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Scale 2591: "Puwian"

Scale 2591: Puwian, 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.

7 (heptatonic)

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: 3851


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

5 (multihemitonic)


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

4 (multicohemitonic)


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: 191


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, 1, 1, 1, 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.

<5, 5, 4, 3, 3, 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> = {1,2,5}
<2> = {2,3,6,7}
<3> = {3,4,7,8}
<4> = {4,5,8,9}
<5> = {5,6,9,10}
<6> = {7,10,11}

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.

(62, 25, 86)

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}121
Minor Triadsam{9,0,4}210.67
Diminished Triads{9,0,3}121

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

Parsimonious Voice Leading Between Common Triads of Scale 2591. Created by Ian Ring ©2019 am am a°->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.

Central Verticesam
Peripheral Verticesa°, A


Modes are the rotational transformation of this scale. Scale 2591 can be rotated to make 6 other scales. The 1st mode is itself.

2nd mode:
Scale 3343
Scale 3343: Vajian, Ian Ring Music TheoryVajian
3rd mode:
Scale 3719
Scale 3719: Xofian, Ian Ring Music TheoryXofian
4th mode:
Scale 3907
Scale 3907, Ian Ring Music Theory
5th mode:
Scale 4001
Scale 4001: Ziyian, Ian Ring Music TheoryZiyian
6th mode:
Scale 253
Scale 253: Bosian, Ian Ring Music TheoryBosian
7th mode:
Scale 1087
Scale 1087: Gobian, Ian Ring Music TheoryGobian


The prime form of this scale is Scale 191

Scale 191Scale 191: Begian, Ian Ring Music TheoryBegian


The heptatonic modal family [2591, 3343, 3719, 3907, 4001, 253, 1087] (Forte: 7-2) is the complement of the pentatonic modal family [47, 1921, 2071, 3083, 3589] (Forte: 5-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 2591 is 3851

Scale 3851Scale 3851: Yilian, Ian Ring Music TheoryYilian


Only scales that are chiral will have an enantiomorph. Scale 2591 is chiral, and its enantiomorph is scale 3851

Scale 3851Scale 3851: Yilian, Ian Ring Music TheoryYilian


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> 2591       T0I <11,0> 3851
T1 <1,1> 1087      T1I <11,1> 3607
T2 <1,2> 2174      T2I <11,2> 3119
T3 <1,3> 253      T3I <11,3> 2143
T4 <1,4> 506      T4I <11,4> 191
T5 <1,5> 1012      T5I <11,5> 382
T6 <1,6> 2024      T6I <11,6> 764
T7 <1,7> 4048      T7I <11,7> 1528
T8 <1,8> 4001      T8I <11,8> 3056
T9 <1,9> 3907      T9I <11,9> 2017
T10 <1,10> 3719      T10I <11,10> 4034
T11 <1,11> 3343      T11I <11,11> 3973
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1961      T0MI <7,0> 701
T1M <5,1> 3922      T1MI <7,1> 1402
T2M <5,2> 3749      T2MI <7,2> 2804
T3M <5,3> 3403      T3MI <7,3> 1513
T4M <5,4> 2711      T4MI <7,4> 3026
T5M <5,5> 1327      T5MI <7,5> 1957
T6M <5,6> 2654      T6MI <7,6> 3914
T7M <5,7> 1213      T7MI <7,7> 3733
T8M <5,8> 2426      T8MI <7,8> 3371
T9M <5,9> 757      T9MI <7,9> 2647
T10M <5,10> 1514      T10MI <7,10> 1199
T11M <5,11> 3028      T11MI <7,11> 2398

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 2589Scale 2589: Puvian, Ian Ring Music TheoryPuvian
Scale 2587Scale 2587: Putian, Ian Ring Music TheoryPutian
Scale 2583Scale 2583: Purian, Ian Ring Music TheoryPurian
Scale 2575Scale 2575: Pumian, Ian Ring Music TheoryPumian
Scale 2607Scale 2607: Aerolian, Ian Ring Music TheoryAerolian
Scale 2623Scale 2623: Aerylyllic, Ian Ring Music TheoryAerylyllic
Scale 2655Scale 2655: Qojian, Ian Ring Music TheoryQojian
Scale 2719Scale 2719: Zocryllic, Ian Ring Music TheoryZocryllic
Scale 2847Scale 2847: Phracryllic, Ian Ring Music TheoryPhracryllic
Scale 2079Scale 2079: Hexatonic Chromatic 4, Ian Ring Music TheoryHexatonic Chromatic 4
Scale 2335Scale 2335: Epydian, Ian Ring Music TheoryEpydian
Scale 3103Scale 3103: Heptatonic Chromatic 3, Ian Ring Music TheoryHeptatonic Chromatic 3
Scale 3615Scale 3615: Octatonic Chromatic 4, Ian Ring Music TheoryOctatonic Chromatic 4
Scale 543Scale 543: Denian, Ian Ring Music TheoryDenian
Scale 1567Scale 1567: Jobian, Ian Ring Music TheoryJobian

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.