
Scale 1567Scale 1567: Jobian, Ian Ring Music TheoryJobian
Scale 2591Scale 2591: Puwian, Ian Ring Music TheoryPuwian

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.