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Scale 125: "ATWian"

Scale 125: ATWian, 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: 1985


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.

3 (tricohemitonic)


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.

prime: 95


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.

[2, 1, 1, 1, 1, 6]

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, 4, 3, 2, 1, 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.

<0.8, 0.667, 0.6, 0, 0.2, 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> = {1,2,6}
<2> = {2,3,7,8}
<3> = {3,4,8,9}
<4> = {4,5,9,10}
<5> = {6,10,11}

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.

(34, 13, 55)

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
Diminished Triads{0,3,6}000

The following pitch classes are not present in any of the common triads: {2,4,5}

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

2nd mode:
Scale 1055
Scale 1055: GIHian, Ian Ring Music TheoryGIHian
3rd mode:
Scale 2575
Scale 2575: PUMian, Ian Ring Music TheoryPUMian
4th mode:
Scale 3335
Scale 3335: VADian, Ian Ring Music TheoryVADian
5th mode:
Scale 3715
Scale 3715: XICian, Ian Ring Music TheoryXICian
6th mode:
Scale 3905
Scale 3905: YUSian, Ian Ring Music TheoryYUSian


The prime form of this scale is Scale 95

Scale 95Scale 95: ARKian, Ian Ring Music TheoryARKian


The hexatonic modal family [125, 1055, 2575, 3335, 3715, 3905] (Forte: 6-2) is the complement of the hexatonic modal family [95, 1985, 2095, 3095, 3595, 3845] (Forte: 6-2)


The inverse of a scale is a reflection using the root as its axis. The inverse of 125 is 1985

Scale 1985Scale 1985: MEWian, Ian Ring Music TheoryMEWian


Only scales that are chiral will have an enantiomorph. Scale 125 is chiral, and its enantiomorph is scale 1985

Scale 1985Scale 1985: MEWian, Ian Ring Music TheoryMEWian


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> 125       T0I <11,0> 1985
T1 <1,1> 250      T1I <11,1> 3970
T2 <1,2> 500      T2I <11,2> 3845
T3 <1,3> 1000      T3I <11,3> 3595
T4 <1,4> 2000      T4I <11,4> 3095
T5 <1,5> 4000      T5I <11,5> 2095
T6 <1,6> 3905      T6I <11,6> 95
T7 <1,7> 3715      T7I <11,7> 190
T8 <1,8> 3335      T8I <11,8> 380
T9 <1,9> 2575      T9I <11,9> 760
T10 <1,10> 1055      T10I <11,10> 1520
T11 <1,11> 2110      T11I <11,11> 3040
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1355      T0MI <7,0> 2645
T1M <5,1> 2710      T1MI <7,1> 1195
T2M <5,2> 1325      T2MI <7,2> 2390
T3M <5,3> 2650      T3MI <7,3> 685
T4M <5,4> 1205      T4MI <7,4> 1370
T5M <5,5> 2410      T5MI <7,5> 2740
T6M <5,6> 725      T6MI <7,6> 1385
T7M <5,7> 1450      T7MI <7,7> 2770
T8M <5,8> 2900      T8MI <7,8> 1445
T9M <5,9> 1705      T9MI <7,9> 2890
T10M <5,10> 3410      T10MI <7,10> 1685
T11M <5,11> 2725      T11MI <7,11> 3370

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 127Scale 127: Heptatonic Chromatic, Ian Ring Music TheoryHeptatonic Chromatic
Scale 121Scale 121: ASOian, Ian Ring Music TheoryASOian
Scale 123Scale 123: ASUian, Ian Ring Music TheoryASUian
Scale 117Scale 117: ANBian, Ian Ring Music TheoryANBian
Scale 109Scale 109: AMSian, Ian Ring Music TheoryAMSian
Scale 93Scale 93: ANUian, Ian Ring Music TheoryANUian
Scale 61Scale 61: AJUian, Ian Ring Music TheoryAJUian
Scale 189Scale 189: BEFian, Ian Ring Music TheoryBEFian
Scale 253Scale 253: BOSian, Ian Ring Music TheoryBOSian
Scale 381Scale 381: Kogian, Ian Ring Music TheoryKogian
Scale 637Scale 637: Debussy's Heptatonic, Ian Ring Music TheoryDebussy's Heptatonic
Scale 1149Scale 1149: Bydian, Ian Ring Music TheoryBydian
Scale 2173Scale 2173: NEHian, Ian Ring Music TheoryNEHian

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.