The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 2583: "Purian"

Scale 2583: Purian, 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
Purian

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

{0,1,2,4,9,11}

Forte Number

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

6-8

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.

[0.5]

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.

no

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.

2 (dicohemitonic)

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.

5

Prime Form

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

no
prime: 189

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, 2, 5, 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, 4, 3, 2, 3, 0>

Interval Spectrum

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

p3m2n3s4d3

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

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.

1.866

Polygon Perimeter

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

5.485

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.

[1]

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.

(30, 16, 59)

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}110.5
Minor Triadsam{9,0,4}110.5

The following pitch classes are not present in any of the common triads: {2,11}

Parsimonious Voice Leading Between Common Triads of Scale 2583. Created by Ian Ring ©2019 am 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.

Diameter1
Radius1
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 2583 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3339
Scale 3339: Smuian, Ian Ring Music TheorySmuian
3rd mode:
Scale 3717
Scale 3717: Xidian, Ian Ring Music TheoryXidian
4th mode:
Scale 1953
Scale 1953: Macian, Ian Ring Music TheoryMacian
5th mode:
Scale 189
Scale 189: Befian, Ian Ring Music TheoryBefianThis is the prime mode
6th mode:
Scale 1071
Scale 1071: Gorian, Ian Ring Music TheoryGorian

Prime

The prime form of this scale is Scale 189

Scale 189Scale 189: Befian, Ian Ring Music TheoryBefian

Complement

The hexatonic modal family [2583, 3339, 3717, 1953, 189, 1071] (Forte: 6-8) is the complement of the hexatonic modal family [189, 1071, 1953, 2583, 3339, 3717] (Forte: 6-8)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2583 is 3339

Scale 3339Scale 3339: Smuian, Ian Ring Music TheorySmuian

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> 2583       T0I <11,0> 3339
T1 <1,1> 1071      T1I <11,1> 2583
T2 <1,2> 2142      T2I <11,2> 1071
T3 <1,3> 189      T3I <11,3> 2142
T4 <1,4> 378      T4I <11,4> 189
T5 <1,5> 756      T5I <11,5> 378
T6 <1,6> 1512      T6I <11,6> 756
T7 <1,7> 3024      T7I <11,7> 1512
T8 <1,8> 1953      T8I <11,8> 3024
T9 <1,9> 3906      T9I <11,9> 1953
T10 <1,10> 3717      T10I <11,10> 3906
T11 <1,11> 3339      T11I <11,11> 3717
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1953      T0MI <7,0> 189
T1M <5,1> 3906      T1MI <7,1> 378
T2M <5,2> 3717      T2MI <7,2> 756
T3M <5,3> 3339      T3MI <7,3> 1512
T4M <5,4> 2583       T4MI <7,4> 3024
T5M <5,5> 1071      T5MI <7,5> 1953
T6M <5,6> 2142      T6MI <7,6> 3906
T7M <5,7> 189      T7MI <7,7> 3717
T8M <5,8> 378      T8MI <7,8> 3339
T9M <5,9> 756      T9MI <7,9> 2583
T10M <5,10> 1512      T10MI <7,10> 1071
T11M <5,11> 3024      T11MI <7,11> 2142

The transformations that map this set to itself are: T0, T1I, T4M, T9MI

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 2581Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
Scale 2579Scale 2579: Pupian, Ian Ring Music TheoryPupian
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian
Scale 2591Scale 2591: Puwian, Ian Ring Music TheoryPuwian
Scale 2567Scale 2567: Puhian, Ian Ring Music TheoryPuhian
Scale 2575Scale 2575: Pumian, Ian Ring Music TheoryPumian
Scale 2599Scale 2599: Malimic, Ian Ring Music TheoryMalimic
Scale 2615Scale 2615: Thoptian, Ian Ring Music TheoryThoptian
Scale 2647Scale 2647: Dadian, Ian Ring Music TheoryDadian
Scale 2711Scale 2711: Stolian, Ian Ring Music TheoryStolian
Scale 2839Scale 2839: Lyptian, Ian Ring Music TheoryLyptian
Scale 2071Scale 2071: Moxian, Ian Ring Music TheoryMoxian
Scale 2327Scale 2327: Epalimic, Ian Ring Music TheoryEpalimic
Scale 3095Scale 3095: Tivian, Ian Ring Music TheoryTivian
Scale 3607Scale 3607: Wopian, Ian Ring Music TheoryWopian
Scale 535Scale 535: Dejian, Ian Ring Music TheoryDejian
Scale 1559Scale 1559: Jowian, Ian Ring Music TheoryJowian

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.