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Scale 3267: "Urfian"

Scale 3267: Urfian, 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.

6 (hexatonic)

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: 2151


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

4 (multihemitonic)


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

2 (dicohemitonic)


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: 207


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, 5, 1, 3, 1, 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.

<4, 2, 2, 2, 3, 2>

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

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.

(30, 10, 55)

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 TriadsF♯{6,10,1}110.5
Diminished Triads{7,10,1}110.5

The following pitch classes are not present in any of the common triads: {0,11}

Parsimonious Voice Leading Between Common Triads of Scale 3267. Created by Ian Ring ©2019 F# F# F#->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.



Modes are the rotational transformation of this scale. Scale 3267 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 3681
Scale 3681: Xahian, Ian Ring Music TheoryXahian
3rd mode:
Scale 243
Scale 243: Bomian, Ian Ring Music TheoryBomian
4th mode:
Scale 2169
Scale 2169: Nefian, Ian Ring Music TheoryNefian
5th mode:
Scale 783
Scale 783: Etuian, Ian Ring Music TheoryEtuian
6th mode:
Scale 2439
Scale 2439: Pagian, Ian Ring Music TheoryPagian


The prime form of this scale is Scale 207

Scale 207Scale 207: Beqian, Ian Ring Music TheoryBeqian


The hexatonic modal family [3267, 3681, 243, 2169, 783, 2439] (Forte: 6-5) is the complement of the hexatonic modal family [207, 963, 2151, 2529, 3123, 3609] (Forte: 6-5)


The inverse of a scale is a reflection using the root as its axis. The inverse of 3267 is 2151

Scale 2151Scale 2151: Natian, Ian Ring Music TheoryNatian


Only scales that are chiral will have an enantiomorph. Scale 3267 is chiral, and its enantiomorph is scale 2151

Scale 2151Scale 2151: Natian, Ian Ring Music TheoryNatian


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> 3267       T0I <11,0> 2151
T1 <1,1> 2439      T1I <11,1> 207
T2 <1,2> 783      T2I <11,2> 414
T3 <1,3> 1566      T3I <11,3> 828
T4 <1,4> 3132      T4I <11,4> 1656
T5 <1,5> 2169      T5I <11,5> 3312
T6 <1,6> 243      T6I <11,6> 2529
T7 <1,7> 486      T7I <11,7> 963
T8 <1,8> 972      T8I <11,8> 1926
T9 <1,9> 1944      T9I <11,9> 3852
T10 <1,10> 3888      T10I <11,10> 3609
T11 <1,11> 3681      T11I <11,11> 3123
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2277      T0MI <7,0> 1251
T1M <5,1> 459      T1MI <7,1> 2502
T2M <5,2> 918      T2MI <7,2> 909
T3M <5,3> 1836      T3MI <7,3> 1818
T4M <5,4> 3672      T4MI <7,4> 3636
T5M <5,5> 3249      T5MI <7,5> 3177
T6M <5,6> 2403      T6MI <7,6> 2259
T7M <5,7> 711      T7MI <7,7> 423
T8M <5,8> 1422      T8MI <7,8> 846
T9M <5,9> 2844      T9MI <7,9> 1692
T10M <5,10> 1593      T10MI <7,10> 3384
T11M <5,11> 3186      T11MI <7,11> 2673

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 3265Scale 3265: Urrian, Ian Ring Music TheoryUrrian
Scale 3269Scale 3269: Raga Malarani, Ian Ring Music TheoryRaga Malarani
Scale 3271Scale 3271: Mela Raghupriya, Ian Ring Music TheoryMela Raghupriya
Scale 3275Scale 3275: Mela Divyamani, Ian Ring Music TheoryMela Divyamani
Scale 3283Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
Scale 3299Scale 3299: Syptian, Ian Ring Music TheorySyptian
Scale 3203Scale 3203: Etrian, Ian Ring Music TheoryEtrian
Scale 3235Scale 3235: Pothimic, Ian Ring Music TheoryPothimic
Scale 3139Scale 3139: Towian, Ian Ring Music TheoryTowian
Scale 3395Scale 3395: Vepian, Ian Ring Music TheoryVepian
Scale 3523Scale 3523, Ian Ring Music Theory
Scale 3779Scale 3779, Ian Ring Music Theory
Scale 2243Scale 2243: Noyian, Ian Ring Music TheoryNoyian
Scale 2755Scale 2755: Rivian, Ian Ring Music TheoryRivian
Scale 1219Scale 1219: Hidian, Ian Ring Music TheoryHidian

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.