The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 545: "Dewian"

Scale 545: Dewian, 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
Dewian

Analysis

Cardinality

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

3 (tritonic)

Pitch Class Set

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

{0,5,9}

Forte Number

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

3-11

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: 137

Hemitonia

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

0 (anhemitonic)

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.

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.

2

Prime Form

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

no
prime: 137

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.

[5, 4, 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.

<0, 0, 1, 1, 1, 0>

Interval Spectrum

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

pmn

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

Spectra Variation

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

1.333

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.

1.183

Polygon Perimeter

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

5.078

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".

Strictly Proper

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.

(0, 0, 6)

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}000

Since there is only one common triad in this scale, there are no opportunities for parsimonious voice leading between triads.

Modes

Modes are the rotational transformation of this scale. Scale 545 can be rotated to make 2 other scales. The 1st mode is itself.

2nd mode:
Scale 145
Scale 145: Raga Malasri, Ian Ring Music TheoryRaga Malasri
3rd mode:
Scale 265
Scale 265: Boxian, Ian Ring Music TheoryBoxian

Prime

The prime form of this scale is Scale 137

Scale 137Scale 137: Ute Tritonic, Ian Ring Music TheoryUte Tritonic

Complement

The tritonic modal family [545, 145, 265] (Forte: 3-11) is the complement of the enneatonic modal family [1775, 1915, 1975, 2935, 3005, 3035, 3515, 3565, 3805] (Forte: 9-11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 545 is 137

Scale 137Scale 137: Ute Tritonic, Ian Ring Music TheoryUte Tritonic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 545 is chiral, and its enantiomorph is scale 137

Scale 137Scale 137: Ute Tritonic, Ian Ring Music TheoryUte Tritonic

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> 545       T0I <11,0> 137
T1 <1,1> 1090      T1I <11,1> 274
T2 <1,2> 2180      T2I <11,2> 548
T3 <1,3> 265      T3I <11,3> 1096
T4 <1,4> 530      T4I <11,4> 2192
T5 <1,5> 1060      T5I <11,5> 289
T6 <1,6> 2120      T6I <11,6> 578
T7 <1,7> 145      T7I <11,7> 1156
T8 <1,8> 290      T8I <11,8> 2312
T9 <1,9> 580      T9I <11,9> 529
T10 <1,10> 1160      T10I <11,10> 1058
T11 <1,11> 2320      T11I <11,11> 2116
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 515      T0MI <7,0> 2057
T1M <5,1> 1030      T1MI <7,1> 19
T2M <5,2> 2060      T2MI <7,2> 38
T3M <5,3> 25      T3MI <7,3> 76
T4M <5,4> 50      T4MI <7,4> 152
T5M <5,5> 100      T5MI <7,5> 304
T6M <5,6> 200      T6MI <7,6> 608
T7M <5,7> 400      T7MI <7,7> 1216
T8M <5,8> 800      T8MI <7,8> 2432
T9M <5,9> 1600      T9MI <7,9> 769
T10M <5,10> 3200      T10MI <7,10> 1538
T11M <5,11> 2305      T11MI <7,11> 3076

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 547Scale 547: Pyrric, Ian Ring Music TheoryPyrric
Scale 549Scale 549: Raga Bhavani, Ian Ring Music TheoryRaga Bhavani
Scale 553Scale 553: Rothic 2, Ian Ring Music TheoryRothic 2
Scale 561Scale 561: Phratic, Ian Ring Music TheoryPhratic
Scale 513Scale 513: Major Sixth Ditone, Ian Ring Music TheoryMajor Sixth Ditone
Scale 529Scale 529: Raga Bilwadala, Ian Ring Music TheoryRaga Bilwadala
Scale 577Scale 577: Illian, Ian Ring Music TheoryIllian
Scale 609Scale 609: Docian, Ian Ring Music TheoryDocian
Scale 673Scale 673: Estian, Ian Ring Music TheoryEstian
Scale 801Scale 801: Fahian, Ian Ring Music TheoryFahian
Scale 33Scale 33: Honchoshi, Ian Ring Music TheoryHonchoshi
Scale 289Scale 289: Valian, Ian Ring Music TheoryValian
Scale 1057Scale 1057: Sansagari, Ian Ring Music TheorySansagari
Scale 1569Scale 1569: Jocian, Ian Ring Music TheoryJocian
Scale 2593Scale 2593: Puxian, Ian Ring Music TheoryPuxian

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.