The Exciting Universe Of Music Theory

more than you ever wanted to know about...

Scale 167: "Barian"

Scale 167: Barian, 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.

enantiomorph: 3233


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.



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, 3, 2, 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.

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

(9, 9, 38)

Common Triads

There are no common triads (major, minor, augmented and diminished) that can be formed using notes in this scale.


Modes are the rotational transformation of this scale. Scale 167 can be rotated to make 4 other scales. The 1st mode is itself.

2nd mode:
Scale 2131
Scale 2131: Nahian, Ian Ring Music TheoryNahian
3rd mode:
Scale 3113
Scale 3113: Tigian, Ian Ring Music TheoryTigian
4th mode:
Scale 901
Scale 901: Bofian, Ian Ring Music TheoryBofian
5th mode:
Scale 1249
Scale 1249: Howian, Ian Ring Music TheoryHowian


This is the prime form of this scale.


The pentatonic modal family [167, 2131, 3113, 901, 1249] (Forte: 5-14) is the complement of the heptatonic modal family [431, 1507, 1933, 2263, 2801, 3179, 3637] (Forte: 7-14)


The inverse of a scale is a reflection using the root as its axis. The inverse of 167 is 3233

Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian


Only scales that are chiral will have an enantiomorph. Scale 167 is chiral, and its enantiomorph is scale 3233

Scale 3233Scale 3233: Unbian, Ian Ring Music TheoryUnbian


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> 167       T0I <11,0> 3233
T1 <1,1> 334      T1I <11,1> 2371
T2 <1,2> 668      T2I <11,2> 647
T3 <1,3> 1336      T3I <11,3> 1294
T4 <1,4> 2672      T4I <11,4> 2588
T5 <1,5> 1249      T5I <11,5> 1081
T6 <1,6> 2498      T6I <11,6> 2162
T7 <1,7> 901      T7I <11,7> 229
T8 <1,8> 1802      T8I <11,8> 458
T9 <1,9> 3604      T9I <11,9> 916
T10 <1,10> 3113      T10I <11,10> 1832
T11 <1,11> 2131      T11I <11,11> 3664
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3107      T0MI <7,0> 2183
T1M <5,1> 2119      T1MI <7,1> 271
T2M <5,2> 143      T2MI <7,2> 542
T3M <5,3> 286      T3MI <7,3> 1084
T4M <5,4> 572      T4MI <7,4> 2168
T5M <5,5> 1144      T5MI <7,5> 241
T6M <5,6> 2288      T6MI <7,6> 482
T7M <5,7> 481      T7MI <7,7> 964
T8M <5,8> 962      T8MI <7,8> 1928
T9M <5,9> 1924      T9MI <7,9> 3856
T10M <5,10> 3848      T10MI <7,10> 3617
T11M <5,11> 3601      T11MI <7,11> 3139

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 165Scale 165: Genus Primum, Ian Ring Music TheoryGenus Primum
Scale 163Scale 163: Bapian, Ian Ring Music TheoryBapian
Scale 171Scale 171: Pruian, Ian Ring Music TheoryPruian
Scale 175Scale 175: Bewian, Ian Ring Music TheoryBewian
Scale 183Scale 183: Bebian, Ian Ring Music TheoryBebian
Scale 135Scale 135: Armian, Ian Ring Music TheoryArmian
Scale 151Scale 151: Bahian, Ian Ring Music TheoryBahian
Scale 199Scale 199: Raga Nabhomani, Ian Ring Music TheoryRaga Nabhomani
Scale 231Scale 231: Bifian, Ian Ring Music TheoryBifian
Scale 39Scale 39: Afuian, Ian Ring Music TheoryAfuian
Scale 103Scale 103: Apuian, Ian Ring Music TheoryApuian
Scale 295Scale 295: Gyritonic, Ian Ring Music TheoryGyritonic
Scale 423Scale 423: Sogimic, Ian Ring Music TheorySogimic
Scale 679Scale 679: Lanimic, Ian Ring Music TheoryLanimic
Scale 1191Scale 1191: Pyrimic, Ian Ring Music TheoryPyrimic
Scale 2215Scale 2215: Ranimic, Ian Ring Music TheoryRanimic

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.