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Scale 819: "Augmented Inverse"

Scale 819: Augmented Inverse, 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

Western Modern
Augmented Inverse
Six Tone Symmetrical
Named After Composers
Ode To Napoleon Hexachord
Liszt's Prometheus
Messiaen 3rd Mode Truncated
T2 Prime Mode
Messiaen Mode 3 Truncated



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.

[4, 8]

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.

[0.5, 2.5, 4.5]


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.



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.

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 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, 3, 1, 3, 1, 3]

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, 0, 3, 6, 3, 0>

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, 0.6, 1, 0.6, 0>

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}
<2> = {4}
<3> = {5,7}
<4> = {8}
<5> = {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 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".

Strictly Proper

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.

(0, 0, 27)

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
Major TriadsC♯{1,5,8}331.5
Minor Triadsc♯m{1,4,8}331.5
Augmented TriadsC+{0,4,8}331.5
Parsimonious Voice Leading Between Common Triads of Scale 819. Created by Ian Ring ©2019 C+ C+ c#m c#m C+->c#m fm fm C+->fm am am C+->am C# C# c#m->C# A A c#m->A C#+ C#+ C#->C#+ C#->fm F F C#+->F C#+->A fm->F F->am am->A

view full size

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.


Triad Polychords

Also known as Bi-Triadic Hexatonics (a term coined by mDecks), and related to Generic Modality Compression (a method for guitar by Mick Goodrick and Tim Miller), these are two common triads that when combined use all the tones in this scale.

There are 4 ways that this hexatonic scale can be split into two common triads.

Augmented: {0, 4, 8}
Augmented: {1, 5, 9}

Minor: {1, 4, 8}
Major: {5, 9, 0}

Major: {1, 5, 8}
Minor: {9, 0, 4}

Minor: {5, 8, 0}
Major: {9, 1, 4}


Modes are the rotational transformation of this scale. Scale 819 can be rotated to make 1 other scale. The 1st mode is itself.

2nd mode:
Scale 2457
Scale 2457: Augmented, Ian Ring Music TheoryAugmented


This is the prime form of this scale.


The hexatonic modal family [819, 2457] (Forte: 6-20) is the complement of the hexatonic modal family [819, 2457] (Forte: 6-20)


The inverse of a scale is a reflection using the root as its axis. The inverse of 819 is 2457

Scale 2457Scale 2457: Augmented, Ian Ring Music TheoryAugmented


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> 819       T0I <11,0> 2457
T1 <1,1> 1638      T1I <11,1> 819
T2 <1,2> 3276      T2I <11,2> 1638
T3 <1,3> 2457      T3I <11,3> 3276
T4 <1,4> 819       T4I <11,4> 2457
T5 <1,5> 1638      T5I <11,5> 819
T6 <1,6> 3276      T6I <11,6> 1638
T7 <1,7> 2457      T7I <11,7> 3276
T8 <1,8> 819       T8I <11,8> 2457
T9 <1,9> 1638      T9I <11,9> 819
T10 <1,10> 3276      T10I <11,10> 1638
T11 <1,11> 2457      T11I <11,11> 3276
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 819       T0MI <7,0> 2457
T1M <5,1> 1638      T1MI <7,1> 819
T2M <5,2> 3276      T2MI <7,2> 1638
T3M <5,3> 2457      T3MI <7,3> 3276
T4M <5,4> 819       T4MI <7,4> 2457
T5M <5,5> 1638      T5MI <7,5> 819
T6M <5,6> 3276      T6MI <7,6> 1638
T7M <5,7> 2457      T7MI <7,7> 3276
T8M <5,8> 819       T8MI <7,8> 2457
T9M <5,9> 1638      T9MI <7,9> 819
T10M <5,10> 3276      T10MI <7,10> 1638
T11M <5,11> 2457      T11MI <7,11> 3276

The transformations that map this set to itself are: T0, T4, T8, T1I, T5I, T9I, T0M, T4M, T8M, T1MI, T5MI, T9MI

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 817Scale 817: Zothitonic, Ian Ring Music TheoryZothitonic
Scale 821Scale 821: Aeranimic, Ian Ring Music TheoryAeranimic
Scale 823Scale 823: Stodian, Ian Ring Music TheoryStodian
Scale 827Scale 827: Mixolocrian, Ian Ring Music TheoryMixolocrian
Scale 803Scale 803: Loritonic, Ian Ring Music TheoryLoritonic
Scale 811Scale 811: Radimic, Ian Ring Music TheoryRadimic
Scale 787Scale 787: Aeolapritonic, Ian Ring Music TheoryAeolapritonic
Scale 851Scale 851: Raga Hejjajji, Ian Ring Music TheoryRaga Hejjajji
Scale 883Scale 883: Ralian, Ian Ring Music TheoryRalian
Scale 947Scale 947: Mela Gayakapriya, Ian Ring Music TheoryMela Gayakapriya
Scale 563Scale 563: Thacritonic, Ian Ring Music TheoryThacritonic
Scale 691Scale 691: Raga Kalavati, Ian Ring Music TheoryRaga Kalavati
Scale 307Scale 307: Raga Megharanjani, Ian Ring Music TheoryRaga Megharanjani
Scale 1331Scale 1331: Raga Vasantabhairavi, Ian Ring Music TheoryRaga Vasantabhairavi
Scale 1843Scale 1843: Ionygian, Ian Ring Music TheoryIonygian
Scale 2867Scale 2867: Major Romani, Ian Ring Music TheoryMajor Romani

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.