The Exciting Universe Of Music Theory

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Scale 175: "BEWian"

Scale 175: BEWian, 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).



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: 3745


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

3 (trihemitonic)


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 Starr/Rahn algorithm.



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, an indicator of maximum hierarchization.


Interval Structure

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

[1, 1, 1, 2, 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.

<3, 4, 2, 2, 3, 1>

Proportional Saturation Vector

First described by Michael Buchler (2001), this is a vector showing the prominence of intervals relative to the maximum and minimum possible for the scale's cardinality. A saturation of 0 means the interval is present minimally, a saturation of 1 means it is the maximum possible.

<0.6, 0.667, 0.4, 0, 0.6, 0.333>

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> = {2,3,4,6,7}
<3> = {3,4,5,7,8,9}
<4> = {5,6,8,9,10}
<5> = {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 Distribution Spectra have 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.

(29, 19, 65)

Coherence Quotient

The Coherence Quotient is a score between 0 and 1, indicating the proportion of coherence failures (ambiguity or contradiction) in the scale, against the maximum possible for a cardinality. A high coherence quotient indicates a less complex scale, whereas a quotient of 0 indicates a maximally complex scale.


Sameness Quotient

The Sameness Quotient is a score between 0 and 1, indicating the proportion of differences in the heteromorphic profile, against the maximum possible for a cardinality. A higher quotient indicates a less complex scale, whereas a quotient of 0 indicates a scale with maximum complexity.



This scale has no generator.

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
Minor Triadscm{0,3,7}000

The following pitch classes are not present in any of the common triads: {1,2,5}

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 175 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2135
Scale 2135: NAKian, Ian Ring Music TheoryNAKian
3rd mode:
Scale 3115
Scale 3115: TIHian, Ian Ring Music TheoryTIHian
4th mode:
Scale 3605
Scale 3605: OLKian, Ian Ring Music TheoryOLKian
5th mode:
Scale 1925
Scale 1925: LUMian, Ian Ring Music TheoryLUMian
6th mode:
Scale 1505
Scale 1505: JEPian, Ian Ring Music TheoryJEPian


This is the prime form of this scale.


The hexatonic modal family [175, 2135, 3115, 3605, 1925, 1505] (Forte: 6-9) is the complement of the hexatonic modal family [175, 1505, 1925, 2135, 3115, 3605] (Forte: 6-9)


The inverse of a scale is a reflection using the root as its axis. The inverse of 175 is 3745

Scale 3745Scale 3745: XUVian, Ian Ring Music TheoryXUVian


Only scales that are chiral will have an enantiomorph. Scale 175 is chiral, and its enantiomorph is scale 3745

Scale 3745Scale 3745: XUVian, Ian Ring Music TheoryXUVian


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> 175       T0I <11,0> 3745
T1 <1,1> 350      T1I <11,1> 3395
T2 <1,2> 700      T2I <11,2> 2695
T3 <1,3> 1400      T3I <11,3> 1295
T4 <1,4> 2800      T4I <11,4> 2590
T5 <1,5> 1505      T5I <11,5> 1085
T6 <1,6> 3010      T6I <11,6> 2170
T7 <1,7> 1925      T7I <11,7> 245
T8 <1,8> 3850      T8I <11,8> 490
T9 <1,9> 3605      T9I <11,9> 980
T10 <1,10> 3115      T10I <11,10> 1960
T11 <1,11> 2135      T11I <11,11> 3920
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3115      T0MI <7,0> 2695
T1M <5,1> 2135      T1MI <7,1> 1295
T2M <5,2> 175       T2MI <7,2> 2590
T3M <5,3> 350      T3MI <7,3> 1085
T4M <5,4> 700      T4MI <7,4> 2170
T5M <5,5> 1400      T5MI <7,5> 245
T6M <5,6> 2800      T6MI <7,6> 490
T7M <5,7> 1505      T7MI <7,7> 980
T8M <5,8> 3010      T8MI <7,8> 1960
T9M <5,9> 1925      T9MI <7,9> 3920
T10M <5,10> 3850      T10MI <7,10> 3745
T11M <5,11> 3605      T11MI <7,11> 3395

The transformations that map this set to itself are: T0, T2M

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 173Scale 173: Raga Purnalalita, Ian Ring Music TheoryRaga Purnalalita
Scale 171Scale 171: PRUian, Ian Ring Music TheoryPRUian
Scale 167Scale 167: BARian, Ian Ring Music TheoryBARian
Scale 183Scale 183: BEBian, Ian Ring Music TheoryBEBian
Scale 191Scale 191: BEGian, Ian Ring Music TheoryBEGian
Scale 143Scale 143: BACian, Ian Ring Music TheoryBACian
Scale 159Scale 159: BAMian, Ian Ring Music TheoryBAMian
Scale 207Scale 207: BEQian, Ian Ring Music TheoryBEQian
Scale 239Scale 239: BIKian, Ian Ring Music TheoryBIKian
Scale 47Scale 47: AGOian, Ian Ring Music TheoryAGOian
Scale 111Scale 111: AROian, Ian Ring Music TheoryAROian
Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic
Scale 431Scale 431: Epyrian, Ian Ring Music TheoryEpyrian
Scale 687Scale 687: Aeolythian, Ian Ring Music TheoryAeolythian
Scale 1199Scale 1199: Magian, Ian Ring Music TheoryMagian
Scale 2223Scale 2223: Konian, Ian Ring Music TheoryKonian

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow and Lilypond, graph visualization by Graphviz, audio by TiMIDIty and FFMPEG. 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.