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Scale 147: "Bafian"

Scale 147: Bafian, 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.

4 (tetratonic)

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: 2337


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

1 (unhemitonic)


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

0 (ancohemitonic)


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.



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, 3, 3, 5]

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.

<1, 0, 2, 1, 1, 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> = {1,3,5}
<2> = {4,6,8}
<3> = {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.

(2, 0, 15)

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 TriadsC{0,4,7}110.5
Diminished Triadsc♯°{1,4,7}110.5
Parsimonious Voice Leading Between Common Triads of Scale 147. Created by Ian Ring ©2019 C C c#° c#° C->c#°

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 147 can be rotated to make 3 other scales. The 1st mode is itself.

2nd mode:
Scale 2121
Scale 2121: Nabian, Ian Ring Music TheoryNabian
3rd mode:
Scale 777
Scale 777: Empian, Ian Ring Music TheoryEmpian
4th mode:
Scale 609
Scale 609: Docian, Ian Ring Music TheoryDocian


This is the prime form of this scale.


The tetratonic modal family [147, 2121, 777, 609] (Forte: 4-18) is the complement of the octatonic modal family [879, 1779, 1947, 2487, 2937, 3021, 3291, 3693] (Forte: 8-18)


The inverse of a scale is a reflection using the root as its axis. The inverse of 147 is 2337

Scale 2337Scale 2337: Ogoian, Ian Ring Music TheoryOgoian


Only scales that are chiral will have an enantiomorph. Scale 147 is chiral, and its enantiomorph is scale 2337

Scale 2337Scale 2337: Ogoian, Ian Ring Music TheoryOgoian


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> 147       T0I <11,0> 2337
T1 <1,1> 294      T1I <11,1> 579
T2 <1,2> 588      T2I <11,2> 1158
T3 <1,3> 1176      T3I <11,3> 2316
T4 <1,4> 2352      T4I <11,4> 537
T5 <1,5> 609      T5I <11,5> 1074
T6 <1,6> 1218      T6I <11,6> 2148
T7 <1,7> 2436      T7I <11,7> 201
T8 <1,8> 777      T8I <11,8> 402
T9 <1,9> 1554      T9I <11,9> 804
T10 <1,10> 3108      T10I <11,10> 1608
T11 <1,11> 2121      T11I <11,11> 3216
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2337      T0MI <7,0> 147
T1M <5,1> 579      T1MI <7,1> 294
T2M <5,2> 1158      T2MI <7,2> 588
T3M <5,3> 2316      T3MI <7,3> 1176
T4M <5,4> 537      T4MI <7,4> 2352
T5M <5,5> 1074      T5MI <7,5> 609
T6M <5,6> 2148      T6MI <7,6> 1218
T7M <5,7> 201      T7MI <7,7> 2436
T8M <5,8> 402      T8MI <7,8> 777
T9M <5,9> 804      T9MI <7,9> 1554
T10M <5,10> 1608      T10MI <7,10> 3108
T11M <5,11> 3216      T11MI <7,11> 2121

The transformations that map this set to itself are: T0, T0MI

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 145Scale 145: Raga Malasri, Ian Ring Music TheoryRaga Malasri
Scale 149Scale 149: Eskimo Tetratonic, Ian Ring Music TheoryEskimo Tetratonic
Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian
Scale 155Scale 155: Bakian, Ian Ring Music TheoryBakian
Scale 131Scale 131: Atoian, Ian Ring Music TheoryAtoian
Scale 139Scale 139: Ayoian, Ian Ring Music TheoryAyoian
Scale 163Scale 163: Bapian, Ian Ring Music TheoryBapian
Scale 179Scale 179: Beyian, Ian Ring Music TheoryBeyian
Scale 211Scale 211: Bisian, Ian Ring Music TheoryBisian
Scale 19Scale 19: Acuian, Ian Ring Music TheoryAcuian
Scale 83Scale 83: Amuian, Ian Ring Music TheoryAmuian
Scale 275Scale 275: Dalic, Ian Ring Music TheoryDalic
Scale 403Scale 403: Raga Reva, Ian Ring Music TheoryRaga Reva
Scale 659Scale 659: Raga Rasika Ranjani, Ian Ring Music TheoryRaga Rasika Ranjani
Scale 1171Scale 1171: Raga Manaranjani I, Ian Ring Music TheoryRaga Manaranjani I
Scale 2195Scale 2195: Zalitonic, Ian Ring Music TheoryZalitonic

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.