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Scale 2437: "Pafian"

Scale 2437: Pafian, 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
Pafian

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

{0,2,7,8,11}

Forte Number

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

5-Z18

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: 1075

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.

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.

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: 179

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.

[2, 5, 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.

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

Interval Spectrum

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

p2m2n2sd2t

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> = {3,4,6,7}
<3> = {5,6,8,9}
<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.

3.2

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.

(9, 5, 36)

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 TriadsG{7,11,2}110.5
Diminished Triadsg♯°{8,11,2}110.5

The following pitch classes are not present in any of the common triads: {0}

Parsimonious Voice Leading Between Common Triads of Scale 2437. Created by Ian Ring ©2019 Parsimonious Voice Leading Between Common Triads of Scale 2437. Created by Ian Ring ©2019 G g#° g#° G->g#°

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 2437 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 1633
Scale 1633: Kapian, Ian Ring Music TheoryKapian
3rd mode:
Scale 179
Scale 179: Beyian, Ian Ring Music TheoryBeyianThis is the prime mode
4th mode:
Scale 2137
Scale 2137: Nalian, Ian Ring Music TheoryNalian
5th mode:
Scale 779
Scale 779: Etrian, Ian Ring Music TheoryEtrian

Prime

The prime form of this scale is Scale 179

Scale 179Scale 179: Beyian, Ian Ring Music TheoryBeyian

Complement

The pentatonic modal family [2437, 1633, 179, 2137, 779] (Forte: 5-Z18) is the complement of the heptatonic modal family [755, 815, 1945, 2425, 2455, 3275, 3685] (Forte: 7-Z18)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 2437 is 1075

Scale 1075Scale 1075: Gotian, Ian Ring Music TheoryGotian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 2437 is chiral, and its enantiomorph is scale 1075

Scale 1075Scale 1075: Gotian, Ian Ring Music TheoryGotian

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> 2437       T0I <11,0> 1075
T1 <1,1> 779      T1I <11,1> 2150
T2 <1,2> 1558      T2I <11,2> 205
T3 <1,3> 3116      T3I <11,3> 410
T4 <1,4> 2137      T4I <11,4> 820
T5 <1,5> 179      T5I <11,5> 1640
T6 <1,6> 358      T6I <11,6> 3280
T7 <1,7> 716      T7I <11,7> 2465
T8 <1,8> 1432      T8I <11,8> 835
T9 <1,9> 2864      T9I <11,9> 1670
T10 <1,10> 1633      T10I <11,10> 3340
T11 <1,11> 3266      T11I <11,11> 2585
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3217      T0MI <7,0> 295
T1M <5,1> 2339      T1MI <7,1> 590
T2M <5,2> 583      T2MI <7,2> 1180
T3M <5,3> 1166      T3MI <7,3> 2360
T4M <5,4> 2332      T4MI <7,4> 625
T5M <5,5> 569      T5MI <7,5> 1250
T6M <5,6> 1138      T6MI <7,6> 2500
T7M <5,7> 2276      T7MI <7,7> 905
T8M <5,8> 457      T8MI <7,8> 1810
T9M <5,9> 914      T9MI <7,9> 3620
T10M <5,10> 1828      T10MI <7,10> 3145
T11M <5,11> 3656      T11MI <7,11> 2195

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 2439Scale 2439: Pagian, Ian Ring Music TheoryPagian
Scale 2433Scale 2433: Pacian, Ian Ring Music TheoryPacian
Scale 2435Scale 2435: Raga Deshgaur, Ian Ring Music TheoryRaga Deshgaur
Scale 2441Scale 2441: Kyritonic, Ian Ring Music TheoryKyritonic
Scale 2445Scale 2445: Zadimic, Ian Ring Music TheoryZadimic
Scale 2453Scale 2453: Raga Latika, Ian Ring Music TheoryRaga Latika
Scale 2469Scale 2469: Raga Bhinna Pancama, Ian Ring Music TheoryRaga Bhinna Pancama
Scale 2501Scale 2501: Ralimic, Ian Ring Music TheoryRalimic
Scale 2309Scale 2309: Ocuian, Ian Ring Music TheoryOcuian
Scale 2373Scale 2373: Dyptitonic, Ian Ring Music TheoryDyptitonic
Scale 2181Scale 2181: Nemian, Ian Ring Music TheoryNemian
Scale 2693Scale 2693: Rajian, Ian Ring Music TheoryRajian
Scale 2949Scale 2949: Sikian, Ian Ring Music TheorySikian
Scale 3461Scale 3461: Vodian, Ian Ring Music TheoryVodian
Scale 389Scale 389: Cixian, Ian Ring Music TheoryCixian
Scale 1413Scale 1413: Iruian, Ian Ring Music TheoryIruian

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.