The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 2571: "Pukian"

Scale 2571: Pukian, 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




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.



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 Rahn/Ring formula.

prime: 93


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.


Interval Structure

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

[1, 2, 6, 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, 3, 2, 2, 0, 1>

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

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 Interval Spectra has 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, 5, 32)

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
Diminished Triads{9,0,3}000

The following pitch classes are not present in any of the common triads: {1,11}

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


Modes are the rotational transformation of this scale. Scale 2571 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 3333
Scale 3333: Vacian, Ian Ring Music TheoryVacian
3rd mode:
Scale 1857
Scale 1857: Liwian, Ian Ring Music TheoryLiwian
4th mode:
Scale 93
Scale 93: Anuian, Ian Ring Music TheoryAnuianThis is the prime mode
5th mode:
Scale 1047
Scale 1047: Gician, Ian Ring Music TheoryGician


The prime form of this scale is Scale 93

Scale 93Scale 93: Anuian, Ian Ring Music TheoryAnuian


The pentatonic modal family [2571, 3333, 1857, 93, 1047] (Forte: 5-8) is the complement of the heptatonic modal family [381, 1119, 2001, 2607, 3351, 3723, 3909] (Forte: 7-8)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2571 is itself, because it is a palindromic scale!

Scale 2571Scale 2571: Pukian, Ian Ring Music TheoryPukian


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> 2571       T0I <11,0> 2571
T1 <1,1> 1047      T1I <11,1> 1047
T2 <1,2> 2094      T2I <11,2> 2094
T3 <1,3> 93      T3I <11,3> 93
T4 <1,4> 186      T4I <11,4> 186
T5 <1,5> 372      T5I <11,5> 372
T6 <1,6> 744      T6I <11,6> 744
T7 <1,7> 1488      T7I <11,7> 1488
T8 <1,8> 2976      T8I <11,8> 2976
T9 <1,9> 1857      T9I <11,9> 1857
T10 <1,10> 3714      T10I <11,10> 3714
T11 <1,11> 3333      T11I <11,11> 3333
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 681      T0MI <7,0> 681
T1M <5,1> 1362      T1MI <7,1> 1362
T2M <5,2> 2724      T2MI <7,2> 2724
T3M <5,3> 1353      T3MI <7,3> 1353
T4M <5,4> 2706      T4MI <7,4> 2706
T5M <5,5> 1317      T5MI <7,5> 1317
T6M <5,6> 2634      T6MI <7,6> 2634
T7M <5,7> 1173      T7MI <7,7> 1173
T8M <5,8> 2346      T8MI <7,8> 2346
T9M <5,9> 597      T9MI <7,9> 597
T10M <5,10> 1194      T10MI <7,10> 1194
T11M <5,11> 2388      T11MI <7,11> 2388

The transformations that map this set to itself are: T0, T0I

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 2569Scale 2569: Pujian, Ian Ring Music TheoryPujian
Scale 2573Scale 2573: Pulian, Ian Ring Music TheoryPulian
Scale 2575Scale 2575: Pumian, Ian Ring Music TheoryPumian
Scale 2563Scale 2563: Pofian, Ian Ring Music TheoryPofian
Scale 2567Scale 2567: Puhian, Ian Ring Music TheoryPuhian
Scale 2579Scale 2579: Pupian, Ian Ring Music TheoryPupian
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian
Scale 2603Scale 2603: Gadimic, Ian Ring Music TheoryGadimic
Scale 2635Scale 2635: Gocrimic, Ian Ring Music TheoryGocrimic
Scale 2699Scale 2699: Sythimic, Ian Ring Music TheorySythimic
Scale 2827Scale 2827: Runian, Ian Ring Music TheoryRunian
Scale 2059Scale 2059: Moqian, Ian Ring Music TheoryMoqian
Scale 2315Scale 2315: Orkian, Ian Ring Music TheoryOrkian
Scale 3083Scale 3083: Rehian, Ian Ring Music TheoryRehian
Scale 3595Scale 3595: Wihian, Ian Ring Music TheoryWihian
Scale 523Scale 523: Debian, Ian Ring Music TheoryDebian
Scale 1547Scale 1547: Jopian, Ian Ring Music TheoryJopian

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 ( 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.