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Scale 2579: "PUPian"

Scale 2579: PUPian, 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).



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


Forte Number

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


Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.


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.



A palindromic scale has the same pattern of intervals both ascending and descending.



A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

enantiomorph: 2315


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

2 (dihemitonic)


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

1 (uncohemitonic)


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.



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.


Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

prime: 157


Indicates if the scale can be constructed using a generator, and an origin.


Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.


Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 3, 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.

<2, 2, 2, 2, 2, 0>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.5, 0.5, 0.5, 0.333, 0.5, 0>

Interval Spectrum

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


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


Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.


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.


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.


Polygon Perimeter

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


Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.



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.


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.



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".


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.

(14, 8, 38)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.


Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.



This scale has no generator.

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: {11}

Parsimonious Voice Leading Between Common Triads of Scale 2579. 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.



Modes are the rotational transformation of this scale. Scale 2579 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3337
Scale 3337: VAFian, Ian Ring Music TheoryVAFian
3rd mode:
Scale 929
Scale 929: FUJian, Ian Ring Music TheoryFUJian
4th mode:
Scale 157
Scale 157: BALian, Ian Ring Music TheoryBALianThis is the prime mode
5th mode:
Scale 1063
Scale 1063: GOMian, Ian Ring Music TheoryGOMian


The prime form of this scale is Scale 157

Scale 157Scale 157: BALian, Ian Ring Music TheoryBALian


The pentatonic modal family [2579, 3337, 929, 157, 1063] (Forte: 5-11) is the complement of the heptatonic modal family [379, 1583, 1969, 2237, 2839, 3467, 3781] (Forte: 7-11)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2579 is 2315

Scale 2315Scale 2315: ORKian, Ian Ring Music TheoryORKian


Only scales that are chiral will have an enantiomorph. Scale 2579 is chiral, and its enantiomorph is scale 2315

Scale 2315Scale 2315: ORKian, Ian Ring Music TheoryORKian


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> 2579       T0I <11,0> 2315
T1 <1,1> 1063      T1I <11,1> 535
T2 <1,2> 2126      T2I <11,2> 1070
T3 <1,3> 157      T3I <11,3> 2140
T4 <1,4> 314      T4I <11,4> 185
T5 <1,5> 628      T5I <11,5> 370
T6 <1,6> 1256      T6I <11,6> 740
T7 <1,7> 2512      T7I <11,7> 1480
T8 <1,8> 929      T8I <11,8> 2960
T9 <1,9> 1858      T9I <11,9> 1825
T10 <1,10> 3716      T10I <11,10> 3650
T11 <1,11> 3337      T11I <11,11> 3205
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 929      T0MI <7,0> 185
T1M <5,1> 1858      T1MI <7,1> 370
T2M <5,2> 3716      T2MI <7,2> 740
T3M <5,3> 3337      T3MI <7,3> 1480
T4M <5,4> 2579       T4MI <7,4> 2960
T5M <5,5> 1063      T5MI <7,5> 1825
T6M <5,6> 2126      T6MI <7,6> 3650
T7M <5,7> 157      T7MI <7,7> 3205
T8M <5,8> 314      T8MI <7,8> 2315
T9M <5,9> 628      T9MI <7,9> 535
T10M <5,10> 1256      T10MI <7,10> 1070
T11M <5,11> 2512      T11MI <7,11> 2140

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

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 2577Scale 2577: PUNian, Ian Ring Music TheoryPUNian
Scale 2581Scale 2581: Raga Neroshta, Ian Ring Music TheoryRaga Neroshta
Scale 2583Scale 2583: PURian, Ian Ring Music TheoryPURian
Scale 2587Scale 2587: PUTian, Ian Ring Music TheoryPUTian
Scale 2563Scale 2563: POFian, Ian Ring Music TheoryPOFian
Scale 2571Scale 2571: PUKian, Ian Ring Music TheoryPUKian
Scale 2595Scale 2595: Rolitonic, Ian Ring Music TheoryRolitonic
Scale 2611Scale 2611: Raga Vasanta, Ian Ring Music TheoryRaga Vasanta
Scale 2643Scale 2643: Raga Hamsanandi, Ian Ring Music TheoryRaga Hamsanandi
Scale 2707Scale 2707: Banimic, Ian Ring Music TheoryBanimic
Scale 2835Scale 2835: Ionygimic, Ian Ring Music TheoryIonygimic
Scale 2067Scale 2067: MOVian, Ian Ring Music TheoryMOVian
Scale 2323Scale 2323: Doptitonic, Ian Ring Music TheoryDoptitonic
Scale 3091Scale 3091: TISian, Ian Ring Music TheoryTISian
Scale 3603Scale 3603: WOMian, Ian Ring Music TheoryWOMian
Scale 531Scale 531: DEGian, Ian Ring Music TheoryDEGian
Scale 1555Scale 1555: JOTian, Ian Ring Music TheoryJOTian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler ( 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.