 The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

# Scale 2457: "Augmented" This highly symmetrical scale is constructed from two interlocked augmented triads. Often referred to as just "The Augmented Scale" or "Augmented Hexatonic", it was used prominently used in Liszt's "Faust Symphony". It is sometimes confused with another scale which is nicknamed Major Augmented, the 3rd mode of the Harmonic Minor.

### 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
Messiaen
Messiaen Mode 3 Truncated Inverse
T2 First Rotation
Messiaen Mode 3 Truncated Rotation 1
Ancient Greek
Genus Tertium
Carnatic
Raga Devamani
Zeitler
Ionythimic
Dozenal
PARian

## Analysis

#### Cardinality

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

{0,3,4,7,8,11}

#### Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-20

#### 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.

[1.5, 3.5, 5.5]

#### Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

#### Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

no

#### Hemitonia

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

3 (trihemitonic)

#### Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

0 (ancohemitonic)

#### Imperfections

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.

3

#### Modes

Modes are the rotational transformations of this scale. This number includes the scale itself, so the number is usually the same as its cardinality; unless there are rotational symmetries then there are fewer modes.

2

#### Prime Form

Describes if this scale is in prime form, using the Starr/Rahn algorithm.

no
prime: 819

#### Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

#### Deep Scale

A deep scale is one where the interval vector has 6 different digits, an indicator of maximum hierarchization.

no

#### Interval Structure

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

[3, 1, 3, 1, 3, 1]

#### 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.

p3m6n3d3

#### 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.

1

#### Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

#### 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.

no

#### 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.

2.25

#### Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.796

#### Myhill Property

A scale has Myhill Property if the Distribution Spectra have exactly two specific intervals for every generic interval.

no

#### Balanced

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.

yes

#### 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.

[3,7,11]

#### Propriety

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.

1

#### 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.

0.64

## Generator

This scale has no generator.

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

E{4,8,11}331.5
G♯{8,0,3}331.5
em{4,7,11}331.5
g♯m{8,11,3}331.5
D♯+{3,7,11}331.5

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.

Diameter 3 3 yes

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.

 Minor: {0, 3, 7}Major: {4, 8, 11} Major: {0, 4, 7}Minor: {8, 11, 3} Augmented: {0, 4, 8}Augmented: {3, 7, 11} Minor: {4, 7, 11}Major: {8, 0, 3}

## Modes

Modes are the rotational transformation of this scale. Scale 2457 can be rotated to make 1 other scale. The 1st mode is itself.

 2nd mode:Scale 819 Augmented Inverse This is the prime mode

## Prime

The prime form of this scale is Scale 819

 Scale 819 Augmented Inverse

## Complement

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

## Inverse

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

 Scale 819 Augmented Inverse

## Hierarchizability

Based on the work of Niels Verosky, hierarchizability is the measure of repeated patterns with "place-finding" remainder bits, applied recursively to the binary representation of a scale. For a full explanation, read Niels' paper, Hierarchizability as a Predictor of Scale Candidacy. The variable k is the maximum number of remainders allowed at each level of recursion, for them to count as an increment of hierarchizability. A high hierarchizability score is a good indicator of scale candidacy, ie a measure of usefulness for producing pleasing music. There is a strong correlation between scales with maximal hierarchizability and scales that are in popular use in a variety of world musical traditions.

kHierarchizabilityBreakdown PatternDiagram
11(1001)(1001)(1001)
21(1001)(1001)(1001)
31(1001)(1001)(1001)
41(1001)(1001)(1001)
51(1001)(1001)(1001)

## Transformations:

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. A note about the multipliers: multiplying by 1 changes nothing, multiplying by 11 produces the same result as inversion. 5 is the only non-degenerate multiplier, with the multiplier 7 producing the inverse of 5.

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 2457       T0I <11,0> 819
T1 <1,1> 819      T1I <11,1> 1638
T2 <1,2> 1638      T2I <11,2> 3276
T3 <1,3> 3276      T3I <11,3> 2457
T4 <1,4> 2457       T4I <11,4> 819
T5 <1,5> 819      T5I <11,5> 1638
T6 <1,6> 1638      T6I <11,6> 3276
T7 <1,7> 3276      T7I <11,7> 2457
T8 <1,8> 2457       T8I <11,8> 819
T9 <1,9> 819      T9I <11,9> 1638
T10 <1,10> 1638      T10I <11,10> 3276
T11 <1,11> 3276      T11I <11,11> 2457
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2457       T0MI <7,0> 819
T1M <5,1> 819      T1MI <7,1> 1638
T2M <5,2> 1638      T2MI <7,2> 3276
T3M <5,3> 3276      T3MI <7,3> 2457
T4M <5,4> 2457       T4MI <7,4> 819
T5M <5,5> 819      T5MI <7,5> 1638
T6M <5,6> 1638      T6MI <7,6> 3276
T7M <5,7> 3276      T7MI <7,7> 2457
T8M <5,8> 2457       T8MI <7,8> 819
T9M <5,9> 819      T9MI <7,9> 1638
T10M <5,10> 1638      T10MI <7,10> 3276
T11M <5,11> 3276      T11MI <7,11> 2457

The transformations that map this set to itself are: T0, T4, T8, T3I, T7I, T11I, T0M, T4M, T8M, T3MI, T7MI, T11MI

## 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 2459 Ionocrian Scale 2461 Sagian Scale 2449 Zacritonic Scale 2453 Raga Latika Scale 2441 Kyritonic Scale 2473 Raga Takka Scale 2489 Mela Gangeyabhusani Scale 2521 Mela Dhatuvardhani Scale 2329 Styditonic Scale 2393 Zathimic Scale 2201 Ionagitonic Scale 2713 Porimic Scale 2969 Tholian Scale 3481 Katathian Scale 409 Laritonic Scale 1433 Dynimic

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 (http://allthescales.org) 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.