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Scale 1415: "IMPian"

Scale 1415: IMPian, 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: 3125


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.

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 Starr/Rahn algorithm.

prime: 215


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, 5, 1, 2, 2]

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, 3, 2, 2, 3, 2>

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, 0.5, 0.4, 0, 0.6, 0.667>

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}
<3> = {4,5,7,8}
<4> = {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.

(22, 14, 61)

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 Triadsgm{7,10,2}110.5
Diminished Triads{7,10,1}110.5

The following pitch classes are not present in any of the common triads: {0,8}

Parsimonious Voice Leading Between Common Triads of Scale 1415. Created by Ian Ring ©2019 gm gm g°->gm

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 1415 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 2755
Scale 2755: RIVian, Ian Ring Music TheoryRIVian
3rd mode:
Scale 3425
Scale 3425: VIHian, Ian Ring Music TheoryVIHian
4th mode:
Scale 235
Scale 235: BIHian, Ian Ring Music TheoryBIHian
5th mode:
Scale 2165
Scale 2165: NECian, Ian Ring Music TheoryNECian
6th mode:
Scale 1565
Scale 1565: JOZian, Ian Ring Music TheoryJOZian


The prime form of this scale is Scale 215

Scale 215Scale 215: BIVian, Ian Ring Music TheoryBIVian


The hexatonic modal family [1415, 2755, 3425, 235, 2165, 1565] (Forte: 6-Z12) is the complement of the hexatonic modal family [335, 965, 1265, 2215, 3155, 3625] (Forte: 6-Z41)


The inverse of a scale is a reflection using the root as its axis. The inverse of 1415 is 3125

Scale 3125Scale 3125: TONian, Ian Ring Music TheoryTONian


Only scales that are chiral will have an enantiomorph. Scale 1415 is chiral, and its enantiomorph is scale 3125

Scale 3125Scale 3125: TONian, Ian Ring Music TheoryTONian


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> 1415       T0I <11,0> 3125
T1 <1,1> 2830      T1I <11,1> 2155
T2 <1,2> 1565      T2I <11,2> 215
T3 <1,3> 3130      T3I <11,3> 430
T4 <1,4> 2165      T4I <11,4> 860
T5 <1,5> 235      T5I <11,5> 1720
T6 <1,6> 470      T6I <11,6> 3440
T7 <1,7> 940      T7I <11,7> 2785
T8 <1,8> 1880      T8I <11,8> 1475
T9 <1,9> 3760      T9I <11,9> 2950
T10 <1,10> 3425      T10I <11,10> 1805
T11 <1,11> 2755      T11I <11,11> 3610
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 3125      T0MI <7,0> 1415
T1M <5,1> 2155      T1MI <7,1> 2830
T2M <5,2> 215      T2MI <7,2> 1565
T3M <5,3> 430      T3MI <7,3> 3130
T4M <5,4> 860      T4MI <7,4> 2165
T5M <5,5> 1720      T5MI <7,5> 235
T6M <5,6> 3440      T6MI <7,6> 470
T7M <5,7> 2785      T7MI <7,7> 940
T8M <5,8> 1475      T8MI <7,8> 1880
T9M <5,9> 2950      T9MI <7,9> 3760
T10M <5,10> 1805      T10MI <7,10> 3425
T11M <5,11> 3610      T11MI <7,11> 2755

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 1413Scale 1413: IRUian, Ian Ring Music TheoryIRUian
Scale 1411Scale 1411: IROian, Ian Ring Music TheoryIROian
Scale 1419Scale 1419: Raga Kashyapi, Ian Ring Music TheoryRaga Kashyapi
Scale 1423Scale 1423: Doptian, Ian Ring Music TheoryDoptian
Scale 1431Scale 1431: Phragian, Ian Ring Music TheoryPhragian
Scale 1447Scale 1447: Mela Ratnangi, Ian Ring Music TheoryMela Ratnangi
Scale 1479Scale 1479: Mela Jalarnava, Ian Ring Music TheoryMela Jalarnava
Scale 1287Scale 1287: HUTian, Ian Ring Music TheoryHUTian
Scale 1351Scale 1351: Aeraptimic, Ian Ring Music TheoryAeraptimic
Scale 1159Scale 1159: HASian, Ian Ring Music TheoryHASian
Scale 1671Scale 1671: KEMian, Ian Ring Music TheoryKEMian
Scale 1927Scale 1927: LUNian, Ian Ring Music TheoryLUNian
Scale 391Scale 391: CIYian, Ian Ring Music TheoryCIYian
Scale 903Scale 903: FOSian, Ian Ring Music TheoryFOSian
Scale 2439Scale 2439: PAGian, Ian Ring Music TheoryPAGian
Scale 3463Scale 3463: VOFian, Ian Ring Music TheoryVOFian

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.